CASE-160: Interacting accumulators on an influence network: if each perspective's continuous support for one of two interpretations follows dxᵢ = [dᵢ − λxᵢ + κ Σⱼ Wᵢⱼxⱼ]dτ + noise, with W irreducible and row-stochastic and ℓ its left eigenvector, then ℓᵀ(κW − λI) = (κ − λ)ℓᵀ, so the ℓ-weighted support L = ℓ·x is exactly an Ornstein–Uhlenbeck process with drift ℓ·d, rate κ − λ and variance ℓᵀΣℓ, and a drift-diffusion process at κ = λ; with tanh coupling and member noise the transition is at κ_c = λ/E[sech² ε], not at κ = λ

The case

M perspectives and two interpretations, H₊ and H₋. xᵢ is perspective i’s support for H₊ over H₋, “a continuous-valued opinion or decision variable; it need not be a calibrated log likelihood” (A36):

dxᵢ = [dᵢ − λxᵢ + κ Σⱼ Wᵢⱼ xⱼ] dτ + noiseᵢ,   Σⱼ Wᵢⱼ = 1,
ℓᵀW = ℓᵀ,  Σᵢ ℓᵢ = 1,   L = Σᵢ ℓᵢ xᵢ.

dᵢ is the evidence drift perspective i receives, λ its leak, κ the strength with which it takes up its neighbours’ support, and W the influence network. Σ is the covariance of the members’ noise.

What follows

1. Exact closure. Since ℓᵀ(κW − λI) = (κ − λ)ℓᵀ, dL = [ℓ·d + (κ − λ)L]dτ + ℓ·noise. The same identity holds step by step for an Euler scheme, so a simulated L is exactly a Gaussian AR(1) whatever the step size. A36: “This reduction is exact for the specified linear system.” It is THEORY-733’s condition for exact dynamical independence: the coarse-graining’s row space is invariant under the transpose of the drift. CLAIM-388’s uniform sum is the case ℓ = 1/M.
2. Drift diffusion at balance. At κ = λ, dL = ℓ·d dτ + σ_eff dB, the standard drift-diffusion model. With κ < λ, L is a leaky Ornstein–Uhlenbeck process. With κ > λ it is unstable, and early evidence is amplified. That is THEORY-068’s net leak, carried from two competing units to a network. The other modes of κW − λI decay at rates λ − κ Re μ_k > 0 at balance, since Re μ_k < 1 for every other eigenvalue of an irreducible W. So the members settle around L with a stationary spread of order σ/√λ.
3. Variance from influence concentration. σ_eff² = ℓᵀΣℓ. Only for independent noises is this Σℓᵢ²σᵢ², and with equal σ it is σ²/M_eff, M_eff = 1/Σℓᵢ². A36’s example, one perspective with weight ½ and 23 sharing the rest, gives M_eff = 3.83. Since max ℓᵢ² ≤ Σℓᵢ² ≤ max ℓᵢ, the collective noise vanishes exactly when the largest influence weight does, which is THEORY-321’s condition (Golub and Jackson) in variance form. A36: “a small effective population doesn’t necessarily mean poor decisions. A highly weighted specialist might be more accurate than everyone else.” Under correlated evidence, as in A43’s typed workspace with its correlation_group, the diagonal formula understates the noise: by a factor of about 12 in a check with pairwise correlation ½.
4. The hidden and the visible tally. In CASE-074’s terms, voters who cannot see the tally integrate independently, the balanced case. A visible tally that draws votes to the leader adds κ above λ. At a fixed bound on L, raising κ − λ above 0 shortens decisions and lowers accuracy. That is the direction of A37’s reported public-tally run (85.4% correct in 49.5 steps balanced, 67.1% in 18.5 with feedback; reported from a sandbox, parameters not given).

The nonlinear version, corrected

A36 §6 replaced κ Σⱼ Wᵢⱼxⱼ by κ Σⱼ Wᵢⱼ tanh(xⱼ) and wrote the mean field as dL = [d̄ − λL + κ tanh L]dτ + …, with a pitchfork at κ = λ. It called this “a plausible mean-field equation”. But Σⱼ ℓⱼ tanh(xⱼ) ≠ tanh(L) when the members disagree. With member noise the members spread about L by ε, the slope of the coupling at the origin is κ E[sech² ε], and the symmetric state loses stability at κ_c = λ/E[sech² ε] > λ. For uniform W and λ = σ = 1, ε has variance ½ and κ_c ≈ 1.38. A36’s κ = λ is the limit of no member noise.

