A formal note · VI

Coevolution of memes and cognitive architectures

Reflexive semantic dynamics, evolving observer modalities, and joint replicator–mutator systems in an ∞-topos.

Grothendieck construction / identity transport / reflexive selection / lumpability

Abstract

We extend memetic replication to the case in which cognitive architectures themselves evolve under memetic influence. The semantic equivalence relation defining what counts as the same meme is therefore no longer fixed.

Let $(C, J)$ be a cognitive ∞-site, and let $M : C^{op} \to S$ be a memetic field. Its Grothendieck construction $\int_{C} M$ has objects $(c, m)$ , an architecture together with a memetic state as realized by that architecture. The joint population state space is the maximal ∞-groupoid

Ω_{M} := (\int_{C} M)^{≃} .

A population is $ν \in P (Ω_{M})$ , where $P$ is a compatible probability monad. Fitness and transmission may depend on the complete joint population:

w : Ω_{M} \times P (Ω_{M}) \times E \to R_{> 0},

K : Ω_{M} \times P (Ω_{M}) \times E \to P (Ω_{M}) .

The evolutionary equation is

R (ν, e) = (K_{ν, e})_{*} \frac{w ( - , ν , e ) ν}{\int _{Ω_{M}} w ( x , ν , e ) d ν ( x )} .

Because the kernel may change the architecture coordinate, memes can modify the observer modality through which subsequent memetic identity is evaluated. If architecture $c$ carries a semantic localization $L_{c}$ , then $m ≃_{c} n$ need not imply $m ≃_{d} n$ after the architecture changes from $c$ to $d$ . Memetic evolution and semantic identity become mutually coupled.

Memes alter cognition, cognition alters memetic identity, and identity alters subsequent selection.

We formulate this coupled dynamics, derive its finite-state and continuous-time forms, characterize its equilibria, establish exact conditions under which architecture or meme marginals evolve autonomously, and prove an identity-transport theorem specifying when semantic identity is preserved under cognitive change. The fixed-architecture theory is Note V.

Notation

$Ω_{M}$: The joint state space: an architecture together with a meme as that architecture realizes it.
$≃_{c}$: Semantic equivalence as judged by architecture c. It can change when c changes.
$ν$: A joint population, a probability state on architectures and memes together.
$A$: A cognitive plasticity kernel. Memes may change the next architecture.
$q$: The total semantic quotient, from joint states to joint semantic classes.
$T_{c d}$: A translation of memes from architecture c to architecture d.

1Cognitive architectures

Let $C$ be a small ∞-category whose objects are cognitive architectures. A morphism $u : c \to d$ represents an admissible semantic translation, abstraction, simulation, or architectural transformation. Equip $C$ with a Grothendieck topology $J$ , producing the cognitive ∞-site $(C, J)$ . Its semantic ∞-topos is

H = Sh_{\infty} (C, J) .

2Memetic fields

Definition2.1Memetic field

A memetic field is an ∞-sheaf

M : C^{op} ⟶ S .

For every architecture $c$ , $M (c)$ is the space of memetic states representable at $c$ . For every architecture morphism $u : c \to d$ , there is a restriction or translation

u^{*} : M (d) \to M (c) .

3The joint architecture–meme state space

The appropriate state is no longer merely a meme $m$ . It is a pair $(c, m)$ , encoded by the Grothendieck construction.

Definition3.1Memetic Grothendieck construction

Let $\int_{C} M$ denote the ∞-category obtained by unstraightening $M : C^{op} \to S$ . There is a canonical right fibration

p : \int_{C} M ⟶ C .

Its objects are pairs $(c, m)$ with $m \in M (c)$ .

Definition3.2Joint population space

Define

Ω_{M} := (\int_{C} M)^{≃}

to be the maximal ∞-groupoid of the Grothendieck construction. The architecture projection

π : Ω_{M} \to C^{≃}

satisfies $π (c, m) = c$ .

