Network Memetics: Formalizing the Theory through Network Science

Introduction

This document formalizes the concepts of Extended Meme Theory through the language of network science. The goal is to transition from a descriptive model to a predictive one, using the ready-made mathematical apparatus of graph theory.

Conceptual foundation: For the underlying ideas, examples, and case studies, see Extended Meme Theory – the main document of the project (30 Parts).

Table of Contents

• Part I. Why Network Science
• Part II. Basic Concepts of Network Science
• Part III. Heterogeneous Networks with Hubs and Zipf’s Law
• Part IV. Small-World Networks
• Part V. The Meme as a Node, the Association as an Edge
• Part VI. Meme “Strength” Is Centrality
• Part VII. Memeplex as a Cluster (Community)
• Part VIII. Dynamics: How Memes Activate and Compete
• Part IX. The Immune System – a Network View
• Part X. Inter-Level Transitions: Multilayer Networks
• Part XI. Communicative Asymmetry of the Graph
• Part XII. What Can Now Be Predicted
• Part XIII. Consciousness Level – a Metric of the Level of Consciousness
• Part XIV. Triple Binding – the Unity of Conscious Experience
• Part XV. The Diffusion Engine and Embedding Space
• Part XVI. Inter-Agent Exchange, Scarcity, and Critical Periods
• Part XVII. Subsumption of Competing Theories of Consciousness
• Part XVIII. Retrodiction: COGITATE (2025)
• Appendix: Key Formulas
• Key Works

Part I. Why Network Science

The problem: the theory is descriptive

The current theory of memes explains phenomena post-hoc:

• “The memeplex defends itself” – but how do we measure defense?
• “One meme is stronger than another” – in what units?
• “Clusters of memes” – where are the boundaries?

We need a transition from metaphors to measurable quantities.

The solution: network science

Network science is an interdisciplinary field that studies complex systems as networks of nodes and links. It has already been successfully applied to:

The key idea

The memeplex is literally a network, not a metaphor.

• Memes – nodes (fractal: each meme consists of sub-memes)
• Associations – edges (connections between memes, $w \in [-1, +1]$)
• Strength of association – edge weight
• “Strength” of a meme – node centrality
• Clusters – communities (memeplexes)
• Hub – role: an element with anomalously high centrality

This gives us:

1. Rigorous definitions
2. Measurable metrics
3. Testable predictions
4. Ready-made algorithms

In particular, the formalism allows the construction of a quantitative Consciousness Level metric CL (Part XIII), based on $\sigma_{SW}$, the activity of the Self-Model Cluster (SMC), and the balance of the memeplex, as well as the formalization of triple binding (Part XIV) – a mechanism explaining the unity of conscious experience through structural, temporal, and competitive binding.

Part II. Basic Concepts of Network Science

Graph: nodes and edges

A graph $G = (V, E)$ consists of:

• $V$ – a set of nodes (vertices)
• $E$ – a set of edges (links between nodes)

In the context of memetics:

• Node = meme
• Edge = association between memes
• Edge weight = strength of association

Key metrics

Numerical example: a network of 5 memes

Consider a simple network:

    A --- B
    |     |
    C --- D --- E

Adjacency matrix:

Computing degrees:

Node D is central: it connects parts of the network.

Part III. Heterogeneous Networks with Hubs and Zipf’s Law

Why some memes become hubs

Many real-world networks – from the internet to social connections – exhibit heavy-tailed degree distributions:

Important clarification (Broido & Clauset, 2019): A strict power law is found in only ~4% of studied networks. Alternative distributions include log-normal, power law with exponential cutoff, and Zipf-polylog (Lee et al., 2024). However, what matters for the theory of memeplexes is not the mathematical strictness of the power law, but the presence of hubs – nodes with a disproportionately large number of connections. Hubs are empirically observed in all networks with heavy tails.

As a first approximation, such networks obey a power-law distribution:

where $k$ is the node degree (number of connections), and $\gamma$ is the exponent (typically $2 < \gamma < 3$).

What this means: Most nodes have few connections, but a small number of nodes (hubs) have orders of magnitude more connections. This is not coincidence – it is a mathematical inevitability.

Mechanism: preferential attachment

Why do hubs arise? Because of the preferential attachment mechanism:

The formula means: The probability of acquiring a new connection is proportional to the number of existing connections. “The rich get richer.”

Zipf’s Law: rank vs. frequency

Zipf’s law is a special case of the power-law distribution:

where $r$ is the rank of the element. The most popular element occurs twice as frequently as the second, three times as frequently as the third, and so on.

In a memeplex, this means:

How to verify that a network is scale-free

It is not sufficient to plot a log-log graph and see a “straight line.”

Why a log-log plot is needed: On a linear scale, the “tail” of the distribution (rare hubs with high degree) is practically invisible. Only a log-log scale reveals that the data follow a power law.

Correct procedure (Clauset, Shalizi & Newman, 2009):

1. Estimate the parameter $\gamma$ using MLE (maximum likelihood estimation), not OLS regression
2. Determine $x_{min}$ – the lower bound of the power-law regime (where the “tail” begins)
3. Goodness-of-fit test (Kolmogorov-Smirnov): compare empirical and theoretical distributions
4. Compare with alternatives: log-normal, exponential, stretched exponential
5. Likelihood ratio test between competing models

Methodological position of BMC (heavy-tailed, not strict scale-free). The debate about “scale-free networks” continues: Broido & Clauset (2019) showed that only ~4% of networks pass strict tests for pure power-law; Voitalov et al. (2019) using a less strict (regularly varying) definition obtained ~65%; Serafino et al. (2021) via finite-size scaling – ~50%.

BMC does not require a strict power law. A heavy-tailed distribution is sufficient – of any form (power-law, log-normal, stretched exponential), as long as: (1) hubs exist ($k_i \gg \langle k \rangle$), (2) preferential attachment operates, (3) hub displacement is possible. The specific form of the tail is in the red zone of the Structural Equivalence Test (SET) (does not transfer between levels); the existence of hubs and their dynamics is in the green zone. A useful distinction is heavy-tailed vs. thin-tailed, not power-law vs. log-normal (Holme, 2019).

For semantic (cognitive) networks – the most relevant for BMC L1 – the power law has been confirmed more rigorously: Steyvers & Tenenbaum (2005) showed $\gamma \approx 3.0$–$3.2$. Formalization of this position: SM, Part I.

Graph interpretation:

• Steps 1–2: Find $x_{min}$ – the point where the power-law regime begins. Data to the left of it (gray zone) do not follow the power law
• Step 3: The KS test compares the empirical and theoretical distributions. A small KS distance and $p > 0.1$ means we do not reject the power law
• Steps 4–5: Compare with alternatives. A positive LLR means the power law describes the data better

Practical tool: The Python package powerlaw (Alstott et al., 2014) implements the entire procedure.

Comparison of network types

Numerical example: network evolution

Consider network growth with preferential attachment:

Start (t=0): 2 nodes, 1 connection

A — B
Degrees: A=1, B=1

After adding 8 nodes (t=8):

Conclusion: Node B, which appeared earlier and gained a small initial advantage, became a hub. This is not merit – it is mathematics.

Implications for memetics

See also: Conceptual explanation of hubs in the memeplex – EMT, Part XI.

Percolation threshold: resilience to attacks

Percolation is the transition of a network from a connected state to a fragmented one upon the removal of nodes or edges. The key parameter is the percolation threshold $f_c$: the fraction of nodes that must be removed to destroy the giant component.

For targeted attack (removing nodes in order of decreasing degree):

where $\kappa = \frac{\langle k^2 \rangle}{\langle k \rangle}$ is the network heterogeneity coefficient.

For networks with heavy tails: the more heterogeneous the network (larger $\kappa$), the higher the resilience to random failures and the lower the resilience to targeted attacks.

Note: The exact values of $f_c$ depend on the distribution form. For strict power law with $\gamma < 3$: $\kappa \to \infty$, $f_c \to 1$. For log-normal and other heavy tails, the effect is qualitatively the same but quantitatively weaker.

This means: to destroy a heterogeneous network with hubs, one must remove nearly all hubs.

Practical implication for memeplexes: Destabilizing a memeplex requires a precise attack on hubs. Random criticism of peripheral beliefs will not destroy a memeplex – but an attack on central identity memes can trigger a cascading fragmentation.

Percolation in signed networks. Standard percolation assumes that removing a hub always reduces connectivity. In signed networks ($w \in [-1, +1]$), this is not necessarily the case: removing a hub with a large number of negative edges can increase effective connectivity, since inhibition of neighboring nodes is lifted. Thus, a targeted attack on an “antibody” hub (one actively suppressing a group of memes) can produce a paradoxical result – instead of fragmentation, previously suppressed nodes integrate into the main cluster. This means that for signed memeplexes, $f_c$ depends not only on topology but also on the sign distribution of edges: a memeplex with a high proportion of negative connections (polarized, “paranoid”) can be simultaneously topologically connected and functionally fragmented.

Source: Nature Reviews Physics (2024) – review of percolation theory in complex networks.

Part IV. Small-World Networks

The “small world” phenomenon

Small-world – a class of networks combining two properties:

1. High clustering – neighbors of a node are often connected to each other
2. Short average paths – any two nodes are connected through a small number of steps

This is the formalization of the “six degrees of separation” phenomenon (Milgram, 1967).

The Watts-Strogatz model (1998)

Starting from a regular lattice, each edge is randomly rewired with probability $p$:

At small $p$ ($\approx 0.01$–$0.1$), the network remains highly clustered, but the average path length drops sharply.

The small-worldness metric

Small-worldness coefficient (Humphries & Gurney, 2008):

where:

• $C$ – the clustering coefficient of the network
• $L$ – the average path length
• $C_{random}$, $L_{random}$ – values for a random network of the same size

Numerical example

A network of 100 nodes with average degree $k=6$:

Small-world in memeplexes

Hypothesis: Personal memeplexes are small-world networks.

Why this matters:

• Rapid spreading activation (information does not get stuck in a cluster)
• Modularity is preserved (clusters do not merge)
• Explains the simultaneous integration and differentiation of consciousness
• The coefficient $\sigma_{SW}$ enters as a multiplier in the Consciousness Level (CL) metric: $CL = \sigma_{SW} \cdot A_{SMC} \cdot f(Balance)$ (see Part XIII)

Source: Han et al. (2025), Brain-X – a review of small-world organization of the brain and its connection to cognitive functions.

Neuroanatomical confirmation at all scales. Small-world + hubs is not a metaphor but a measured property of the brain from the cellular to the connectome level (Bassett & Bullmore, 2006, The Neuroscientist). The Watts-Strogatz model shows: rewiring just ~1–5% of connections in a lattice radically shortens the average path while preserving clustering – explaining how a memeplex combines modularity (subpersonalities = dense clusters) with rapid global integration. Hubs (nodes with heavy-tailed $P(k) \sim k^{-\gamma}$) serve as bridges between modules; their damage is catastrophic ($f_c \approx 0.05$–$0.18$), while random losses of ~50% of nodes are survived without loss of connectivity – directly matching our predictions of hub attack vs. random failure. The universality of this pattern (gene networks, social networks, the internet) confirms cross-level isomorphism: the same topology is reproduced at all levels of BMC.

Part V. The Meme as a Node, the Association as an Edge

Concept mapping

Meme activation: a continuous variable with functional thresholds

Meme activation is a continuous quantity:

where $a_i(t)$ is the activation level of node $i$ at time $t$. Discrete “states” of a meme do not exist – a meme’s behavior is determined by the continuous $a_i$ and the orthogonal property SIT.

Important note: Node activation is always non-negative ($a_i \geq 0$), but edge weights can be negative: $w_{ij} \in [-1, +1]$. A negative weight means an inhibitory (suppressive) connection: activation of node $j$ reduces activation of node $i$.

Two functional thresholds define qualitatively distinguishable behavior:

Between the thresholds ($\theta_{low} < a_i < \theta_{act}$), a meme does not drive behavior but can perform a narrative function – creating a sense of personal continuity, serving as material for nostalgia (for details, see EMT, Part VIII).

These thresholds are not boundaries of discrete states but rather points at which the character of a meme’s influence on behavior changes.

Open meme (SIT > 0) – an orthogonal property independent of activation level. A meme-question (structural gap) has $SIT_i > 0$: SIT periodically raises $a_i$ above $\theta_{act}$ (the thought “surfaces”), then WM switches away and $a_i$ drops, but SIT prevents edges from decaying ($\lambda_{open} = 0.5\lambda$). A meme can be open at any current $a_i$ – SIT and activation are independent. Formally: edge decay is suppressed as long as $SIT_i > 0$ (see Part VIII).

Neural substrate of nodes and edges

The formalization of a meme as a graph node is not merely an abstraction. Behind each element of the network model lies a concrete neurobiological object.

Node = cell assembly (Hebb, 1949): an ensemble of $~10^{3}$–$10^{5}$ neurons that activate together. The ranges of meme activation correspond to physical regimes of the ensemble:

Edge = ensemble overlap: $w_{ij} \propto |ensemble_i \cap ensemble_j| \cdot f(co\text{-}activation)$ (Cai et al., 2016, Nature). Positive weights = excitatory glutamatergic connections. Negative weights ($w < 0$) = GABAergic inhibition via inhibitory interneurons.

Winner-takes-all: lateral inhibition in the competition formula (see below, Part VIII) is realized through inhibitory circuits – basket cells and chandelier cells provide rapid suppression of competing ensembles.

For more detail: Full description of cell assemblies, neural reuse, overlapping engrams – see BM, Part III.

Sigmoid justification from the Hodgkin-Huxley model: The sigmoid $\sigma(x)$ in the spreading activation formula is not an arbitrary choice but an approximation of the nonlinear I-O curve of a real neuron. The HH model (Hodgkin & Huxley, 1952, J. Physiol.) describes the spike through 4 coupled ODEs, where voltage-gated channels create threshold dynamics: below threshold – silence, above – full action potential (all-or-nothing). The coupled feedback (V <-> gates) in HH is isomorphic to the coupling of activation <-> edge weights in the memeplex: meme activity changes weights (Hebb), weights determine activity (spreading). The refractory period after a spike explains the impossibility of continuous meme activation and the inevitable decline of $a_i$ after a peak. Our model operates at the level of cell assemblies (thousands of HH neurons), and HH justifies why the substrate S has precisely those properties (threshold behavior, nonlinearity, discrete states) that we take as given.

Ambivalence is not neutrality

With the introduction of signed weights, an important distinction arises:

• Neutrality: $\bar{w}_m \approx 0$ with weak connections – indifference
• Ambivalence: simultaneously strong + and – connections to different parts of the memeplex

High ambivalence at $\bar{w}_m \approx 0$ is an unstable state, tending toward resolution through structural balance (see Part IX).

Visualization: the memeplex as a network

The following diagram shows a memeplex of 6 memes forming two clusters:

• Cluster 1 (blue): faith, ritual, congregation – religious practice
• Cluster 2 (red): morality, taboo, myth – ethics and narrative

Numbers on edges are connection weights (association strength).

See also: Memeplexes, the life cycle of a meme – EMT, Parts V and VIII.

Part VI. Meme “Strength” Is Centrality

The problem of “strength” in the original theory

We used to say that some memes are “stronger” than others, without defining this rigorously. Now we can:

The “strength” of a meme = its centrality in the memeplex network.

Three types of centrality

1. Degree Centrality: ease of activation

where $k_i$ is the node degree and $n$ is the number of nodes in the network.

Interpretation: A meme with high degree centrality is easily activated – it has many connections, and any of its neighbors can “wake it up.”

Example: The meme “Russia” in the mind of a Russian person – connected to hundreds of other memes (language, history, family, work).

2. Betweenness Centrality: the bridge role

where $\sigma_{st}$ is the number of shortest paths between $s$ and $t$, and $\sigma_{st}(v)$ is the number of such paths through $v$.

Interpretation: A bridge-meme connects different clusters. If it is removed, parts of the memeplex become isolated.

Example: The meme “family” can be a bridge between the “work” cluster and the “religion” cluster.

3. Eigenvector Centrality: connection to the important

Interpretation: It is important not merely to have many connections, but to be connected to important memes. A node’s centrality is proportional to the sum of its neighbors’ centralities.

Example: The meme “status” may have few direct connections, but all of them are with key identity memes.

Numerical example: a network of 6 memes

Consider the network:

A — B — C — D — E — F
        |
        G

Adjacency matrix and computations:

Conclusions:

• C is the key meme (high betweenness and eigenvector)
• An attack on C fragments the network
• G is isolated, despite being connected to the central C

Step-by-step betweenness centrality calculation

Let us examine in detail how betweenness is computed for node C in the network above.

Step 1: Find all shortest paths between pairs of nodes

Step 2: Count paths passing THROUGH C (not as an endpoint)

Paths through C: A–D, A–E, A–F, B–D, B–E, B–F, D–G, E–G, F–G

Total: 9 paths

Step 3: Compute betweenness

Total number of pairs (excluding C): $\binom{6}{2} = 15$

Interpretation: 60% of all shortest paths in the network pass through node C. This makes C a critical “bridge” – its removal splits the network into isolated parts.

Correspondence table

See also: Meme energetics, replication as a selection criterion – EMT, Part VI.

Assortativity: who are hubs connected to?

Assortativity is a measure of the correlation between degrees of neighboring nodes. It answers the question: are hubs preferentially connected to other hubs or to the periphery?

Assortativity coefficient (Newman, 2002):

where:

• $e_{jk}$ – the fraction of edges connecting nodes of degrees $j$ and $k$
• $q_k$ – the degree distribution at the ends of a randomly chosen edge
• $\sigma_q^2$ – the variance of that distribution

Range: $r \in [-1, 1]$

Hypothesis for personal memeplexes: Memeplexes are more likely assortative – the identity core (central memes) is densely interconnected, forming a stable “elite club.”

Methodological note: For networks with heavy tails, the standard coefficient $r$ can be biased. Modern approaches use rank-based assortativity (J. Complex Networks, 2025), which is robust to heterogeneity in degree distributions.

Part VII. Memeplex as a Cluster (Community)

Modularity: partitioning into groups

Modularity $Q$ measures how well a network is partitioned into dense clusters with weak connections between them:

where:

• $m$ – total number of edges
• $A_{ij}$ – element of the adjacency matrix
• $k_i, k_j$ – node degrees
• $\delta(c_i, c_j) = 1$ if nodes are in the same cluster

Range: $Q \in [-0.5, 1]$. High values ($Q > 0.3$) indicate a clear cluster structure.

Clarification for signed networks: Modularity $Q$ is computed on the subgraph of positive edges ($w > 0$). Negative edges encode rejection, not community membership, and including them distorts the cluster structure. This is consistent with the approach of Traag & Bruggeman (2009) to modularity in signed networks.

Memeplexes within a memeplex

A personal memeplex is not a monolith but a hierarchy of clusters:

Dashed lines represent weak ties between clusters (bridges).

Modularity visualization

Numerical example: computing modularity

A network of 8 nodes, 2 clusters:

Cluster 1: {A, B, C, D} – 5 internal edges Cluster 2: {E, F, G, H} – 5 internal edges Inter-cluster edge: D–E

Computation:

• Total edges: $m = 11$
• Within clusters: 10 edges
• Expected randomly: ~5.5 edges

This is high modularity – a clear separation.

Prediction: modularity and memeplex behavior

Hypothesis: Modularity increases with age, explaining age-related rigidity.

Network motifs: recurring connection patterns

Network motifs are small subgraphs that occur in a network significantly more often than in random networks. They represent the “building blocks” of complex networks.

Main motif types:

Hypothesis: Specific motifs correspond to “thinking templates”:

• Excess of triangles -> rigid, dogmatic thinking (everything mutually supports everything)
• Excess of feed-forward loops -> analytical, causal thinking
• Excess of bi-fans -> integrative, contextual thinking

Clinical application: Studies show that motif patterns in cognitive networks differ in neurodegenerative diseases (Alzheimer’s) and in healthy individuals (Battiston et al., multilayer motif analysis).

Source: Milo et al. (2002), Science – the first description of network motifs; for modern methods see the literature on multilayer networks.

Modularity dynamics: splitting and merging of memeplexes

Modularity $Q$ is not a static parameter but a dynamic variable describing processes of memeplex splitting and merging (see EMT, Part XIX).

Splitting: $Q$ rises -> bifurcation

When a memeplex spreads into an environment with a different G-layer (different geography, culture, political context), some memes cease to match observed reality -> dissonance grows -> inter-cluster connections weaken -> $Q$ rises.

Splitting = the moment when $Q > Q_{crit}$ and two subclusters become independent memeplexes.

Formalization:

SPLITTING(Memplex P, Environment G_new):
  for each meme m in P:
    D_m = dissonance(m, G_new)    // mismatch with local reality
    if D_m > theta:
      weaken cross-cluster edges of m: w_ij *= (1 - eta * D_m)

  Q_new = compute_modularity(P)
  if Q_new > Q_crit:
    {P1, P2} = partition(P, max_modularity)  // optimal partition
    // P1, P2 retain a shared core (shared hubs) but have different peripheries
    return P1, P2

Merging: $Q$ falls -> boundary dissolves

When two memeplexes $A$ and $B$ encounter a reality that neither covers adequately, but $A \cup B$ does -> cross-connections strengthen -> $Q(A, B)$ falls -> the boundary dissolves.

If $D(A \cup B) \ll D(A) + D(B)$ (together they cover more), merging accelerates.

Formalization:

MERGING(Memplex A, Memplex B, Environment):
  for each pair (a, b) where a in A, b in B:
    if compatible(a, b):                    // positive connection possible
      w_ab += eta * relevance(a) * relevance(b)  // cross-connections

  Q_AB = compute_modularity(A union B)
  if Q_AB < Q_merge:
    C = merge(A, B)
    // Hubs of A and B are encapsulated as subclusters of C
    return C

Encapsulation: inheritance without loss

During merging, hubs of the original memeplexes are not destroyed but encapsulated – they become subclusters of the new memeplex. This means that the $Q$ of the new memeplex > 0 (it does not become a monolith), but $Q < Q_{crit}$ (subclusters are connected, not isolated).

If the dominant memeplex weakens ($Q$ of the new memeplex rises), suppressed hubs reactivate: the encapsulated subcluster becomes an autonomous memeplex again -> “resurrection.”

Predictions:

Counter-intuitive predictions:

1. Splitting increases CL. A rigid memeplex (high $Q$) has poor small-world properties: clusters are isolated, $L$ is large, $\sigma_{SW}$ is low (relationship between modularity and functional integration: Bertolero, Yeo & D’Esposito, 2015, PNAS; Sporns & Betzel, 2016, Annual Review of Psychology). During splitting, each fragment is a dense subgraph with good internal connectivity and short average path -> $\sigma_{SW}$ of each fragment is higher than that of the original rigid memeplex -> CL rises. Prediction: people who have experienced a worldview collapse (loss of religious faith, exit from an ideological organization) report increased subjective clarity of consciousness during the transitional period, not confusion. Testable: clarity-of-thinking scales (Mindful Attention Awareness Scale) + PCI in people undergoing deconversion. Counter-intuitive: fragmentation is associated with confusion, but BMC predicts clarity.

2. Merging temporarily decreases CL. When two memeplexes merge, the combined structure initially has a high average path $L$ (no short routes between former subclusters) and suboptimal clustering -> $\sigma_{SW}$ drops -> CL decreases. Prediction: cultural integration (immigrants assimilating a second culture; scientists mastering an adjacent discipline) is accompanied by a temporary cognitive “fog” – a decrease in cognitive flexibility and subjective clarity, followed by recovery as cross-cluster connections form. Testable with longitudinal cognitive tests (Stroop, task switching) in bicultural individuals during acculturation. Counter-intuitive: broadening one’s horizons is associated with enrichment, not temporary decline.

3. Critical memeplex size $N_{crit}$ for splitting. Splitting requires that two viable sub-communities can form within the memeplex (each with a sufficient number of hubs and connections for autonomous existence). At $N < N_{crit}$, this is impossible: the memeplex is too small to divide into viable fragments. Prediction: simple belief systems (few memes, few clusters) are more resistant to fragmentation than complex ones. A religion with 5 dogmas will split with more difficulty than one with 50. A small sect is more stable than a large denomination. Testable: correlation between $N$ (number of doctrinal elements) and the frequency of historical schisms for religious movements. Counter-intuitive: intuition suggests that complexity = stability (more connections = stronger). BMC predicts the opposite: complexity = more degrees of freedom for $Q > Q_{crit}$.

Part VIII. Dynamics: How Memes Activate and Compete

Spreading Activation: propagation along connections

When an external stimulus (trigger) activates a meme, activation spreads along connections:

where:

• $a_i(t)$ – activation of node $i$ at time $t$
• $w_{ij} \in [-1, +1]$ – the weight of the connection between $i$ and $j$ (can be negative)
• $\lambda$ – decay coefficient (applied to the node’s own state)
• $\sigma$ – sigmoid: $\sigma(x) = \frac{1}{1 + e^{-\beta(x - x_0)}}$, clamping the result to $[0, 1]$

When $w_{ij} < 0$: activation of node $j$ suppresses node $i$. This models not only competition (lateral inhibition) but also active rejection – aversion to an idea, negative priming (Tipper, 1985).

Note: Decay $(1-\lambda)$ is applied to the node’s own state, not to the result of activation. The sigmoid guarantees $a_i \in [0, 1]$ (activation cannot be negative but can be suppressed to zero). This corresponds to the classic spreading activation model (Collins & Loftus, 1975).

