A formal note · V

Endogenous fitness, evolutionary games, and stochastic lenses

Frequency-dependent selection, interacting populations, open systems, and nonlinear evolutionary dynamics in an ∞-topos.

endogenous fitness / replicator equation / Nash / stochastic lens / lumpability

Abstract

We extend stochastic memetic replication from exogenously specified fitness to endogenous, frequency-dependent evolutionary dynamics.

Let

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

be a cognitive semantic ∞-topos equipped with a commutative probability monad

P : H \to H .

Let $S \in H$ denote an observer-relative semantic state object, so that a population is a probability state $μ \in P (S)$ .

Rather than assuming a fixed fitness map $w : S \to R_{> 0}$ , we introduce an endogenous fitness field

w : S \times P (S) \times E ⟶ R_{> 0},

where $E$ is an environment object. Mutation and transmission may themselves depend on the population and environment through kernels

K : S \times P (S) \times E ⟶ P (S) .

The induced nonlinear evolutionary operator is

R (μ, e) = (K_{μ, e})_{*} \frac{w ( - , μ , e ) μ}{\int _{S} w ( s , μ , e ) d μ ( s )} .

Its equilibria satisfy a nonlinear eigenmeasure equation

L_{μ_{*}, e} μ_{*} = λ μ_{*} .

For finite state spaces, pairwise payoff interactions recover the classical frequency-dependent replicator equation

x ˙_{i} = x_{i} ((A x)_{i} - x^{⊤} A x),

and mutation gives the coupled mutation–selection equation.

We derive a generalized Price identity, characterize interior equilibria, prove a Lyapunov theorem for symmetric interaction matrices, extend the construction to multiple interacting populations, and formulate open evolutionary components as stochastic lenses. Memetic evolution is then not a fixed transfer operator but a nonlinear feedback system in which the population changes the fitness landscape that subsequently changes the population.

Semantic population, interaction, fitness field, replication, and then a new semantic population.

The loop is naturally expressible using stochastic lenses and open-game semantics. The exogenous theory is Note IV.

Notation

$w (s, μ, e)$: Endogenous fitness: the value of a semantic state can depend on the whole population and the environment.
$K_{μ, e}$: A mutation kernel that may itself depend on the population and the environment.
$R (μ, e)$: The nonlinear replicator–mutator step at population μ and environment e.
$L_{μ, e}$: The transfer operator frozen at (μ, e). Equilibria are its eigenmeasures.
$A$: A payoff matrix. Frequency-dependent replicator dynamics is the weak-selection limit.
$P$: A tuple of populations. Each population’s fitness can depend on all of them.
$(L^{\to}, L^{\leftarrow})$: A stochastic lens: a forward exposure and a backward probability kernel.
$BR$: Best-response correspondence. It is not the same map as the evolutionary update.

1Ambient semantic probability theory

Let $(C, J)$ be a cognitive ∞-site and let

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

Assume $H$ is equipped with a commutative probability monad $P : H \to H$ . Let $S \in H$ be a semantic state object. A population state is $μ : 1 \to P (S)$ , and $E \in H$ is an environment object. Expressions

\int_{S} f (s) d μ (s)

are used under the standing assumption that the required internal integration theory exists.

2Endogenous fitness

Definition2.1Endogenous fitness field

An endogenous fitness field is a morphism

w : S \times P (S) \times E ⟶ R_{> 0} .

For a fixed population $μ$ and environment $e$ , define $w_{μ, e} (s) := w (s, μ, e)$ . The reproductive value of a semantic state $s$ may depend on $s$ itself, on the complete current population $μ$ , and on the environment $e$ .

Definition2.2Mean endogenous fitness

Define

w ˉ (μ, e) = \int_{S} w (s, μ, e) d μ (s) .

We assume $0 < w ˉ (μ, e) < \infty$ .

3Population-dependent mutation

The transmission law itself may depend on the population state.

Definition3.1Endogenous mutation kernel

An endogenous mutation kernel is a morphism

K : S \times P (S) \times E ⟶ P (S) .