4Architecture-dependent semantics

Suppose each architecture possesses its own semantic resolution. Let $Mod (E)$ denote an appropriate ∞-category of left-exact accessible modalities on an ambient ∞-topos $E$ .

Definition4.1Cognitive modality assignment

A cognitive modality assignment is a functor

L : C ⟶ Mod (E) .

For $c \in C$ , write $L_{c}$ for the corresponding semantic localization.

Definition4.2Architecture-relative semantic equivalence

For ambient memetic realizations $m, n$ , write $m ≃_{c} n$ when

Id_{L_{c} M} (η_{c} (m), η_{c} (n))

is inhabited. The architecture $c$ participates directly in the definition of memetic identity.

5Dynamic semantic identity

If the architecture changes, the equivalence relation itself may change.

Definition5.1Semantic refinement

A transition $c ⇝ d$ is semantically refining on $M$ if $m ≃_{d} n$ implies $m ≃_{c} n$ . Equivalence at $d$ is then at least as discriminating as equivalence at $c$ .

Definition5.2Semantic coarsening

A transition $c ⇝ d$ is semantically coarsening if $m ≃_{c} n$ implies $m ≃_{d} n$ .

Definition5.3Identity-changing transition

The transition changes semantic identity if there exist $m, n$ such that either $m ≃_{c} n$ but $m \neq ≃_{d} n$ , or $m \neq ≃_{c} n$ but $m ≃_{d} n$ . Architecture evolution can split or merge semantic equivalence classes.

6Identity transport along architecture morphisms

Suppose $u : c \to d$ induces a semantic translation $u^{*} : M (d) \to M (c)$ .

Theorem6.1Identity transport

If $x ≃_{d} y$ , then $u^{*} x ≃_{c} u^{*} y$ .

Proof.

A witness of $x ≃_{d} y$ is a path $p : x ≃ y$ in $M (d)$ . Applying $u^{*}$ gives a path $u^{*} p : u^{*} x ≃ u^{*} y$ . Hence $u^{*} x ≃_{c} u^{*} y$ .

This concerns functorial translation. It does not assert that arbitrary architecture changes preserve the same equivalence relation on an underlying presentation space.

7Identity-preserving cognitive change

Definition7.1Semantically conservative transition

Suppose an architecture transition $c \to d$ comes with a translation $T_{c d} : M (c) \to M (d)$ . It is semantically conservative if

x ≃_{c} y ⟺ T_{c d} (x) ≃_{d} T_{c d} (y) .

Theorem7.2Identity invariance along conservative paths

Let $c_{0} \to c_{1} \to \dots \to c_{n}$ be a sequence of semantically conservative architecture changes. Then for all $x, y \in M (c_{0})$ ,

x ≃_{c_{0}} y

if and only if

T_{0 n} (x) ≃_{c_{n}} T_{0 n} (y),

where

T_{0 n} = T_{c_{n - 1} c_{n}} \circ \dots \circ T_{c_{0} c_{1}} .

Proof.

Each transition preserves and reflects semantic equivalence. Apply this equivalence successively along the path.

Corollary7.3Where identity can change

Semantic identity can change along a cognitive trajectory only if at least one architecture transition fails to be conservative.

8Reflexive memetic influence

A meme may influence the architecture through which it is interpreted.

Definition8.1Architecture transition kernel

A cognitive plasticity kernel is a Markov kernel

A : Ω_{M} ⟶ P (C^{≃}) .

For $(c, m) \in Ω_{M}$ , the distribution $A (c, m)$ describes possible successor cognitive architectures.

Definition8.2Memetically induced cognitive plasticity

The cognitive transition is memetically induced if $A (c, m)$ depends nontrivially on $m$ : there exist $m, n \in M (c)$ with $A (c, m) \neq = A (c, n)$ . Different memes can alter the future cognitive architecture differently.

9Joint transition kernel

Rather than factor architecture and meme transitions separately, the invariant formulation uses one kernel on the total state space.