Neural substrate of the formula:

Connection to Hopfield networks: Our spreading activation is a continuous version of discrete Hopfield inference: $x_i \leftarrow \text{sign}(\sum_j w_{ij} x_j)$ (Hopfield, 1982, PNAS). The difference: we use a sigmoid instead of the sign function, yielding continuous states $a_i \in [0,1]$. The memeplex can be described as an energy landscape: stable meme configurations (subpersonalities) = attractors (local energy minima), recall = “rolling down” to the nearest attractor, SIT = high energy of an unclosed gap. The capacity of a Hopfield network is ~0.14N (Amit et al., 1985), imposing a theoretical ceiling on the number of stable memeplex configurations. False minima (spurious states) – a direct analog of false closure: the memeplex can “roll down” to an explanation that never existed in the training data – superstitions, conspiracy theories, pathological beliefs.

Spreading visualization

Competition: winner-takes-all

When multiple memes are activated simultaneously, lateral inhibition occurs:

where $\alpha$ is the inhibition coefficient. The function $\max(0, \cdot)$ guarantees non-negative activation.

Alternative formulation (softmax normalization):

where $T$ is a “temperature” parameter (as $T \to 0$, we obtain hard winner-takes-all).

Result: the most activated meme suppresses the rest.

Delta-modulated salience: Absolute activation $a_i(t)$ is insufficient for WM selection. A meme on a stable plateau ($a_i(t) \approx a_i(t-1)$) is in long-term memory, not working memory. For correct WM selection, softmax is modulated by a phasic factor and habituation:

where $\Delta a_i(t) = a_i(t) - a_i(t-1)$ is the change in activation per step (phasic signal), $\tau_i(t)$ is the number of consecutive steps the meme has been in WM without significant $|\Delta a_i|$ (tenure), $\beta$ is the phasic response gain, and $\eta$ is the habituation rate. The phasic factor ensures attention capture by a node whose activation has changed (neurobiological analog: phasic dopamine, prediction error). Habituation ensures eviction of nodes on a plateau (neurobiological analog: synaptic adaptation). Whenever there is significant $|\Delta a_i| > \epsilon$, the counter $\tau_i$ resets (dishabituation). Salience determines selection into Active WM (~3–4 pointers); memes evicted from Active WM retain a synaptic trace $\psi_i$ and remain in Latent WM (see Activity-silent WM). Softmax is applied to $salience_i$ instead of $a_i$:

Interaction with negative neighbors: In a signed network, competition is more complex. A meme with positive neighbors receives support; a meme with negative neighbors receives active resistance. The ultimate “strength” of a meme in competition depends on the balance of its positive and negative connections, not merely on the absolute number of connections.

Stochasticity and temperature (Boltzmann machine): The parameter $T$ in softmax is not merely a technical detail but a fundamental property of the memeplex (Ackley, Hinton & Sejnowski, 1985, Cognitive Science). The Boltzmann distribution $P(s) = e^{-E(s)/T} / Z$ connects the energy of a configuration with the probability of its activation. As $T \to 0$ – hard winner-takes-all (rigidity, getting stuck in false closure); at high $T$ – stochastic exploration (escape from local minima). Ontogeny: the Sponge stage = high $T$, the Museum stage = low $T$; $T$ decreases with age in parallel with the growth of modularity $Q$. Contrastive learning ($\Delta w \propto \langle reality \rangle - \langle fantasy \rangle$) is isomorphic to our wake/sleep cycle: wakefulness = positive phase (data correct the model), sleep = negative phase (free-running model, BLEND). Hidden neurons in the Boltzmann machine correspond to abstract memes (values, principles) – they have no direct sensory input but arise automatically from optimization.

Numerical example: activation from a trigger

Initial conditions:

• The trigger activates meme A with strength 1.0
• Spreading coefficient: $\beta = 0.7$
• Decay coefficient: $\lambda = 0.3$

Connection network:

Step-by-step computation:

Result: After 3 steps, the cluster A-B-C is activated, and activation has reached D.

Temporal dynamics

Meme synthesis: graph growth

A memeplex grows not only through externally incoming memes but also through the synthesis of new memes from existing ones. Three formal operations on the graph:

1. Mutation (already described): During transmission/storage, a meme is distorted (schema storage, Fidelity < 1). This is a change to an existing node.

2. Recombination (BLEND):

The new node $m_{new}$ inherits some of the connections of both parents. The inheritance probability $P(n)$ depends on the connection strength in the parent node. Example: “democratic socialism” inherits some connections from “democracy” and some from “socialism.”

3. Abduction/insight:

A new node is created in a zone of high betweenness potential – a structural “hole” between clusters:

where $d(c_1, c_2)$ is the distance between clusters. Abduction fills a “bridge” between previously unconnected areas of knowledge.

Sources: Fauconnier & Turner (2002) – Conceptual Blending; Gabora (1997) – MAV model.

Conceptual basis: Three mechanisms of synthesis – see EMT, Part I.

Neurobiological basis: BLEND during sleep – see BM, Part IV.

Heterogeneous topology and meme displacement

In a network with hubs, meme competition is not a “war” but a redistribution of connections. When a powerful new meme appears, it does not destroy old ones but “pulls” their connections toward itself.

Formula for connection loss (hypothesis of this theory):

where:

• $\Delta k_i$ – change in degree of node $i$
• $k_j$ – degree of the new hub (competitor)
• $\beta$ – redistribution coefficient

Methodological note: This formula is a hypothesis of this theory, not a standard result of network science. In the literature on adaptive networks (Gross & Blasius, 2008), an analogous mechanism is described through rewiring – redirecting connections depending on node states. The exact functional form requires empirical verification.

Signed edges and hub displacement. With $w \in [-1, +1]$, the degree $k_i$ of a node counts all edges, including negative ones. This creates a non-trivial effect: a hub with a large number of negative connections (many “enemies”) formally has a high degree, but its displacement by a competitor can increase network connectivity – removing inhibitory edges lifts suppression of surrounding nodes. On the other hand, an “antibody” meme (high Fidelity, $w < 0$) may be resistant to displacement precisely because of its negative connections: the memeplex needs it as an immune tool, and its displacement cost is higher than what the formula for neutral networks predicts.

Interpretation: The stronger the new hub ($k_j$) and the weaker the old meme ($k_i$), the more connections the old meme loses.

Practical implication: Displacing a meme does not require refuting it. It is sufficient to offer a new meme with more connections – and the old one will gradually be marginalized.

Edge decay (forgetting)

Edge weights between memes are not static – they decay without activation. This is the network formalization of Ebbinghaus’s forgetting curve.

Decay formula with reactivation:

where:

• $w_{ij}(t_0)$ – edge weight at the initial moment
• $\lambda$ – decay coefficient (typically 0.1–0.5 per day)
• $\delta(t - t_k)$ – delta function: activation at time $t_k$
• $\Delta w$ – weight increment upon activation

Empirical basis:

• Ebbinghaus (1885): forgetting curve – exponential memory decay
• Thorndike (1914): decay theory, memory trace
• Temporal networks (Holme & Saramaki, 2012): formalization of edge decay in dynamic networks

Implications for memetics:

Consolidation through spaced repetition:

Each co-activation of two connected memes (both above $\theta_{high}$) reduces the effective decay rate of their connection. This models the transition from early LTP to late LTP and further to structural LTP (formation of stable dendritic spines):

where:

• $\lambda$ – base decay coefficient (accounting for centrality and sign)
• $\kappa$ – consolidation factor (spaced repetition factor, typically 2.0)
• $n_{react}$ – number of co-activations accounting for the refractory period

Constraints:

• Refractory period ($\Delta t_{min} \geq 10$ steps): continuous co-presence in working memory does not count as separate repetitions. Models the neurobiological refractory period between LTP-induction sessions
• Positive edges only: consolidation ($n_{react}$) applies only to $w > 0$. Negative edges (antagonism) have different plasticity (iLTD/iLTP) and are not “learned” through repetition
• Open meme exclusion: edges incident on open memes (SIT gaps) do not accumulate $n_{react}$ – their resilience is ensured by decay resistance ($\lambda_{open} = \lambda \cdot 0.5$), not consolidation

Neurobiological basis:

Modulation by consolidation level $\kappa_i$: The base $\lambda$ additionally depends on the meme’s consolidation level – sensory memes ($\kappa = 0$) decay an order of magnitude faster, while LTM memes ($\kappa = 2$) decay an order of magnitude slower (see the section on consolidation level $\kappa_i$).

Differential decay:

Not all connections decay equally. The coefficient $\lambda$ depends on:

where $C(i)$, $C(j)$ are node centralities. Connections between central memes decay more slowly.

Asymmetric decay (negativity bias):

Additionally, $\lambda$ depends on the sign of the weight:

Negativity bias (Baumeister et al., 2001): negative information is processed more deeply and remembered better. Negative connections decay more slowly than positive ones. This explains the persistence of prejudices and phobias.

Molecular basis of edge decay:

Exponential weight decay is not merely a mathematical convenience but reflects concrete neurobiological processes:

Synaptic downscaling during sleep (Tononi & Cirelli, 2003, SHY) is the main mechanism of PRUNE: proportional weakening of all synapses -> strong pathways survive, weak ones are erased. This is the neurobiological realization of exponential decay.

Asymmetric decay ($\lambda_{neg} < \lambda_{pos}$): the negativity bias has a molecular basis – stress hormones (cortisol, norepinephrine) enhance the consolidation of negative associations via the amygdala. Adrenaline potentiates LTP in the basolateral amygdala, ensuring more durable encoding of negative connections.

Sleeper effect (Hovland et al., 1949): when $w_{ij}(t_0) < 0$ and there is no reinforcement of the negation, the absolute value $|w_{ij}|$ decays toward zero -> renewed contact in a new context can shift the weight into positive territory. This explains why rejected ideas may be accepted later.

Numerical example:

Parameters: $\lambda = 0.3$, $\Delta w = 0.6$.

Sign inversion

The sleeper effect is a special case of a more general mechanism of sign inversion, in which a connection abruptly changes polarity while preserving its absolute magnitude (see EMT, Part XXI).

Mechanism:

1. Meme $m$ has a negative connection to meme $X$: $w_{mX} < 0$ (rejected, antibody)
2. New memes $n_1, n_2, \ldots$ form positive connections to $m$ (new context, new information)
3. Cumulative pressure builds: $\sum_{k} w_{n_k, m} \cdot a_{n_k}$ grows
4. When $\sum w_{positive} > |w_{negative}|$ -> bifurcation: the sign flips
5. Fidelity is preserved (the meme is well-studied!) -> $|w|$ remains high
6. Result: the former “enemy” becomes a “hub” with the same strength

Formalization:

Inversion occurs when:

Key property – bifurcation, not gradient. Inversion occurs abruptly (phase transition), not gradually (Strogatz, 2015, Nonlinear Dynamics and Chaos – saddle-node bifurcation as a model of abrupt switching). This is confirmed by psychology: a sudden change of attitude toward a person, idea, or party – it “clicks” in a single moment rather than drifting gradually.

Connection to structural balance. Sign inversion is a mechanism for restoring structural balance (Heider, 1946, J. Psychology; Cartwright & Harary, 1956, Psychological Review – generalization to signed graphs). If a triple $(m, n, X)$ has an odd number of negative edges (an unstable configuration), the inversion of $w_{mX}$ restores parity -> balance.

Predictions:

Counter-intuitive predictions:

1. Fidelity preservation is quantitative: $|w_{after}| = |w_{before}|$. A former enemy becomes an ally of equal intensity, not a neutral acquaintance. “The convert’s zeal” is not a cognitive bias but a direct consequence of the mechanism: inversion changes the sign but not the absolute value of the edge (fidelity = a measure of how well-studied, not of attitude). Prediction: on attitude scales (feeling thermometer, semantic differential), the intensity of attitude toward the inverted object does not decrease when the sign changes. Converted anti-communists become equally fervent communists; former atheists become equally convinced believers. Testable with longitudinal attitude measurements before/after conversion.

2. Inversion via intermediaries is more effective than direct contact. The formula $\sum_{k \in N(m)} w_{mk} \cdot w_{kX} \cdot a_k > |w_{mX}^{direct}|$ means: inversion is triggered by indirect positive evidence (through mutual acquaintances, intermediary memes), not direct positive experience with the “enemy.” Counter-intuitive: common sense says “get to know your enemy personally, and the prejudice will fade” (contact hypothesis; Allport, 1954, The Nature of Prejudice). BMC predicts that direct contact without prior reframing through intermediaries is ineffective: it increases $|w_{mX}^{direct}|$ (familiarity) but does not change the sign. This is consistent with meta-analyses of the contact hypothesis: the effect of contact is significantly enhanced in the presence of institutional support and mutual acquaintances (Pettigrew & Tropp, 2006, J. Personality and Social Psychology) – in BMC terms, these are precisely “positive intermediaries.” Testable: compare the effectiveness of (a) direct contact with an outgroup vs. (b) stories from friends/authorities about positive experience with the outgroup. BMC predicts (b) > (a).

3. Hubs invert last, but the cascade from them is strongest. A hub $h$ (high centrality, many neighbors) has a higher inversion threshold: $|w_{hX}^{direct}|$ is large (the hub is well-“studied”), and more positive intermediaries are needed to overcome it. But if the hub inverts, it itself becomes a powerful intermediary for all of its $k_h$ neighbors -> avalanche-like cascading inversion. Prediction: (a) radical worldview shifts start from peripheral beliefs and cascade toward hubs, not the reverse; (b) a direct “attack on the central belief” (frontal argumentation) is the least effective persuasion strategy; (c) the distribution of inversion times in the network has a heavy tail: a long silent period (peripheral inversions) -> a sudden cascade (the hub has inverted). Testable by analyzing temporal patterns of attitude change in social networks.

Verification on the prototype: Scenario S3 (direct attack on the hub Science_trust: $-0.3$ stimulus + FEAR/DISGUST) confirms (a): the memeplex activates an immune response (FEAR up, DISGUST up), the hub reduces activation but does not invert – the frontal attack is ineffective. In scenario S8 (sleeper effect), inversion occurs through intermediaries (CARE, SEEKING -> positive context) – an indirect path, as the model predicts. MC results: S3 pass rate 50/50 (hub held), S8 pass rate 50/50 (inversion through intermediaries occurred).

Structural Incompleteness as a Source of Persistent Activation (SIT)

Edge decay explains forgetting: unused connections weaken according to $w(t) = w_0 \cdot e^{-\lambda t}$. But unsolved problems – “what is consciousness?”, “did I lock the door?”, “how to prove the Riemann hypothesis?” – do not obey this law. They return after days, months, years without an external trigger. The current formulas do not explain this phenomenon.

Structural Incompleteness Tension (SIT) – the tension of structural incompleteness – supplements edge decay with a second mechanism: structural gaps in the memeplex generate their own SEEKING activation, independent of external stimuli.

Definition of a structural gap

Definition. A structural gap $g$ in cluster $C$ is a position in the topology of $C$ possessing high betweenness potential (many pairs of nodes whose shortest paths would pass through $g$ if $g$ were a real node), but not filled by an actual meme.

Formally: let $B^*(g, C)$ be the betweenness centrality that node $g$ would receive if it were realized. Then:

A gap with high $relevance$ is a central “hole” in the cluster, without which many connections remain broken.

The SIT formula

where:

• $relevance(g) \in [0, 1]$ – the relative betweenness potential of position $g$
• $centrality(C) \in [0, 1]$ – eigenvector centrality of cluster $C$ in the full memeplex network (the more “important” the cluster, the higher the tension from its incompleteness)
• $closure(g) \in [0, 1]$ – the degree of gap closure (0 = fully open, 1 = closed)

The Learning Progress (LP) filter

Not all unfilled gaps generate activation. A problem on which there is no progress gradually “lets go”:

When $LP > 0$ – a sense of progress, SIT amplifies SEEKING. When $LP \to 0$ – stagnation, the SIT contribution fades. When $LP \gg 0$ suddenly (new information) – a return to the unsolved problem.

Effective SIT contribution to SEEKING:

where $f(LP)$ is a sigmoidal filter: $f(LP) = \sigma(k \cdot LP)$, providing rapid engagement when progress appears and smooth fading during stagnation.

False closure: why humanity invented gods

SIT generates persistent cognitive tension, and the brain strives to resolve it at any cost – even by filling the gap with a node of zero empirical validity. This is false closure:

With false closure, SIT formally decreases and tension drops – but the cluster becomes vulnerable to falsification. The mechanism explains:

Evolutionary logic: False closure is evolutionarily more advantageous than persistent SIT, because chronic SIT consumes cognitive resources. For survival, a wrong answer (low cognitive load) is better than an eternal question (constant drain on SEEKING). Science is the systematic replacement of false closure with valid closure, which requires cultural institutions that suppress the natural impulse to fill gaps quickly.

Contrast: edge decay vs. SIT

Numerical example

Consider two clusters: one with an open gap, the other fully closed.

Cluster A (unsolved problem “how does gravity work”):

• $gaps(A) = \{g_1\}$, $relevance(g_1) = 0.8$ (central question)
• $centrality(A) = 0.7$ (important cluster)
• $closure(g_1) = 0.3$ (partial answers exist)
• $LP(A, t) = 0.1$ (slow progress)

Cluster B (mastered skill “riding a bicycle”):

• $gaps(B) = \emptyset$
• $SIT(B) = 0$ – no gaps, no tension, normal edge decay operates

Cluster A generates SEEKING activation proportional to $SIT_{eff}(A, t)$, even without external stimuli. A physicist may not read about quantum gravity for years, but when a new paper appears ($LP$ spikes) – they immediately return to the problem.

See also: The biosubstrate of SIT (DMN, the extended SEEKING formula) – BM, Parts III–IV; conceptual exposition – EMT, Part VII; engineering implementation – AGI_F, Part III.

Motor learning example: tennis

Edge decay + selection manifest clearly in motor learning. Consider how a tennis player refines a stroke.

Motor variability as exploration (Dhawale et al., 2017, Annual Review of Neuroscience, DOI: 10.1146/annurev-neuro-072116-031548): the beginner generates many movement variants – this is analogous to creating edges between the meme “stroke” and various motor patterns. Variability is not noise but exploration of the solution space.

Reward-based retention (Galea et al., 2015, Current Biology, DOI: 10.1016/j.cub.2015.06.030): a dopaminergic signal upon a successful hit strengthens LTP for a specific pattern -> that variant receives a higher initial weight $w_0$ and decays more slowly.

Generalization gradient:

where $d$ is the “distance” of the variant from the successful pattern in motor space, and $\sigma$ is the generalization width.

Connection to Fidelity: Successful motor patterns -> dopaminergic reinforcement -> enhanced consolidation during sleep (Yang et al., 2014) -> high Fidelity. Unsuccessful ones -> no reinforcement -> edge decay -> variant lost.

Openness-gated reactivation

During Hebbian reactivation of edges (both nodes active above $\theta_{high}$), cross-cluster reactivation is modulated by openness $O$:

Justification: Openness to experience (McCrae & Costa, 1999) reflects readiness to integrate “foreign” memes. With age, $O$ decreases (Roberts et al., 2006), suppressing cross-cluster reactivation. This explains the growth of modularity with age ($Q_{old} > Q_{young}$): within-cluster connections continue to strengthen, while between-cluster connections lose their restoration mechanism.

Empirical basis: Wulff et al. (2019) showed that the semantic networks of older adults have higher modularity and fewer “bridge” connections – consistent with the selective suppression of cross-cluster reactivation at low $O$.

Associative memory vs. isolated memorization

Problem: Why are some facts remembered easily and for a long time, while others are forgotten within days? Why do all mnemonic techniques work through creating associations?

Network answer: A connected meme has higher centrality -> its connections decay more slowly according to the differential decay formula:

An isolated meme (no connections) -> low centrality -> high $\lambda$ -> rapid decay. A connected meme (many connections) -> high centrality -> low $\lambda$ -> slow decay.

Empirical basis:

• Craik & Lockhart (1972), “Levels of Processing”: deep (semantic) processing -> better memorization than shallow (phonetic). Network interpretation: semantic processing creates more connections.
• Bower et al. (1969): organized material is remembered 2–3 times better than a random list. Organization = structuring connections.
• Collins & Loftus (1975), spreading activation model: memory is a network where activation spreads along connections.

Neurobiological mechanism of association:

Overlapping engrams (Cai et al., 2016, Nature) – the physical mechanism of associative linking: memories formed in close temporal windows (~6 hours) share neurons, creating a causal link. Hippocampal structures provide two complementary processes:

• Dentate gyrus = pattern separation: splitting similar inputs into unique representations (creating distinct nodes)
• CA3 = pattern completion: restoring the full pattern from partial input (reactivation by trigger)

Method of loci (memory palace) works through hippocampal spatial coding: place cells and grid cells of the entorhinal cortex provide spatial anchoring, creating additional edges to spatial memes. Each “locus” = a spatial ensemble overlapping with the target meme.

Mnemonic techniques as connection creation:

Numerical example: influence of connections on forgetting time

Using the differential decay formula with parameters: $\lambda_0 = 0.3$ per day, $\alpha = 0.5$.

For simplicity: $C(i) \approx k_i$ (degree centrality).

Calculation: $\lambda_{eff} = \frac{0.3}{1 + 0.5 \cdot k}$; time $t^* = \frac{-\ln(0.1)}{\lambda_{eff}} = \frac{2.3}{\lambda_{eff}}$.

Conclusion: Mnemonic techniques increase degree -> decrease effective $\lambda$ -> slow decay -> increase Fidelity.

Practical takeaway: If you want to remember something for a long time – do not try to “memorize by rote.” Instead:

1. Create connections to what you already know (semantic)
2. Add spatial connections (method of loci)
3. Add emotional connections (vivid imagery)
4. Maintain connections through activation (spaced repetition)

Fidelity is orthogonal to weight

Fidelity (storage completeness) and weight (sign of the relationship) are independent characteristics. A meme can be well-studied and yet rejected.

Consequence for the Fidelity formula: Degree $k_m$ counts all connections of a meme – both positive and negative. A rejected meme with many negative connections can have high Fidelity because the memeplex stores it in detail for immune defense purposes.

Differentiated storage: completeness of preservation (Fidelity)

Problem: The brain has limited capacity. Storing thousands of memes in full is impossible.

Solution: Memes are stored with varying completeness (Fidelity) depending on rank and usage.

Relationship to consolidation level $\kappa_i$: Fidelity correlates with $\kappa$ but is not identical to it. A meme can be in LTM ($\kappa = 2$) with skeletal Fidelity – knowledge that has long been unused but is structurally stable (example: a forgotten foreign language with a preserved core). Conversely, a meme in STM ($\kappa = 1$) can have high initial Fidelity – detail-encoded but not yet consolidated. See the section on consolidation level $\kappa_i$.

Fidelity function:

where:

• $k_m$ – degree of meme $m$
• $k_{max}$ – maximum degree in the network
• $\gamma$ – preferential preservation exponent (typically 0.5–1.0)
• $\lambda_f$ – completeness decay rate (typically 0.01–0.1 per month)
• $t_{last}$ – time of last use
• $\beta$ – consolidation rate (typically 0.1–0.5)
• $age$ – time since first acquisition

Three components of the formula:

1. Rank factor $\frac{k_m^{\gamma}}{k_{max}^{\gamma}}$ – hubs are preserved more completely
2. Usage factor $e^{-\lambda_f (t - t_{last})}$ – what is unused degrades
3. Consolidation factor $(1 - e^{-\beta \cdot age})$ – older memes are more stable than new ones

Three storage modes

Key example: a foreign language

Scenario: 5 years of study -> 10 years without use -> one week in the environment

Why is reactivation fast (days) rather than slow (years)?

• During learning: new synapses are created (slow)
• During reactivation: existing weakened connections are strengthened (fast)
• The core (grammatical structures) is preserved — only “warming up” connections is needed

Reactivation Dynamics

where:

• $\rho$ — reactivation rate (typically 0.1–0.5 per day⁻¹)
• $exposure(t)$ — exposure intensity (0–1)

Important property: Reactivation rate depends on current Fidelity:

• At Fidelity = 0.2: $\frac{dF}{dt} = 0.3 \cdot 0.8 \cdot 1.0 = 0.24$ per day
• At Fidelity = 0.8: $\frac{dF}{dt} = 0.3 \cdot 0.2 \cdot 1.0 = 0.06$ per day

The lower the current completeness — the faster the initial reactivation (given a preserved core).

Adaptive Value of Differential Storage

Neurobiological basis: Physical storage structure and neurobiology of modes — see BM, Part IV.

Meme structure: Core, connections, details — see EMT, Part I.

Application to AGI: Skeletal storage in artificial systems — see AGI_F, Part III.

Consolidation Level $\kappa_i$: Multi-Level Memory as an Emergent Property

The mechanisms of spreading activation, edge decay, Fidelity, and SIT give rise to multi-level memory as an emergent property of graph dynamics — without a separate memory module. Memory levels emerge from combinations of existing meme parameters.

Definition. The consolidation level $\kappa_i(t) \in \{0, 1, 2\}$ of meme $m_i$ is a derived discrete variable:

$\kappa_i$ is not an independent parameter but a classifying function of $(I_{passed}, n_{react}, Fidelity, age)$. Analogy: thermodynamic state of matter (gas / liquid / solid) — derived from temperature and pressure, not an independent variable.

Notation remark. The symbol $\kappa$ in the formula $\lambda_{eff} = \lambda/(1 + \kappa \cdot n_{react})$ denotes the scalar consolidation factor (spaced repetition factor). Here $\kappa_i(t)$ is the consolidation level of meme $i$. Context distinguishes: $\kappa$ without subscript — scalar, $\kappa_i(t)$ — per-meme level.