For fixed $(μ, e)$ , write $K_{μ, e} : S \to P (S)$ . Then $K_{μ, e} (s)$ is the conditional distribution of semantic descendants of $s$ under population state $μ$ and environment $e$ .

4Nonlinear replicator–mutator operator

Definition4.1Endogenous selection

Define $Sel_{w} (μ, e)$ by

d Sel_{w} (μ, e) (s) = \frac{w ( s , μ , e )}{w ˉ ( μ , e )} d μ (s) .

Definition4.2Endogenous evolutionary operator

Define

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

Equivalently, for an event $A$ ,

R (μ, e) (A) = \frac{\int _{S} w ( s , μ , e ) K _{μ, e} ( s , A ) d μ ( s )}{w ˉ ( μ , e )} .

Unlike the exogenous case, $R : P (S) \times E \to P (S)$ is generally nonlinear in $μ$ .

5Population-indexed transfer operators

For each $(μ, e)$ , define $L_{μ, e} : M_{+} (S) \to M_{+} (S)$ by

(L_{μ, e} ν) (A) = \int_{S} w (s, μ, e) K_{μ, e} (s, A) d ν (s) .

For fixed $(μ, e)$ the operator is linear in $ν$ . The assignment $μ \mapsto L_{μ, e}$ is generally nonlinear.

Proposition5.1Projectivization at a frozen operator

The evolutionary operator satisfies

R (μ, e) = \frac{L _{μ, e} μ}{L _{μ, e} μ ( S )} .

Proof.

Since $K_{μ, e} (s, S) = 1$ ,

L_{μ, e} μ (S) = \int_{S} w (s, μ, e) d μ (s) = w ˉ (μ, e) .

Substitution gives the result.

Endogenous evolution remains a projectivized positive operator, but the operator itself now depends on the point being evolved.

6Nonlinear eigenmeasure equilibrium

Definition6.1Endogenous equilibrium

For a fixed environment $e$ , a population $μ_{*}$ is an equilibrium if $R (μ_{*}, e) = μ_{*}$ .

Theorem6.2Nonlinear eigenmeasure characterization

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

L_{μ_{*}, e} μ_{*} = λ μ_{*},

where $λ = w ˉ (μ_{*}, e)$ .

Proof.

By Proposition 5.1, $R (μ_{*}, e) = μ_{*}$ if and only if $L_{μ_{*}, e} μ_{*} = L_{μ_{*}, e} μ_{*} (S) μ_{*}$ . But that total mass equals $w ˉ (μ_{*}, e)$ . Hence the equilibrium equation is exactly $L_{μ_{*}, e} μ_{*} = λ μ_{*}$ .

The difference from the exogenous case is substantial. Previously one solved an eigenproblem for one fixed positive operator. Now one solves the self-consistent nonlinear problem

L_{μ} μ \propto μ .

7Generalized Price identity

Let $φ : S \to R$ be an observable. Define

(K_{μ, e} φ) (s) := \int_{S} φ (t) K_{μ, e} (s, d t) .

Theorem7.1Generalized Price equation

For one evolutionary step $μ^{'} = R (μ, e)$ ,

E_{μ^{'}} [φ] = E_{μ} [φ] + \frac{Cov _{μ} ( w _{μ, e} , φ )}{w ˉ ( μ , e )} + \frac{E _{μ} [ w _{μ, e} ( K _{μ, e} φ - φ ) ]}{w ˉ ( μ , e )} .

Proof.

By definition,

E_{μ^{'}} [φ] = \frac{E _{μ} [ w _{μ, e} K _{μ, e} φ ]}{w ˉ ( μ , e )} .

Add and subtract $E_{μ} [w_{μ, e} φ]$ in the numerator:

E_{μ^{'}} [φ] = \frac{E _{μ} [ w φ ]}{w ˉ} + \frac{E _{μ} [ w ( K φ - φ ) ]}{w ˉ} .

The covariance identity

E_{μ} [w φ] = Cov_{μ} (w, φ) + w ˉ E_{μ} [φ]

then yields the stated formula.