Definition9.1Joint evolutionary kernel

A joint architecture–meme transition kernel is

K : Ω_{M} ⟶ P (Ω_{M}) .

For a state $x = (c, m)$ , the distribution $K (x)$ describes the joint law of successor architectures and successor memetic realizations. The architecture kernel is recovered by pushforward:

A = P (π) \circ K .

10Conditional factorization

Under a suitable disintegration, the joint kernel factors schematically as an architecture choice followed by a conditional memetic choice:

K ((c, m), d (c^{'}, m^{'})) = A ((c, m), d c^{'}) B ((c, m), c^{'}, d m^{'}) .

Here $A ((c, m), d c^{'})$ chooses the successor architecture, while $B ((c, m), c^{'}, d m^{'})$ chooses the successor memetic realization conditional on that architecture. The factorization is useful conceptually but is not needed for the invariant theory.

11Joint populations

Assume a compatible probability monad $P$ .

Definition11.1Joint cognitive–memetic population

A population is

ν \in P (Ω_{M}) .

It describes the joint distribution of architectures and memetic states.

12Architecture marginal

Define

a = π_{*} ν \in P (C^{≃}) .

This is the distribution of cognitive architectures.

13Conditional memetic populations

Where disintegration exists, write

d ν (c, m) = a (d c) μ_{c} (d m),

where $μ_{c} \in P (M (c))$ is the conditional memetic distribution within architecture $c$ . A joint population is architecture frequencies together with architecture-conditioned meme distributions.

14Endogenous joint fitness

Let $E$ be an environment object.

Definition14.1Joint fitness field

A joint fitness field is

w : Ω_{M} \times P (Ω_{M}) \times E \to R_{> 0} .

Thus $w ((c, m), ν, e)$ may depend on the architecture, the meme, the entire joint population, and the environment.

15Joint selection

Define mean fitness

w ˉ (ν, e) = \int_{Ω_{M}} w (x, ν, e) d ν (x) .

The selected population is

d Sel_{w} (ν, e) (x) = \frac{w ( x , ν , e )}{w ˉ ( ν , e )} d ν (x) .

16Coevolutionary operator

Allow the transition kernel itself to depend on $(ν, e)$ :

K_{ν, e} : Ω_{M} \to P (Ω_{M}) .

Definition16.1Cognitive–memetic evolutionary operator

Define

R (ν, e) = (K_{ν, e})_{*} Sel_{w} (ν, e) .

Explicitly, for an event $B$ ,

R (ν, e) (B) = \frac{\int _{Ω_{M}} w ( x , ν , e ) K _{ν, e} ( x , B ) d ν ( x )}{w ˉ ( ν , e )} .

This is the central evolution equation.

17Population-dependent transfer operator

For fixed $(ν, e)$ , define $L_{ν, e}$ on finite positive measures by

(L_{ν, e} ρ) (B) = \int_{Ω_{M}} w (x, ν, e) K_{ν, e} (x, B) d ρ (x) .

Then

R (ν, e) = \frac{L _{ν, e} ν}{L _{ν, e} ν ( Ω _{M} )} .

18Coevolutionary equilibrium

Definition18.1Joint equilibrium

A population $ν_{*}$ is an equilibrium at environment $e$ if $R (ν_{*}, e) = ν_{*}$ .

Theorem18.2Nonlinear eigenmeasure theorem

A joint population $ν_{*}$ is an equilibrium if and only if

L_{ν_{*}, e} ν_{*} = λ ν_{*},

where $λ = w ˉ (ν_{*}, e)$ .

Proof.

As in the fixed-architecture theory, $R (ν_{*}, e) = ν_{*}$ if and only if $L_{ν_{*}, e} ν_{*} = L_{ν_{*}, e} ν_{*} (Ω_{M}) ν_{*}$ . Since the kernel is normalized,

L_{ν_{*}, e} ν_{*} (Ω_{M}) = \int_{Ω_{M}} w (x, ν_{*}, e) d ν_{*} (x) = w ˉ (ν_{*}, e) .

