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CLAIM-444: A replicator dynamic on the distribution of authority among an organization's processes is the special case of the meaning–authority–legibility model in which the organization's interpretation is the authority-weighted mixture of its processes' and evidence reaches scoring without a maintained regime of its own; the replicator tools (Bayesian reading, potential and harmonic games, stochastic stability) then apply, and the record's separate interpretation and legibility states earn their place exactly where interpretation departs from that mixture or legibility keeps a memory of past authority

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Properties11
StatusProposed
Version2
Rolethesis
Warrantinterpretive
Tagsself-governance, learning-theory, complex-systems
Linedistributed-agency
Rests onCLAIM-400 — In the meaning–authority model linearized at symmetry, a third, maintained legibility state that authority moves (ε) and that moves interpretation (ν) or endorsement (μ), with persistence λ < 1 and nonnegative couplings, destabilizes the symmetric state exactly when the two-variable condition holds with β + νε/(1 − λ) in place of β and δ + με/(1 − λ) in place of δ: at the threshold the regime is indistinguishable from stronger two-variable couplings, and the slower it is the more it adds, while away from the threshold it adds a mode that no two-variable model has, so a lean in the regime can outlast the advantages it was built from
GroundsTHEORY-899 — In an infinite, well-mixed population whose members each sample the quality of their own type, imitation of success has the expected update of a single Cross Learning bandit agent (the Taylor replicator dynamic), and the weighted voter model, in which members broadcast their type in proportion to their sample and adopt the first type they perceive, has the expected update of Maynard-Cross Learning, π_a ← π_a + (r_k / v^π)(1[a = k] − π_a), the Maynard-Smith replicator dynamic; the population vector is the policy and no member learns, CASE-155 — Authority as a replicator population: for a distribution π of control over processes with mean qualities q, the expected Maynard-Cross update is π_a q_a / v, Bayes' rule when q is each process's likelihood; the Maynard dynamic is the Taylor dynamic slowed by 1/v, so the two share their orbits; mean quality climbs at the rate Var_π(q); and when a process's score rises with its own share, ℓ′ = ℓ + h + c(2π − 1) has an unstable threshold π* = (1 − h/c)/2, above which a process the evidence disfavours captures the organization, CASE-124 — A maintained legibility lean as a third state of the linearized meaning–authority model: authority moves the lean r of the categories through which evidence is read, and r feeds interpretation or authority, with its own persistence λ, THEORY-894 — A flow of pairwise comparisons on a graph splits orthogonally and uniquely into a gradient part, the l2-best global ranking, and a cyclic part, which splits into curl (local) and harmonic (global) inconsistency; a finite game's flow of unilateral payoff differences has no curl part, so every finite game is uniquely a potential game plus a harmonic game plus a nonstrategic remainder, and potential games are exactly those with no harmonic part; the decomposition is cardinal, THEORY-896 — A finite game in strategic form has the finite improvement property, every path of strict unilateral improvements being finite, exactly when it has a generalized ordinal potential, a real function on strategy profiles that rises with every strict unilateral improvement; so in a finite game an ordering along which every improving move climbs exists exactly when there is no improvement cycle, and every sequence of improving moves ends at a pure equilibrium
ComplementsCLAIM-443 — A learned prior over which processes govern, P(g | z, H) ∝ exp(βQ(g, z) + b_H(g, z)), its bias b_H trained by reward-modulated Hebbian updates to the relations by which one process recruits or endorses another, is a candidate mechanism for durable authority and for amortized governance; but persistence is not amortization, which is shown only when earlier work replaces later computation, communication or recruitment, and a prior over authority differs from a prior over actions only when what is appropriate changes, CLAIM-448 — An ordinal potential describes only the winning-move relation it is computed on; where an organization's legibility regime selects which contests are admitted or recorded, an acyclic record, even a persistent potential, can be the product of selective contestation, so observed coherence is not evidence that the organization's perspectives are integrated, and an intervention that improves its self-governance may at first increase the disagreement observed, CLAIM-369 — In symmetric resource-mediated competition with interaction matrix A = UᵀU and growth rates r = Uᵀb − c, Lotka–Volterra population dynamics are multiplicative gradient flow on the convex feature-selection objective ½‖b − Un‖² + cᵀn over the nonnegative orthant: the invasion fitness of an absent component is the negative partial derivative of that objective, and the uninvadable equilibria are its KKT points
Illustrated byCASE-155 — Authority as a replicator population: for a distribution π of control over processes with mean qualities q, the expected Maynard-Cross update is π_a q_a / v, Bayes' rule when q is each process's likelihood; the Maynard dynamic is the Taylor dynamic slowed by 1/v, so the two share their orbits; mean quality climbs at the rate Var_π(q); and when a process's score rises with its own share, ℓ′ = ℓ + h + c(2π − 1) has an unstable threshold π* = (1 − h/c)/2, above which a process the evidence disfavours captures the organization, CASE-157 — Boosting the vote: in the owner's 2014 model, voters of fixed accuracy drawn from Beta(2, 2) vote in a weight-averaged election, and every voter's weight is multiplied by 1 + f when their own vote was right and 1 − f when not, with a fraction replaced each round at weight 1; with full-information feedback the weighted result rises from 0.50 toward 0.98 as f grows, and churn only costs in a world that never changes; the model escapes the hive mind's absorbing states because every voter is scored, heeded or not, and it assumes away the capture its author predicted, CASE-159 — Authority geometry and response geometry: when an organization's response is the authority-weighted mixture p_π(y | c) = Σ πᵢ pᵢ(y | c), the Fisher metric of its responses is the pullback JᵀD⁻¹J of the categorical (Shahshahani) metric on the shares π and never exceeds it; it vanishes along any redistribution between processes that respond alike under the probes used, so two specialists who both answer yes with probability 0.8 in routine contexts carry no information about their shares there, while a rare context in which they answer 0.9 and 0.1 carries 0.64/(p(1 − p)) per observation, p = 0.1 + 0.8π
Cited as “complements” byCLAIM-452 — For an organization whose response in each probe context is the authority-weighted mixture of its processes' known response profiles, the authority shares are identifiable from a set of contexts exactly when the processes' stacked profiles over those contexts are affinely independent; because the response is linear in the shares, local and global identifiability coincide, the condition is that the Fisher information of any design supported on those contexts is positive definite on the simplex's tangent space, and with binary outcomes at least one context fewer than there are processes is needed