The two correction terms are selection covariance and transmission change. The decomposition remains valid even though fitness depends on the population.

8Pairwise interaction games

A particularly important endogenous fitness model is induced by pairwise interaction. Let

a : S \times S \times E ⟶ R

be a payoff function. For a state $s$ , population $μ$ , and environment $e$ , define

f (s, μ, e) = \int_{S} a (s, t, e) d μ (t) .

This is the expected payoff obtained by $s$ when interacting with a random member of the population $μ$ .

9From payoff to positive fitness

Discrete-time fitness should be positive. A canonical choice is

w (s, μ, e) = exp (β f (s, μ, e)),

where $β \geq 0$ is selection strength. For weak selection,

w = 1 + β f + O (β^{2}) .

Thus additive payoff dynamics appears infinitesimally.

10Finite evolutionary game

Let $S = {1, \dots, n}$ and let $A = (A_{ij})$ be a payoff matrix. For a population $x = (x_{1}, \dots, x_{n})$ , the payoff of type $i$ is

f_{i} (x) = (A x)_{i} = j \sum A_{ij} x_{j} .

The population mean payoff is

f ˉ (x) = i \sum x_{i} f_{i} (x) = x^{⊤} A x .

11Frequency-dependent replicator equation

Assume no mutation. Under the continuous-time weak-selection limit,

x ˙_{i} = x_{i} (f_{i} (x) - f ˉ (x)) .

Hence

x ˙_{i} = x_{i} ((A x)_{i} - x^{⊤} A x) .

This is the classical frequency-dependent replicator equation.

12Conservation theorem

Theorem12.1Invariance of the simplex

The simplex

Δ = {x_{i} \geq 0, i \sum x_{i} = 1}

is invariant under replicator dynamics.

Proof.

i \sum x ˙_{i} = i \sum x_{i} (f_{i} - f ˉ) = f ˉ - f ˉ = 0.

Thus total mass remains 1. Moreover, if $x_{i} = 0$ , then $x ˙_{i} = 0$ . Trajectories cannot cross a boundary face into negative coordinates.

13Interior equilibria

Theorem13.1Equal-payoff characterization

Let $x_{*}$ be an interior population: $x_{*, i} > 0$ for every $i$ . Then $x_{*}$ is a fixed point of the pure replicator equation if and only if

f_{i} (x_{*}) = f ˉ (x_{*}) for every i .

Proof.

At equilibrium,

x_{*, i} (f_{i} (x_{*}) - f ˉ (x_{*})) = 0.

Because $x_{*, i} > 0$ , we must have $f_{i} (x_{*}) = f ˉ (x_{*})$ . Conversely, if every payoff equals the mean payoff, every derivative vanishes.

Thus every semantic type appearing with positive frequency at an interior equilibrium has equal reproductive payoff.

14Relation to Nash equilibrium

For a symmetric population game, a population $x_{*}$ is a symmetric Nash equilibrium if

f_{i} (x_{*}) \leq x_{*}^{⊤} A x_{*}

for every pure type $i$ , with equality for every type in the support of $x_{*}$ . Every interior replicator equilibrium satisfies the Nash condition because all pure strategies have the same payoff. Boundary replicator equilibria need not be Nash equilibria, since an absent strategy may have strictly greater payoff without affecting the instantaneous derivative of its zero frequency.

15Symmetric games and a Lyapunov function

Assume $A = A^{⊤}$ . Define $V (x) = x^{⊤} A x$ .

Theorem15.1Mean-payoff monotonicity

For symmetric $A$ , along the replicator equation,

\frac{d}{d t} (x^{⊤} A x) = 2 Var_{x} (f) \geq 0.

Proof.

Since $A = A^{⊤}$ ,

\frac{d}{d t} (x^{⊤} A x) = 2 x ˙^{⊤} A x .

With $(A x)_{i} = f_{i}$ ,

x ˙^{⊤} A x = i \sum x_{i} (f_{i} - f ˉ) f_{i} = i \sum x_{i} f_{i}^{2} - f ˉ i \sum x_{i} f_{i} = i \sum x_{i} f_{i}^{2} - f ˉ^{2} = Var_{x} (f) .