19Evolution of the architecture marginal

Let $a^{'} = π_{*} ν^{'}$ , and write $A_{ν, e} = P (π) \circ K_{ν, e}$ .

Theorem19.1Architecture marginal equation

The successor architecture distribution is

a^{'} (D) = \frac{\int _{Ω_{M}} w ( x , ν , e ) A _{ν, e} ( x , D ) d ν ( x )}{w ˉ ( ν , e )} .

Proof.

By definition $a^{'} = π_{*} R (ν, e)$ , so $a^{'} (D) = R (ν, e) (π^{- 1} (D))$ . Substitute Definition 16.1:

a^{'} (D) = \frac{\int w ( x , ν , e ) K _{ν, e} ( x , π ^{- 1} ( D ) ) d ν ( x )}{w ˉ} .

The inner kernel probability is $A_{ν, e} (x, D)$ .

20Memes can select cognitive architectures

Even if cognitive architectures have no fitness term independent of memes, architecture frequencies may change because fitness is correlated with the memes carried within each architecture. Suppose architecture never mutates, so $A ((c, m), D) = δ_{c} (D)$ . Then

a^{'} (c) = \frac{\int _{M (c)} w (( c , m ) , ν , e ) d ν ( c , m )}{w ˉ} .

Architectures correlated with highly reproducing memes increase in frequency.

21Memes can directly transform architectures

If $A ((c, m), d)$ depends on $m$ , then architecture evolution occurs even without differential survival between architectures. This is a stronger feedback mechanism:

m \to c^{'} \to L_{c^{'}} \to new semantic equivalence .

The meme changes the modality under which subsequent memes are classified.

22Reflexive memes

Definition22.1Reflexive meme

A memetic state $m \in M (c)$ is reflexive if its transmission can alter the architecture in a way that changes semantic distinctions relevant to future memetic transmission. Formally, there exists an architecture $d$ in the support of $A (c, m)$ and states $x, y$ such that $x ≃_{c} y$ but $T_{c d} (x) \neq ≃_{d} T_{c d} (y)$ , or conversely. A reflexive meme modifies the equivalence relation used to classify later memes.

23Semantic self-modification

The resulting causal structure is

m_{t} ⟶ c_{t + 1} ⟶ ≃_{c_{t + 1}} ⟶ classification of m_{t + 1} .

This is stronger than ordinary frequency-dependent fitness. The state changes not merely its future reproduction probability but the semantic state space in which reproduction is evaluated.

24Identity drift

Definition24.1Identity drift

For two persistent presentations $x, y$ , define their time-dependent semantic relation

I_{t} (x, y) = Id_{L_{c_{t}} M} (η_{c_{t}} (x), η_{c_{t}} (y)) .

Identity drift occurs if the homotopy type of $I_{t} (x, y)$ changes with $t$ . Identity can therefore itself be a dynamical variable.

25Identity-drift theorem

Theorem25.1Conservative paths do not drift

Suppose every architecture transition along a trajectory is semantically conservative. Then no semantic identity drift occurs along that trajectory.

Proof.

By Theorem 7.2, every conservative transition preserves and reflects semantic equivalence. Therefore the truth and homotopy class of $x ≃_{c_{t}} y$ is invariant along the trajectory, up to the coherent transport supplied by the transitions.

Corollary25.2Drift requires a nonconservative step

Every identity-changing trajectory contains a nonconservative cognitive transition.

26Dynamic coarse-graining

Suppose every architecture $c$ has a semantic quotient $q_{c} : M (c) \to S (c)$ . The semantic state at time $t$ is not merely $q (m_{t})$ , but

q_{c_{t}} (m_{t}) .

Both arguments evolve.

27Total semantic field

Let $S : C^{op} \to S$ be the semantic quotient field, with a natural transformation $q : M \Rightarrow S$ . Taking Grothendieck constructions yields

\int_{C} M ⟶ \int_{C} S .

Passing to cores gives

q : Ω_{M} \to Ω_{S} .