Independence of $\kappa_i$, $a_i$, and SIT

Three meme variables — $\kappa_i$ (consolidation depth), $a_i$ (current activation), and $SIT_i$ (structural incompleteness) — are independent:

• $\kappa_i$ is determined by meme history: $(I_{passed}, n_{react}, Fidelity, age)$
• $a_i$ is determined by current dynamics: spreading activation, incoming signal
• $SIT_i$ is determined by the presence of an unclosed gap

Examples of combinations:

• $\kappa = 2$, $a_i \approx 0$: long-consolidated knowledge, currently inactive (“studied French 10 years ago”)
• $\kappa = 1$, $a_i > \theta_{act}$: fresh idea, not yet consolidated, but currently driving behavior
• $\kappa = 2$, $SIT > 0$: a question a person has been pondering for years (“what is consciousness?”)

Limitation: A meme with $\kappa = 0$ (in the S-layer) has not yet passed the I-filter — it can only have high current $a_i$ (sensory input) and cannot carry SIT.

$\kappa$-Transition Conditions

How sleep replay leads to fulfilling these conditions: Each nightly replay increases $n_{react}$; cumulatively over days–weeks, core memes reach $N_{crit}$. The full process is formalized below — see Consolidation Process. | $2 \to 1$ | $F < F_{STM}$ AND $n_{react}$ not growing; OR reconsolidation destabilization ($\Delta_{PE} > \theta_{destab}$); OR sustained I-suppression ($\tau_{supp} \geq T_{suppress}$) | Deconsolidation | Spine retraction, late-LTP degradation; anisomycin-blockable lability | | $1 \to \varnothing$ | $a_i < \theta_{prune}$ for $T_{prune}$ AND $F < F_{trace}$ | Pruning | Synaptic downscaling (SHY) |

Dependence of Decay Rate on $\kappa_i$

The base $\lambda$ from the edge decay formula is modulated by consolidation level:

Sensory memes ($\kappa = 0$) decay 10× faster than the base rate; LTM memes ($\kappa = 2$) — 10× slower. The full decay formula accounting for all modulators:

where $c_{sr}$ is the spaced repetition factor (denoted $\kappa$ in the edge decay formula above, renamed to avoid collision with $\kappa_i$), $s(w) = 1$ for $w > 0$, $s(w) = \lambda_{neg}/\lambda_{pos} < 1$ for $w < 0$ (negativity bias).

Engram Allocation: Competition for Consolidation

Not all STM memes ($\kappa = 1$) transition to LTM. Memes compete for limited consolidation resources — analogous to CREB/excitability allocation in neuroscience (Josselyn & Frankland, 2015):

where $C_E$ is eigenvector centrality (hubs have priority), $G_{align}$ is alignment with G-drives, $E_{emot}$ is emotional intensity. Highly central memes with emotional G-linkage “win” the competitive selection.

Consequence: memory linking — memes created within a close temporal window ($\Delta t < T_{link} \sim 6$ h) share co-activation → acquire joint edges → consolidate together (Cai et al., 2016, Nature).

Schema-Congruence and $\kappa$-Transition Speed

The speed of the $1 \to 2$ transition depends on the meme’s compatibility with the existing memplex:

U-shaped curve: both congruent and very novel material is remembered well — through different mechanisms. This coincides with the SLIMM model (van Kesteren et al., 2012): congruent → mPFC → fast integration; incongruent → hippocampus → detailed encoding.

SIT / Prediction Error as $\kappa$-Transition Gate

SIT directly determines the type of encoding:

WM as a Pointer System over $\kappa$-Levels

Working memory (top-k by salience) is not a separate storage level but a pointer system to memes at any $\kappa$-level. A WM pointer binds a meme to the current context:

The capacity bottleneck is not information volume but the number of content-independent pointers (~3–4 primary + chunking). Chunking = grouping related memes (memplex) under one pointer (Oberauer, 2025, Trends in Cognitive Sciences). The next level is automatization: $wm\_cost$ decreases from 1 (chunked) to 0 (automatic), fully freeing the pointer.

WM Ontogenesis: $k_{active}(t_{dev})$

The ceiling of ~4 active pointers is a property of the mature substrate. An immature substrate has a lower ceiling: theta-gamma coupling in PFC matures gradually as myelination progresses, and the number of gamma cycles fitting within one theta cycle increases with age (Bola & Bhatt, 2024, Consciousness and Cognition; see BM: PFC myelination). Convergence at ~4 active slots in crows, macaques, and rats (Hahn et al., 2021) points to an evolutionary optimum, not a Homo sapiens artifact.

Formula:

where $k_{min} \approx 1$ (neonatal substrate), $k_{max} \approx 4$ (mature substrate), $t_{0.5}$ is the half-saturation point, $\lambda_{dev}$ is the steepness. Parameters $t_{0.5}$ and $\lambda_{dev}$ are species-dependent and individually variable (analogous to $T_i$).

Empirical anchors:

Sources: Cowan, 2001; Gathercole et al., 2004, Memory; Gogtay et al., 2004, PNAS.

Consequences:

1. Auto(S) is impossible at $k = 1$: $|S| \geq 2$ requires $k \geq 2$ for chain learning → automatization is impossible before ~3 years.
2. Age barrier of transmission: A meme requiring $n$ simultaneous elements cannot be transmitted when $k_{active} < n$. Hence the age stratification of culture.
3. Critical threshold ~7 years: $k \approx 3$ = minimum for Piaget’s concrete operations + tertiary binding. The “age of reason” is not a metaphor but a $k$ threshold.
4. $k$ is a G-parameter: set by the substrate (PFC maturation), not by learning. Training improves chunking and strategies, but not $k_{active}$.
5. Decline: PFC atrophy (>65) → theta-gamma decoupling → $k_{active}$↓ → compensated by stigmergy (notebooks, rituals) + highly automatized skills ($habit \to 1$) are preserved.

G→WM Competition: $k_{eff}(t)$

$k_{active}(t_{dev})$ sets the ceiling, but available WM capacity may be lower. Two channels reduce available capacity:

1. G-capture (affective, subcortical): G-programs at high activation capture PFC computational resources, reducing the number of free pointers (Pessoa, 2009, TiCS). Empirically: CDA K-score drops from ~3.5 to ~2 under negative affect — a loss of ~50% WM capacity (Figueira et al., 2017, SCAN).
2. Signal-capture (communicative, cortical): Signal memes (grounding, routing, fidelity maintenance) occupy WM slots, displacing navigational and survival-relevant memes. At $k_{eff} \approx 3\text{–}4$, even one signal meme = 25–33% loss of survival-relevant capacity.

Formula:

$n_{captured}^{signal}(t)$ = number of WM slots occupied by active signal memes (grounding associations, routing state, fidelity maintenance). Determined by the number of active communication processes — not analogous to $w^{capture}$ (no fixed weights; depends on communicative load).

Constraint: $k_{eff} \geq 1$. At $k_{eff} \to 0$ — panic freeze: loss of deliberation, Self(t) narrows to $\{u: a_u > \theta\}$ (nearly pure G without M).

Evolutionary consequence (Language Emergence Threshold): Language = permanent M-capture of WM. It arises only when 4 conditions are simultaneously met: (1) resource surplus (rich environment), (2) $k_{eff}$ sufficient for survival + signal, (3) executive planning (converting signal to navigation), (4) memetic pressure (social density). See EMT Part XXVIII.

Capture weights:

Remark: $T$ is already accounted for — $a_u(t)$ includes T-modulation ($a_{FEAR}$ depends on $T_{FEAR}$). $w^{capture}$ is a property of the G-program, not of the individual.

$\psi$-traces for utility nodes. G-programs leave their own synaptic traces. Dynamics are analogous to memes, but with different parameters:

where $\lambda_\psi^G < \lambda_\psi^M$: emotional traces decay slower than cognitive ones (amygdala-mediated consolidation). Difference from meme $\psi$: utility nodes have no consolidation level ($\kappa$ is not applicable — G-programs are innate, not consolidated). $\psi_u$ describes temporary sensitization of the circuit, not storage.

PTSD loop. Traumatic FEAR: $a_{FEAR} \gg \theta_{act}$ → $\psi_{FEAR}$ is anomalously high. After deactivation: $\psi_{FEAR}$ decays slowly ($\lambda_\psi^G < \lambda_\psi^M$). Contextual trigger → ping → FEAR reactivation → WM-capture ($n_{captured}$↑) → $k_{eff}$↓ → inability to deliberate → no rational override → $\psi_{FEAR}$ refresh → vicious cycle. Formally: PTSD = $\psi_u$ stuck above $\theta_\psi$ with $\lambda_\psi^G \to 0$.

Link to $P_{auto}$: G-activation → $k_{eff}$↓ → $WM\_load$↑ → $P_{auto}$↑ → relapse to the most entrenched habit (see automatization). This closes the chain: affect → WM-capture → automatic behavior.

Interaction with Self(t). $Self(t) = SMC \cup WM(t) \cup \{u: a_u > \theta\}$. Utility nodes remain in the third set (not in WM), but their capture reduces $|WM(t)|$. In panic, $Self(t)$ narrows to $\{u: a_u > \theta\}$ — nearly pure G without M.

Prediction P-WM1: FEAR induction reduces CDA (K-score) by ~50%; PLAY induction does not reduce or slightly increases it. Test: CDA change-detection under fear-induction vs play-induction vs neutral (Eysenck et al., 2007, Cognition & Emotion; Stout et al., 2017, Scientific Reports).

Neurobiological basis: Theta-gamma coupling, PFC myelination, dual competition — see BM: PFC myelination, BM: G-programs and WM.

Engineering implementation: k_eff pseudocode, capture weights — see AGI_F: k_eff.

Activity-Silent WM: Synaptic Trace $\psi$

The pointer system (top-k by salience) describes active working memory — memes in the focus of attention. However, neuroscientific data show that information is stored in WM in two ways — through sustained neural activation (active WM) and through short-term synaptic traces (activity-silent WM; Stokes, 2015, TiCS; Wolff et al., 2017, Nature Neuroscience). This explains the classic discrepancy: ~3–4 active items (Cowan, 2001) + ~3–4 latent = 7±2 (Miller, 1956).

Computationally verified. Working memory capacity $k \approx 4$ has been reproduced as an evolutionary equilibrium in the BMC engine: populations with heritable WM capacity and metabolic brain cost in a delayed-reward foraging environment converge to $k \approx 4.4$ from both directions — Cowan’s $4 \pm 1$ range. Three-point sweep ($\beta_{wm}$: 0.04, 0.06, 0.08) confirmed the inverted U-curve. See DOI: 10.5281/zenodo.19310012.

Definition. For each meme $m_i$ we introduce a third continuous variable — the synaptic trace:

Three axes of a meme:

• $a_i(t) \in [0,1]$ — activation (access to attention)
• $\kappa_i(t) \in \{0, 1, 2\}$ — consolidation depth (sensory / STM / LTM)
• $\psi_i(t) \in [0,1]$ — synaptic trace (short-term synaptic potentiation)

All three variables are independent: a meme with $a_i \approx 0$ can have $\psi_i > 0$ (latent in WM — not in consciousness, but recoverable) or $\psi_i = 0$ (not in WM). Limitation: $\psi_i > 0$ is impossible at $\kappa_i = 0$ (meme has not passed the I-filter — no encoding in WM).

Dynamics of $\psi$:

where $\lambda_\psi$ is the trace decay rate ($T_{1/2} \sim$ minutes). When a meme enters Active WM through top-k selection: $\psi_i \leftarrow 1$. Upon repeated pinging: $\psi_i$ is refreshed ($\psi_i \leftarrow \min(1, \psi_i + \delta_{refresh})$), the meme remains latent without full reactivation.

Two-compartment WM model:

Pinging — reactivation of latent memes. A contextual signal (spreading activation from active memes) can reactivate a latent meme:

Reactivation condition:

where $salience_i^{ping}$ is computed using the standard salience formula with boosted activation. Reactivation is not automatic: the meme must pass top-k competition, displacing a less salience-relevant meme from Active WM. If $a_i + a_i^{ping} < \theta_{act}$, but $\psi_i > \theta_\psi$ — the meme remains latent, and $\psi_i$ receives a refresh ($\psi_i \leftarrow \min(1, \psi_i + \delta_{refresh})$).

SIT-gap driven pinging (theoretical extension): an open SIT-gap with non-zero activation periodically generates spreading activation reaching latent memes linked to the gap’s theme. This is the BMC explanation of the Zeigarnik effect for WM: unfinished tasks are maintained in Latent WM because gap pulsation keeps $\psi_i > \theta_\psi$.

Pointer lifecycle:

$\psi$ and chunking: When a chunk (memplex linked by a single WM pointer) is deactivated, all member memes receive $\psi_i = \psi_{chunk}$. When one meme from the chunk is pinged — reactivation spreads to the entire chunk through strong internal edges. Chunking effectively increases Latent WM capacity: a single $\psi$-trace addresses an entire memplex.

Latent WM and coherence. The coherence constraint ($Binding_{comp}$, Part XIII) applies only to Active WM — memes in the focus of attention must be mutually non-contradictory. Latent WM is not constrained by coherence: latent memes are not in consciousness, and contradictions between them are not experienced. Upon pinging and reactivation, a latent meme enters Active WM and is subjected to competitive binding — it may be rejected if $D(m_i, m_j) > \theta_D$ with memes in Active WM. This explains “intrusive thoughts” — a latent meme with high $\psi$ and strong ping is reactivated despite conflict with the current Active WM, causing a sensation of dissonance.

Predictions:

Neurobiological basis of $\kappa$ and $\psi$: Molecular markers of $\kappa$-levels, transition mechanisms, and the synaptic substrate of activity-silent WM ($\psi$) — see BM, Part III. Engram allocation: Josselyn & Frankland (2015). Memory linking: Cai et al. (2016), Nature. Schema-congruence: van Kesteren et al. (2012), SLIMM. Activity-silent WM: Stokes (2015), TiCS; Rose et al. (2016), JoCN; Wolff et al. (2017), Nature Neuroscience. Generative model of memory: Spens & Burgess (2024), Nature Human Behaviour.

Conceptual framework: Multi-level memory and activity-silent WM — see EMT, Part VIII.

Engineering implementation: $\kappa$ and $\psi$ as node attributes, two-compartment WM — see AGI_F, Part III.

Episodic Memory: Discretization of Experience

$\kappa$-levels determine the depth of meme storage. But memory is organized not only by depth — it is discretized into episodes: time-bounded sets of co-active memes indexed by content-independent markers.

Episode as a Formal Structure

Definition. An episode $\varepsilon_k$ is a tuple:

where:

• $B_k \in \{0,1\}^{|V|}$ — barcode (sparse binary vector, $\|B_k\|_0 / |V| \approx 0.05{-}0.10$)
• $M_k(t) = \{m_i : a_i(t) > \theta_{episode}\}$ — set of memes active in episode $k$
• $\tau_k = [t_{start}^k, \; t_{end}^k]$ — temporal interval
• $\partial\varepsilon_k$ — boundary conditions (what triggered the beginning/end)

The barcode $B_k$ is a content-independent index: a random sparse pattern generated by the hippocampus at the start of a new episode. It does not encode content — it addresses it, similar to a hash code (Chettih et al., 2024, Cell).

Difference from $\kappa$: $\kappa$ is a property of an individual meme (how consolidated it is). An episode is a property of a group of memes (what was co-active in one temporal interval). A meme can participate in multiple episodes, and its $\kappa$ does not depend on which episode it first appeared in.

Event Boundary Detection: When One Episode Ends and Another Begins

An episode boundary arises upon abrupt change of memplex state. Formally, the system monitors a context vector $\mathbf{c}(t)$ — the aggregate of current activations:

A new episode $\varepsilon_{k+1}$ begins when at least one of the following conditions is met:

When a boundary fires:

1. The current $\varepsilon_k$ is closed: $t_{end}^k = t$
2. A new barcode $B_{k+1}$ is generated (sparse random co-activation)
3. $\varepsilon_{k+1}$ opens with $t_{start}^{k+1} = t$

Link to $\kappa$-gating (see above): PE serves a dual function — it determines both the fate of an individual meme ($\kappa$-transition) and an episode boundary. But the thresholds differ: $\theta_{boundary} > \theta_{\kappa}$ — not every creation of a new meme opens a new episode.

Temporal Order within an Episode: Time Cells

Within a single $\varepsilon_k$, memes are not merely co-active — they are temporally ordered. In addition to theta-binding (~125 ms, synchronizing components within a single moment, see Part XIV), there exists a slower scale: time cells encode the meme’s position on the episode’s temporal axis (seconds — minutes).

Formally, each meme $m_i$ in episode $\varepsilon_k$ receives an ordinal label:

The label $\phi_k$ is computed upon episode closure (when $t_{end}^k$ is known) or during replay; during an active episode, the absolute timestamp $t_{activate}(m_i)$ is used.

This allows retrieval to reconstruct not only what was in the episode but also in what order — a necessary condition for narrative memory.

Two levels of temporal encoding:

Retrieval: Pattern Completion via Barcode

Episode retrieval = partial reactivation of its barcode. It suffices to activate a portion of $B_k$ so that spreading activation restores the entire pattern:

where $\rho_{min} \approx 0.3$ — the minimum fraction of reactivated barcode for full reconstruction.

A cue activates memes that through shared edges activate part of the barcode → the barcode through pattern completion restores the remaining memes of the episode. Hence:

• Retrieval specificity: different barcodes are orthogonal (sparse + random), confusion between episodes is minimal
• Contextual cue: the same stimulus in different episodes is linked to different barcodes → one can recall the specific episode needed
• Partial recall: at $\rho < \rho_{min}$ — only fragments of the episode, order is lost (tip-of-the-tongue effect)

Temporal Chaining: Sequence of Episodes

Sequential episodes are linked through context overlap at the boundary:

The more memes remain active across the boundary — the stronger the associative link between episodes. Also: overlapping engrams (Cai et al., 2016) create a structural bridge — episodes within ~6 hours literally share graph nodes.

Barcode Lifecycle

The barcode is not a permanent structure. Its dynamics:

Ontological status of the barcode: $B_k$ is not a meme (graph node) but an index structure (episode metadata). A barcode has no $\kappa$ of its own; its persistence is determined by replay frequency. Without reactivation, the pattern $B_k$ fades — not through $\kappa$-transitions, but through loss of co-activation coherence among nodes in $\text{supp}(B_k)$.

This is trace transformation: episodic memory (detailed, with barcode) → semantic memory (gist, without barcode). BMC predicts this transformation from first principles: content strengthened through replay transitions to $\kappa = 2$; the barcode as an index structure without G-linkage and without replay loses coherence and disappears.

Exception: Emotionally significant episodes ($E_{emotional} > \theta_E$) retain their barcode longer — through the amygdala-hippocampal loop. Hence vivid flashbulb memories: barcode is preserved → retrieval is complete, with details and order.

SIT-Gaps and Episodes: Orthogonal Timescales

A SIT-gap (open meme, Part V) lives across episodes:

ε₁ ──── ε₂ ──── ε₃ ──── ε₄ ──── ε₅
  gap_A: ━━━━━━━━━━━━━━━━━━━━━━━ (closure in ε₅)
  gap_B: ━━━━━━━━━ (closure in ε₂)
  gap_C:                ━━━━━━━━━━━━━ (open)

Connections:

• A gap causes thematic coherence between episodes: $\varepsilon_1$ and $\varepsilon_3$ are linked not only chronologically but also through shared gap_A
• Partial closure of a gap within $\varepsilon_k$ → PE → $\varepsilon_k$ is tagged for consolidation
• Full closure ($closure \geq 1.0$) → strong event boundary → $\varepsilon_k$ terminates

Zeigarnik in episodic terms: An open gap prevents associated episodes from being forgotten — barcodes $B_k$ with an unclosed gap do not decay because the gap generates periodic reactivation (SIT-driven pulsation → partial replay → barcode is maintained).

Predictions

Neurobiological basis: Barcodes (Chettih et al., 2024), time cells, event boundary neurons — see BM, Part IV. Hippocampal indexing theory: Teyler & DiScenna (1986).

Conceptual framework: Episodes as discrete units of experience — see EMT, Part VIII.

Engineering implementation: Episode struct, barcode generation, boundary detection — see AGI_F, Part III.

Overnight Network Optimization: Sleep as a Network Process

Network interpretation of sleep:

The unit of overnight consolidation is the episode $\varepsilon_k$ (see Episodic Memory). Sleep SWR reactivates barcode $B_k$ → through pattern completion restores $M_k$ → the following operations are performed:

1. Decomposition (DECOMPOSE): Episode $\varepsilon_k$ → set of components (features) — SWS
2. Binding (CONNECT): New edges between features and existing categories — SWS→neocortex
3. Recombination (BLEND): Recombination of components from different clusters — REM. Formal operation: $m_{new} = blend(m_i, m_j)$ for nodes from different communities (see Meme Synthesis)
4. Pruning (PRUNE): Removal of weak edges ($|w| < \theta_{min}$) — synaptic homeostasis
5. Strengthening (STRENGTHEN): Increasing weights of emotional and frequently reactivated edges

Effect of sleep on edge decay:

where $\eta \approx 0.3$ — sleep efficiency.

Consolidation process: How operations on episodes cumulatively produce $\kappa$-transitions, SWR-tagging, trace transformation — see Consolidation Process below.

Neurobiological basis: Triple coupling and the decomposition mechanism — see BM: Sleep as Meme Consolidation Mechanism.

Verified sources: Staresina et al. (2015, Nature Neuroscience) — hierarchical nesting in humans; Latchoumane et al. (2017, Neuron) — causal proof via optogenetics; Yang et al. (2014, Science) — spine formation during sleep; Tononi & Cirelli (2014, Neuron) — SHY review: synaptic downscaling as the basis of PRUNE.

Consolidation Process: From Tagging to κ-Transition

The κ-levels section defines the conditions for transitions, the overnight optimization section defines the operations on the graph. This section formalizes the process linking them: how operations on episodes cumulatively produce $\kappa$-transitions.

SWR-Tagging: Selection of Episodes for Consolidation

Consolidation begins not during sleep, but during wakefulness. During pauses in activity (task completion, distraction), awake SWRs occur that reactivate recent episodes and tag them for overnight consolidation (Buzsaki, 2024, Science: 5–20 SWR after significant experience).

Tagging is a two-level process:

1. Meme → engram allocation (defined above): $P_{cons}(m_i) \propto C_E(m_i) \cdot (1 + \gamma_G \cdot G_{align}) \cdot (1 + \gamma_E \cdot E_{emot})$

2. Episode → SWR-tag: the episode inherits the priority of its best meme with a recency correction:

where $\lambda_r$ is the recency decay rate ($T_{1/2} \sim 1$ h), $gap\_rel(\varepsilon_k) = \max\{1 - closure_j : gap_j \text{ active in } \varepsilon_k\}$.

The second argument of $\max$ is the Zeigarnik correction: an episode with an unclosed SIT-gap ignores recency decay because the gap maintains periodic reactivation.

3. Sleep → probability of successful replay: determined by oscillation quality (SO–spindle–SWR triple coupling, defined in BM):

where $\alpha_{SO}, \alpha_{spindle}, \alpha_{SWR} \in [0,1]$ are normalized oscillation amplitudes (defined in BM).

Full pipeline: $P_{cons}(m_i) \to P_{tag}(\varepsilon_k) \to P_{cons}^{night}$ — three filters, each of which can block consolidation.

Consolidation Cascade

Each sleep cycle (SWS + REM) performs a sequence of operations on tagged episodes. Operations are defined in overnight optimization; here their ordering and cumulative effect are formalized:

FOR each sleep cycle (n = 1, 2, ...):
  FOR each ε_k with P_tag > θ_tag (typically ~ 0.2):
    1. REPLAY:      reactivate B_k → pattern completion → M_k
    2. DECOMPOSE:    extract features from M_k (SWS)
    3. CONNECT:      link features to existing schemas (SWS → neocortex)
    4. BLEND:        recombine with other ε (REM)
    5. PRUNE:        weaken peripheral connections (SHY)
    6. STRENGTHEN:   strengthen core connections (replay-induced L-LTP)
    7. κ-ASSESS:     recompute κ_i for each m_i ∈ M_k
  STOP replay ε_k WHEN:
    — all core memes have reached κ = 2, OR
    — P_tag(ε_k) has fallen below θ_tag (recency faded, gap closed)

At step 1, replay increases $n_{react}$ for co-active memes. At step 7, memes are evaluated against standard $\kappa$-transition conditions. Cumulative effect: after $N$ cycles, core memes accumulate $n_{react} \geq N_{crit}$ → $\kappa: 1 \to 2$. Cessation of replay → barcode $B_k$ loses support → fades.

The capacity of a single night is limited: the number of replay slots is $\sim 10{-}20$ episodes per night. Episodes compete by $P_{tag}$ for these slots — the strongest win.

Trace Transformation: Detail → Gist

Repeated replay transforms the episode — it does not copy but restructures (Moscovitch, 2024; Nature, 2025). Formalization through per-meme dynamics within $\varepsilon_k$:

Core memes ($C_E(m_i) > \theta_{core}$ within $M_k$, where $\theta_{core}$ is the median $C_E$ across $M_k$): each replay → $n_{react}$++ → STRENGTHEN → weight++ → $Fidelity$ grows:

Peripheral memes ($C_E(m_i) < \theta_{core}$): SHY downscaling → weight– → $Fidelity$ decays:

where $n$ is the number of replay cycles, $\rho_c \sim 0.3$ is the consolidation strengthening rate (not to be confused with $\rho$ of reactivation from Fidelity), $\beta \sim 0.5$ is the consolidation rate, $\lambda_{SHY} \sim 0.2$ is the homeostatic downscaling rate.