Multiplying by 2 gives the result.

Symmetric frequency-dependent selection performs ascent on mean interaction payoff, except at states where all supported strategies have equal payoff.

16Shahshahani gradient formulation

On the interior of the simplex define the Shahshahani metric

g_{ij} = \frac{δ _{ij}}{x _{i}} .

For the potential $Φ (x) = \frac{1}{2} x^{⊤} A x$ with symmetric $A$ , the replicator vector field is the constrained gradient of $Φ$ under this metric:

x ˙ = \nabla_{Sh} Φ.

The geometry of frequency-dependent selection is not ordinary Euclidean gradient flow but information-geometric gradient flow on the probability simplex.

17Mutation–selection games

Let $L (x)$ be a population-dependent Markov generator satisfying $L_{ij} (x) \geq 0$ for $i \neq = j$ , and $L (x) 1 = 0$ . The continuous-time endogenous mutation–selection dynamics is

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

Proposition17.1Probability conservation

j \sum x ˙_{j} = 0.

Proof.

The replicator contribution sums to zero by Theorem 12.1. For the mutation term,

j \sum i \sum x_{i} L_{ij} (x) = i \sum x_{i} j \sum L_{ij} (x) = 0,

because each row of $L (x)$ sums to zero.

18Nonlinear Markov processes

The state-dependent mutation law $μ \mapsto K_{μ}$ turns the Markov process into a nonlinear Markov process. The transition law is no longer $K_{*} μ_{t}$ for one fixed $K$ , but

(K_{μ_{t}})_{*} μ_{t} .

Adding selection gives

(K_{μ_{t}})_{*} (\frac{w _{μ_{t}} μ _{t}}{⟨ w _{μ_{t}} , μ _{t} ⟩}) .

The population modifies its own transition operator.

19Interacting populations

Let $A$ be a finite set of population classes. For every $a \in A$ , let $S_{a}$ be its semantic state space. The joint population object is

P = a \in A \prod P (S_{a}) .

A joint state is $(μ_{a})_{a \in A}$ .

20Coupled fitness fields

Definition20.1Coupled fitness and mutation

For each population $a$ , let

w_{a} : S_{a} \times P \times E \to R_{> 0} .

The fitness of a state in population $a$ may depend on all populations simultaneously. Likewise let

K_{a} : S_{a} \times P \times E \to P (S_{a}) .

21Coupled evolutionary operator

Define

R_{a} (μ, e) = (K_{a, μ, e})_{*} (\frac{w _{a} ( - , μ , e ) μ _{a}}{w ˉ _{a} ( μ , e )}) .

The full coupled operator is

R = a \in A \prod R_{a},

so $R : P \times E \to P$ .

22Pairwise multi-population games

Suppose population $a$ has states $i = 1, \dots, n_{a}$ . Let $A_{ij}^{ab}$ be the payoff to type $i$ in population $a$ from interacting with type $j$ in population $b$ . Then

f_{i}^{a} (x) = b \in A \sum j \sum A_{ij}^{ab} x_{j}^{b} .

Mean payoff within population $a$ is

f ˉ^{a} = i \sum x_{i}^{a} f_{i}^{a} .

23Coupled replicator equations

The continuous dynamics becomes

x ˙_{i}^{a} = x_{i}^{a} (f_{i}^{a} (x) - f ˉ^{a} (x)) .

With mutation,

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

Theorem23.1Separate population normalization

For every $a$ , the condition $\sum_{i} x_{i}^{a} = 1$ is invariant.

Proof.

For each fixed $a$ , the replicator contribution sums to $f ˉ^{a} - f ˉ^{a} = 0$ . The mutation generator satisfies $\sum_{i} L_{j i}^{a} = 0$ . Hence

\frac{d}{d t} i \sum x_{i}^{a} = 0.

24Environment as a dynamical object

So far $e \in E$ has been treated parametrically. Let

G : E \times P (S) ⟶ E

be an environmental update. The closed coupled system is

F : P (S) \times E ⟶ P (S) \times E,

given by

F (μ, e) = (R (μ, e), G (e, μ)) .