The observer-relative semantic state of a joint architecture–meme state is a point of the total semantic space $Ω_{S}$ .

28Semantic population

The semantic population induced by $ν$ is

σ = q_{*} ν \in P (Ω_{S}) .

A central question is whether $σ$ evolves autonomously.

29Dynamic semantic lumpability

Definition29.1Dynamic semantic lumpability

The joint evolutionary system is lumpable through $q : Ω_{M} \to Ω_{S}$ if there exists an induced operator $R_{S} : P (Ω_{S}) \to P (Ω_{S})$ such that

R_{S} \circ q_{*} = q_{*} \circ R .

30Closure theorem

Theorem30.1Exact semantic closure

Suppose fitness factors through the total semantic quotient,

w (x, ν, e) = w_{S} (q (x), q_{*} ν, e),

and the transition kernel is lumpable,

P (q) \circ K_{ν, e} ≃ K_{q_{*} ν, e}^{S} \circ q .

Then there exists a closed evolutionary operator $R_{S}$ on semantic populations satisfying

q_{*} R (ν, e) = R_{S} (q_{*} ν, e) .

Proof.

The first condition ensures that selection weights depend only on the coarse semantic state and the coarse population, so semantic pushforward commutes with selection. The second ensures that pushforward commutes with transition. Composing these two commutative operations gives the stated equation.

31Failure of semantic closure

If two joint populations satisfy $q_{*} ν = q_{*} ρ$ but

q_{*} R (ν) \neq = q_{*} R (ρ),

then no autonomous first-order semantic dynamics exists on $P (Ω_{S})$ . The observer has discarded information needed to predict semantic evolution.

32Architecture as hidden state

A common source of nonclosure is failure to retain architecture. Suppose two agents currently assign the same semantic class to a meme, $q_{c} (m) ≃ q_{d} (n)$ . If $c$ and $d$ induce different future distinctions or transition laws, then the current semantic class alone is not sufficient to predict the next semantic state. Architecture functions as a hidden dynamical variable.

The same current semantic state need not have the same future semantic dynamics.

33Minimal sufficient semantic state

One may seek an augmented quotient $r : Ω_{M} \to Z$ such that the intended semantic observable factors through $r$ and the dynamics is lumpable through $r$ . Then $Z$ is a dynamically sufficient semantic state object:

P (Ω_{M}) ⟶ P (Z) ⟶ P (Z)

closes. This generalizes the earlier principle: add hidden state until the dynamics becomes Markovian.

34Finite architecture–meme model

Let cognitive architectures be indexed by $α = 1, \dots, r$ and memetic states by $i = 1, \dots, n$ . Let $x_{α i}$ be the population fraction in the joint state $(α, i)$ . Then

x_{α i} \geq 0, α, i \sum x_{α i} = 1.

35Joint fitness matrix

Let $w_{α i} (x) > 0$ be endogenous fitness, and let $Q_{α i, β j} (x)$ be the probability that a reproducing state $(α, i)$ produces successor state $(β, j)$ . Thus

Q_{α i, β j} \geq 0, β, j \sum Q_{α i, β j} = 1.

36Finite coevolutionary replicator–mutator equation

Theorem36.1Finite joint update

The joint update is

x_{β j}^{'} = \frac{α , i \sum x _{α i} w _{α i} ( x ) Q _{α i, β j} ( x )}{α , i \sum x _{α i} w _{α i} ( x )} .

Proof.

This is the finite-state expression of Definition 16.1 on the state space ${1, \dots, r} \times {1, \dots, n}$ .

37Architecture marginal in the finite theory

Define $a_{β} = \sum_{j} x_{β j}$ . Then

a_{β}^{'} = \frac{α , i \sum x _{α i} w _{α i} ( j \sum Q _{α i, β j} )}{w ˉ} .

Architecture frequencies change by both differential fitness and transition into new architectures.