Checked when filed

By script (numpy), M = 24 and a random row-stochastic W:

• the identity ℓᵀ(κW − λI) = (κ − λ)ℓᵀ holds exactly at (κ, λ) = (1, 1), (0.3, 0.9) and (1.5, 0.5);

• at κ = λ = 0.5, 400 runs of 2,000 steps at dτ = 0.01: variance rate 0.0448 predicted and 0.0448 observed; drift −0.0089 predicted and −0.0118 observed, drift being the noisier estimate;

• the regression slope of the increment on L was −0.008, −0.48 and 0.50 against κ − λ = 0, −0.5 and 0.5;

• with pairwise noise correlation ½, the observed variance rate was 0.997 of ℓᵀΣℓ, which was 12.4 times Σℓᵢ²;

• M_eff = 3.833 for A36’s example;

• with drift 0.5, σ = 1 and bounds at ±1 on L, 20,000 runs each:

  κ − λ	0	0.5	1	2
  P(correct)	0.732	0.717	0.708	0.677
  mean decision time	0.96	0.82	0.72	0.56

  against the analytic drift-diffusion value 0.731 at κ = λ;

• for the tanh version, E[sech² ε] = 0.726 for ε ∼ N(0, ½), so κ_c = 1.377. A simulation (M = 50, 200 runs, d = 0) gave a mean |L| of 0.12, 0.16, 0.20, 0.37, 0.99 and 1.77 at κ = 0.8, 1.0, 1.2, 1.4, 1.6 and 2.0.

A37’s own check, “residual variance 1.003 against 1” over 24 perspectives and 200 trials, tests an identity that holds at every Euler step by construction. It verifies the code, not the theory, as A37 itself said: “That verifies the implementation of this benchmark; it does not yet establish the broader organizational hypothesis.”

What it can show

That a collective can be an exact drift-diffusion decider when none of its members is one. That its noise is set by how concentrated influence is. That the condition is a balanced linear mode, and that a public tally is a dial on it.

What it cannot show

• Emergence. A35’s self-audit of its own app applies here too: “we obtain the result because Gaussian additive evidence is built into the update rule”. A36: “a unit test for the population simulator, not an empirical test of our philosophical theory.”
• A decision. The result is the marginal law of L. The collective decision is a drift-diffusion decision only if commitment is a bound on L. A bound on one member’s support, or on a majority count, is not.
• Anything for a readout not weighted by ℓ, such as a uniform average or one member’s support. Those carry a stationary transverse term.
• Anything for a W with several closed classes. Several closed groups give several balanced modes, and L is then one neutral coordinate among several. A W with one closed class and transient members is covered: ℓ is unique and zero off the class, and the closure and the decay of the other modes hold as for an irreducible W (checked, THEORY-960). The title’s “W irreducible” is sufficient, not necessary.
• Settling in discrete time without damping. The closure in part 1 holds for any step. But if the members update as x ← Wx at each step (DeGroot’s iteration, an Euler step with hλ = 1 at balance), the other modes decay only if W is also aperiodic: a periodic two-cycle, W = [[0, 1], [1, 0]], does not converge (checked). With hλ < 1, and in the continuous-time equation, irreducibility is enough. That ℓ is DeGroot’s consensus weight is his Theorem 3 (THEORY-960), and the need for aperiodicity in discrete time is his own statement for a periodic closed class (LIT-1527, §5).
• Anything for binary positions. A37 called xᵢ “each perspective’s binary support variable”. If the members broadcast sign(xⱼ) in place of xⱼ, the closure fails: the drift of L then depends on the members’ positions, not on L alone, and a check at κ = λ gave a nonzero slope.
• Endogenous influence. With W(τ), dL gains (dℓ)ᵀx (A36). That term is where governance enters, reflected appraisal among others (THEORY-916), and it is not covered here.