A closed equilibrium is a pair $(μ_{*}, e_{*})$ satisfying

μ_{*} = R (μ_{*}, e_{*}), e_{*} = G (e_{*}, μ_{*}) .

This is a simultaneous population–environment fixed point.

25Open evolutionary systems

Instead of closing the environment internally, one can expose it as an interface. Let $(X, X^{'})$ denote a system with forward state type $X$ and backward feedback type $X^{'}$ .

26Stochastic lenses

Definition26.1Stochastic lens

A stochastic lens $L : (X, X^{'}) ⟶ (Y, Y^{'})$ consists of a forward map

L^{\to} : X \to Y

and a stochastic backward update

L^{\leftarrow} : X \times Y^{'} \to P (X^{'}) .

The forward map exposes an observable output. The backward kernel propagates downstream feedback back into the upstream state or control variable.

27Identity stochastic lens

The identity on $(X, X^{'})$ has forward part $id_{X}$ and backward kernel $δ_{x^{'}}$ .

28Composition of stochastic lenses

Let $L : (X, X^{'}) \to (Y, Y^{'})$ and $M : (Y, Y^{'}) \to (Z, Z^{'})$ . The composite has forward part

(M \circ L)^{\to} = M^{\to} \circ L^{\to} .

For a backward feedback $z^{'}$ , first sample

y^{'} \sim M^{\leftarrow} (L^{\to} (x), z^{'}),

and then sample

x^{'} \sim L^{\leftarrow} (x, y^{'}) .

Hence the backward kernel is

(M \circ L)^{\leftarrow} (x, z^{'}) = \int_{Y^{'}} L^{\leftarrow} (x, d y^{'}) M^{\leftarrow} (L^{\to} (x), z^{'}) .

More invariantly, this is Kleisli composition using the strength of the commutative probability monad.

Theorem28.1Associativity

Composition of stochastic lenses is associative up to the canonical coherence of the probability monad.

Proof.

The forward components compose associatively because morphism composition does. The backward components compose by iterated Kleisli integration. Associativity therefore follows from associativity of the monad multiplication $P^{3} \to P$ .

Stochastic lenses form a category, or an appropriate ∞-categorical enhancement when higher morphisms are retained.

29Evolution as a stochastic lens

Take $X = P (S)$ . Suppose a population exposes an observable $O : P (S) \to Y$ . The environment returns a fitness or interaction signal $r \in Y^{'}$ . An evolutionary response kernel has the form

U : P (S) \times Y^{'} \to P (P (S)) .

Then

(O, U) : (P (S), P (S)) ⟶ (Y, Y^{'})

is a stochastic evolutionary lens. If the evolutionary update is deterministic, the backward kernel is $δ_{R_{r} (μ)}$ .

30Feedback closure

Suppose an environment lens converts the exposed population observable into a fitness signal. Composing the population and environment lenses closes the loop

Population, environment, fitness signal, and then the population update.

Endogenous fitness is a feedback phenomenon. The equation $w = w_{μ}$ is the algebraic shadow of this closed interaction.

31Open evolutionary games

Definition31.1Evolutionary open game

An evolutionary open game consists of:

a semantic strategy object $S$ ;
a population object $P (S)$ ;
an environment interface $E$ ;
a payoff field $u : S \times P (S) \times E \to R$ ;
an evolutionary response operator $R : P (S) \times E \to P (S)$ .

The payoff field need not itself prescribe the update. Different evolutionary rules may act on the same game.

32Best-response correspondence

For fixed $(μ, e)$ , define

BR (μ, e) = s \in S argmax u (s, μ, e) .

In a homotopical setting, $BR$ should more properly be treated as a subobject or correspondence rather than an ordinary set.

Definition32.1Nash population

A population $μ_{*}$ is a Nash population at environment $e$ if

supp (μ_{*}) \subseteq BR (μ_{*}, e) .

Every semantic strategy represented with nonzero population weight must be a best response to the current population.