38Meme marginal

If the same meme label $i$ is meaningful across architectures, define $m_{j} = \sum_{β} x_{β j}$ . Then

m_{j}^{'} = \frac{α , i \sum x _{α i} w _{α i} ( β \sum Q _{α i, β j} )}{w ˉ} .

Neither marginal generally closes independently.

39Marginal nonclosure theorem

Theorem39.1Marginals do not close

The architecture marginal $a_{t}$ generally does not determine $a_{t + 1}$ , and the meme marginal $m_{t}$ generally does not determine $m_{t + 1}$ .

Proof.

The formulas above depend on the full joint quantities $x_{α i}$ . Two joint distributions can have the same architecture marginal but different conditional meme distributions $P (i ∣ α)$ . If fitness or transition depends on $i$ , they produce different architecture updates. The dual argument applies to the meme marginal.

Architecture and meme must generally be modeled jointly.

40Continuous-time coevolution

Let $z = (α, i)$ index joint states. Let $f_{z} (x)$ be Malthusian fitness and $L_{z z^{'}} (x)$ a population-dependent Markov generator on joint states. Then

x ˙_{z^{'}} = x_{z^{'}} (f_{z^{'}} (x) - f ˉ (x)) + z \sum x_{z} L_{z z^{'}} (x) .

Equivalently,

x ˙_{β j} = x_{β j} (f_{β j} (x) - f ˉ (x)) + α, i \sum x_{α i} L_{α i, β j} (x) .

41Conservation theorem

Theorem41.1Conservation of total probability

If $\sum_{z^{'}} L_{z z^{'}} (x) = 0$ , then $\sum_{z} x_{z} = 1$ is preserved.

Proof.

Summing the replicator term gives $f ˉ - f ˉ = 0$ . Summing the transition term gives

z^{'} \sum z \sum x_{z} L_{z z^{'}} (x) = z \sum x_{z} z^{'} \sum L_{z z^{'}} (x) = 0.

Hence $\frac{d}{d t} \sum_{z} x_{z} = 0$ .

42Decomposition of architecture change

Let $ϕ : Ω_{M} \to R$ be any architecture observable depending only on $c$ . The generalized Price equation gives

Δ E [ϕ] = \frac{Cov ( w , ϕ )}{w ˉ} + \frac{E [ w ( K ϕ - ϕ ) ]}{w ˉ} .

Architecture change decomposes into

Differential reproduction of architectures, plus within-lineage cognitive transformation.

43Decomposition of semantic change

For a semantic observable $ψ : Ω_{S} \to R$ , pull it back along $q$ and apply the same identity:

Δ E [ψ \circ q] = \frac{Cov ( w , ψ \circ q )}{w ˉ} + \frac{E [ w ( K ( ψ \circ q ) - ψ \circ q ) ]}{w ˉ} .

The second term now includes both ordinary memetic mutation and changes caused by altered cognitive architecture.

44Semantic mutation versus architectural reinterpretation

A change in semantic class may occur in at least two distinct ways.

Memetic mutation. $(c, m) ⟶ (c, m^{'})$ with $q_{c} (m) \neq ≃ q_{c} (m^{'})$ .

Cognitive reinterpretation. $(c, m) ⟶ (d, T_{c d} m)$ with $q_{c} (m) \neq ≃ q_{d} (T_{c d} m)$ , despite preservation of the underlying presentation. Semantic change need not require a change in the presentation itself.

45Semantic conversion by architecture change

Definition45.1Pure reinterpretation event

A transition is a pure reinterpretation event if the presentation is transported without substantive alteration but its semantic class changes:

m^{'} ≃ T_{c d} m while [m]_{c} \neq = [m^{'}]_{d} .

This is impossible in a theory with a fixed semantic quotient, but natural in the present framework.

46Memetic niche construction

A population can alter the cognitive architectures that determine its future fitness landscape. This is a semantic analogue of niche construction. The loop is

ν_{t} \to A_{ν_{t}} \to a_{t + 1} \to L_{a_{t + 1}} \to Q_{t + 1} \to w_{t + 1} \to ν_{t + 1} .