Barcode $B_k$: maintained as long as replay continues. When core memes are integrated into the neocortex ($\kappa = 2$), replay of $\varepsilon_k$ ceases → $B_k$ fades → retrieval via $B_k$ becomes impossible → memory transitions from episodic to semantic.

Result: gist = surviving core memes with $\kappa = 2$; details = pruned peripheral memes; barcode = lost. Mature memory = semantic knowledge without the episodic “wrapper.”

Schema-Congruence: Modulation of Consolidation Speed

The number of replay cycles required for $\kappa: 1 \to 2$ depends on the meme’s I-compatibility with the existing memplex (continuation of the SLIMM analysis from the $\kappa$ section):

U-shaped curve: both the familiar (many anchor points in the schema) and the shockingly novel (emotional boost) consolidate quickly; slowest of all is the moderately novel without emotion.

Temporal Linking: Co-Consolidation of Close Episodes

Episodes within the temporal linking window ($\Delta t < T_{link}$) share engram neurons (Cai et al., 2016, Nature) → during replay of one $\varepsilon_j$, $\varepsilon_k$ is partially reactivated → co-consolidation:

where $T_{link} \sim 6$ h. The first factor is temporal proximity; the second is content overlap (Jaccard index).

Consequence: events of the same day consolidate as a cluster → forming “daily narratives” in autobiographical memory.

Temporal Scale of Consolidation

Exception: Flashbulb memories — the emotional tag (amygdala) protects the barcode from fading → episodic details are preserved for years.

Predictions

$\kappa$-transition conditions: engram allocation, decay rates — see $\kappa$-levels above.

Operations on the graph: DECOMPOSE/BLEND/PRUNE/STRENGTHEN — see overnight optimization above.

Neurobiological basis: Triple coupling, SWR-tagging, SHY, spine formation — see BM, Part IV. Trace transformation at the molecular level — see BM: Trace transformation.

Conceptual framework: Consolidation as active reconstruction — see EMT, Part VIII.

Engineering implementation: sleep_cycle(), swr_tag(), trace_transform() — see AGI_F, Part III.

Active Forgetting and Reconsolidation

Passive decay ($w_{ij} \to 0$ in the absence of activation) is the only forgetting mechanism formalized above. However, neuroscience shows that forgetting can be active: the I-system deliberately suppresses specific memes (Anderson & Green, 2001; Yi Zhong et al., 2012 — Rac1/Cdc42 cascade). Furthermore, memories retrieved from LTM become labile — reconsolidation (Nader et al., 2000).

Two Types of Forgetting

I-Suppression Signal

Summation over all nodes with negative edges to $m_i$. The sign is inverted ($-w_{ki} > 0$) so that $I_{sig} \geq 0$. The more active the I-meme and the stronger the negative connection, the stronger the suppression.

When $I_{sig}(m_i, t) > \theta_I$:

Sustained suppression → fidelity damage. We introduce a counter $\tau_{supp}(m_i)$ — the number of consecutive steps in which both conditions are met:

When $\tau_{supp} \geq T_{suppress}$ (sustained suppression):

Prolonged suppression does not merely deactivate the meme — it destroys its consolidation (analogous to Rac1-induced actin depolymerization, AMPAR internalization).

Gap archiving upon sustained suppression: If I-suppression has led to $a_i < \theta_{prune}$ and the meme has an associated GAP-node → $archived\_gaps.add(gap_i)$. Without this, open_memes would contain “dead” gaps, inflating SIT.

Four Mechanisms of Active Forgetting

Mechanisms 2–4 were formalized earlier (WM displacement, engram interference, overnight pruning). Mechanism 1 (I-mediated suppression) is new, formalized above.

Retrieval-Induced Forgetting (RIF)

Recall of meme $m_i$ strengthens $m_i$, but suppresses competing memes through lateral inhibition (Anderson et al., 1994; Anderson, 2003). A competitor = a meme with high semantic similarity but not an ally ($w_{ij} < w_{ally}$; ally memes are protected from RIF):

where $w_{ally} \approx 0.3$ — threshold: edges above this value indicate alliance.

Link to hub protection:

Peripheral memes (low eigenvector centrality) are suppressed under RIF more strongly than hubs. Hubs are protected by virtue of multiple supporting connections.

Reconsolidation: Plasticity Window

Definition of recall. In the context of reconsolidation, recall is the reactivation of a consolidated meme:

Only LTM memes ($\kappa = 2$) are subject to reconsolidation. Activation crossing $\theta_{act}$ from below = recall.

Lability trigger: $recalled(m_i, t_r) = true$ AND prediction error $\Delta_{PE}(t_r) > \theta_{recon}$.

Lability function:

$\tau_{recon}$ — window duration (biology: ~6 hours; AGI: N steps). While $Labile > \theta_{labile}$, the meme is in a labile state: its Fidelity naturally degrades (analogous to the need for protein resynthesis per Nader):

Reconsolidation outcomes:

Boundary Conditions of Reconsolidation

Relationship between I-Suppression and Reconsolidation

I-suppression and reconsolidation are two independent mechanisms for weakening a meme:

• I-suppression: deliberate suppression via $I_{sig}$ without recall; the meme is deactivated, with sustained suppression — fidelity damage.
• Reconsolidation: Fidelity degradation upon recall + PE; without I-involvement.

The mechanisms can coincide: sustained I-suppression weakens a meme → recall in this state → lability is facilitated (low $F_i$ → less resistance to reconsolidation per boundary conditions).

Predictions

Conceptual framework: Forgetting as an immune function, therapeutic reconsolidation — see EMT, Part VIII.

Neurobiological basis: Rac1/Cdc42, GABA inhibition, Nader (2000) — see BM, Part IV.

Engineering implementation: i_suppress_cycle(), check_reconsolidation(), active_forget_cycle() — see AGI_F, Part III.

Automatization: Transition from WM-Dependent to WM-Independent Execution

Blocks 1–5 formalized storage, access, and forgetting of memes. Automatization is the next level: the transition from expensive WM-dependent execution of behavioral sequences to WM-independent execution. This is not an additional memory system but a necessary consequence of M » G with limited WM — if all pointers are occupied by routine tasks, there are no resources for new memes, and the memplex stagnates. Automatization removes this constraint.

Automatic Chain Auto(S)

Let $S = (m_1, m_2, \ldots, m_k)$ be a sequence of memes with $w(m_i, m_{i+1}) > 0$ for all $i$. We define:

where $w_{auto} \approx 0.7$ is the threshold edge weight, $N_{auto} \approx 20$ is the minimum number of successful executions.

Interpretation: A sequence is considered automatic if (a) all edges in the chain are sufficiently strong — no “gaps” requiring WM control, and (b) the chain has been executed sufficiently often — routineness is confirmed. Chunking (Block 4) groups memes under one pointer — automatization goes further: the chain requires no pointer at all.

Semantics: Activation of the trigger $m_1$ cascades along strong edges through $m_2, \ldots, m_k$ without involvement of the WM pointer system. WM does not track intermediate steps — this is not “unconscious monitoring” but the absence of monitoring: the sequence unfolds autonomously.

Habit Dynamics

For each meme $m_i$ participating in behavioral chains, we introduce a continuous variable:

Update after each execution:

where $\alpha_h \approx 0.05$ is the learning rate, $\mathbb{1}[exec\_success]$ is the indicator of successful execution (errors do not strengthen the habit), and:

$\beta_{motor} \approx 0.3$ is the motor consolidation bonus during sleep (~30% enhancement; Solano et al., 2024, J Neurosci), $T_{crit} \approx 1h$ is the critical window between training and sleep.

Full automatization threshold: $\theta_{habit} \approx 0.8$. When $habit_i > \theta_{habit}$ for all $m_i \in S$ — the chain transitions to automatic mode. Relationship to Auto(S): successful execution strengthens edges via the Hebbian mechanism ($w(m_i, m_{i+1}) \uparrow$ upon joint activation), so habit growth usually accompanies weight growth, and $habit > \theta_{habit}$ is a sufficient condition for satisfying the edge-weight clause of Auto(S).

Orthogonality: $F_i \perp habit_i$ — Fidelity (content accuracy) and habit (execution fluency) are independent axes. A meme can be high-fidelity but not automatized (a new skill), or automatized but with degraded content (a bad habit with an outdated world model).

WM-Release Rule

Three WM-cost regimes for a sequence of length $k$:

Formally:

When $Auto(S) = true$: activation of the trigger $m_1$ launches the cascade $m_1 \to m_2 \to \ldots \to m_k$ through strong edges ($w > w_{auto}$) without allocating WM pointers. Freed pointers are available for parallel tasks — an expert drives “on autopilot” and simultaneously carries on a conversation.

DMS/DLS Competition: Deliberative vs Automatic

Probability of automatic execution:

where $habit(S) = \min_i habit_i$ for $m_i \in S$, $\gamma_h \approx 2$ is the transition steepness coefficient.

Critical property: Turner et al. (2022, J Neurosci): the deliberative system (DMS) functionally competes with the automatic system (DLS) at the beginning of learning — DMS engagement slows the transition to DLS-dominant mode; the novice cannot “let go of control” even if individual steps are already strengthened. High $WM\_load$ accelerates the transition to automatic mode: when WM is overloaded, deliberative inhibition weakens, and DLS “takes over” control.

Distinction of PE types (Lakshminarasimhan et al., 2024): Goal-directed = reward prediction error (mismatch of expected reward); habitual = action prediction error (mismatch of expected action). The two systems use different error signals, which allows them to operate in parallel on different chains.

MDL / Policy Distillation

Moskovitz et al. (2024, NeurIPS): automatization as behavioral compression following the Minimum Description Length (MDL) principle. The deliberative policy $\pi$ (complete, expensive) is gradually “distilled” into the automatic policy $\pi_0$ (compressed, cheap). S1/S2 (Kahneman, 2011) are not two separate modules but two ends of a continuum of $habit$:

• $habit \approx 0$: System 2 (deliberative, $\pi$, high WM cost)
• $habit \approx 1$: System 1 (automatic, $\pi_0$, zero WM cost)

The brain minimizes the description length of behavior — automatization = compression. This is consistent with BMC: the system strives for minimal WM cost while maintaining adequacy of execution.

Sleep Motor Consolidation

Consolidation of automatic chains occurs during sleep through spindle-SO coupling (distinct from declarative SWR-driven consolidation):

• Spindle-SO coupling → strengthening of edges $w(m_i, m_{i+1})$ in Auto(S)
• Critical window: sleep within ~1h after motor training → $\beta_{motor} \approx 0.3$ (Solano et al., 2024)
• Mechanism: motor consolidation is mediated by cortical spindle-SO coupling, not hippocampal SWR (as in declarative)

De-Automatization

Cost of switching from automatic to deliberative execution:

where $c_0$ is the base override cost. Cost is quadratic in habit: the deeper the habit, the more expensive it is to abandon. The square root of $n_{exec}$ adds a “momentum” effect — long-practiced skills are harder to retrain even at equal habit.

Two failure modes:

1. Rigidity ($habit \to 1$, $n_{exec} \gg 1$): $Cost_{override}$ is so high that the deliberative system cannot take over control. The habit is resistant to PE → behavior does not adapt to changed conditions. Clinically: OCD-like repetitive behaviors.

2. Relapse (stress → $WM\_load \uparrow$ → $P_{auto} \uparrow$): under stress, WM is overloaded, deliberative inhibition weakens, and the old automatic chain “takes over” control. Mechanism of addiction relapse: stress → WM overload → P_auto for substance-seeking chain ↑ → relapse.

Note: Mastery and Addiction — One Mechanism

$P_{auto}$ describes both mastery and addiction. The difference is in the content of $S$, not the mechanism: for a pianist $S$ = sequence of movements, for an addict $S$ = substance-seeking chain. $WM\_load$ is the bridge: stress increases $P_{auto}$ for all automatic chains simultaneously (through $k_{eff}$↓ → $WM\_load$↑). Under stress, the chain with the highest $habit$ wins — regardless of desirability. This is why stress triggers relapse to the most entrenched habit.

Non-trivial prediction: $Cost_{override}$ for retraining a master ≈ $Cost_{override}$ for breaking an addiction at equal $habit$ and $n_{exec}$. The parameters are identical — only the evaluation of $S$ through utility nodes differs. A pianist retraining fingering and an addict breaking a habit are solving one and the same de-automatization problem.

Predictions

Conceptual framework: Derivation from first principles, two escape-valves of one tension, evolutionary confirmation — see EMT, Part VIII.

Neurobiological substrate: DLS-SNr-PF-DLS loop, DMS/DLS competition, spindle-SO coupling — see BM, Part IV.

Engineering implementation: check_auto_chain(), update_habit(), execute_auto() — see AGI_F, Part III.

Network Modulation by Character

Effect of the T vector on spreading activation:

For utility nodes (emotional nodes), the spreading activation formula is modified to account for the individual character vector:

where $T_g$ is the character vector component for emotional node $g$.

Consequence: With high $T_{FEAR}$, the FEAR node activates more strongly given the same input signal → more often “wins” in competition for attention → forms more connections with new memes.

Effect on connection formation:

Neurobiological basis: Character vector and its influence on memplex formation — see BM: Character.

Affective Space: Blend Formula

The seven G-programs are discrete circuits. But subjective affective experience is a continuous point in (valence, arousal) space, emergent from the blend of several simultaneously active G-programs:

where $\mathbf{v}_g$ is a fixed vector in (valence, arousal) coordinates per G-program: $\mathbf{v}_{SEEKING} = (+0.6, +0.7)$, $\mathbf{v}_{FEAR} = (-0.8, +0.9)$, $\mathbf{v}_{RAGE} = (-0.7, +0.8)$, $\mathbf{v}_{LUST} = (+0.5, +0.6)$, $\mathbf{v}_{CARE} = (+0.8, +0.2)$, $\mathbf{v}_{GRIEF} = (-0.9, -0.3)$, $\mathbf{v}_{PLAY} = (+0.9, +0.5)$.

$E(t)$ is a continuous point in the circumplex (Russell, 1980, J Pers Soc Psychol). “Nostalgia” = $E$ near GRIEF + SEEKING + CARE; “Schadenfreude” = PLAY + RAGE blend. Emotional memes are M-nodes clustering certain zones of E-space. Cultures differ in clustering: amae (Jpn.), saudade (Port.) = different M-clusters given identical G. Barrett is right for the M-level, Panksepp for the G-level.

Neurobiological basis: G = discrete (PAG, hypothalamus), M = continuous neocortical construction, dementia as a test — see BM: Affective Space.

Engineering implementation: E(t) from utility activations — see AGI_F: Blend.

Note. BMC uses 7 Panksepp G-programs: SEEKING, FEAR, RAGE, LUST, CARE, PANIC/GRIEF, PLAY. DISGUST is an I-layer mechanism (marking “foreign”), not an 8th G-program. DISGUST has no vector $v_g$ in E(t) and is not included in capture weights. In engineering implementations, it may be represented as a utility node for convenience.

Part IX. Immune System — Network View

Entry Filter: Compatibility with Central Nodes

A new meme X is evaluated by compatibility with already existing memes. Central nodes play the key role:

where:

• $C(i)$ — centrality of meme $i$
• $compat(X, i)$ — compatibility of the new meme with $i$

Three-zone acceptance rule:

A rejected meme is not merely “not accepted.” It is stored with negative weights as an antibody (high-fidelity negative meme), ready for automatic activation upon re-encounter with the threat.

The I-system is not limited to filtering incoming memes. The same negative edges provide retroactive suppression of already accepted memes: if a previously compatible meme becomes conflicting with the current memplex, the I-suppression signal ($I_{sig}$) deliberately extinguishes it — see Active Forgetting and Reconsolidation.

Visualization of the Threshold Mechanism

Cognitive Dissonance: Simultaneous Activation of Incompatibles

Network interpretation: Two memes are incompatible if their simultaneous activation causes “tension” in the network. With signed edges:

Incompatibility is now directly determined by the negative weight of the edge: the stronger the negative connection, the greater the dissonance upon simultaneous activation.

High $D$ → cognitive dissonance → motivation to resolve.

Defensive circuit tension (Tension): An additional metric beyond dissonance — load on the utility/meme boundary. For signed edges:

where $gap(u, m) = \max(0, a_u - a_m)$ when $w_{um} > 0$ (utility “pushes,” meme does not support) and $gap(u, m) = \max(0, a_m - \theta_{suppress})$ when $w_{um} < 0$ (meme is active despite utility rejection).

High $Tension$ → the defensive circuit is overloaded. Unlike $D$ (incompatibility within memes), $Tension$ measures misalignment between layers (utility vs memetic).

Hub Protection Hypothesis: Which Belief Will Change?

Standard theory (Festinger, 1957) describes the motivation to reduce D, but does not predict which meme will change. BMC gives a precise answer through eigenvector centrality:

where $C_E(i)$ is the eigenvector centrality of node $i$. The meme with the lowest $C_E$ will be suppressed first: its change minimizes cascade restructuring.

Formal justification: Consider a pair $(i, j)$ with $w_{ij} < 0$ and $a_i, a_j > \theta_{high}$. Suppressing $i$ causes a cascade of length $L_i = \sum_{k \in N(i)} |a_k - a'_k|$, suppressing $j$ — a cascade $L_j$. Minimizing total restructuring → the meme with smaller $L$ is suppressed, which correlates with $C_E$:

Testable prediction: In experiments with induced cognitive dissonance: semantic network degree (as a proxy for $C_E$) of the target belief predicts which of two beliefs will change. The belief with higher degree is preserved.

Boundary of applicability: Hub Protection operates when $D < \theta_{crisis}$ (everyday dissonance). When $Tension > \theta_{crisis}$ (life crises, prolonged accumulation of discrepancies) hub displacement dominates — the hub yields, cascade restructuring occurs. The approximation $\arg\min(L_i, L_j) \approx \arg\min(C_E(i), C_E(j))$ holds at moderate $a_i, a_j$ and not too high overall $Tension$.

Conceptual discussion of hub advantage in dissonance resolution — see EMT, Part IV.

Second Type of Tension: Structural Incompleteness (SIT)

$Tension$ and $D$ describe conflicts between existing nodes. But there is a second type of tension — from missing nodes. $SIT$ (Structural Incompleteness Tension) measures the total tension from structural gaps in the memplex (definition and formulas — see Part VIII: SIT).

Two types of tension:

Combined metric of total tension:

where $\gamma_{SIT}$ is the coefficient of SIT’s relative contribution (individual, depends on $T_{SEEK}$ — the Panksepp character component). High $T_{SEEK}$ → high $\gamma_{SIT}$ → researchers and scientists experience more tension from unsolved problems.

$Tension_{total}$ is a unified metric of cognitive load on the system. High $Tension_{total}$ → motivation to act (resolving conflicts or filling gaps). This explains why a scientist might leave a comfortable job for an unsolved problem: the $SIT$ component outweighs all other utility.

Defense Mechanism: Isolation of the Threatening Node

When a new meme threatens central nodes, the memplex “isolates” it:

1. Connection blocking: Edge weights to the threatening meme are reduced
2. Counter-argumentation: Antagonist memes are activated
3. Rationalization: New connections are created that devalue the threat

Numerical Example: Meme Acceptance Threshold

Situation: New meme X (scientific fact) encounters a memplex with a religious core.

Central memes and compatibility with X:

Sum: $S(X) = 0.08 + 0.24 + 0.18 + 0.05 = 0.55$

Threshold: $\theta = 0.50$

Result: $S(X) = 0.55 > 0.50$ → meme is accepted, but with modification (low compatibility with “Faith” will cause adaptation).

See also: Immune system of the person and memeplexes — EMT, Parts XIII–XIV.

Structural Balance of the Memplex

With the introduction of signed edges ($w \in [-1, +1]$), the memplex becomes a signed network. Structural balance theory describes stable configurations of such networks.

Formalization: A signed graph $G = (V, E^+, E^-)$ is balanced if all cycles contain an even number of negative edges.

Dynamics: The memplex tends toward weak balance — this corresponds to a modular structure with several clusters. Pathological strict balance ($k = 2$) — black-and-white thinking, fanaticism.

Sources: Heider (1946) — P-O-X model; Cartwright & Harary (1956) — Structure Theorem; Davis (1967) — weak balance; Dalege, Borsboom et al. (2016) — CAN model; Brandt & Sleegers (2021) — belief systems as signed networks.

Conceptual description: Structural balance in the context of the immune system — see EMT, Part XV.

Ambivalence Metric

Predictive power: High ambivalence predicts instability: the meme is prone to abrupt transition (acceptance → rejection or vice versa) upon small external perturbation.

Fractal Structure of the I-Layer

The Problem of Scaling Immunity

The entry filter $S(X) = \sum_i C(i) \cdot compat(X, i)$ (above) operates at a single level of abstraction — this is a special case for a memplex without hierarchical structure. But a real memplex contains memes at different levels: from sensory micro-memes to abstract values. A filter tuned for semantic compatibility will not detect a perceptual artifact; a perceptual noise filter will not evaluate an ideological threat. Hierarchical filtering at different processing levels is described in attention research (Broadbent, 1958) and predictive processing (Friston, 2005). BMC formalizes it as per-level immune subsystems.

Formalization: $I^{(k)}$ — Immune Subsystem of Level $k$

Let the memplex be organized into a hierarchy of levels $k \in \{0, 1, \ldots, K\}$ (from sensory patterns to abstractions). At each level, an immune subsystem is defined:

Entry filter of level $k$:

where $C^{(k)}(i)$ is the centrality of immune meme $i$ at level $k$, $compat^{(k)}$ is the compatibility function defined for memes of level $k$.

Filtering rule: Meme $X$ of level $k$ is checked by filter $I^{(k)}$, not by filters of other levels:

The threshold $\theta^{(k)}$ may differ at each level (stricter at L3, softer at L0).

Immunogenesis: Formation of $I^{(k)}$ through Rejection Experience

$I^{(k)}$ is formed by the same mechanism as the main memes of level $k$ — through stabilization of recurring patterns:

Fidelity of immune memes grows with the number of rejections (spacing effect):

where $n_{reject}$ is the number of rejections of memes of type $Y$.

Constitutional G-Invariants as Hard Core

At level $K$ (abstract memes), $I^{(K)}$ includes immutable immune patterns — constitutional G-invariants:

These invariants are an analog of immunological tolerance: the I-system is trained not to attack “self” (the G-core). No experience and no level of $I^{(k)}$ can modify them.

Rate limitation of G-modification:

Violation → I-rejection (blocked as “infection”).

Predictions

1. $|I^{(k)}| / |M^{(k)}|$ — optimal fraction of immune memes. Speculative analogy: in biology ~5–10% of cells are immune. Direct transfer of this proportion to BMC is not theoretically justified. Prediction: when $|I^{(k)}| / |M^{(k)}| < \rho_{min}$ (threshold not determined), the memplex is vulnerable to “infections” at level $k$. Empirical calibration is required.

2. Vulnerability window during memogenesis. When the M-layer reaches a new level $k+1$ but $I^{(k+1)}$ is not yet formed, the memplex is vulnerable to memes at level $k+1$. Analog: immature immune system of the newborn; vulnerability of adolescents to ideological capture (PFC not myelinated → $I^{(3)}$ is weak).

3. Ablation: A flat I-system ($I = I^{(0)}$ for all levels) → degraded filtering at high levels. Metric: $R(t) = Q(t)/H(t) \cdot (1 - SIT(t))$ grows faster with flat I than with fractal I.

Conceptual justification: EMT, Part XV. Engineering specification: AGI_F, Part IV. Neurobiology: BM, Part IV.

Cognitive Biases as Consequences of Network Topology

Six already formalized BMC mechanisms give rise to ~200 cognitive biases as emergent side effects. Here we provide the formal definition of bias strength function for each mechanism.

Conceptual overview and mapping: EMT, Part XXIV.

Bias Strength Functions

1. H-bias (hub inertia): Bias strength is proportional to eigenvector centrality:

where $\alpha_H$ is a scaling coefficient. When $C_E \to 0$ (peripheral meme): $B_H \to 0$ — no bias, the meme is easily updated. When $C_E \gg 1$ (hub): $B_H \to 1$ — maximum bias, the meme is nearly immutable. Confirmation bias, belief perseverance, backfire effect are consequences of $B_H \approx 1$ for core memes of the memplex.

Link to cognitive dissonance: $P(\Delta a_i < 0 \mid D > \theta) \propto 1/C_E(i)$ (defined above, Hub Protection Hypothesis). H-bias is the steady-state manifestation of the same mechanism: hubs do not wait for dissonance — they preemptively filter incompatible information through I.

2. I-bias (immune filtration): Bias strength is determined by the filter threshold:

where $S(X) = \sum_i C_i \cdot compat(X, m_i)$ — compatibility of the incoming meme $X$ with the current memplex (defined in the Entry Filter section). When $S(X) < \theta$, the meme is rejected → in-group bias, hostile media effect, not-invented-here. I-bias depends on memplex structure: changing core memes → changing the threshold → previously rejected memes are accepted (conversion).

3. W-bias (WM constraints): Bias strength is inversely proportional to free WM slots:

where $k_{eff}(t) = k_{active} - n_{captured}(t)$ (defined in the WM constraints section). When $k_{eff} = k_{active}$ (calm state, no G-capture): $B_W = 0$. When $k_{eff} = 1$ (strong G-capture or WM load): $B_W = 1 - 1/k_{active} \approx 0.75$. Anchoring, framing, base rate neglect are amplified proportionally to $B_W$: fewer free slots → the decision is determined by what is already loaded.

