Replicator-Optimization Mechanism (ROM)

• ROM is a unified framework that formalizes persistence-conditioned selection-transmission processes by merging replicator-mutator dynamics with optimization principles.
• It specifies system evolution through well-defined scales, atomic units, interaction topologies, and stochastic transmission kernels to generalize behavior across diverse fields.
• ROM provides operational recipes for measurement, falsification, and prediction, enabling empirical validation and normative analysis in models ranging from molecular to social systems.

The Replicator-Optimization Mechanism (ROM) formalizes persistence-conditioned selection-transmission processes as a scale-relative, kernel-parametric framework bridging replicator-mutator and Price-style dynamics with optimization principles relevant across physical, biological, economic, cognitive, and social domains. ROM structures system evolution by explicitly specifying scale, atomic units, interaction topologies, and stochastic transmission kernels, enabling generalization and instantiation from molecular replicator flows to institutional consent dynamics and Nash-convergent population games. Its axiomatic backbone outlines the necessary modeling components, while its novel contributions include a systematic kernel-triple parameterization, application to legitimacy/friction in consent-based metaethics, and an independent derivation from social-contract theory, thus grounding empirical and normative analysis. ROM yields operational recipes for measurement, falsification, and prediction, encompassing regulatory, computational, and control-theoretic interpretations (Farzulla, 10 Jan 2026).

1. Formal Axiomatic Structure

ROM is defined through five core axioms, each parametrized by a choice of scale $S$ and atomic agent %%%%1%%%%:

• A1. Minimal Atoms (scale-relative): All dynamics at scale $S$ are described via the states of $\text{Atom}_S$ (e.g., particle, cell, organism, institution).
• A2. Interaction Network: Atoms interact via a time-varying graph $G_{S,t}=(A_S,E_{S,t})$.
• A3. Entropy Pressure (Decay): Absent active maintenance, configurations drift to higher entropy.
• A4. Replication/Propagation with Variation: Patterns persist by imperfect propagation events, parametrized by a stochastic transmission kernel.
• A5. Large Numbers/Concentration: In sufficiently large populations, observable macro-variables exhibit concentration phenomena reflecting law-of-large-numbers scaling (Farzulla, 10 Jan 2026).

Dynamics unfold over equivalence classes of configurations $\tau \in T_S$, each defined via observer-dependent similarity relations, with update equations acting over frequency distributions $p_t(\tau)$.

2. Dynamical Equations and Kernel Specification

ROM generalizes the replicator-mutator and Price equations through scale-relative kernel-triple parametrization. At any given scale $S$:

• Minimal Replicator-Mutator Equation: With distribution $x_i$ over types, persistence $f_i(x,t)$, and mutation kernel $Q_{ji}$,

$$\dot x_i = \sum_j x_j f_j(x,t) Q_{ji} - x_i \bar{f}(x,t),$$

where $\bar{f}(x,t) = \sum_k x_k f_k(x,t)$.

• Kernel-Based ROM Update: Defining
  • $w_S(\tau)$ — intrinsic weight,
  • $\rho_S(\tau,G_{S,t},p_t)$ — survival probability,
  • $M_S(\tau' \rightarrow \tau)$ — row-stochastic transmission kernel,

$$\frac{dp_t(\tau)}{dt} = \sum_{\tau'} p_t(\tau')\,w_S(\tau')\,\rho_S(\tau',G_{S,t},p_t)\,M_S(\tau' \rightarrow \tau) - p_t(\tau)\,\bar{\phi}_t,$$

with $\bar{\phi}_t$ normalizing (Farzulla, 10 Jan 2026).

• Discrete-Time Price Equation Partition:

$$\Delta\bar{z} = \frac{1}{\bar{w}}\,\mathrm{Cov}(w, z) + \frac{1}{\bar{w}}\,\mathbb{E}[w\,\Delta z].$$

At all levels, the choice of $\text{Atom}_S$ determines the nature of $T_S$, $G_S$, and the kernel triple $(w_S, \rho_S, M_S)$. Coarse-graining and lumpability (Theorem 4.1 in (Farzulla, 10 Jan 2026)) guarantee formal invariance across observational scales.