Selection acts on a landscape partly constructed by earlier memetic evolution.

47Semantic niche

Definition47.1Semantic niche

The semantic niche of a joint population $ν$ is the architecture-weighted family of modalities

{L_{c} : c \in supp (π_{*} ν)},

together with their frequencies and translation structure. A change in architecture distribution changes the semantic niche.

48Reflexive selection

Definition48.1Reflexive selection

Selection is reflexive when present memetic states influence future cognitive modalities in a manner that changes their own or their descendants’ future fitness. Schematically,

m \to L^{'} \to [m]_{L^{'}} \to w^{'} (m) .

This creates a higher-order feedback loop:

State, observer, classification, fitness, and then state again.

49Fixed semantic theory as a special case

Suppose $A (c, m) = δ_{c}$ for every $(c, m)$ . Then architecture never changes. If the initial population is concentrated on one architecture $c_{0}$ , then $ν_{t}$ remains supported on $M (c_{0})$ . The theory reduces to the endogenous memetic evolutionary system of Note V on the fixed semantic state space $M (c_{0})$ .

50Architecture-only evolution as a special case

If every fiber $M (c)$ is contractible, then no nontrivial memetic state remains. The total state space reduces to $C^{≃}$ , and the theory becomes ordinary evolutionary dynamics on cognitive architectures. The coupled theory contains both meme evolution and architecture evolution as limiting cases.

51Joint stability

Let $F (ν) = R (ν, e)$ . At equilibrium $F (ν_{*}) = ν_{*}$ , perturbations satisfy

δ ν_{t + 1} = D F_{ν_{*}} δ ν_{t} .

Unlike fixed-architecture dynamics, the derivative contains architecture–meme cross terms. Schematically,

D F_{ν_{*}} = (D_{meme \to meme} D_{meme \to arch} D_{arch \to meme} D_{arch \to arch}) .

The off-diagonal blocks encode the coevolutionary feedback.

52Coevolutionary instability

Even if the meme subsystem and the architecture subsystem would each be stable when considered separately, the coupled system can be unstable because of the off-diagonal interaction terms. For a linearized block system

(A C B D) (m a),

stability depends on the spectrum of the full block operator, not merely the spectra of $A$ and $D$ . Feedback $m \to a \to m$ can create new unstable modes.

53Higher-order cognitive evolution

Architecture itself may be represented by a hierarchy

c^{(0)}, c^{(1)}, c^{(2)}, \dots

where $c^{(1)}$ controls transformations of $c^{(0)}$ , $c^{(2)}$ controls transformations of $c^{(1)}$ , and so forth. One may encode this by an iterated fibration $C_{n + 1} \to C_{n}$ . The total evolutionary state then lies in the iterated Grothendieck construction

\int_{C_{n}} \dots \int_{C_{1}} M .

This is a formal route toward evolution of meta-cognitive architecture.

54Self-modeling

Suppose some memetic states encode aspects of the architecture $c$ itself. Then there exists a representation map

r_{c} : M (c) \to Rep (c),

where $Rep (c)$ is an object of internal models of architecture $c$ . A state may influence architecture through an internally represented model:

m \to r_{c} (m) \to c^{'} \to M (c^{'}) .

This is the natural categorical setting for self-model-mediated cognitive modification.

55Self-referential semantic dynamics

The coupled structure may now take the form

M (c) \to Rep (c) \to C \to M (c^{'}) \to Rep (c^{'}) \to \dots .

The state modifies the interpreter, and the modified interpreter changes the meaning assigned to later states. This should be distinguished from ordinary self-replication. It is semantic self-modification.

56Main theorem

Theorem56.1Cognitive–memetic coevolution theorem

Let $(C, J)$ be a cognitive ∞-site, $M : C^{op} \to S$ a memetic field, $Ω_{M} = (\int_{C} M)^{≃}$ , and $P$ a compatible probability monad. Suppose given

w : Ω_{M} \times P (Ω_{M}) \times E \to R_{> 0}

and

K : Ω_{M} \times P (Ω_{M}) \times E \to P (Ω_{M}) .