4. G-bias (affective capture): Bias strength is proportional to affective pressure:

where $w_{capture,g}$ is the number of WM slots captured by G-system $g$ (FEAR=1.0, RAGE=0.8, PANIC/GRIEF=0.7, …; defined in the G→WM capture section). $B_G \in [0, 1]$: when $a_g = 0$ for all G: $B_G = 0$; at full activation of the dominant G: $B_G \to 1$. Loss aversion, optimism bias, affect heuristic are proportional to $B_G$.

Loss asymmetry: FEAR ($w_{capture} = 1.0$) captures more WM slots than PLAY ($w_{capture} = 0$) → $B_G$ is higher for threats than for opportunities. This produces loss aversion as a network effect rather than an ad hoc parameter.

5. A-bias (automatization): Bias strength is proportional to the square of habit strength:

Normalized to $[0, 1]$. $Auto(S)$ and $habit(S)$ are defined in the Automatization section. Cost of switching to an alternative:

Status quo bias, functional fixedness, Einstellung effect are consequences of $Cost_{override} \gg 0$: even when a better strategy $S_{alt}$ is available, switching requires WM resources that may be insufficient (intersection with $B_W$).

6. R-bias (reconsolidation): Bias strength is active only in the labile window:

where $Labile(m_i, t) \in \{0, 1\}$ is defined in the Reconsolidation section (4 outcomes: strengthen, update, destabilize, erase). $\delta_{context}$ is the measure of context mismatch at retrieval. Hindsight bias: $Labile = 1$ upon recall + outcome is known ($\delta_{context}$ is high) → update incorporating the outcome. Misinformation effect: $Labile = 1$ + external meme with high $compat$ → update with false content.

Cross-Modulation: Negativity Bias as a Parameter

Negativity bias ($\lambda_{neg} < \lambda_{pos}$) is not a separate mechanism but a parameter amplifying H- and I-biases for negative content:

with $\gamma > 0$ and $\lambda_{neg} < \lambda_{pos}$. Negative hubs are more inert → confirmation bias for negative beliefs is stronger than for positive ones.

Prediction P-CB1 (Formal Version)

If biases are generated by 6 independent mechanisms, bias correlations should cluster:

where $B_X(i)$, $B_X(j)$ are two biases from the same group $X$, and $B_Y(j)$ is a bias from another group $Y$.

Specifically: $\rho(\text{confirmation bias}, \text{belief perseverance}) > \rho(\text{confirmation bias}, \text{anchoring})$, because the first two are H-mechanism, while anchoring is W-mechanism.

Existing data (Ceschi et al., 2019) found 3 factors in 17 biases. BMC predicts: expanding the battery to 30+ biases (covering all 6 groups) → 6 factors, with H and I possibly merging (both are hub-dependent, but differ in mechanism: inertia vs filtration).

Neurodata: Confirmation Bias as Failure of Readout

Park et al. (2025, Nature Communications) showed: in a sequential categorization task, evidence is accurately encoded in parietal cortex regardless of compatibility with the current choice, but readout for behavior is greater for consistent evidence. This directly corresponds to the I-mechanism: $S(X)$ is lower for memes incompatible with the current memplex — not because the information is not encoded (the sensory S-layer works), but because the I-filter blocks readout for behavior.

Conceptual mapping: EMT, Part XXIV. Neurosubstrate: BM, Part IX. AGI engineering: AGI_F, Part VI.

Part X. Inter-Level Transitions: Multilayer Networks

The Scaling Problem

The theory applies at different levels:

• Memplex in the head (neural networks)
• Organization memplex (social bonds)
• State memplex (institutions, media)

The question: how to transfer concepts between levels?

N scales, not two. The hierarchy of scales is not limited to two or three levels: memes-in-agent ↔ agents-in-swarm ↔ teams-in-organization ↔ organizations-in-industry ↔ cultures-in-SMR. At each level, BMC = (G, M, I, S) is reproduced with the same formal apparatus (heavy-tailed distribution, Q, σ_SW, PA, hub displacement), but on a different substrate and with different timescales — renormalization invariance. SMR as a fractal BMC is formalized in SM, Part VII. All levels are in the green transfer zone for topological properties (Part XV).

Multilayer Networks: Unified Formalism

A multilayer network allows modeling all levels within a single framework.

Mathematical Formalization

A multilayer network is described by an adjacency tensor:

where:

• $i, j$ — node indices
• $\alpha, \beta$ — layer indices
• $\mathcal{A}_{ij}^{[\alpha\alpha]}$ — connections within layer $\alpha$ (intra-layer)
• $\mathcal{A}_{ii}^{[\alpha\beta]}$ — connections between layers for node $i$ (inter-layer)

Intra-layer vs Inter-layer dynamics:

General spreading dynamics:

The first sum is the influence of neighbors in the same layer; the second is the influence of “copies” of the node in other layers.

Localized vs Delocalized Spreading

In multilayer networks, two regimes are possible (Tey & Cozzo, 2024):

Transition between regimes is a phase transition; the critical parameter is the inter/intra connection ratio.

Application to memeplexes:

• Localized regime: A thought “gets stuck” in one context (work does not affect family)
• Delocalized regime: A thought permeates all spheres of life (conversion, obsession)

Complex Contagion vs Simple Contagion

Key difference between memes and diseases:

Complex contagion formula:

where $\theta$ is the adoption threshold (fraction threshold).

Consequence: For spreading complex memes (ideologies, practices), bridges with wide overlap are important — not just a connection between clusters, but multiple connections. A single bridge is insufficient.

Source: Kivela et al. (2014), J. Complex Networks — review of multilayer networks; Behavior Research Methods (2025) — cognitive multiplex networks.

Replication Pressure: Individual Mechanism of Transmission

Simple and complex contagion describe meme reception (adoption): under what conditions an agent accepts a meme from others. But contagion is a two-sided process. What drives transmission (expression) from the speaker’s side?

The standard answer — “social influence,” “communicative intention” — describes the phenomenon but does not explain it. In the BMC formalism, the answer is direct: a meme is a replicator, and an activated meme creates pressure on the communication channel. This is not a metaphor: the pressure is computable and predicts observed behavior.

Definition. The replication pressure (expression pressure) of meme $m_i$ at time $t$ with an open communication channel:

where:

• $a_i(t)$ — current activation of the meme
• $F_i$ — fidelity (a consolidated meme is expressed more accurately → more frequently)
• $\text{rel}(m_i, \text{context})$ — relevance to the current communication context (cosine similarity with the conversation topic)
• $C_E(m_i)$ — eigenvector centrality (hub bonus: central memes are expressed more frequently)
• $\alpha_C$ — hub amplification coefficient (≈ 0.3)

Threshold condition: A meme is expressed if $R_{expr}(m_i, t) > \theta_{expr}$ and the communication channel is open (the agent is in dialogue, not in inner speech).

Candidate selection: Among memes with $R_{expr} > \theta_{expr}$, the top-$k$ with the highest pressure are selected. Parameter $k$ is limited by channel throughput (one person cannot express 20 thoughts simultaneously; speech bottleneck ≈ 150 bits/s, see EMT, Part XX).

Two types of communicative impulse. In BMC, communication has two independent sources:

These are not the same thing. SEEKING is a G-program (innate drive, genetic layer). Replication pressure is an emergent property of the M-layer (memes replicate because it is their nature, just as genes replicate through DNA polymerase). In real dialogue, both impulses compete for the communication channel:

A healthy dialogue is an alternation of listen and share. Monopoly of one impulse produces dysfunction:

Replication success. Meme expression is not the end of replication. Successful replication = the meme is accepted by the recipient (adoption, see complex contagion above). Feedback:

Upon success: $w_{ij}$ is strengthened (link between the meme and the context in which it successfully replicated). Upon failure: $R_{expr}$ decreases for this particular recipient (not fidelity — the meme did not become less “accurate,” it’s just that this recipient did not accept it). This is adaptive modulation: an experienced communicator learns which memes work with which interlocutor.

Prediction P-E1: People with high hub centrality in a particular semantic domain spend more time talking on that topic (controlling for profession). Testable: analysis of speech corpora + extraction of individuals’ semantic networks → correlation of eigenvector centrality with share of speech on the topic.

Prediction P-E2: In one-sided communication (one talks, the other only listens) the speaker is subjectively satisfied (replication pressure is discharged), the listener is not (their $R_{expr}$ is not discharged). In bilateral communication — both are satisfied. Testable: experimental manipulation of dialogue structure + satisfaction self-report.

Link to neurobiology: see BM, Part V — Broca as substrate, inner speech, tip-of-the-tongue as evidence of $R_{expr}$.

Architectural specification for AGI: see AGI_F, Part IV — expression candidates, pseudocode.

Philosophical justification: see EMT, Part XX — analogy with DNA polymerase, bidirectional communication.

Stigmergy: Coordination through the Environment

Simple and complex contagion describe direct meme transmission (agent → agent). But in multilayer networks, there exists a fundamentally different coordination mechanism — stigmergy (Grasse, 1959): indirect coordination through traces in the environment.

Definition: An agent leaves a trace in a shared environment (M-layer). Another agent reads the trace and modifies behavior. There is no direct contact between agents.

In the multilayer network formalism, stigmergy is an inter-layer connection of a special type: an agent in layer $\alpha$ modifies a node in layer $\beta$ (environment), and an agent from layer $\gamma$ reads the modification. The adjacency tensor $\mathcal{A}_{ij}^{[\alpha\beta]}$ describes not only direct connections between layers but also environment-mediated ones.

Why stigmergy scales while direct coordination does not:

• Direct coordination requires $O(N^2)$ connections (or hubs, creating a single point of failure)
• Stigmergy scales to $O(N)$: each agent interacts with the environment, not with every other agent
• Traces are persistent (books, laws, code) — coordination does not require simultaneous presence of participants

Stigmergy in BMC terms: the environment = an externalized M-layer. All four BMC subsystems function identically:

• G — utilitarian drives determine which traces an agent leaves and which it responds to (SEEKING → scouting new trails, CARE → maintaining structures, FEAR → marking dangerous zones)
• M — traces in the environment = memes with activation (trace intensity), fidelity (preservation), and connections (the trail leads to food)
• I — the immune system filters foreign traces (an ant from another colony ignores foreign pheromones; a Wikipedia user reverts vandalism)
• S — information deficits in the environment (a broken trail, an unfinished article) activate SIT and motivate the agent to fill the gap

Replication in stigmergic systems. Objection: “a meme is a replicator, but a trace in the environment does not replicate.” Response: in stigmergy, what replicates is not the trace but the behavior. An ant sees the trail → follows it → deposits pheromone. The behavior “follow and reinforce” replicates from agent to agent. The trace is the cumulative result of replicated behavior. The analogy with the neural level is exact: in the brain, what replicates is the activation pattern (neuron behavior), and synaptic weight is the cumulative result. There is no fundamental difference between synapse reinforcement and trail reinforcement.

Formally, a stigmergic trace satisfies all criteria of Darwinian selection:

Stigmergy in SMR. Stigmergic traces in SMR form their own heavy-tailed graph: frequently used “knowledge trails” are reinforced (PA), rarely used ones fade (decay), hub-paradigms and Q-modules (cultural branches) form. This is renormalization invariance: neural synapses ($L_1$) → social bonds ($L_2$) → corporate processes → industry standards → stigmergic traces in SMR ($L_N$). Formalization: SM, Part VII.

Prediction: Stigmergic systems (ant colonies, open-source projects, wiki platforms) demonstrate the same BMC patterns as individual memeplexes: competition of traces for resources (attention), immune selection (moderation), SIT-driven innovation (gap filling), and phase transitions at critical trace mass.

Source: Grasse (1959) — the term “stigmergy” for termite coordination; Theraulaz & Bonabeau (1999), Artificial Life — formalization of stigmergy in multi-agent systems; Heylighen (2016), Cognitive Systems Research — stigmergy as the basis of collective intelligence.

Correspondence Table between Levels

What Transfers and What Does Not

Transfers (isomorphism):

• Logic of spreading activation
• Competition for limited resources
• Cluster structure
• Defense mechanisms
• Network effects (cascades, phase transitions)

Does not transfer (substrate differences):

• Timescales (ms vs years)
• Intentionality (neurons vs people)
• External memory (only societies have books, archives)
• Reflexivity (people can consciously change connections)

Formal Criteria for Scale Transfer

The assertion “one algorithm on different substrates” requires qualification: formal isomorphism (matching equations) does not guarantee causal isomorphism (matching mechanisms). Physics solves this problem through universality classes: systems with different micro-mechanisms but identical critical exponents belong to one class (Kadanoff, 1966; Wilson, 1971; Batterman, 2002).

Empirical fact: neuronal avalanches (Beggs & Plenz, 2003) and information cascades in social networks (Notarmuzi et al., 2022, Nature Communications) obey power laws, but with different critical exponents:

Therefore, the neural and social levels belong to different universality classes. This does not refute the isomorphism — it refines its boundaries.

Transfer Criterion through Renormalization

Geometric renormalization of networks (Garcia-Perez, Boguna & Serrano, 2018, Nature Physics; Villegas et al., 2023, Nature Physics) provides a formal tool: under coarse-graining (scale enlargement), we verify invariant preservation.

Transfer of prediction $P$ from level $\alpha$ to level $\beta$ is valid if:

where $\text{dep}(P)$ is the set of topological invariants on which prediction $P$ depends; $\mathcal{I}^{[\alpha]}$ is the invariant value at level $\alpha$; $\epsilon$ is the tolerance.

Principal invariants:

Consequence: Predictions depending only on the first three invariants (topology, degree distribution, modularity) transfer. Predictions depending on critical exponents or timescale require a scaling correction.

Three Transfer Zones

Operationalization of the Yellow/Red Zone Boundary: Structural Equivalence Test (SET)

Mechanism $\mathcal{M}$ is in the red zone for the level pair $(L_1, L_2)$ if and only if:

where $F$ is the dynamic equation of the mechanism, $\mathbf{x}$ is the state vector, $\boldsymbol{\theta}$ are parameters. If the mapping $\varphi$ exists and the equations are structurally equivalent (same functional form $F$, different $\boldsymbol{\theta}$), the mechanism is transferable.

Decision procedure:

1. $\nexists$ functional analog at the target level → red
2. $\exists$ analog, but $F_{L_1} \not\equiv F_{L_2}$ (different equation form) → red
3. $F_{L_1} \equiv F_{L_2}$, prediction depends only on topological invariants → green
4. $F_{L_1} \equiv F_{L_2}$, prediction depends on $\boldsymbol{\theta}$ → yellow (parameter recalculation)

Sources: Garcia-Perez, Boguna & Serrano (2018), Nature Physics 14, 583 — geometric RG for networks; Villegas, Gili, Caldarelli & Gabrielli (2023), Nature Physics 19, 445 — Laplacian RG for heterogeneous networks; Notarmuzi et al. (2022), Nature Communications 13, 1308 — universality classes in social vs neural avalanches; Batterman (2002), The Devil in the Details, OUP — formal vs causal isomorphism across scales.

Part XI. Communicative Asymmetry of the Graph

11.1. Motivation: From Symmetric Edges to Directed Flow

In Part V, edges are formalized as $w_{ij} = w_{ji}$ — an undirected graph with symmetric weights. This is correct for structure: the association between two memes reflects the overlap of their neural ensembles (Cai et al., 2016), and overlap is symmetric by definition ($|ens_i \cap ens_j| = |ens_j \cap ens_i|$).

However, the effective flow through an edge is asymmetric. A hub meme with activation $a = 0.9$ transmits strong spreading activation along all its edges; a peripheral meme with $a = 0.1$ transmits weak activation. Yet the edge weight between them is the same in both directions. The result: the overwhelming majority of edges in BMC function as unidirectional broadcast, although structurally they remain bidirectional.

What is needed is a formalism describing this functional directionality without violating the biologically correct weight symmetry.

11.2. Formal Definitions

Definition 11.1 (Effective flow). For each edge $(i, j)$ at time $t$, we define the directed flow:

Since $a_i(t) \neq a_j(t)$ in general, $\phi_{i \to j}(t) \neq \phi_{j \to i}(t)$ even when $w_{ij} = w_{ji}$.

Definition 11.2 (Effective directionality). Averaged over a window of $T$ ticks (by default $T = \tau_{generation}$):

where $\bar{\phi}_{i \to j} = \frac{1}{T}\sum_{t=1}^{T} \phi_{i \to j}(t)$.

• $D_{eff} = 0$: purely bidirectional (flows are equal)
• $D_{eff} = 1$: purely unidirectional (flow only in one direction)

Definition 11.3 (Bidirectional degree). The number of edges of node $i$ with $D_{eff}$ below the threshold:

Analogously, $k_{uni}(i, \theta) = |\{j : D_{eff}(i,j) \geq \theta\}|$ — unidirectional degree.

Definition 11.4 (Bidirectional capacity). $N_{bid}(i) = k_{bid}(i, \theta_{bid})$ — the number of effectively bidirectional partners of node $i$.

11.3. Three Layers of Asymmetry

Conclusion: asymmetry in BMC is functional (from activation magnitude), not structural (weights remain symmetric). This is biologically correct: synaptic overlap = symmetric, but firing rate = asymmetric.

11.4. Law $N_{bid}(L) \approx \text{bandwidth}(L) - 1$

Proposition 11.1. Bidirectional capacity at level $L$ is determined by the throughput of the cognitive channel at that level.

Mechanism: Bidirectional = requires mutual co-activation / contact. Co-activation requires WM co-presence. WM holds 3–4 slots → maximum 3 bidirectional partners per tick. Over $T$ ticks, only persistent “neighbors” from the same cluster accumulate enough co-activations for $D_{eff} \to 0$.

At level L2, the WM slot analog is the number of bonds an agent can maintain (Dunbar number, $\sim 150$; Dunbar, 1992). At level L3 — the number of active alliances a culture can simultaneously maintain (diplomatic bandwidth, $\sim 10$–$20$).

11.5. Empirical Data (Level 1)

Results from bid_analysis.py (seed=42, N=505 main memes + utility, 1000 ticks):

• 1094 unique edges, 99% weight < 0.17, mean weight = 0.01
• Top-10 hubs: $k_{total} = 13$–$16$, $k_{bid} = 0$–$3$ (ratio $\sim 5\%$)
• Middle tier: $k_{total} = 8$–$9$, $k_{bid} = 0$–$3$
• Bottom tier: $k_{total} = 4$, $k_{bid} = 0$
• $N_{bid} \approx 3$ at ANY threshold ($0.1$–$0.7$) — does not depend on the choice of $\theta$

Note: $k_{bid}$ was determined through weight threshold (proxy for $D_{eff}$): $w > \theta$ → regular co-activation → bidirectional flow. Full $D_{eff}$ analysis is a separate validation (Level 3).

11.6. Edge Lifecycle (Directed)

Transition formula bid → uni: if over $T$ ticks the co-activation count $< \theta_{maintain}$ → $D_{eff}$ grows → the edge becomes unidirectional. Formally:

11.7. Consequences for Network Dynamics

1. Hub = broadcaster. A hub “communicates” bidirectionally with $\sim 3$ memes (nearest neighbors in the cluster). The remaining $10+$ edges = broadcasting (one-sided spreading activation). This explains why LOCAL normalization (shared neighbors $\sim 5$) is correct for hub displacement (Part VIII, S6): competition is for $\sim 3$ bidirectional slots, not for the entire $k_{total}$.

2. Modularity from asymmetry. Bidirectional edges are concentrated within clusters (co-activation frequency is higher within communities). Consequence: $Q$ is amplified — the bidirectional core forms a dense module nucleus, cross-cluster edges remain unidirectional and unstable.

3. Hierarchical encoding. Inner core ($k_{bid}$ — bidirectional partners) → middle ($k_{uni}$ — frequent activation recipients) → periphery (rare activation). Fractally repeated at L2: inner core $\sim 5$–$15$ (close contacts) → middle $\sim 150$ (Dunbar) → periphery $\sim 1000+$ (acquaintances). See SM, Part II.

11.8. Predictions

P-CA1 ($N_{bid}$ is determined by $k_{active}$). When $k_{active}$ increases from 4 to 6 → $N_{bid}$ grows to $\sim 5$. Verification: change WM_CAPACITY in the simulator, recompute bidirectional degree.

P-CA2 (Locality of hub displacement). Competition for $\sim 3$ bidirectional slots → displacement effect is local (shared neighbors), not global (total degree). Verification: S6 test with local vs global normalization.

P-CA3 (Bimodality of $D_{eff}$). The distribution of $D_{eff}$ across all edges is bimodal (peaks near 0 and near 1). Intermediate values are rare due to the WM bottleneck: an edge is either regularly co-activated (bidirectional) or not (unidirectional). Verification: $D_{eff}$ histogram from bid_analysis.py.

Sources: Dunbar R.I.M. (1992). “Neocortex size as a constraint on group size in primates.” Journal of Human Evolution; Cai D.J. et al. (2016). “A shared neural ensemble links distinct contextual memories encoded close in time.” Nature; Hill R.A., Bentley R.A. & Dunbar R.I.M. (2008). “Network scaling reveals consistent fractal pattern in hierarchical mammalian societies.” Biology Letters; Cowan N. (2001). “The magical number 4 in short-term memory.” Behavioral and Brain Sciences.

Part XII. What Can Now Be Predicted

Hypothesis → Network Formulation → Verification Method

Specific Predictions

1. Age-related rigidity = growth of modularity

Prediction: In elderly individuals, semantic networks exhibit higher modularity than in younger individuals.

Mechanism: With age, intra-cluster connections are strengthened while inter-cluster connections are weakened.

Formula:

Verification: Compare the modularity of word association networks across different age groups.

Differential plasticity at low openness. When $O < 0.3$, a differential plasticity regime is activated: intra-cluster connections are strengthened while inter-cluster connections are weakened with intensity proportional to “rigidity” $r = 1 - O$:

Weight update at each step:

This mechanism formalizes “cognitive entrenchment”: as openness decreases, the memplex actively reinforces internal coherence at the expense of inter-cluster connectivity.

2. Schizophrenia = network fragmentation

Prediction: In schizophrenia, betweenness centrality of bridge nodes is reduced.

Mechanism: “Bridges” between clusters are destroyed, leading to fragmentation of thought.

Formula:

Verification: Compare network metrics of semantic networks.

3. Regression under stress = k-core activation

Prediction: Under stress, the network’s “core” is activated — nodes with maximum connectivity.

Mechanism: The k-core is a subgraph where each node has at least $k$ connections. These are the most “ancient” and deeply rooted memes.

Formula:

Verification: fMRI: under stress, brain regions associated with basic beliefs are activated.

4. Meme virality = exceeding the epidemic threshold

Prediction: A meme becomes viral when the basic reproductive number $R_0 > 1$.

Formula:

where:

• $\beta$ — probability of transmission per contact
• $\gamma$ — rate of “recovery” (forgetting)
• $\langle k \rangle$ — average node degree in the social network

Verification: Analysis of meme propagation dynamics on social media.

4a. Virality in networks with hubs

In heterogeneous networks with hubs, the epidemic threshold is significantly lower than in homogeneous networks:

Mathematical detail: For a strict power law with $\gamma \leq 3$ in an infinite network, $\lambda_c \to 0$. For finite networks and other heavy-tailed forms (log-normal, truncated power law), the threshold exists but remains low due to the high degree variance $\langle k^2 \rangle$.

What this means: In a network with hubs (including memplexes and social networks), a meme with moderate “contagiousness” can spread — if it starts from the right entry point. Exact threshold values depend on the specific topology, but the qualitative conclusion is robust.

Key implication: The entry point is critical.

Practical conclusion: For meme propagation, it matters less what the meme’s “quality” is and more through whom it enters. A meme that starts from a hub has a chance even with weak “contagiousness.” A meme from the periphery may be “perfect” but will never spread.

4b. Radicalization: trajectory in BMC parameter space

Radicalization is not a discrete event but a continuous trajectory in the space $(Q, \sigma_{SW}, SIT, D)$.

Formalization:

Let $\mathbf{x}(t) = (Q(t), \sigma_{SW}(t), SIT(t), H(t))$ be the state vector of a memplex, where $H(t) = -\sum_c p_c \log p_c$ is the entropy of the activation distribution across clusters (a measure of diversity).

Radicalization = trajectory:

Characteristics:

• $Q(t) \uparrow$ (modularity increases — the memplex fragments into “us” and “them”)
• $H(t) \downarrow$ (diversity of active clusters drops — a single ideological module dominates)
• $SIT(t) \to 0$ (all gaps are “closed” by false closure — the ideology explains everything)
• $\sigma_{SW}$ may remain $\approx 1$ (the radical does not lose cognitive capabilities; they lose content)

Critical point — I-layer recalibration:

Immune system thresholds shift: in-group memes are accepted at low compatibility, while out-group memes are rejected at high compatibility.

Radicalization metric:

$R \to \infty$ when $H \to 0$ (monopoly of a single cluster) and $SIT \to 0$ (no doubts). Normal state: $R \approx 1$. Preliminary danger zone threshold: $R > 5$ (requires empirical calibration).

Comparison with other trajectories:

Testable prediction: Semantic network analysis of radicals will show: (1) elevated Q (modularity), (2) reduced H (entropy), (3) reduced SIT-proxy (Need for Cognition Scale). All three are quantitative metrics accessible through standard methods.