33Interior replicator equilibria and Nash equilibrium

Theorem33.1Interior equilibria are Nash

For a finite symmetric population game, every interior equilibrium of the pure replicator equation is a Nash equilibrium.

Proof.

By Theorem 13.1, $f_{i} (x_{*}) = f ˉ (x_{*})$ for every $i$ . Since $x_{*}$ is interior, every pure strategy appears in its support. Hence every pure strategy obtains exactly the same payoff, every pure strategy is a best response, and $supp (x_{*}) \subseteq BR (x_{*})$ .

34Evolutionary dynamics versus rational optimization

The framework distinguishes two maps:

BR : P (S) ⇉ S

and

R : P (S) \to P (S) .

The first is an optimization correspondence. The second is a population dynamical law. Replicator dynamics need not jump to best responses. It continuously amplifies strategies according to relative payoff.

A game equilibrium is not an evolutionary update rule.

The two become linked through equilibrium theorems, not by definition.

35Observer-relative games

Let $q : M \to S$ be a capability-relative semantic quotient. Suppose microscopic interaction payoff is

u : M \times M \times E \to R .

A closed game exists on $S$ only if the payoff descends: there is $u : S \times S \times E \to R$ with

u (m, n, e) = u (q (m), q (n), e) .

If this fails, two states indistinguishable to the observer may nevertheless have different interaction consequences. The coarse observer then does not possess a closed game description.

36Game-theoretic lumpability

Definition36.1Strategic lumpability

A semantic quotient $q : M \to S$ is strategically lumpable if payoff factors through $q$ , mutation and transmission are $q$ -lumpable, and environmental coupling factors through the coarse state. Under these conditions, the observer-relative population dynamics on $S$ closes autonomously.

Theorem36.2Observer-relative evolutionary closure

If $q$ is strategically lumpable, then there exists an induced evolutionary game on $S$ whose population trajectory is exactly the pushforward of the microscopic trajectory:

q_{*} \circ R ≃ R_{S} \circ q_{*} .

Proof.

Payoff factorization implies that the induced fitness depends only on the coarse state. Kernel lumpability implies that stochastic transmission commutes with $q_{*}$ . Environmental factorization ensures that feedback signals are also determined by the coarse population. The coarse-graining theorem for replicator–mutator dynamics therefore applies at every feedback step.

37Failure of strategic lumpability

If strategic lumpability fails, coarse population state is not sufficient to predict its own future. One must augment the state with hidden variables. Formally, if

q_{*} μ = q_{*} ν

but

q_{*} R (μ) \neq = q_{*} R (ν),

then no map $F : P (S) \to P (S)$ can satisfy

F \circ q_{*} = q_{*} \circ R

for all microscopic states. The observer’s coarse semantics is dynamically incomplete.

38Memory from hidden semantic variables

One can repair non-Markovian coarse dynamics by enlarging the observed state. Suppose

h : M \to H

records additional hidden semantic information. Replace $q : M \to S$ by

(q, h) : M \to S \times H .

If the enlarged quotient is lumpable, then $(S, H)$ is a sufficient state representation. This yields a formal criterion for semantic state augmentation:

Add enough hidden structure until the dynamics closes.

39Higher-order frequency dependence

Pairwise interaction is not essential. Let

a_{n} : S^{n + 1} \times E \to R .

Define

f_{n} (s, μ, e) = \int_{S^{n}} a_{n} (s, s_{1}, \dots, s_{n}, e) d μ (s_{1}) \dots d μ (s_{n}) .

Total payoff may be

f (s, μ, e) = n \geq 0 \sum λ_{n} f_{n} (s, μ, e) .

This is the stochastic-population analogue of the earlier operadic theory. Fitness can depend on genuinely multi-agent semantic interactions rather than only pairwise games.

40Polynomial interaction laws

An even more categorical formulation uses a polynomial

I s E p B t I .

Each operation $b \in B$ specifies a finite interaction pattern. Given a population $μ$ , one samples inputs according to the corresponding fibers $E_{b}$ , evaluates their interaction, and produces a fitness contribution. Polynomial functors specify the combinatorial shape of interactions, while the probability monad integrates over randomly sampled participants.