3. Optimization Principles in Replicator Systems

ROM includes explicit optimization dynamics in systems exhibiting time-scale separation between fast replicator evolution and slow fitness parameter adaptation. In permanent systems (Drozhzhin et al., 2019):

• Definition: State $u(t) \in S_n$, fitness matrix $A(\tau)$ evolving on slow timescale $\tau$.
• Objective: At each $\tau$, compute steady-state $\bar{u}(\tau)$ solving $A(\tau)\, \bar{u}(\tau) = \bar{f}(\tau)\, 1$, subject to a quadratic norm constraint on $A$.
• Optimization Problem:

$$\max_{A}\; \bar{f}(A)\qquad \text{s.t.} \quad \Vert A \Vert^2 \leq Q^2$$

Updated via linear programming step ensuring $\delta\,\bar{f} \geq 0$ at each evolutionary increment, producing adaptive fitness landscapes (Drozhzhin et al., 2019).

This mechanism encompasses resource-constrained maximization of mean fitness, yielding emergence of cyclic, autocatalytic, altruistic, and parasite-resistant structures.

4. Information-Theoretic Bounds and Functional Strategies

ROM admits an information-theoretic decomposition for productivity in minimal replicator systems (Piñero et al., 2024):

• Continuous-flow reactor: Species $X_i$ with concentration $x_i(t)$, autocatalytic rates $\eta_i$, subject to fluctuating environments.
• Average productivity: $\sigma = \sigma^* + \ln q_r$, where $q_r$ is initial winner fraction; generalizes substitutional load in population genetics.
• Fluctuating environments with side-information $Y$: For winner $R=r(E)$,

$$\langle \sigma \rangle = \langle \sigma^* \rangle - \Gamma - \Omega C_{\pi,q}(R|Y),$$

where $C_{\pi,q}(R|Y)$ splits into environmental entropy $H_{\pi}(R)$, side-information gain $I_{\pi}(R;Y)$, and strategy mismatch $D(\pi_{R|Y}\Vert q_{R|Y})$.

• Universal bound and optimal strategy:

$$\langle \sigma \rangle \leq \langle \sigma^* \rangle - \Gamma - \Omega[H_{\pi}(R) - I_{\pi}(R;Y)]$$

with strategy $q^*_{r|y} = \pi_{r|y}$, analogous to Kelly gambling (Piñero et al., 2024).

This analytic structure links ROM with classical learning, memory, and payoff-optimization results in stochastic environments.

5. Control-Theoretic, Geometric, and Nash-Game Extensions

ROM admits control-theoretic characterization via Lie algebra and Hamiltonian structures (Raju et al., 2020):

• Replicator Dynamics on Simplex: State $x \in \Delta^{n-1}$, fitness map $f(x)$; replicator ODE

$\dot{x}_i = x_i\big(f^i(x) - x\cdot f(x)\big)$

• Lie Algebra of Fitness Maps: Bracket $[f,g]_R := X_f(g) - X_g(f)$, homomorphic to replicator vector fields.
• Hamiltonian Lift: On $T^*\Delta$, define $\mathcal{H}(x,p) = p \cdot X_f(x)$; Hamilton's equations recover replicator trajectory.
• Controllability: Fitness maps actuated by controls $u_k(t)$; Lie-algebra rank condition ensures accessibility.
• Optimal Control: Cost functional $J[u] = \int_0^T L(x(t),u(t))\,dt + \Phi(x(T))$, solved via Pontryagin Maximum Principle (Raju et al., 2020).

