nucleation

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

6 min read

Properties10
StatusActive
Version3
Standingstipulated
Kindconstruction
Tagslearning-theory, self-governance, game-theory
Linedistributed-agency
SupportsCLAIM-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, CLAIM-453 — A population of perspectives, each advocating one of two interpretations, makes its collective choice by drift diffusion when the coordinate the decision is read from integrates independent evidence with no net leak or self-reinforcement and commitment is a bound on that coordinate; two interpretations make the reduction one-dimensional, but they do not by themselves give it
IllustratesCLAIM-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, CLAIM-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
Variant ofCASE-099 — The two-hypothesis coupled meaning–influence model linearized at its symmetric state: a meaning bias m and an authority bias a, each stable alone, coupled by authority shaping interpretation (β) and interpretation shaping endorsement (γ)
VariantsCASE-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-162 — Capture and selectively admitted evidence in two lab realizations: in a four-variable model of authority, record and competence the equilibrium fold under self-scoring is at c ≈ 0.4247 while the declared history is trapped from c ≈ 0.4469, and in a 32-member event-level population, admission of cues that depends on the record reverses the sign of the admitted evidence and leaves the worse alternative in charge in 41 of 64 worlds, which exact inverse-propensity weighting brings to 0 of 64 below full selectivity

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The case

An organization’s processes a = 1, …, n hold shares π_a of practical authority, a point on the simplex. Authority passes between them by a revision protocol of the kind THEORY-899 treats: in the weighted-voter form, each unit of control goes to a process with probability proportional to the score it has just earned. With q_a the expected score and v = Σ π_b q_b, the mean dynamic is the population’s.

1. Evidence-tracking authority is Bayesian. The expected Maynard-Cross update is

π_a + π_a(q_a − v)/v = π_a q_a / v.

If q_a is the likelihood process a assigns to the evidence just observed, this is Bayes’ rule over the processes as hypotheses. Authority allocated by success-weighted recruitment is then, in expectation, Bayesian model averaging. This was checked numerically when filed. It is the replicator dynamic read as inference, which Harper and Shalizi have made before for other purposes.

2. Speed, not direction. In a single population the Maynard dynamic is the Taylor dynamic divided by v > 0, a scalar, so the two trace the same orbits at different speeds. Which protocol the organization uses changes how fast authority settles, not where. The equivalence fails for coupled populations, each with its own v.

3. Climb rate is variance. For fixed q, along the Taylor dynamic dv/dt = Σ q_a π_a(q_a − v) = Var_π(q) ≥ 0, which is Fisher’s fundamental theorem. Mean quality improves only as fast as the processes differ, and not at all where they agree. This is cardinal: q must be a score whose differences mean something (ADR-052).

4. Authority-dependent scoring gives lock-in. Two processes, A and B. Let ℓ be the log-odds of A’s share π, h the log likelihood ratio per step of evidence scored independently (h < 0 favours B), and c > 0 the part of A’s score advantage that grows with A’s own share. That is the CASE-156 filter in its simplest form, a scoring channel that admits more of what favours the process already in charge. Then

ℓ′ = ℓ + h + c(2π − 1).

The increment rises with π, so the one interior rest point, π* = (1 − h/c)/2, which lies in (0, 1) when |h| < c, is unstable. Below π*, B wins as the evidence says. Above it, A’s share goes to one against the evidence. With h = −0.2 and c = 1, π* = 0.6: runs from 0.55 and 0.59 went to B, and runs from 0.61 and 0.65 to A, checked by script when filed. The threshold is a correction threshold. An intervention that leaves h and c alone works only if it pushes A’s share below π*. Raising |h|, more diagnostic evidence, or lowering c, independent scoring, moves the threshold, and |h| ≥ c removes the lock-in altogether.

What it can show

That the record’s evidence-tracking authority has an exact form (1); that protective lock-in (CLAIM-446) needs no bias in any process, only scoring that depends on who governs (4); and where a correction has to reach.

