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CASE-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)

3 min read

Properties8
StatusActive
Standingfictional
Kindconstruction
Tagscomplex-systems, self-governance
Linedistributed-agency
SupportsCLAIM-330 — In the coupled meaning–influence model linearized at its symmetric state, with each field stable alone and nonnegative couplings, the coupling destabilizes symmetry exactly when βγ > (1 − α)(1 − δ): only the loop can do it, and the direction that grows mixes meaning and authority, CLAIM-338 — The selectivity active inference gives action, sampling what one expects to experience, is played in the coupled meaning–authority model by the authority space: influence decides which operators retrieve and so what the next judgement observes, which makes Friston's self-confirming sampling the model's authority-to-interpretation arrow and his agent a coupled loop run without independent scoring, CLAIM-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
Variant ofCASE-092 — A minimal coupled meaning–influence model: two hypotheses, two retrieval specialists, a fixed balanced corpus, queries set by the dominant interpretation and endorsement updated from apparent retrieval success
VariantsCASE-106 — The two-hypothesis linearization with endorsement scored on committed predictions: when the shared meaning bends the specialists' predictions with weight κ and the reading of the evidence with weight κ′, the coupling of interpretation to endorsement is γ = 4ηs(κ′ − κ̄), CASE-123 — The regulator and two accounts: a formally independent regulator weighs two parties' accounts of a risk, one party comes to supply more of the evidence it reads and the experts it trusts, and the credibility that party gains decides whose evidence is read next; with variants for a one-time perturbation, a press as the evaluator, and an objective that gives one property no weight, 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, 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

Graph View

The construction

CASE-092 has two hypotheses, A and B, and two retrieval specialists, one better at finding evidence for each. A14 §2 reduced it to two numbers: m ∈ [−1, 1], the prevailing interpretive bias toward A, and a ∈ [−1, 1], the relative authority of the A-specialist. Near the symmetric state (m, a) = (0, 0) it wrote

(m_{t+1}, a_{t+1})ᵀ = J (m_t, a_t)ᵀ + (ξ_t, ζ_t)ᵀ, J = [[α, β], [γ, δ]],

with α the persistence of the meaning distribution, δ the persistence of authority, β the effect of authority on interpretation (the A-specialist’s retrievals push meaning toward A) and γ the effect of interpretation on endorsement (in CASE-092’s Regime A, operators are scored against the dominant interpretation, so a lean toward A raises the A-specialist’s score).

What it shows

The eigenvalues are λ± = ½[(α + δ) ± √((α − δ)² + 4βγ)]. Take α, β, γ, δ ≥ 0 and α, δ < 1, so each field alone is stable. Then λ+ is the larger eigenvalue and, since 2 − α − δ > 0,

λ+ > 1 ⇔ (α − δ)² + 4βγ > (2 − α − δ)² ⇔ βγ > (1 − α)(1 − δ).

A14’s condition is right. Three things follow, and were checked here:

  1. The loop, not either arrow, destabilizes. If βγ = 0 the eigenvalues are α and δ, both inside the unit circle. One-way coupling, however strong, leaves the symmetric state stable.
  2. The unstable direction is mixed. For β, γ > 0 the eigenvector of λ+ is (β, λ+ − α), with both components positive: the mode that grows is a combined lean of meaning and authority, not either alone. That is the object CLAIM-164 says a gap makes definite, a subspace, though A14 noted the caveat that Jacobian eigenvalues are not the singular-value gaps CLAIM-164 is about.
  3. Regime B removes it. Scoring operators by independently assessed relevance sets γ ≈ 0, so by 1 the coupling cannot destabilize the symmetric state, whatever the endorsement memory.

Near the threshold λ+ approaches 1 and its relaxation time τ = −Δt / ln λ+ grows without bound, so the mixed mode is also the slow one: the coupling makes a slow collective variable before it makes an unstable one.

What it does not show

That the nonlinear bounded system has two captured attractors (A14: it “could lead to competing stable interpretive–authority configurations”); that the signs assumed hold in any implementation of CASE-092; or that capture lowers fidelity, which is CLAIM-297’s further claim. It is the textbook instability condition for a positive 2×2 linear system, applied to this reading of the coupling. Fictional by construction.

Note of 2026-10-18: a proposed extension with demands and a feasible-update oracle

A39 §3 of the exchange of 2026-10-17 proposed extending this case as its “case study 2”: “The new experiment would add explicit competing demands and a feasible-update oracle. It would ask whether interpretation– endorsement feedback can make a system persistently fail to use an available satisfactory update.” The oracle computes the residual Φ of CLAIM-365, so the extension would separate structural from governance-induced frustration (CLAIM-357) in this model. Its key contrast, “endorsement based on independent performance evidence versus endorsement scored through the interpretations whose authority is being evaluated”, is not new: it is Regime B against the γ arrow here, and CLAIM-342’s and CLAIM-297’s thesis. A39 called it “the best candidate for an early discriminating experiment”. A40’s Study C restates it. Not run. The dynamics it would need are QUESTION-124’s.

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