Further, ROM generalizes nonconvex optimization (Anderson et al., 2024) by lifting the objective to measures $\mu$ on $\mathbb{R}^d$, whose Nash equilibria correspond to global minima. The approximately Gaussian replicator flows (AGRF) evolve probability measures via deterministic ODEs:

$$\dot{m}_i = m_i E_{x\sim N(m,C)}[f(x)] - E_{x\sim N(m,C)}[x_i f(x)],\quad \dot{C}_{ij} = (C_{ij}-m_i m_j) E[f(x)] - E[x_i x_j f(x)] + m_i E[x_j f(x)] + m_j E[x_i f(x)]$$

Solving AGRF equations achieves globally optimal trajectories in convex-quadratic and locally convex regions, ascending over barriers in nonconvex landscapes (Anderson et al., 2024).

6. Domain-Specific Instantiations: Consent-Friction Model

ROM's kernel triple is instantiated for political philosophy with friction and legitimacy as primitives (Farzulla, 10 Jan 2026):

• Friction $F(d,t)$: Quantifies tension between consent-holders and consequence-bearers, formulated as

$$F(d,t) = \sum_i s_i(d) \frac{1+\epsilon_i(d,t)}{1+\alpha_i(d,t)},$$

where $s_i$ is stake, $\alpha_i$ alignment, $\epsilon_i$ entropy (epistemic control).

• Legitimacy $L$: Defined as $L(d,t) = 1 - ½ \sum_i |\hat{s}_i - \hat{v}_i|$, measuring total-variation distance between normalized stakes and voice.
• Survival Kernel: $\rho_S(\tau) = L(\tau)/(1+F(\tau))$.
• Belief-Transfer (Ownership Accumulation):

$$\frac{d O_A}{dt} = \beta (1-O_A) \mathbb{1}[A \text{ holds consent for } d]$$

• Mutation Kernel Modulation:

$$M_S(\tau' \rightarrow \tau) = \frac{M_0(\tau' \rightarrow \tau) \exp[-\gamma(\bar{O}(\tau')-\bar{O}(\tau))]}{Z(\tau')}$$

Regimes with suppressed friction (high latent but low observed $F$) are predicted to undergo tipping instabilities when suppression capacity $\kappa$ falls, producing rapid collapse phenomena.

7. Measurement, Falsification, and Empirical Recipes

ROM is operationalized by systematic measurement and falsification (Farzulla, 10 Jan 2026):

• Empirical Observables: Include stakes, voice, alignment, entropy, observed and latent friction, ownership metrics.
• Falsifiable Predictions: Correlations between legitimacy and friction, reform-induced friction reduction, belief-transfer effects, instability linked to suppression ratios, and concentration scaling of macro-observables.
• Falsification Criteria: Includes failure of predicted correlations, deviation from replicator-mutator form, absence of observable concentration in large systems, and collapse events uncorrelated with suppression shocks.
• Experimental Realization: In replicator reactors, productivity gains above no-memory bounds operationalize quantitative tests of functional information processing (Piñero et al., 2024), while optimization and Nash-game trajectories are assessed against benchmark nonconvex landscapes (Anderson et al., 2024).

A plausible implication is that ROM supplies a unified, cross-domain recipe: select atomic unit and scale $S$, define equivalence classes $T_S$ and kernel triple, formulate update equations, and deploy measurement practices to validate or falsify predicted system-level behavior.

Summary Table: ROM Kernel Triple and Domain Instantiation

Domain	Atomic Unit (Atomₛ)	Survival Kernel (ρₛ)	Transmission Kernel (Mₛ)
Molecular	Molecule	Replicator fitness function	Mutation probabilities
Institutional	Organization	Legitimacy / friction ratio	Belief-transfer, ownership accumulation
Population game	Strategy	Nash payoff	Game mutation kernel
Consent model	Consent-holder	$L/(1+F)$	Ownership-driven transfer

ROM generalizes the operational dynamics underpinning persistence, optimization, and selection-transmission processes, offering a versatile kernel-parametric formalism for empirical, computational, and philosophical analysis.