What it cannot show

  • Anything about finite populations. There every pure state is absorbing, which is the absorbing lower state of TERM-078’s revision, and leaving needs mutation or exploration. Stochastic-stability results (Young; and Kandori, Mailath and Rob), named here and not registered, would give the escape time from each regime, of order exp(N × barrier). That is the next step, not derived here.
  • A separate meaning or legibility state. Here belief and authority are one vector (CLAIM-444).
  • That any organization’s scoring has this form.

Note of 2026-10-24: what the learning-in-games readings change

Read the same day (curation entry of 2026-10-24, 2nd).

  • Expectation, not runs. Börgers and Sarin (THEORY-938, THEORY-934) show that Cross learning follows the replicator dynamic in expectation, and in actual runs only over finite horizons as learning slows. At every learning speed, a learner ends on one pure profile with probability one, matching pennies included. So part 4’s lock-in has a cousin that needs no self-scoring c at all: reinforcement of whatever was chosen, with no way back.
  • Bandit feedback breaks the reduction. Where only the governing process’s score is updated, as in plain Q-learning, the expected change carries an extra factor x_i (THEORY-932). The dynamic is then not a replicator on π. Part 1’s Bayes reading assumes every process is scored on every step, BoostTheVote’s full-information case.
  • Forgetting is exploration. In Sato and Crutchfield’s softmax learners with decaying memory (THEORY-928, THEORY-936), the forgetting rate becomes an entropy term pulling toward equal shares. The rest points are then logit equilibria, and the edges stop being absorbing. That is the principled form of the mutation this case left as a next step.
  • α-Rank cannot express part 4. Its fitness does not depend on a strategy’s share (NOTE-1156).

Note of 2026-10-25: two processes as a drift-diffusion decider, and the conformity arm

  • The SPRT walk. For two processes, part 1’s Bayes form gives ℓ′ = ℓ + log(q_A/q_B) for the log odds of A’s share. With q the likelihood each process gives a fresh observation, the increment is that observation’s log likelihood ratio h. That is Wald’s sequential probability ratio walk (Wald’s Z_m, THEORY-952; starting from the prior log odds, it is his §3.1 posterior form; the SPRT proper adds fixed bounds), and its continuum limit is THEORY-068’s drift-diffusion model (not stated by Wald; his overshoot-free formulas are exactly the diffusion’s, THEORY-951). For Gaussian evidence N(±δ, σ²), h has mean 2δ²/σ² and variance 4δ²/σ², so the drift is half the variance. Part 4’s ℓ′ = ℓ + h + c(2π − 1) is the same walk with feedback from the share. Since 2π − 1 = tanh(ℓ/2), it is ℓ′ = ℓ + h + c·tanh(ℓ/2), whose linear part (c/2)ℓ is the self-reinforcement of A35’s dial dL = [μ + (β − λ)L]dt + σ dW. With forgetting that shrinks ℓ by (1 − α) per step (a form suggested by the Sato–Crutchfield note above, assumed here and not derived from THEORY-928), the symmetric state loses stability at c = 2α. In a finite population of N, resampling adds noise of order 1/(Nπ(1 − π)) to the log odds, the voter-model term. Checked when filed, with δ = 0.3 and σ = 1: increment mean 0.178 and variance 0.360 against 0.18 and 0.36; the logit variance of a binomial share matches 1/(Nπ(1 − π)) at N = 100 and 1,000; and the forgetting threshold at α = 0.1 between c = 0.15 and c = 0.25. This answers the owner’s U34 hypothesis of the exchange of 2026-10-24, continued, inside the record’s own model (CLAIM-453). The exchange did not see it, though U28 had asked for this material.
  • The conformity arm. A47’s conformity scoring, which rewards agreement with the prevailing collective proposal, is part 4’s self-scoring c, and it held the old allocation after a reversal of expertise, as part 4 predicts (CASE-158; reported, not reproduced).

The case is unchanged.

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