Polynomial interaction together with probability yields endogenous fitness.

41Operadic game interaction

If $O$ is an ∞-operad of interaction forms, one may assign to $o \in O (n)$ a payoff map $u_{o} : S^{n + 1} \to R$ . The population-level expected payoff is obtained by integrating the other $n$ arguments against $μ$ :

\int_{S^{n}} u_{o} (s, s_{1}, \dots, s_{n}) d μ^{\otimes n} .

Operadic recombination and evolutionary game interaction can be encoded by the same arity structure.

42Endogenous semantic fidelity

Semantic fidelity relative to the class $[s]_{c}$ , in the exogenous theory, was $K (s, [s]_{c})$ . With endogenous transmission,

F_{c} (s, μ, e) = K_{μ, e} (s, [s]_{c}) .

Replication fidelity itself becomes frequency dependent. A meme may copy accurately when rare and inaccurately when common, or conversely.

43Effective endogenous reproduction number

Define

R_{c} (s, μ, e) = w (s, μ, e) F_{c} (s, μ, e) .

This quantity measures expected production of descendants that remain in the same observer-relative semantic class. Unlike a fixed reproduction number, $R_{c}$ changes with the population itself.

44Frequency-dependent semantic selection

Suppose two semantic classes $a$ and $b$ satisfy

R_{c} (a, μ) > R_{c} (b, μ)

for one population state, but

R_{c} (a, ν) < R_{c} (b, ν)

for another. Then there is no absolute fitness ordering between $a$ and $b$ . Fitness is relational: a property of a state relative to a population and an environment. This is the categorical analogue of frequency-dependent selection.

45Nonlinear categorical dynamics

The exogenous theory had, for a fixed positive operator $L$ ,

μ_{t + 1} = \frac{L μ _{t}}{L μ _{t} ( S )} .

The endogenous theory has

μ_{t + 1} = \frac{L _{μ_{t}} μ _{t}}{L _{μ_{t}} μ _{t} ( S )} .

The underlying geometry is not projective linear dynamics but a nonlinear self-map of the probability object. Equilibria solve

L_{μ_{*}} μ_{*} = λ μ_{*} .

46Linearization around equilibrium

Let $F (μ) = R (μ, e)$ for fixed $e$ . At an equilibrium $F (μ_{*}) = μ_{*}$ , local stability is controlled by the derivative $D F_{μ_{*}}$ . If $T_{μ_{*}} P (S)$ denotes the tangent space of normalized perturbations, then infinitesimal perturbations evolve by

δ μ_{t + 1} = D F_{μ_{*}} δ μ_{t} .

Definition46.1Local asymptotic stability

The equilibrium is linearly asymptotically stable if the spectral radius of

D F_{μ_{*}}_{T_{μ_{*}} P (S)}

is less than 1 in discrete time. For continuous time, stability requires the relevant eigenvalues of the Jacobian to have negative real part. Second-order structure re-enters through the derivative of the population-dependent fitness field.

47Feedback derivatives

Since $w = w (s, μ, e)$ , the derivative of the evolutionary operator contains two contributions:

D F = D_{direct} + D_{feedback} .

The direct term changes population frequencies while holding the fitness field fixed. The feedback term arises because perturbing $μ$ changes $w_{μ}$ and potentially $K_{μ}$ . This term has no analogue in a fixed-fitness linear transfer system.

48Second-order interaction structure

For a finite game with payoff field $f_{i} (x)$ , define

J_{ij} = \frac{\partial f _{i}}{\partial x _{j}} .

Then $J$ measures how increasing population type $j$ changes the reproductive payoff of type $i$ . For the linear game $f (x) = A x$ , one has $J = A$ . For nonlinear interaction games, $D^{2} f$ measures how those interaction effects themselves change with population composition. Higher derivatives classify increasingly high-order ecological or memetic feedback.

49Stochastic-lens interpretation of endogenous fitness

The complete feedback structure can be written schematically as

P (S) O Y env. Y^{'} ⟶ Fit (S) ⟶ P (S) .