Summary Table of Predictions

Prediction #9: SIT and persistent activation of incomplete clusters

Formulation: Clusters with $SIT(C) > 0$ generate persistent SEEKING activation correlating with Default Mode Network (DMN) activity, even in the absence of external stimuli.

Test: Construct semantic networks of subjects (using free association methods), identify clusters with high gap density, conduct fMRI resting-state and measure the correlation:

Accompanying test (Zeigarnik extension): Give subjects a set of tasks, interrupt a random subset. Measure: (a) free recall (classic Zeigarnik), (b) DMN activation for clusters containing interrupted tasks.

Prediction: $DMN_{activation}$ for interrupted tasks is higher by $\delta \propto centrality(C)$ — the more important the cluster, the stronger the effect.

Falsification: If gap density of semantic networks does not correlate with DMN activation during resting-state (at $p > 0.05$, $N > 50$), the SIT mechanism is refuted or requires revision.

See also: DMN as substrate for SIT — BM, Part III; SIT-specific predictions — BM, Part XI.

See also: Theory falsifiability, what would refute it — EMT, Part XXIX.

Part XIII. Consciousness Level — A Metric for the Level of Consciousness

Problem: no scale for states of consciousness

The theory describes consciousness through SMC (Self-Model Cluster) (see EMT, Part XVI), but does not formalize levels of consciousness: wakefulness vs sleep vs coma vs psychedelics. A metric computable through network parameters is needed.

CL Formula

where:

• $\sigma_{SW}$ — small-worldness (integration + differentiation, analog of Massimini’s PCI)
• $A_{SMC}$ — activity of the Self-Model Cluster: $A_{SMC} = \frac{1}{|SMC|}\sum_{m \in SMC} a_m(t)$
• $f(Balance)$ — a bell-shaped function, maximum at $Balance \in [1, 2]$

Definition of Balance(t):

where $\bar{a}_M = \frac{1}{|V_m|}\sum_{i \in V_m} a_i(t)$ is the mean activation of meme-nodes, $\bar{a}_G = \frac{1}{|V_u|}\sum_{g \in V_u} a_g(t)$ is the mean activation of utility nodes, and $\epsilon$ is a small constant ($\epsilon \sim 10^{-3}$) preventing division by zero. Neurobiological proxy: $A_{PFC}/A_{limbic}$ (see BM, Part III).

Interpretive invariance. CL(t) and its extension $CL_{full}(t) = CL \cdot I_{intero}$ are defined exclusively through network metrics ($\sigma_{SW}$, $A_{SMC}$, Balance; $I_{intero}$ is added only in $CL_{full}$) — no component contains variables dependent on one’s position regarding the Hard Problem. All empirical predictions of BMC (proxy indices, phase transitions, component-wise analysis of states) are identical under realism, illusionism, dualism, and neutral monism. The formalism is interpretationally invariant (details — EMT, Part IV: Ontological Neutrality of the BMC Formalism).

Proxy Metrics for Experimental In Vivo Validation

CL is directly computable only in the model (the full memplex graph is known). For experimental verification, each CL component requires a measurable proxy indicator:

Layer-by-layer verification of $A_{SMC}$ via DMN:

Composite proxy index (unique BMC prediction):

where $DMN_{SMC}$ is the normalized activity of DMN nodes, $EC_{ratio}$ is the ratio of top-down to bottom-up effective connectivities. Prediction: $CL_{proxy}$ correlates with subjective consciousness scales (PCS, ASC) more strongly than any single component. PCI alone $\approx$ IIT’s prediction; DMN alone $\approx$ HOT/AST — each captures one aspect; BMC asserts that consciousness = the product of integration, self-model, and balance. Testable with standard methods: simultaneous TMS-EEG + fMRI + subjective reports.

Limitation of the proxy approach: The mapping CL $\to$ {PCI, DMN, EC} is an approximation, not an identity. The memplex graph does not coincide with the functional connectome: a meme $\neq$ an ROI, meme activation $\neq$ BOLD signal. Proxies allow ranking states by CL and verifying the direction of predictions ($\uparrow$/$\downarrow$), but absolute CL(t) values are computable only within the model.

Criticality: $\sigma \approx 1$ as operating point

The parameter $\sigma_{SW}$ is related to network criticality. At the critical point ($\sigma \approx 1$), the network optimally balances integration and differentiation:

• $\sigma < 1$ (subcriticality): memes cannot spread $\to$ reduced consciousness
• $\sigma \approx 1$ (criticality): optimal regime $\to$ normal wakefulness
• $\sigma > 1$ (supercriticality): avalanche-like propagation $\to$ altered states

Neurophysical basis of criticality: The branching ratio $\sigma$ (mean number of offspring neurons per active neuron) is a direct analog of our spreading activation regime (Beggs & Plenz, 2003, J. Neuroscience). At $\sigma = 1$, neuronal avalanches (activation cascades) follow a power-law size distribution $P(x) \propto x^{-\gamma}$, which coincides with the heavy-tailed degree distribution in our memplex. The E/I (excitation/inhibition) balance determines $\sigma$: enhanced inhibition (GABA) $\to$ $\sigma < 1$, weakened inhibition $\to$ $\sigma > 1$; this is pharmacologically confirmed (Shew et al., 2011, J. Neuroscience). In BMC terms: $\sigma < 1$ (depression, memes “offline”), $\sigma > 1$ (mania, panic — uncontrolled activation), $\sigma \approx 1$ = healthy balance of lateral inhibition.

Bistability and hysteresis: The memplex is a bistable system: the same information can lead to different stable states depending on history (Izhikevich, 2007, Dynamical Systems in Neuroscience). Hysteresis explains rigidity: returning the memplex from a new state requires more effort than the transition into it — deconversion is harder than conversion. Bifurcations (saddle-node, Andronov-Hopf) describe worldview collapse: smooth parameter change (stress, informational pressure) can cause a qualitative jump — the old equilibrium disappears (Strogatz, 2015, Nonlinear Dynamics and Chaos). This is the mathematical foundation of “the straw that breaks the camel’s back”: subthreshold tension accumulation $\to$ sudden memplex restructuring.

Map of Consciousness States

Degenerate case BMC_G = (G, $\emptyset$, $\emptyset$, S). When $M \to \emptyset$, the CL formula reduces: Balance $\to$ 0, $f(0) \approx 0.10$, $CL \approx 0.10 \cdot \sigma_{SW} \cdot A_{proto-SMC}$. Real organisms lie on a continuum of M/G (chimpanzee: M < G, but M > 0; human: M $\geq$ G). BMC_G $\approx$ IIT ($\sigma_{SW}$) + proto-AST ($A_{proto-SMC}$): BMC subsumes simpler theories as special cases.

Non-trivial predictions of BMC_G (absent from IIT and AST):

• G-1 (Ceiling): $CL_G \leq f(0) \cdot \max(\sigma_{SW}) \cdot \max(A_{proto-SMC}) \approx 0.10 \cdot \sigma^* \cdot A^*$. G-only consciousness has a hard upper bound; IIT does not predict a ceiling for $\Phi$. Test: PCI of large-brained G-only animals $\to$ plateau.
• G-2 (Phase transition): When $M: 0 \to \varepsilon$, $f(Balance)$ undergoes a discontinuity: $f(0) = 0.10$, $f(\varepsilon / A_G) > 0.10$, and simultaneously proto-immune system and proto-SIT mechanisms are activated. The result is a nonlinear CL jump upon the appearance of proto-memes. IIT and AST predict monotonic growth. Test: PCI of crows (proto-M > 0) » PCI of pigeons (proto-M $\approx$ 0) at comparable brain size. Operational criterion: proto-M > 0 $\equiv$ $\exists$ at least one of: (a) documented social learning of novel behavior (Hoppitt & Laland 2013), (b) tool manufacture (Shumaker et al. 2011), (c) cultural variation between populations (Whiten et al. 1999). Detailed species table — EMT, Part XVI.
• G-3 (Topology-first): $CL_G \propto \sigma_{SW}$, not $\propto |N|$ (number of neurons). Organisms with convergent neural network topology should demonstrate comparable $CL_G$ regardless of phylogenesis. Test: PCI of octopus $\approx$ PCI of crow at comparable $\sigma_{SW}$.

Detailed formalization and mechanism table — EMT, Part XVI.

Prediction G-4: M » G as a necessary condition for reflection (formalized). Define recursion levels: $SMC^{(1)} = \{m \in V_m : target(m) \in V_u \cup I\}$ (narrative self), $SMC^{(k)} = \{m \in V_m : target(m) \in SMC^{(k-1)}\}$ (meta-memes). Theorem: $|SMC^{(2)}| > 0$ requires $|V_m| \geq (\alpha + \beta + \gamma\beta) \cdot |V_u|$, where $\alpha \geq 1$ (environment modeling), $\beta \geq 1$ (L1 self-model), $\gamma > 0$ (L2 meta-memes). Lower bound: M/G $\geq$ 3; in real systems $\alpha \gg 1$, therefore M/G $\gg$ 3. Threshold estimate: $M/G_{crit} \sim \mathcal{O}(10)$ — metacognition appears at proxy M/G $\sim$ 20–50 (macaques), is absent at M/G < 1 (pigeons). $M/G_{crit}$ depends on $\alpha$ (environmental complexity). Spectrum: M/G $\approx$ 0 $\to$ L0, M/G < 1 $\to$ L1, M/G $\sim$ 1 $\to$ proto-L2 (chimpanzees), M/G » 1 $\to$ full reflection (humans). IIT, AST, GWT do not distinguish M and G — they cannot formulate a threshold condition. Detailed formalization, threshold table, and re-analysis #3 — EMT, Part XVI.

Relationship to EMT definition. The flat definition $SMC = \{m \in M : target(m) \in M \cup G \cup I\}$ (EMT) is equivalent to $SMC = \bigcup_{k \geq 1} SMC^{(k)}$ — the union of all levels of the recursive hierarchy.

Detailed Altered States: Component-Wise CL Analysis

The table above provides an overview map. Below is a component-wise analysis of each altered state with model predictions and empirical anchors.

1. Psychedelics: $\sigma > 1$, but CL is non-monotonic

Components:

• $\sigma_{SW}$: significantly elevated (> 1.5). Connectome entropy increases (Carhart-Harris et al., 2014); homological scaffolds are restructured (Petri et al., 2014, J. Royal Society Interface). In BMC terms: lateral inhibition is weakened $\to$ memes from different clusters are activated simultaneously.
• $A_{SMC}$: non-monotonic. Low doses $\to$ hyperreflection ($A_{SMC}$ $\uparrow$, “everything is about me”). High doses $\to$ ego dissolution ($A_{SMC} \to 0$, the “I” dissolves). The transition is not gradual but a bifurcation: the threshold depends on the baseline rigidity of the memplex.
• $Balance$: unstable. The G-system (emotions, bodily signals) is amplified, the M-system is restructured $\to$ Balance oscillations.
• $f(Balance)$: with unstable Balance, the function oscillates $\to$ CL “flickers.”

Model prediction (unique to BMC): CL under psychedelics follows an inverted U-curve by dose:

• Low dose: $\sigma$ $\uparrow$ moderately, $A_{SMC}$ $\uparrow$ $\to$ CL $\uparrow$
• Medium dose: $\sigma$ $\uparrow\uparrow$, $A_{SMC}$ begins to oscillate $\to$ CL unstable
• High dose: $\sigma$ $\uparrow\uparrow\uparrow$, $A_{SMC} \to 0$ (ego dissolution) $\to$ CL $\downarrow$ despite high $\sigma$

This distinguishes BMC from theories predicting monotonic increase of consciousness with dose. IIT would predict $\phi$ $\uparrow$ (more integration = more consciousness); BMC predicts: without a self-model there is no subject of experience, therefore CL drops.

Sources: Carhart-Harris & Friston (2019). “REBUS and the anarchic brain”. Pharmacological Reviews; Petri et al. (2014). “Homological scaffolds of brain functional networks”. J. Royal Society Interface; Tagliazucchi et al. (2016). “Increased global functional connectivity correlates with LSD-induced ego dissolution”. Current Biology.

2. Meditation: two modes, two CL profiles

Focused attention (FA, samatha):

• $\sigma_{SW}$: $\approx$ 1.0 (unchanged). The network remains in normal regime.
• $A_{SMC}$: selectively reduced. The practitioner suppresses mind-wandering (DMN $\downarrow$) but preserves meta-awareness “I am meditating” $\to$ $A_{SMC}$ is partially active but narrowed to monitoring.
• $Balance$: M > G (voluntary control over automatic processes).
• $CL$: $\approx$ normal or slightly above (enhanced clarity without expansion).

Open monitoring (OM, vipassana):

• $\sigma_{SW}$: $\approx$ 1.0–1.1 (slight elevation). Cross-cluster connections are strengthened.
• $A_{SMC}$: modified: neither suppressed nor amplified, but the pattern is changed. DMN is reconfigured: narrative self $\downarrow$, minimal self $\uparrow$ (Brewer et al., 2011, PNAS).
• $Balance$: M » G (deep voluntary control).
• $CL$: modified — formally close to normal, but qualitatively different (less narrative “I,” more pure observation).

Prediction: FA and OM are distinguishable by $A_{SMC}$ pattern at the same $\sigma_{SW}$. Testable: fMRI DMN patterns in FA vs OM practitioners at identical overall connectivity.

Sources: Lutz et al. (2008). “Attention regulation and monitoring in meditation”. Trends in Cognitive Sciences; Fox et al. (2016). “Functional neuroanatomy of meditation”. Neuroscience & Biobehavioral Reviews; Brewer et al. (2011). “Meditation experience is associated with differences in default mode network activity and connectivity”. PNAS.

3. Anesthesia: graded CL shutdown

Key mechanism: Anesthetics (propofol, sevoflurane) primarily suppress $\sigma_{SW}$ through enhanced GABA inhibition $\to$ $\sigma < 1$ $\to$ subcritical regime $\to$ memes cannot spread. $A_{SMC}$ fades secondarily: without spreading activation, the self-model loses input.

Prediction (unique to BMC): Loss of consciousness under anesthesia occurs in steps, not gradually:

1. First, level 2 recursion disappears (metacognition: “I know that I know”)
2. Then level 1 recursion (phenomenal consciousness: the feeling of “something”)
3. Last — level 0 (basic reflexes)

Each level is a separate attractor with a bifurcation threshold. Between them lies hysteresis: the dose for falling asleep $\neq$ the dose for awakening (empirically confirmed: Friedman et al., 2010, Anesthesiology).

Sources: Casali et al. (2013). “A theoretically based index of consciousness”. Science Translational Medicine; Mashour & Hudetz (2018). “Neural correlates of unconsciousness in large-scale brain networks”. Trends in Neurosciences.

4. Flow: the paradox of peak performance at reduced CL

Components:

• $\sigma_{SW}$: $\approx$ 1.0 (optimal regime, criticality).
• $A_{SMC}$: reduced (transparent self-model). The athlete does not think “I am running” — they simply run. DMN $\downarrow$, dorsolateral PFC $\downarrow$ (transient hypofrontality: Dietrich, 2004, Consciousness and Cognition).
• $Balance$: optimal ($\approx$ 1.5). The skill is automatized ($habit > \theta_{habit}$, $Auto(S)$ — see Automatization), task difficulty $\approx$ skill level (M-level).
• $f(Balance)$: maximum $\to$ amplifying multiplier.

Paradox: $CL = \sigma \cdot A_{SMC} \cdot f(Balance)$. At $\sigma \approx 1.0$ and $f(Balance) = max$, but $A_{SMC}$ $\downarrow$ $\to$ CL is lower than during ordinary wakefulness. Yet subjectively, flow is described as a peak experience.

Resolution of the paradox (BMC’s theoretical position): CL measures self-awareness as a subject, not quality of performance. Flow is a state in which the self-model becomes transparent: the subject does not observe themselves but acts directly. This is not a deficit of consciousness but its optimal regime for action. Proposed refinement: distinguish $CL_{reflexive}$ (includes $A_{SMC}$) and $CL_{operative}$ (= $\sigma \cdot f(Balance)$, without $A_{SMC}$). Flow = low $CL_{reflexive}$, high $CL_{operative}$.

Formalization of the flow channel via SIT:

Csikszentmihalyi’s flow channel (challenge $\leftrightarrow$ skill) is formalized through SIT:

where:

• $SIT_{bore} = \epsilon$ (minimal structural deficit — no gaps $\to$ boredom)
• $SIT_{anxiety} = SIT_{bore} + \Delta_{LP}$, where $\Delta_{LP}$ is the range in which LP is predicted to be > 0

Flow readiness formula:

where $\hat{x} = \frac{x - x_{min}}{x_{max} - x_{min}}$ are normalized values. $F_{ready} \to 1$ when: SIT is at the optimum (neither 0 nor max), $\sigma \approx 1$, $A_{SMC}$ is minimal, LP is maximal.

Testable prediction: The correlation of $F_{ready}$ with the Flow State Scale (FSS) should be higher than the correlation of any single component. This allows quantitative operationalization of the flow channel.

Sources: Csikszentmihalyi (1990). Flow: The Psychology of Optimal Experience; Dietrich (2004). “Neurocognitive mechanisms underlying the experience of flow”. Consciousness and Cognition; Ulrich et al. (2014). “Neural correlates of experimentally induced flow experiences”. NeuroImage.

5. Lucid dreaming: partial SMC reactivation in REM

Components:

• $\sigma_{SW}$: $\approx$ 0.8 (as in normal REM — reduced connectivity).
• $A_{SMC}$: partially reactivated. The dreamer is aware that “I am dreaming” $\to$ SMC level 1 (phenomenal) is active, sometimes level 2 (metacognition: “I am controlling the dream”).
• $Balance$: G > M (as in REM), but M is partially restored (frontal activation).
• $f(Balance)$: elevated compared to normal REM.

Prediction: $CL_{lucid} > CL_{REM}$ at the same $\sigma_{SW}$, due to $A_{SMC}$ $\uparrow$. Quantitatively: $CL_{lucid} \approx 0.5$–$0.6$ vs $CL_{REM} \approx 0.3$–$0.4$.

Testable prediction: PCI(lucid dreaming) > PCI(normal REM). Verifiable via TMS-EEG in trained lucid dreamers with confirmation through LRLR (Left-Right eye signals).

Neurosubstrate: Frontal 40 Hz activity during lucid dreaming (Voss et al., 2009, Sleep) — in BMC terms: reactivation of frontal SMC nodes over the “background” REM regime.

Sources: Voss et al. (2009). “Lucid dreaming: a state of consciousness with features of both waking and non-lucid dreaming”. Sleep; Voss et al. (2014). “Induction of self awareness in dreams through frontal low current stimulation”. Nature Neuroscience; Baird et al. (2019). “The cognitive neuroscience of lucid dreaming”. Neuroscience & Biobehavioral Reviews.

6. Dissociative states: formally high CL, subjectively “empty”

Types: depersonalization (DP), derealization (DR), dissociative identity disorder (DID).

Components:

• $\sigma_{SW}$: $\approx$ 1.0 (network structurally intact).
• $A_{SMC}$: formally active — the patient is self-aware and can describe their state.
• $Balance$: not disrupted through G/M imbalance.
• Key problem: the CL formula yields a high value ($\sigma$ normal, $A_{SMC}$ normal, Balance normal) — yet subjective experience is described as “unreal,” “observing from the outside,” “I am a robot.”

Diagnostic challenge for the model: Dissociation exposes a limitation of the formula $CL = \sigma \cdot A_{SMC} \cdot f(Balance)$. What is disrupted?

BMC’s answer: The I-layer (immune system) is disrupted — the connection between G-signals (emotions, bodily sensations) and SMC. Depersonalization = emotional deafferentation of the SMC: the self-model is active but does not receive emotional coloring from the G-layer.

Formalization: we introduce a correction coefficient for interoceptive integration:

where $I_{intero}$ is the degree of connectivity between G-signals and SMC:

In depersonalization: $w_{gs} \to 0$ (insula disconnected) $\to$ $I_{intero} \to 0$ $\to$ $CL_{full} \to 0$ despite formally normal $\sigma$, $A_{SMC}$, Balance.

Prediction: In patients with depersonalization:

• PCI $\approx$ normal ($\sigma_{SW}$ is not disrupted)
• fMRI: insula $\downarrow$, somatosensory cortex $\to$ mPFC connectivity $\downarrow$
• Subjective consciousness scales $\downarrow$ (CDS, DES)
• Dissociation between “cognitive” CL (high) and “full” $CL_{full}$ (low)

Sources: Sierra & Berrios (1998). “Depersonalization: neurobiological perspectives”. Biological Psychiatry; Medford et al. (2005). “Emotional experience and awareness of self: functional MRI studies of depersonalization”. Frontiers in Psychology; Simeon et al. (2000). “Feeling unreal: a PET study of depersonalization disorder”. American Journal of Psychiatry.

7. Unified parameter space for consciousness disorders

All consciousness disorders are described as points in the space $(\sigma_{SW}, A_{SMC}, Balance, I_{intero}, Q, SIT, H)$:

Formalization of distance:

where $\mathbf{x}_P$ is the parameter vector in pathology, $\mathbf{x}_N$ is the parameter vector in normal state, and $\boldsymbol{\sigma}_N$ is the standard deviations of the normal state. Comorbidity is predictable:

Disorders with close profiles in BMC space are more often comorbid (ADHD + depression: both deviate in $\sigma_{SW}$; OCD + autism: both deviate in I-parameter).

Testable prediction: The comorbidity matrix (epidemiological data) correlates with the BMC distance matrix. This is formalizable and verifiable on existing data (NCS-R, UK Biobank). For a falsifiable test: numerical parameter values must be fixed before verification against comorbidity data (pre-registration).

Distinction from RDoC/HiTOP: RDoC (NIMH, 2010) and HiTOP use a dimensional approach but without a unified causal mechanism and without a formal comorbidity metric $d(P,N)$. BMC complements them with a dynamic model.

Neurosubstrate: The neuromechanics of each disorder — see BM. Conceptual taxonomy — see EMT, Part XXII.

Relationship with PCI (Perturbational Complexity Index)

PCI (Massimini et al.) measures the complexity of cortical response to TMS stimulation. In our model:

PCI and CL are not identical: PCI measures the network’s potential for integration (one component of CL), while CL measures the current state accounting for SMC and Balance. PCI is the best available proxy for $\sigma_{SW}$, but not for CL as a whole (see the proxy metrics table above). This is precisely why BMC predicts that the composite index $CL_{proxy}$ outperforms PCI alone: states with identical PCI (e.g., wakefulness vs depersonalization) differ in CL due to $A_{SMC}$ and $I_{intero}$.

Predictions

Sources: Massimini et al. (2005). “Breakdown of cortical effective connectivity during sleep”. Science; Carhart-Harris et al. (2014). “The entropic brain”. Frontiers in Human Neuroscience; Carhart-Harris & Friston (2019). “REBUS and the anarchic brain”. Pharmacological Reviews; Petri et al. (2014). J. Royal Society Interface; Lutz et al. (2008). Trends in Cognitive Sciences; Casali et al. (2013). Science Translational Medicine; Dietrich (2004). Consciousness and Cognition; Voss et al. (2009). Sleep; Voss et al. (2014). Nature Neuroscience; Sierra & Berrios (1998). Biological Psychiatry; COGITATE Consortium (2025). Nature.

Part XIV. Triple Binding — Unity of Conscious Experience

Problem: why is experience unified?

Multiple memes are active simultaneously: visual, auditory, emotional, semantic. Yet subjective experience is unified: we see a red cup of hot coffee as a single whole, not as a set of disconnected features. How does a network of memes give rise to unity?

Three Binding Mechanisms

The unity of conscious experience is ensured by three independent but complementary mechanisms (see EMT, Part XVI):

1. Structural binding: ensemble overlap

Two memes are structurally bound if their cell assemblies share common neurons:

The greater the overlap, the stronger the bond. Activating one meme automatically partially activates the other through shared neurons.

Network analog: Shared neurons = a shared connection with high weight. Structural binding = the presence of an edge $w_{ij} > \theta_{bind}$.

2. Temporal binding: synchronization within the theta window

Two memes are considered “bound in time” if their peak activations fall within the same theta window (~125 ms):

where $\phi_\theta$ is the theta rhythm phase (4–8 Hz), and $\delta$ is the tolerance (~30 ms).

Network analog: Synchronous activation = both nodes $a_i(t) > \theta$ and $a_j(t) > \theta$ within the same discrete step $t$.

Neurosubstrate: The hippocampal theta rhythm as the “substrate clock” — groups activations into discrete packets. Memes that fall within the same theta cycle are perceived as part of the same experience (see BM, Part III).

Relationship to event segmentation: Temporal binding operates within an episode — synchronizing components of a single moment. Event segmentation (see Part VIII: Episodic Memory) operates on a different scale — determining boundaries between episodes through PE jumps. Two levels of discretization: theta cycle (~125 ms) within a moment, event boundary (seconds to minutes) between episodes.

3. Competitive (Bayesian) binding: filtration through WM

Active working memory admits only a coherent coalition of memes simultaneously. Memes with high dissonance are evicted:

If $D > \theta_D$ for a pair in the coalition $\to$ one of the memes is evicted through lateral inhibition. The coherence constraint applies only to Active WM ($k_{active} \approx 3$–$4$ pointers); Latent WM (memes with $\psi_i > \theta_\psi$, not in the focus of attention) can contain contradictory memes without conflict — they are not in consciousness. Upon reactivation (pinging) of a latent meme, it enters Active WM and undergoes competitive binding (see Activity-Silent WM).

Network analog: Softmax normalization + lateral inhibition (see Part VIII). Active WM = an attractor in which only a coherent configuration is stable.