Graph View

The claim

The owner, 2026-10-24: “can we use any of the tools here (or from related work) to formalize our more abstract version of the dynamics?”

The identification. In Soma et al. the population vector is the policy (THEORY-899). Read the distribution π of practical authority over an organization’s processes as that vector. If the organization’s interpretation is the π-weighted mixture of the processes’ interpretations, then CASE-124’s meaning m is a function of authority a, and only a, together with the scores, is state. If evidence reaches the scores directly, with no maintained regime r, then the model reduces to a replicator dynamic on π. In the population-dependent version, scores depend on π.

What the reduction brings.

  • Evidence-tracking authority is Bayesian model averaging (CASE-155, part 1).
  • Authority-dependent scoring is a coordination game, with an unstable threshold and lock-in above it (part 4). That is the self-confirming loop of CASE-099 and CASE-124 in one variable, with the correction threshold in closed form.
  • Where the scores form a potential game, the dynamic is a gradient flow in the Shahshahani metric (CLAIM-369’s tool), the cardinal counterpart of a persistent potential. Their harmonic part, in Candogan’s decomposition (THEORY-894), cycles. That is CLAIM-451’s stable cyclic regime. The ordinal versions are finite improvement paths and ordinal potentials (THEORY-896), as ADR-052 asks.
  • In finite populations every pure state is absorbing, and stochastic-stability theory measures how hard each regime is to leave (CASE-155, limits).

What the reduction loses, and so what the record’s states are for.

  • Interpretation apart from authority. When an organization’s prevailing interpretation is not the authority-weighted mixture, m is state of its own. That happens when interpretations combine nonlinearly (List and Pettit’s agendas), or when meaning persists after authority shifts.
  • Legibility with memory. CLAIM-400’s maintained r, an exponentially weighted history of authority (CASE-156), is a slow state the stateless macro-agent cannot hold. Its signature is a lock-in that outlasts the authority that set it.

So the replicator model is the null model for the record’s governance dynamics. A claim that needs m or r should show a response to intervention that the replicator on authority, with population-dependent scores, does not reproduce at matched complexity (CLAIM-427’s standard).

What it does not say

  • That organizations are populations of imitators. The identification is formal: shares of control play the part of type frequencies.
  • That the mixture condition holds in any actual organization.

Note of 2026-10-24: three conditions the reduction needs

From the readings filed the same day, and the tensions the owner asked about. The reduction to a replicator on authority holds only under three conditions, each rejected somewhere in the record’s account:

  1. Every process is scored on every step. That is full-information feedback. With bandit feedback, only the governing process updated, the dynamic gains a factor x_i and is no longer a replicator on π (THEORY-932).
  2. One standard scores every process. A common scale is what CLAIM-436 and CLAIM-437 deny for perspectives with different concerns. Wolpert and Tumer’s factored utilities (THEORY-939) and monotone value mixing (THEORY-929) give the ordinal, member-by-member condition under which no common scale is needed. That is the regime where the reduction is safe.
  3. Aggregation is share-weighted, not by pairwise contest. With fixed scores the mean score is a potential, so the replicator cannot produce CLAIM-451’s Condorcet cycles. Cycles there need scores that depend on the population.

Even within the reduction, the replicator describes expected motion. Actual runs lock on to pure states (THEORY-938). Access and authority, which CLAIM-349 keeps apart, are one share here, and a share of zero never returns without exploration (THEORY-928).

Note of 2026-10-25: the geometry in which evidence arrives

A28 of the exchange of 2026-10-24, continued, offered this claim’s Shahshahani identity as “the unexpected Amari connection” and “perhaps the mathematically cleanest new synthesis”, without citing it here. It was rechecked: for q = ∇Φ, the replicator is the Riesz representer of dΦ in g(v, w) = Σ vᵢwᵢ/πᵢ, and dΦ/dt = Var_π(q). That metric is the one in which authority moves. The metric in which evidence about authority arrives, when only the organization’s responses are seen, is its pullback through the processes’ response profiles, and is never larger (CASE-159). So the mixture condition has a cost: a redistribution between processes that agree under the current probes is a change of state the probes cannot see. The claim is unchanged.

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