Closing the interfaces gives

P (S) ⟶ P (S) .

The endogenous nonlinear map is the closed semantics of an open stochastic system. Lens-like semantics is natural here because forward behavior and backward feedback are separate components before the system is closed.

50Main theorem

Theorem50.1Endogenous memetic evolution theorem

Let $H = Sh_{\infty} (C, J)$ be equipped with a compatible probability monad $P$ . Let $S, E \in H$ , and suppose given

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

and

K : S \times P (S) \times E \to P (S) .

Then:

(i) The formula $R (μ, e) = (K_{μ, e})_{*} (\frac{w _{μ, e} μ}{w ˉ ( μ , e )})$ defines a normalized nonlinear evolutionary operator. See Sections 4–5.
(ii) Equilibria are precisely solutions of $L_{μ_{*}} μ_{*} = λ μ_{*}$ . See Theorem 6.2.
(iii) Changes in any observable satisfy the generalized Price equation. See Theorem 7.1.
(iv) Pairwise linear interaction generates $x ˙_{i} = x_{i} ((A x)_{i} - x^{⊤} A x) .$ See Sections 8–11.
(v) For symmetric $A$ , mean payoff is nondecreasing: $\frac{d}{d t} (x^{⊤} A x) = 2 Var_{x} (A x) \geq 0.$ See Theorem 15.1.
(vi) Multiple semantic populations admit coupled replicator systems. See Sections 19–23.
(vii) Population–environment feedback is the closure of an open stochastic system. See Sections 24–30.
(viii) Stochastic lenses compose such open systems associatively. See Theorem 28.1.
(ix) A semantic quotient inherits an autonomous evolutionary game precisely when payoff, mutation, and environment interactions descend through that quotient. See Theorem 36.2.
(x) Failure of such descent produces hidden-state dependence and generally non-Markovian observer-relative dynamics. See Sections 37–38.

Proof.

Statements (i)–(x) are established in Sections 4–6, Theorem 7.1, Sections 8–17, Sections 19–23, Sections 24–30, Theorem 28.1, and Sections 35–38.

51Full specification

The endogenous theory is summarized by

(C, J, H, P, M, Q, S, E, w, K, R) .

Symbol	Meaning
$M$	Fine memetic realizations.
$Q : M \to S$	Observer-relative semantics.
$P (S)$	Semantic population space.
$E$	Environment.
$w (s, μ, e)$	Endogenous fitness.
$K_{μ, e}$	Endogenous stochastic transmission.
$R$	Nonlinear population evolution.

The fundamental equation is

μ_{t + 1} = \frac{L _{μ_{t}, e_{t}} μ _{t}}{L _{μ_{t}, e_{t}} μ _{t} ( S )} .

If the environment also evolves,

μ_{t + 1} e_{t + 1} = R (μ_{t}, e_{t}), = G (e_{t}, μ_{t}) .

52Conceptual synthesis

The earlier fixed-fitness theory treated the memetic environment as external:

μ L μ^{'} .

The endogenous theory closes the feedback loop:

μ \to w_{μ} \to L_{μ} \to μ^{'} \to w_{μ^{'}} \to \dots .

The population constructs part of its own selective environment. For pairwise interaction,

f (s, μ) = \int_{S} a (s, t) d μ (t),

and consequently

μ \mapsto w_{μ} \mapsto R (μ) .

This turns memetic evolution from a linear transmission problem into a feedback game.

The stochastic-lens perspective separates the open interfaces: observation and feedback. The evolutionary-game perspective supplies interaction, then payoff, then relative reproduction. The ∞-topos layer determines which distinctions count as semantic distinctions for a given cognitive architecture. The probability monad determines stochastic inheritance. The game determines frequency-dependent fitness. The lens structure determines how open components are coupled.

Observer-relative stochastic game dynamics with feedback.

Equivalently, endogenous memetic evolution.

The theory no longer assigns a meme an absolute reproductive fitness. Fitness is a relational functional

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

and replication occurs within the nonlinear closed loop generated by that functional.