Full Binding Formula

All three components are necessary. Binding = absence of dissonance in WM given the presence of structural and temporal connection. The integrity of experience is a result of filtration, not a given.

What Triple Binding Explains

Predictions

Sources: Fries P. (2005). “A mechanism for cognitive dynamics: neuronal communication through neuronal coherence”. Trends in Cognitive Sciences; Lamme V.A.F. (2006). “Towards a true neural stance on consciousness”. Trends in Cognitive Sciences; Lisman J. & Jensen O. (2013). “The theta-gamma neural code”. Neuron.

Part XV. Diffusion Engine and Embedding Space

Part VIII formalizes the Graph Engine — propagation of activation along edges of the graph $G = (V, E)$. However, biological networks use two parallel transmission mechanisms: synaptic (discrete, along connections) and volume (volume transmission — diffusion of neurotransmitters into the extracellular space, without addressing a specific synapse). The second mechanism influences semantically proximate nodes without a direct edge between them. Formalizing this mechanism requires introducing an embedding space and a Diffusion Engine. Architectural integration of the three engines is described in AGI_F, Part VII.

Modulation Vector Mod(t)

The third component of the architecture is the Modulation Engine: a global state that modulates all Graph Engine and Diffusion Engine operations in each computational cycle. Formally:

Each component is a function of active G-programs:

Full G $\to$ Mod table (modulation direction for each G-program) — see AGI_F, Part VII. Key examples: FEAR $\downarrow\downarrow$ $\theta_{act}$ (hypervigilance) + $\uparrow\uparrow$ $\lambda_{speed}$ (accelerated processing); PLAY $\uparrow\uparrow$ $\lambda_{noise}$ (stochastic combinatorics) + $\uparrow$ $\lambda_{inh}$ (controlled chaos); CARE $\uparrow\uparrow$ $\lambda_{soc}$ (oxytocin).

Embedding Space

Definition 14.1. The embedding space is $\mathbb{R}^d$, where $d = \dim(\mathbf{f})$, $\mathbf{f}$ is the feature vector extracted from a sensory signal by a G-level module (feature extraction, hardwired). Each meme $m_i \in V$ possesses a coordinate (embedding) $\mathbf{e}_i \in \mathbb{R}^d$.

Birth. Upon meme creation (Path 1 or Path 2 of memogenesis, see AGI_F, Part VII):

The embedding is fixed at the moment of birth — the meme “inherits” its coordinate from the generating signal.

Drift. Reconsolidation (Part VIII, Reconsolidation section) and recombination (BLEND) shift the embedding:

where $\Delta\mathbf{e}_i$ is the cumulative shift from reconsolidation and BLEND. Drift is bounded: $\|\Delta\mathbf{e}_i\| \leq r_{drift}$, otherwise the meme loses its identity (it semantically becomes a different meme). Biological analog: memory distortion during reconsolidation (Nader et al., 2000).

Definition 14.2. Proximity of two memes in embedding space:

where $\sigma_{emb}$ is the proximity scale (analog of the volume transmission radius). When $\|\mathbf{e}_i - \mathbf{e}_j\| \to 0$: $\text{prox} \to 1$; when $\|\mathbf{e}_i - \mathbf{e}_j\| \gg \sigma_{emb}$: $\text{prox} \to 0$.

Key distinction: proximity $\neq$ edge weight.

Proximity is a latent structure describing semantic similarity; the graph is an explicit structure describing established associations. Semantically close memes ($\text{prox} \gg 0$) may have no edge (not yet connected); distant memes ($\text{prox} \approx 0$) may have a strong edge (contextual association, episodic binding). Diffusion operates on the former; spreading activation (Part VIII) operates on the latter.

Diffusion Engine

Definition 14.3. Diffusion activation of meme $m_j$ at time $t$:

where:

• $\lambda_{diff}(t) \in [0, 1]$ is the modulatory diffusion coefficient (set by the Modulation Engine through G-programs; $\lambda_{diff} \neq \lambda_{noise}$: diffusion is deterministic proximity-based transmission, noise is stochastic perturbation),
• summation is over memes without a direct edge to $j$ ($(i,j) \notin E$), restricted to the top-$K$ nearest in embedding space,
• $K$ is the truncation parameter ($K \sim 10$–$50$), ensuring computational efficiency.

Complexity. Full computation: $O(|V|^2)$. With top-$K$ truncation: $O(|V| \cdot K)$. Using approximate nearest neighbor (ANN) structures to maintain top-$K$: $O(|V| \cdot K \cdot \log |V|)$ for updates after embedding drift.

Three Roles of the Diffusion Engine

Role 1: Priming. An active meme “warms up” semantic neighbors without an edge, facilitating their subsequent activation through the Graph Engine. This explains experimental data on semantic priming (Meyer & Schvaneveldt, 1971) for words without a direct association: “doctor” $\to$ “nurse” (direct edge, Graph Engine) vs “doctor” $\to$ “stethoscope” (no edge, but high proximity $\to$ Diffusion Engine).

Role 2: Crystallization (Path 2 of memogenesis). Diffuse activation creates a “density field” in embedding space (see Crystallization section below). When density in a region without an explicit node exceeds the threshold, a new meme is born.

Role 3: Neural stigmergy. Activation traces in embedding space influence subsequent processing — the agent “leaves marks” for itself. Formalized in BM, Part III.

Comparison of Graph Engine and Diffusion Engine

Both engines operate in parallel in each computational cycle. The Graph Engine determines what is active (focus). The Diffusion Engine determines what is “warmed up” (context, background).

Combined Activation Formula

Definition 14.4. Combined activation of meme $m_i$ at step $t+1$:

where:

• $a_i^{graph}(t+1) = \sigma\!\left((1 - \lambda) \cdot a_i(t) + \sum_j w_{ij} \cdot a_j(t)\right)$ is graph activation (Part VIII); the factor $(1 - \lambda)$ inside $\sigma$ controls the inertia of self-activation (how much of the previous activation the node “remembers”), a Graph Engine parameter,
• $a_i^{diff}(t)$ is diffusion activation (Definition 14.3),
• $\lambda_{decay} \in (0, 1)$ is global energy decay: without external input ($a^{graph} = a^{diff} = 0$), activation tends to zero. Distinction from $(1-\lambda)$: $\lambda_{decay}$ acts after summation of all sources and ensures the energy balance of the system,
• $\text{clip}[x, 0, 1] = \max(0, \min(1, x))$ guarantees the invariant $a_i \in [0, 1]$.

Note. The term $-\lambda_{decay} \cdot a_i(t)$ ensures energy economy: a meme receiving neither graph nor diffusion support decays. This formalizes the resource interpretation of competition: WM slots are limited not conventionally ($k_{active}$ as a parameter) but energetically — each active meme consumes “computational budget” (AGI_F, Part VII).

Crystallization: Density Field and Path 2 of Memogenesis

The Diffusion Engine creates background activation in embedding space. This allows formalizing Path 2 of memogenesis — the birth of a meme not from a single event ($PE > \theta_{PE}$) but from a recurring pattern.

Definition 14.5. The density field at point $\mathbf{x} \in \mathbb{R}^d$:

where $K(r) = \exp(-r^2 / 2\sigma_K^2)$ is a Gaussian kernel (kernel density estimation). Density is high in regions where active memes are concentrated. Regions with high $\rho$ but without an explicit meme node are candidates for crystallization.

Definition 14.6 (Crystallization). A new meme crystallizes at point $\mathbf{x}^*$ if:

The first condition: density exceeds the threshold (sufficient “mass” of activation). The second: no existing meme within the vicinity $r_{min}$ (the meme crystallizes in a “void,” not duplicating an existing node).

The crystallized meme receives $\mathbf{e}_{new} = \mathbf{x}^*$, $\kappa = 0$, $a_0 = a_{init}$. Post-hoc G-check: if at the moment of crystallization a G-program $g$ is active with $\text{rel}(g, \mathbf{x}^*) > \theta_G$, the meme receives an edge to $g$ ($u_{link}$). If no G-program is relevant, the meme is created as G-neutral ($u_{link} = \emptyset$): low priority in WM, subject to rapid pruning, but may survive through subsequent binding.

Corollary 14.1. The Diffusion Engine is necessary for Path 2 of memogenesis.

Argument: Path 2 requires density $\rho(\mathbf{x}^*, t) > \theta_{crystal}$ at point $\mathbf{x}^*$ far from existing memes ($\nexists\, m_i : \|\mathbf{e}_i - \mathbf{x}^*\| < r_{min}$). The kernel $K(\|\mathbf{x}^* - \mathbf{e}_i\|)$ decays exponentially with distance: if $\mathbf{x}^*$ is far from all memes, the contribution of the Graph Engine (which modifies only $a_i$ for existing nodes) to $\rho(\mathbf{x}^*, t)$ is negligible. The Diffusion Engine, by contrast, “warms up” embedding space continuously, creating nonzero activation in inter-node regions, which supports density $\rho$ above the crystallization threshold. Ergo, without the Diffusion Engine, crystallization is practically impossible for points that are not immediate neighbors of existing memes.

Corollary 14.2. Crystallization and PE-memogenesis complement each other.

Memogenesis: Formal Pipeline of Meme Birth

Definitions 14.5–14.6 formalize Path 2 (crystallization). Below, Path 1 (PE $\times$ G-relevance) and the full memogenesis pipeline unifying both paths are formalized. Architectural context — AGI_F, Part VII.

Feature Extraction and S-Layer

Definition 14.7 (Feature extraction). The feature vector $\mathbf{f}(t) \in \mathbb{R}^d$ is extracted from the raw sensory signal by a G-level module:

$\text{extract}$ is a fixed (non-learnable within BMC) function. Analog: primary sensory cortices (V1, A1, S1) — hardwired G-level processors (BM, Part IV). Dimensionality $d = \dim(\mathbf{f})$ defines the embedding space (Definition 14.1). Embedding at birth: $\mathbf{e}_i = \mathbf{f}(t_{birth})$.

Path 1: Event-Driven Memogenesis ($PE \times G_{rel}$)

Definition 14.8 (Prediction Error). The agent maintains a prediction $\hat{\mathbf{f}}(t)$ based on current graph activation (spreading activation $\to$ expected sensory pattern). Prediction Error:

$PE > \theta_{PE}$ is a necessary condition for event-driven memogenesis. A predictable signal ($PE \leq \theta_{PE}$) does not generate a new meme node.

Relationship to event boundary detection (Episodic Memory section, Part VIII): $PE$ is one of 4 episode boundary triggers. High $PE$ simultaneously (a) creates a meme node and (b) may terminate the current episode $\varepsilon_k$.

Definition 14.9 (G-relevance). G-relevance of a signal at time $t$:

where $a_g(t)$ is the activation of G-program $g$, and $\text{rel}(g, \mathbf{f})$ is the relevance function (proximity in embedding space between the G-prototype and the feature vector). $G_{rel} > \theta_G$ is the second necessary condition: not every unexpected signal creates a meme, only G-significant ones.

Definition 14.10 (Event-driven memogenesis — Path 1). A meme node $m_{new}$ is created when the conjunction is satisfied:

Attributes of the new meme:

where $\kappa = 0$ (sensory consolidation level, see Consolidation Level section), $F_{init}$ is the initial fidelity (low), and $u_{link}$ is the edge to the dominant G-program.

Definition 14.11 (Contextual binding). Co-active memes in WM at the time of $m_{new}$’s birth receive edges:

Binding connects the new meme to the context of its origin. Relationship to barcodes $B_k$ (Episodic Memory section): binding includes $m_{new}$ in the current episode, creating a content-independent index. Relationship to SIT: if $m_{new}$ fills a structural gap, $closure(gap)$ increases; if it opens a new question (e.g., an unexpected stimulus without explanation), a new gap is created $\to$ $SIT$ rises.

Phase 0 and the Paradox of Total Novelty

Proposition 14.1 (Self-balancing of memogenesis). With $|V_m| = 0$ (empty graph), memogenesis does not overwhelm the system despite $PE = \|\mathbf{f}(t)\|$ for all signals.

Argument. With $|V_m| = 0$: $\hat{\mathbf{f}} = \mathbf{0}$ (no basis for prediction), therefore $PE(t) = \|\mathbf{f}(t)\|$ — every signal is “novel.” Simultaneously, SEEKING is active by default $\to$ $G_{rel}$ is satisfied. But $S_{bw}(0) \approx S_{bw}^{min}$ — the S-channel bandwidth is minimal at the start of ontogenesis (AGI_F, Part VII). A narrow sensory channel = natural throttle: few signals arrive per unit time $\to$ memogenesis proceeds at a controlled rate. As $|V_m|$ grows:

The system self-balances: channel expansion is compensated by improved predictions. $\square$

Necessity of Both Paths

Proposition 14.2 (Completeness of memogenesis). Neither Path 1 nor Path 2 alone provides complete coverage of the meme space. The conjunction is necessary.

Argument.

• Path 1 alone ($\theta_{crystal} = \infty$): memes arise only when $PE > \theta_{PE}$. Low-PE patterns (habitual, repetitive, emotionally neutral) never generate nodes. The agent does not form representations of stable background regularities (gravity, friction, daily patterns). This contradicts perceptual learning.
• Path 2 alone ($\theta_{PE} = \infty$): memes arise only through crystallization ($\rho > \theta_{crystal}$). Rare but significant events (one-time, high-PE) do not create memes since they cannot accumulate density. The agent does not remember flashbulb events. This contradicts episodic memory.
• Together: Path 1 covers rare + significant events; Path 2 covers frequent + background patterns. $\square$

Note. BLEND (recombination during sleep, see Dream-phase section) creates memes from combinations of existing ones — this is not a third path of memogenesis from S-signal, but M-level recombination: memes are born from memes, not from the sensory stream.

Integration with the Consolidation Pipeline

Memogenesis is the entry to the memory pipeline (see Consolidation Process section):

Each newborn meme ($\kappa = 0$) enters the $P_{tag}$ pipeline (SWR-tagging section): emotionally significant memes ($u_{link} \neq \emptyset$, high $G_{rel}$) receive elevated $P_{tag}$ $\to$ priority consolidation during sleep. G-neutral memes (Path 2, $u_{link} = \emptyset$) go through $P_{tag}$ on general terms and are often pruned before reaching $\kappa = 1$.

Memogenesis Predictions

P-M1 (Double dissociation of memogenesis paths). Ablation of Path 1 ($\theta_{PE} = \infty$) while preserving Path 2 eliminates flashbulb memes (memes created in a single episode under high PE). Ablation of Path 2 ($\theta_{crystal} = \infty$) while preserving Path 1 eliminates background patterns (memes from low-PE repetitive stimuli). Quantitatively: with full ablation of one path, the proportion of that type’s memes $\to$ 0, and the other $\to$ 100%. Verification: ablation study in native BMC simulation, counting memes by birth type (event-born vs crystallized), comparing cognitive deficits.

P-M2 ($G_{rel}$ modulates the rate of memogenesis). Increasing mean activation of G-programs ($\overline{a_g}$ $\uparrow$) increases the frequency of meme creation via Path 1 (given a fixed PE distribution of incoming signals). Consequence: emotionally saturated periods (FEAR, SEEKING) produce more memes per unit time than emotionally neutral ones. Verification: neurobiological data — amygdala activation enhances hippocampal encoding (Dolcos et al., 2004); behaviorally — more details are remembered from emotional events.

P-M3 (Phase 0: memogenesis rate correlates with $S_{bw}$). At the start of ontogenesis ($|V_m| \approx 0$), the rate of memogenesis is limited by $S_{bw}$, not the PE filter. As development progresses, the bottleneck shifts to PE (most signals become predictable). Verification: in native BMC simulation, plot a “memes/cycle” vs $|V_m|$ curve: expect initially linear growth (bottleneck = $S_{bw}$), then plateau and decline (bottleneck = PE).

Formal Properties

Proposition 14.3 (Graph/Diffusion Asymmetry). The Graph Engine and Diffusion Engine are irreducible to each other.

Argument:

• The Graph Engine cannot activate nodes without edges ($(i,j) \notin E \Rightarrow$ no transmission). The Diffusion Engine can.
• The Diffusion Engine cannot transmit negative activation ($\text{prox} \geq 0$). The Graph Engine can ($w_{ij} < 0$).
• Ergo: Graph Engine $\not\supset$ Diffusion Engine, Diffusion Engine $\not\supset$ Graph Engine.

Proposition 14.4 (Steady state). In the absence of external S-input, with $\lambda_{diff} = const$ and $\lambda_{decay} > 0$:

Argument: $a_i(t+1) \leq a_i^{graph} + a_i^{diff} - \lambda_{decay} \cdot a_i$. In a closed system (no external input), total energy $E = \sum_i a_i$ decreases, since $\lambda_{decay}$ subtracts a fraction at each step, while $a^{graph}$ and $a^{diff}$ redistribute but do not create energy. This ensures economy: without stimulation, the system “falls asleep.”

Proposition 14.5 (Diffusion as a pathway to edges). The Diffusion Engine creates prerequisites for edge formation in the Graph Engine.

Argument: Diffusion activation raises $a_j$ for proximity neighbors. If $a_i > 0$ and $a_j > 0$ simultaneously (co-activation), Hebbian learning creates an edge $w_{ij} > 0$. Thus, latent similarity (proximity) can “crystallize” into an explicit connection (edge). Trajectory: $\text{prox}(i,j) \gg 0 \xrightarrow{\text{Diffusion}} \text{co-activation} \xrightarrow{\text{Hebbian}} (i,j) \in E$.

Modulation of $\lambda_{diff}$ and States of Consciousness

The parameter $\lambda_{diff}(t)$ is modulated by the Modulation Engine (AGI_F, Part VII):

Relationship to Part XIII (Consciousness Level): increasing $\lambda_{diff}$ under psychedelics explains the increase in activation pattern entropy ($S_{activation}$ $\uparrow$) — the Diffusion Engine “blurs” the topological boundaries of memplexes, creating the ego dissolution effect through a mechanism distinct from the weakening of competitive binding (Part XIV).

Predictions

P-D1 (Necessity of the Diffusion Engine for crystallization). Ablation of the Diffusion Engine ($\lambda_{diff} = 0$) while preserving the Graph Engine eliminates Path 2 of memogenesis. Quantitatively: the proportion of memes born through crystallization drops to zero. Verification: ablation study in native BMC simulation, counting memes by birth type.

P-D2 (Priming without edges). Semantic priming (reduced reaction latency to a target stimulus following a proximity-close but associatively unrelated prime) correlates with $\text{prox}(i,j)$, not with the presence of an edge. Verification: semantic priming experiments with control for associative norms (Nelson et al., 2004) — the effect persists for pairs with zero associative strength but high cosine similarity (proxy for prox).

P-D3 (Diffusion and creativity). $\lambda_{diff}$ positively correlates with creativity (measured by: divergent thinking, AUT — Alternative Uses Task). Verification: individual differences in volume transmission (DA/5-HT balance) predict AUT scores. Prediction: substances that enhance volume transmission (psychedelics, low-dose DA agonists) increase AUT scores — consistent with experimental data (Mason et al., 2021).

P-D4 (Embedding drift). Repeated reconsolidation shifts $\mathbf{e}_i$ from its original coordinate. Verification: longitudinal fMRI — the representation pattern of a meme (neural decoding accuracy) drifts over time, more strongly for memes that have undergone more reconsolidations (Braga & Buckner, 2017, Neuron).

Sources: Meyer D.E. & Schvaneveldt R.W. (1971). “Facilitation in recognizing pairs of words”. J. Experimental Psychology; Nader K., Schafe G.E. & LeDoux J.E. (2000). “Fear memories require protein synthesis in the amygdala for reconsolidation after retrieval”. Nature; Mason N.L. et al. (2021). “Spontaneous and deliberate creative cognition during and after psilocybin exposure”. Translational Psychiatry; Carhart-Harris R.L. et al. (2014). “The entropic brain: a theory of conscious states informed by neuroimaging research with psychedelic drugs”. Frontiers in Human Neuroscience.

Part XVI. Inter-Agent Exchange, Scarcity, and Critical Periods

Parts I–XV formalize BMC of a single agent: the meme graph, activation, competition, memory, immune system, diffusion space, and memogenesis. But a meme is a replicator (Dawkins, 1976), and replication requires inter-agent transmission. A single agent is a closed system: no memetic selection, no cultural ratchet. This part formalizes three necessary conditions for an evolving memetic system: (1) inter-agent meme exchange, (2) computational scarcity, (3) critical periods of development.

Architectural context — AGI_F, Part VII; coordination through environment — Part X, Stigmergy section; replication pressure formula — Part X, Replication Pressure section; memogenesis — Part XV.

Inter-Agent Meme Exchange

Signal Encoding

Definition 15.1 (Transmission signal). When agent $A$ satisfies the expression drive condition ($R_{expr}(m_{top}, t) > \theta_{expr}$, Part X), a signal is formed:

where $m_{top} = \arg\max_{m_i} R_{expr}(m_i, t)$ is the meme with the greatest replication pressure, and $\text{Mod}_A(t)$ is the modulatory state of the agent (active G-programs, affective tone). Encoding is state-dependent: the same meme under FEAR and under PLAY produces different signals (intonation, intensity, context).

Reception and Memogenesis at the Recipient

Agent $B$ in the communication zone receives $s_A$ through a dedicated sensory channel $S_{social}$. The signal is processed through the standard memogenesis pipeline (Part XV, Definitions 14.7–14.11):

1. Feature extraction: $\mathbf{f}_B(t) = \text{extract}(s_A)$ — feature vector in $B$’s embedding space.
2. PE check: $PE_B(t) = \|\mathbf{f}_B(t) - \hat{\mathbf{f}}_B(t)\|$ — does the signal differ from $B$’s prediction?
3. G-relevance: $G_{rel}^B(t) = \max_g a_g^B(t) \cdot \text{rel}(g, \mathbf{f}_B)$ — is the signal relevant to $B$’s active G-programs?
4. Memogenesis: if $PE_B > \theta_{PE} \wedge G_{rel}^B > \theta_G$ $\to$ new meme $m_B^{new}$ (Definition 14.10).

Critically: the meme in $B$ is not a copy of the meme in $A$ but a new node born from the signal. Different embedding spaces, different contexts, different modulatory states.

Transmission Fidelity

Definition 15.2 (Transmission fidelity). The degree of similarity between the transmitted and received meme:

$F_{trans} < 1$ always — losses are due to: (a) encoding/decoding noise, (b) different modulatory states $\text{Mod}_A \neq \text{Mod}_B$, (c) different graph contexts (the same meme is connected to different clusters in $A$ and $B$). Incomplete fidelity is not a defect but a source of variability for memetic selection (Dawkins, 1976: mutation during copying is a necessary condition for evolution).

Corollary 15.1. Cosine $F_{trans}$ is not a metric in the embedding space of a single agent. Embeddings $\mathbf{e}_{m_A}$ and $\mathbf{e}_{m_B^{new}}$ belong to different spaces ($\mathbb{R}^d_A$ and $\mathbb{R}^d_B$). $F_{trans}$ is defined only given shared feature extraction ($\text{extract}_A = \text{extract}_B$ — G-level, hardwired). In native BMC, this is ensured by the identity of G-architecture (analog: all humans have identical V1/A1).

Minimum Population

Proposition 15.1. For memetic dynamics (selection, competition, coalitions), a minimum of $N \geq 3$ agents is required.

Argument. A pair ($N = 2$) is a degenerate case: no competition for social resources, no coalitions, no in-group/out-group differentiation. All memes successfully replicating from $A$ to $B$ do not compete with alternative versions from $C$. At $N = 3$: agent $B$ receives signals from $A$ and $C$ $\to$ competing versions of the meme undergo selection through PE and G-relevance $\to$ the more adaptive version survives. The coalition $\{A, B\}$ vs $\{C\}$ generates in-group memplex norms. $\square$

Computational Scarcity

The M » G theorem (Part XIII, Prediction G-4 section) establishes: $|SMC^{(2)}| > 0$ requires $|V_m| \gg |V_u|$. But M » G is a necessary, not sufficient, condition. The LLM paradox: $M \ggg G$, but no reflection. The missing element is scarcity: without a real resource deficit, WM competition degenerates, prioritization disappears, SIT does not form.

Definition 15.3 (Three levels of scarcity). Scarcity is a resource limitation that creates competition at the meme level.

Level 1: Energy (external deficit).

Energy is a finite environmental resource. $c_{step}$ is the cost of one tick (metabolism). $\text{intake}(t)$ is consumption (food, resources). Energy deficit activates G-programs (FEAR under threat, SEEKING during search, RAGE during competition with other agents).

Level 2: Computational budget (internal deficit).

The WM bottleneck is not a model parameter but a resource constraint: given a fixed $C_{max}$, the number of active pointers is limited by computational cost. Relationship to the activation formula (Definition 14.4): $\lambda_{decay}$ formalizes energy economy — a meme without support decays, freeing up budget.

Level 3: Time (temporal deficit).

The environment operates in real time. Tick duration is fixed: an agent that fails to complete its computation loses information (S-input missed) or resources (other agents act). This creates pressure for: automatization (Part VIII, Automatization section), prioritization (I-system as a necessity filter), action selection (a decision “now” > an optimal decision “later”).