Then:

(i) The normalized operator of Definition 16.1 defines a joint evolutionary dynamics of memes and cognitive architectures.
(ii) Its equilibria satisfy $L_{ν_{*}} ν_{*} = λ ν_{*}$ . See Theorem 18.2.
(iii) The architecture marginal evolves by the pushforward of the joint selection–transition process. See Theorem 19.1.
(iv) Neither architecture nor meme marginals generally form closed dynamical systems. See Theorem 39.1.
(v) Memetically dependent architecture kernels permit memes to change future cognitive modalities. See Definition 8.2 and Definition 22.1.
(vi) Nonconservative architecture changes permit semantic equivalence classes to split or merge. See Sections 5–7.
(vii) Semantic identity is invariant along trajectories consisting entirely of conservative architecture transitions. See Theorem 25.1.
(viii) The total semantic quotient $q : Ω_{M} \to Ω_{S}$ inherits an autonomous evolutionary dynamics exactly when the joint fitness and transition structure are lumpable through $q$ . See Theorem 30.1.
(ix) Failure of semantic lumpability produces hidden-state dependence and generally non-Markovian coarse semantics. See Section 31.
(x) The fixed-architecture theory is recovered when the architecture transition kernel is concentrated on the identity. See Section 49.

Proof.

Statements (i)–(x) follow from Sections 16–18, Theorem 19.1, Theorem 39.1, Definitions 8.2 and 22.1, Sections 5–7, Theorem 25.1, Theorem 30.1, Section 31, and Section 49.

57Full formal specification

The coevolutionary theory is specified by

(C, J, M, L, Ω_{M}, P, E, w, K) .

Symbol	Meaning
$C$	Cognitive architecture ∞-category.
$J$	Cognitive topology.
$M$	Architecture-indexed memetic field.
$L$	Architecture-indexed semantic modalities.
$Ω_{M}$	Joint architecture–meme state space.
$P$	Probability monad.
$E$	Environment.
$w$	Joint endogenous fitness.
$K$	Joint stochastic transition kernel.

The population equation is

ν_{t + 1} = \frac{L _{ν_{t}, e_{t}} ν _{t}}{L _{ν_{t}, e_{t}} ν _{t} ( Ω _{M} )} .

The semantic state of an individual is $q_{c_{t}} (m_{t})$ . The architecture marginal is $π_{*} ν_{t}$ .

58Conceptual hierarchy

The complete framework developed through the successive extensions is now

memetic presentations ↓ capability-relative equivalence ↓ architecture-indexed sheaf semantics ↓ operadic recombination ↓ network and feedback dynamics ↓ stochastic replication ↓ frequency-dependent selection ↓ coevolution of memes and cognitive architectures.

The final step changes the logical status of semantic identity itself. Previously $≃_{c}$ was fixed while states evolved. Now $c = c_{t}$ also evolves, so $≃_{c_{t}}$ is time dependent.

59Final principle

The strongest formulation is therefore not that memes evolve inside cognition, but that

Memes and the cognitive semantics that classify them coevolve.

A complete state is $(c, m)$ , not merely $m$ . Its semantic interpretation is $q_{c} (m)$ . Its reproduction may produce $(c^{'}, m^{'})$ , and therefore change both $m \mapsto m^{'}$ and $q_{c} \mapsto q_{c^{'}}$ .

The full causal loop is

(c_{t}, m_{t}) ⟶ (c_{t + 1}, m_{t + 1}) ⟶ q_{c_{t + 1}} (m_{t + 1}) ⟶ w_{t + 1} ⟶ (c_{t + 2}, m_{t + 2}) .

Memetic replication is evaluated relative to a semantic modality that can itself be modified by the replicating population. The resulting theory is a dynamics not merely of states, but of states together with their changing equivalence relations: dynamics of objects, dynamics of observers, and dynamics of identity.