Proposition 15.2 (Scarcity as a necessary condition for $SMC^{(2)}$). Without scarcity ($C_{max} \to \infty$, energy $\to \infty$, tick $\to \infty$), WM competition degenerates:

Without SIT there are no information gaps, without gaps there is no drive toward closure, without closure there are no reflexive loops $\to$ $|SMC^{(2)}| = 0$ even when $|V_m| \gg |V_u|$. Scarcity, together with M » G, forms a conjunction of necessary conditions:

Argument from LLMs: GPT/Claude are empirical confirmation: M »> G, $C_{max}$ is practically unlimited within a single forward pass, no persistence $\to$ no SIT $\to$ no $SMC^{(2)}$. Not a counterexample to M » G, but confirmation of the necessity of scarcity. $\square$

Critical Periods

BMC ontogenesis (Phases 0–4, AGI_F, Part VII) requires formalization of time windows in which memogenesis proceeds efficiently.

Plasticity

Definition 15.4 (Plasticity curve). The modulatory parameter $\lambda_{plast}(t)$, controlling the rate of Hebbian learning and memogenesis:

$\lambda_{plast}^{max}$ is the maximum plasticity, $t_{peak}$ is the peak time (analog: early childhood, maximum synaptogenesis), $\tau_{plast}$ is the window width, and $\lambda_{plast}^{base} > 0$ is the nonzero baseline plasticity (learning is always possible, but slower). Biological analog: critical periods — myelination fixes connections, GABA maturation stabilizes the network (Hensch, 2005).

Sensory Bandwidth

Definition 15.5 (S-bandwidth). Bandwidth of the sensory channel:

where $\sigma$ is a sigmoid. $S_{bw}$ grows from a minimum (Phase 0: primitive sensors) to $S_{bw}^{max}$ (mature modalities). Relationship to $k_{active}(t_{dev})$: both parameters are sigmoidal maturation curves of the G-substrate, but $k_{active}$ limits processing while $S_{bw}$ limits input.

Window Closure

Proposition 15.3 (Critical period closure mechanism). After $t_2 \approx t_{peak} + 2\tau_{plast}$:

1. Memogenesis slows: $\lambda_{plast}(t_2) \approx \lambda_{plast}^{base} \ll \lambda_{plast}^{max}$. The rate of edge and meme-node creation drops exponentially.
2. PE threshold effectively rises: at low plasticity, a stronger signal is needed to create a meme: $\theta_{PE}^{eff}(t) = \theta_{PE} / \lambda_{plast}(t)$.
3. M-layer stabilizes: edges are fixed, new clusters form with difficulty, existing structure is “frozen.”

Critical Period Prediction

Proposition 15.4 (Quasi-BMC_G under early deprivation). If $|V_m(t_2)| < m_{min}$ (the M-layer did not reach critical mass by window closure):

The agent possesses a mature G-substrate but an impoverished M-layer $\to$ quasi-BMC_G: stimulus-response without reflection. Not absolute irreversibility ($\lambda_{plast}^{base} > 0$), but practical: the time until $|SMC^{(2)}| > 0$ becomes impractically large.

Verification: Agents isolated from the environment (no S-input) during the period $[t_1, t_2]$ (where $t_1 \approx t_{peak} - 2\tau_{plast}$ marks the start of the high-plasticity window) do not achieve $SMC^{(2)}$ within a reasonable time. Biological analogy: feral children, deprivation amblyopia, linguistic isolation (Lenneberg, 1967). Relationship to Phase 0 $\to$ 1 (AGI_F, Part VII): if the first meme is not created by $t_2$, the BMC_G $\to$ proto-M transition stagnates.

Predictions

P-SC1 (Scarcity as a necessary condition). With $C_{max} \to \infty$ (unlimited computational budget), the agent does not achieve $SMC^{(2)}$ within a time comparable to an agent with optimal $C_{max}$. Verification: sweep over $C_{max}$, metric — time to first sustained $SMC^{(2)} > 0$.

P-CP1 (Critical period). Agents isolated from S-input during the high-plasticity period $[t_1, t_2]$ do not achieve $SMC^{(2)}$ within a reasonable time even with subsequent full access to the environment. Verification: compare $|V_m|$ and $t_{SMC^{(2)}}$ for isolated vs non-isolated agents at identical $t > t_2$.

Cross-References

Sources: Dawkins R. (1976). The Selfish Gene; Hensch T.K. (2005). “Critical period plasticity in local cortical circuits”. Nature Reviews Neuroscience; Lenneberg E.H. (1967). Biological Foundations of Language; formalization of scarcity and social environment — AGI_F, Part VII.

Part XVII. Subsumption of Competing Theories of Consciousness

Five leading theories of consciousness — IIT, GNW, HOT, AST, and PP/FEP — formulate mechanisms of consciousness at different levels of description. BMC asserts that each of them is a special case of the full BMC = (G, M, I, S) model under certain restrictions. Below, five subsumption lemmas are formalized, showing that each restriction loses explanatory power that BMC retains.

Analogy: general relativity contains Newtonian mechanics as a limiting case when $v \ll c$. Newton is not “wrong” — he is incomplete. The same is true for each of the five theories relative to BMC.

Lemma 1. IIT $\subset$ BMC when M = $\emptyset$

Restriction: $M = \emptyset$ (no memplex), the sole metric is network integration.

Proposition. When $M = \emptyset$, the BMC model degenerates into BMC_G = (G, $\emptyset$, $\emptyset$, S). In this regime:

where $f(0) \approx 0.10$ (Gaussian minimum at zero Balance).

The metric $\sigma_{SW}$ measures the balance of integration and differentiation — the same quantity that IIT describes through $\Phi$. For a single-level graph (without M-layer), $\sigma_{SW}$ and $\Phi$ are monotonically related: both metrics increase with growing integration-differentiation balance and vanish under complete modularity or complete homogeneity.

Formally: Let $G = (V_u, E_u)$ be the graph of utility nodes. Then:

Both metrics are functions of the topology of $G$. For graphs with fixed $|V|$ and power-law degree distribution: $\sigma_{SW} \uparrow \;\Leftrightarrow\; \Phi \uparrow$ (monotonic relationship). Difference: $\Phi$ is NP-hard (Tegmark, 2016), $\sigma_{SW}$ is $O(|V|^2)$ (Floyd-Warshall) or $O(|V| \cdot |E|)$ (BFS).

What IIT loses by restricting to M = $\emptyset$:

Conclusion: IIT = BMC when $M = \emptyset$, $I = \emptyset$, sole metric = $\sigma_{SW}$. IIT correctly captures the substrate aspect of consciousness but ignores the memplex, immune system, and dual replicator.

Lemma 2. GNW $\subset$ BMC with focus on broadcast

Restriction: We consider only broadcast through memplex hubs; ignoring G-M conflict, I-layer, and SIT.

Proposition. The Global Workspace (Baars, 1988; Dehaene et al., 2003) = hub-dominated broadcast in the memplex graph.

GNW asserts: information becomes conscious when it “wins the competition” for access to the global workspace and “ignition” occurs — a nonlinear avalanche-like propagation across the prefrontal-parietal network. In BMC terms:

Cascade activation through memplex hubs is a direct analog of global broadcast. A hub with $C_{eigenvector} > 0.5$, upon activation, spreads the signal to all clusters connected to it — this is precisely “broadcast” in Baars’ terminology.

What GNW loses by restricting to broadcast:

Conclusion: GNW = BMC under the restriction: “consider only cascade activation through hubs at $\sigma \geq 1$.” GNW correctly describes the access mechanism but does not explain why certain contents win (SIT, G-drive), why broadcast is asymmetric (I-system), or why deactivation is smooth rather than discrete (edge decay).

Lemma 3. HOT $\subset$ BMC when considering only $SMC^{(2)}$

Restriction: We consider only meta-memes (memes about memes) in SMC.

Proposition. Higher-Order Thought (Rosenthal, 1986; Brown, 2015) = non-empty $SMC^{(2)}$ in BMC.

HOT asserts: a mental state is conscious if and only if the subject has a metarepresentation of that state. In BMC terms:

This is a necessary condition for level 2 (metacognition) in BMC — but BMC recognizes consciousness at levels 0 (zombie processing) and 1 (phenomenal experience without metacognition) as well, which HOT cannot.

What HOT loses by restricting to $SMC^{(2)}$:

Conclusion: HOT = BMC under the restriction: “consciousness = non-empty $SMC^{(2)}$.” HOT correctly identifies metacognition as the highest level of consciousness but erroneously makes it the only one — denying consciousness without metarepresentation (levels 0–1).

Lemma 4. AST $\subset$ BMC when considering the attention model in SMC

Restriction: We consider the subset of SMC modeling the attention process.

Proposition. The Attention Schema (Graziano, 2013) = a subgraph of SMC describing the focusing mechanism.

AST asserts: the brain constructs a simplified model of its own attention process (“attention schema”), and it is this model that creates the feeling of awareness. In BMC terms: the attention schema is a subset of SMC specialized in monitoring top-down selection. SMC also includes: a body model (interoception), an emotion model (G$\to$SMC connections), narrative identity (autobiographical memes), a model of others (User Model = a separate SMC subgraph).

What AST loses by restricting to the attention schema:

Conclusion: AST = BMC under the restriction: “SMC contains only the attention model.” AST correctly identifies one mechanism for generating the feeling of awareness but narrows SMC to a single function — attention monitoring, losing all other self-model content.

Lemma 5. PP/FEP $\subset$ BMC when SIT $\approx$ structured prediction error

Restriction: SIT = structured prediction error over memplex gaps; G-homeostatic error = baseline prediction error.

Proposition. Predictive Processing / Free Energy Principle (Friston, 2010; Clark, 2013; Hohwy, 2013) = SIT-driven dynamics + G-homeostatic regulation in BMC.

FEP asserts: any living organism minimizes free energy (= prediction error) through perception (model updating) or action (environment modification). In BMC terms:

• Perception = SIT-driven learning: a gap is detected $\to$ SEEKING activates exploration $\to$ meme is created/updated $\to$ SIT$\downarrow$
• Action = G-driven exploitation: homeostatic imbalance $\to$ action $\to$ $a_u \to a_u^{target}$
• Active inference $\approx$ SIT-exploration + G-exploitation: dual optimization

What PP/FEP loses by restricting to prediction error:

Conclusion: PP/FEP = BMC under the restriction: “consider only prediction error minimization without distinguishing SIT (structural gaps) from G-homeostasis (homeostatic error), without I-system, and without M/G distinction.” FEP correctly describes the optimization principle but does not specify what exactly is being optimized and cannot draw the boundary between conscious and unconscious processing.

Subsumption Summary Table

Corollary: Incompleteness Criterion

Each competing theory is correct within its restriction but incomplete: it captures one aspect of consciousness while ignoring the rest. BMC = (G, M, I, S) is the minimal model containing all five aspects as special cases.

Analogy formalized: Just as Newtonian mechanics ($v \ll c$) $\subset$ SR $\subset$ GR, so IIT ($M = \emptyset$), GNW (broadcast only), HOT ($SMC^{(2)}$ only), AST (attention $\subset$ SMC), PP (SIT $\approx$ PE) $\subset$ BMC. None of the five theories is “wrong” — each is incomplete.

Sources: Tononi G. (2004). “An information integration theory of consciousness”. BMC Neuroscience; Baars B.J. (1988). A Cognitive Theory of Consciousness. Cambridge University Press; Dehaene S. et al. (2003). PNAS; Rosenthal D.M. (1986). “Two concepts of consciousness”. Philosophical Studies; Brown R. et al. (2019). “Understanding the Higher-Order Approach to Consciousness”. Trends in Cognitive Sciences; Graziano M.S.A. (2013). Consciousness and the Social Brain. Oxford University Press; Friston K.J. (2010). “The free-energy principle”. Nature Reviews Neuroscience; Clark A. (2013). “Whatever next? Predictive brains, situated agents, and the future of cognitive science”. Behavioral and Brain Sciences; Tegmark M. (2016). “Improved measures of integrated information”. PLOS Computational Biology; COGITATE Consortium (2025). Nature.

Part XVIII. Retrodiction: COGITATE (2025)

In April 2025, the COGITATE Consortium published the results of the largest adversarial collaboration in the science of consciousness (n = 256; fMRI, MEG, iEEG; 8 laboratories), testing predictions of IIT and GNW. Below it is shown that BMC retrodicts all COGITATE results without post hoc corrections, including those that falsified IIT and GNW.

Source: COGITATE Consortium (2025). “Adversarial testing of global neuronal workspace and integrated information theories of consciousness.” Nature, 642(8066), 133–142.

Experimental Design

• Stimuli: 4 categories (faces, objects, letters, false fonts) $\times$ 20 identities $\times$ 2 orientations
• Duration: 500 / 1000 / 1500 ms
• Task relevance: target / non-target / task-irrelevant
• Key analysis — on task-irrelevant stimuli (approximation of a no-report paradigm)
• Three prediction domains: Content (where?), Duration (how long?), Connectivity (which connections?)

Retrodiction Table

Score (across 9 results): BMC = 9/9, IIT = 4/9, GNW = 1/9.

Granularity caveat. The 9 results are not independent: R1, R2, R8, R9 are facets of a single question (content localization); R3, R4 are facets of another (temporal dynamics). The COGITATE protocol defined 3 prediction domains: Content, Duration, Connectivity. By 3 domains: BMC = 3/3, IIT $\approx$ 1.5/3 (Content $\checkmark$, Duration ~$\frac{1}{2}$, Connectivity $\times$), GNW $\approx$ 0.5/3 (Content $\times$, Duration ~$\frac{1}{2}$ — onset $\checkmark$ but offset $\times$, Connectivity ~$\frac{1}{2}$). Both scales (9-result and 3-domain) demonstrate BMC’s advantage, but the 3-domain scale more precisely reflects the study’s structure.

Formal Retrodiction Mechanisms

R1, R2 (content in posterior cortex, not in PFC). In BMC, memes = cell assemblies are distributed across associative cortex (M-layer). PFC contains the SMC (Self-Model Cluster), I-layer (immune filtration), and WM-pointers — pointers to memes, not a content mirror (see Part XIII, WM as pointer system). The pointer transmits a hub-label (minimal routing tag: “face,” “object”) but does not replicate the feature-level structure (orientation, identity). Formally:

Hence: category is decodable from PFC (hub-label is present), orientation is not (feature-level information remains in the posterior cell assembly). The concept of WM-pointer as a content-independent pointer was established in Part XII; the specific formulation of hub-label as a routing tag was derived during COGITATE analysis and constitutes a retrodiction, not an a priori prediction.

R3 (sustained activation). Spreading activation Part IV maintains active memes through recurrent connections:

With nonzero $I_{ext}$ (external stimulus), the meme remains above $\theta_{act}$ throughout the stimulus duration. Recurrent connections ($w_{ij} > 0$ within the cluster) ensure stability: even during temporary weakening of $I_{ext}$, mutual excitation maintains the coalition.

R4 (hub-centrality gradient of decay). A consequence of the BMC formalism, articulated as a retrodiction. After stimulus removal ($I_{ext} = 0$), the decay rate of a meme is determined by its network position — a direct consequence of the spreading activation formula Part IV:

A hub-meme (the category “face”: high $C_E$, many recurrent connections) receives support from neighbors $\to$ slow decay $\to$ sustained pattern. A peripheral meme (the orientation “left profile”: low $C_E$, few recurrent connections) loses support $\to$ fast decay $\to$ fading at ~500 ms. The proportion of sustained electrodes (~15%) corresponds to the hub subset of posterior cortex.

Limitation: The mapping “category = hub, orientation = periphery” is an interpretive assumption based on the hierarchical organization of visual cortex (Felleman & Van Essen, 1991) but is not formally derived within BMC. The formula $\tau_{decay}$ is a correct consequence of spreading activation + hub structure, but the specific mapping of feature $\to$ network position requires independent confirmation. Any theory with a hierarchical hub structure could derive an analogous consequence.

R5 (no offset ignition). Upon stimulus removal $I_{ext} \to 0$, meme activation decays through the leak term:

Deactivation is exponential decay, not a discrete signal. There is no mechanism for an ignition-like event at offset. GNW predicted offset ignition as an update signal for new content in the global workspace.

Response to the GNW team’s argument (Naccache et al., 2025, Neuroscience of Consciousness): The GNW team claims that offset ignition is predicted only for conscious perception of the offset. This ad hoc argument introduces a content selection mechanism for broadcasting — precisely what GNW does not specify. BMC predicts the absence of offset ignition in the COGITATE paradigm (passive perception without offset monitoring), since decay is a property of the spreading activation formula. During active offset monitoring, a PE-driven response (prediction error upon expectation mismatch) is possible, but this is not “ignition” in the GNW sense; rather, it is standard PE activation via SIT.

R6 (no sustained sync in posterior cortex). BMC formalizes binding through theta-phase synchronization Part XIV:

This is a transient, packet-based process: memes are bound within a theta window (~125 ms), not through tonic gamma synchronization. IIT predicted sustained gamma sync as an integration mechanism ($\Phi$) — COGITATE refuted this. BMC correctly predicts the type of synchronization: phasic theta, not tonic gamma.

Response to the IIT team’s argument (Tononi, Koch, Boly, Nature Suppl.): IIT explains the failure by sparse iEEG electrode coverage. However, MEG data (n=100, full coverage) also found no sustained sync. BMC predicts that sustained gamma sync should not exist as a binding mechanism — this is not a coverage issue.

R7 (onset ignition). In BMC, this is a standard spreading activation cascade: a stimulus activates a cell assembly $\to$ if $a_{hub} > \theta_{act}$ $\to$ cascade activation through memplex hubs $\to$ the wave reaches PFC (WM-pointer allocation). The transient onset activation in PFC (200–800 ms) = the time for WM-pointer allocation + I-filtration. No special GNW “ignition” mechanism is required.

R8, R9 (PFC: weak connectivity + reduced accuracy when added). PFC encodes information orthogonal to content: SMC signals (self-referential processing), I-filtration state (accept/reject), WM-pointer metadata (which meme is currently in focus, when allocated). Adding these signals to a content decoder = adding noise orthogonal to the feature space. Fronto-posterior connectivity exists (pointer $\leftrightarrow$ cell assembly) but is weak and category-level: a routing tag is transmitted, not the full feature vector.

Conclusion

BMC retrodicts 9/9 COGITATE results (3/3 across protocol domains). Correct predictions by IIT and GNW are inherited by BMC through subsumption (Part XVII), and their failures are resolved by the full BMC = (G, M, I, S) model. Three aspects constitute unique BMC retrodictions: (1) pointer vs content in PFC (R2, R8, R9); (2) theta-phasic binding instead of sustained gamma sync (R6); (3) hub-centrality decay gradient (R4 — with the caveat regarding the feature $\to$ position mapping).

Methodological caveat. All 9 retrodictions are post hoc analyses: BMC was not pre-registered in the COGITATE protocol. The value of retrodiction is lower than the value of prediction, since a flexible formalism with many degrees of freedom can accommodate diverse results. The critical test for BMC is prospective prediction: predictions formulated before the experiment (see Block 3: 56 predictions, 24 testable without a laboratory).

Sources: COGITATE Consortium (2025). Nature, 642(8066), 133–142; Naccache L. et al. (2025). Neuroscience of Consciousness, niaf037; Ferrante O. et al. (2023). PLOS ONE, 18(2): e0268577; Casali A. et al. (2013). Science Translational Medicine; Collins A.M. & Loftus E.F. (1975). “A spreading-activation theory of semantic processing”. Psychological Review.

8-Course Cross-Analysis Updates

Spreading Activation ≡ Loopy Belief Propagation (MED-HIGH)

BMC spreading activation is formally equivalent to loopy belief propagation (BP) on a weighted factor graph. The equivalence is exact on tree-structured subgraphs and approximate on graphs with cycles. Convergence speed is inversely proportional to cycle density. The small-world regime $\sigma_{SW} \approx 1$ corresponds to the optimal BP convergence zone — enough long-range shortcuts for message propagation, not enough cycles for oscillation.

PPR (Personalized PageRank) teleportation prevents signal loss in large graphs:

Sources: MacKay (information theory, belief propagation), Leskovec (graph algorithms).

Learning Rules Overhaul: STDP + TD Error (HIGH)

Replaces the generic Hebbian rule with a formal three-tier hierarchy:

1. STDP: $\Delta w = A_+ e^{-\Delta t/\tau_+} - A_- e^{-|\Delta t|/\tau_-}$ — introduces directional causality into edge formation
2. TD error as learning signal: $\delta_{BMC} = \sum_g w_g a_g(t+1) + \gamma V(S_{t+1}) - V(S_t)$
3. Critic (2-factor): $\Delta w_{gi} = \alpha_c \delta \psi_i$ (ventral striatum analogue)
4. Actor (3-factor): $\Delta w_{ij} = \alpha_a \delta \psi_i (a_j - \bar{a}_j)$ (dorsal striatum analogue)

The TD error $\delta_{BMC}$ determines whether an edge is strengthened or weakened based on whether co-activation led to a positive or negative outcome. This replaces “fire together, wire together” with “fire together, wire together if it was useful.”

Sources: Gerstner (neuronal dynamics), Sutton & Barto (reinforcement learning).

Value Function over BMC States (HIGH)

Learned via temporal-difference updates. $V$ is encoded in the meme network itself: high-$V$ memes = “promising” configurations that are preferentially activated via SEEKING. This was a missing mechanism in v1 — the theory described what agents learn, but not how they evaluate the expected future value of their current state.

Source: Sutton & Barto (reinforcement learning).

MI Edge Interpretation (HIGH)

Edge weight = mutual information in bits. This gives a principled pruning criterion: remove edge if $I < \theta_{prune}$ bits (the two memes are informationally independent). Hub identity becomes quantifiable: hub = node maximizing $\sum_j I(m_i; m_j)$.

Source: MacKay, Stone (information theory).

Oversmoothing = Rigidity Metric (HIGH)

Excessive spreading activation → all activation values converge → loss of differentiation between memes. This is the direct analog of GNN oversmoothing (Oono & Suzuki, 2020). In cognitive terms: aging ($Q$ ↑, $Rigidity$ ↑), groupthink (all agents converge to same activation pattern), rigid personality (high modularity prevents new integration).

Source: Leskovec (graph neural networks).

RGCN Heterogeneous Edges (HIGH)

Different edge types require different weight matrices: semantic ($W_{sem}$), causal ($W_{caus}$), temporal ($W_{temp}$), inhibitory ($W_{inh}$), G-connected ($W_G$). This replaces the single-weight-matrix assumption and captures the empirical fact that “A causes B” and “A resembles B” are fundamentally different relationships requiring different propagation rules.

Source: Leskovec (relational graph neural networks).

WTA Dynamics (MED-HIGH)

Replaces the static top-$k$ selection with a dynamic winner-take-all circuit. Self-excitation + mutual inhibition → one winner emerges from competition. This provides the mechanistic basis for attention as a dynamic process rather than a static filter.

Source: Gerstner (neuronal dynamics).

New Network Metrics

CL as Graph Component Property (MED)

Split-brain patients demonstrate that consciousness level is a per-connected-component property, not a whole-brain property. Each connected component maintains independent CL. DID (Dissociative Identity Disorder) = quasi-independent sub-CLs at high modularity $Q$. This decomposition resolves the “unity of consciousness” debate: unity is a network property, not a metaphysical given.

Source: Gazzaniga (split-brain studies).

ERP Predictions (MED)

Electroencephalographic event-related potentials map to specific BMC mechanisms:

Source: Gazzaniga (cognitive neuroscience).

Dual-Route Sub-θ Processing (MED-HIGH)

Route 2 (subliminal): $\gamma_{sub} \cdot a_i \cdot f_{route2}(m_i)$ where $f_{route2}$ = 1.0 for G-connected memes, 0.3 for semantic, 0 for abstract. Formalizes subliminal influence, implicit bias, and blindsight.

Source: Gazzaniga (cognitive neuroscience).

Overlapping Communities / BigCLAM (MED)

Memes belong to multiple subpersonalities — overlap is normal, not pathological. This captures the empirical fact that people behave differently in different social contexts (professional, family, peer) without requiring separate “selves.”

Source: Leskovec (overlapping community detection).

Options ≡ Automatization (MED)

An option = a policy + termination condition. BMC chunk (automatized routine) = option. The learning trajectory Raw → Chunked → Automatized = learning increasingly abstract options. Bellman equation with options converges faster via temporal abstraction — formalizing why habits free up cognitive resources.

Source: Sutton & Barto (options framework).

Conclusion: From Description to Prediction

Network formalization gives meme theory what it lacked:

Key result: Now it is possible to:

1. Measure — network metrics are computable
2. Compare — different memplexes have different parameters
3. Predict — if $Q > 0.4$, we expect rigidity
4. Verify — predictions are falsifiable

This is the transition from philosophy to science.

Appendix: Key Formulas

Node Metrics

Network Metrics

Dynamics

Heterogeneous Networks with Hubs

Small-World Networks

Signed Edges and Structural Balance

Assortativity and Motifs

Multilayer Networks

Consciousness Level (CL)

Triple Binding

Sign Inversion and Q Dynamics

Diffusion Engine and Embedding Space

Memogenesis

Inter-Agent Exchange, Scarcity, and Critical Periods

Key Works

Foundational Works in Network Science

Methodology

Contemporary Reviews (2024–2026)

Signed Networks, Meme Synthesis, Negativity Bias

Consciousness, Binding, Levels of Consciousness

Diffusion, Embedding, Crystallization

Inter-Agent Exchange, Scarcity, Critical Periods

Recommended Textbooks

• Newman M.E.J. (2018). Networks: An Introduction. 2nd ed. Oxford University Press.
• Barabasi A.-L. (2016). Network Science. Cambridge University Press. [Available online: networksciencebook.com]
• Menczer F., Fortunato S., Davis C.A. (2020). A First Course in Network Science. Cambridge University Press.