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CLAIM-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

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Properties11
StatusActive
Rolethesis
Warrantdemonstrated
Tagscomplex-systems, self-governance
Linedistributed-agency
AnswersQUESTION-241 — In the coupled meaning–influence model linearized at its symmetric state, when is the symmetric state unstable, and does scoring endorsement independently of the interpretation remove the instability?
GroundsCASE-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)
SupportsCLAIM-342 — What keeps a judgement–observation loop from confirming itself is evidence whose arrival or scoring does not pass through the interpretation it tests; action is one source of such evidence, not a necessary one, and a loop that selects its own evidence can lock in with action available, CLAIM-340 — Scoring operators on predictions committed before the evidence is not by itself Regime B: in the linearized operator model the coupling of interpretation to endorsement has the sign of κ′ − κ̄, so commitment removes capture only when the interpretation bends the reading of the evidence no more than it bends the predictions, and it keeps the learning that revising after the evidence would lose, CLAIM-404 — An organization can be governed from outside without any of its decisions being made for it, through the evidence its own procedures receive or through the criteria by which they weigh evidence; the two routes are distinct and each is possible without the other, so informational and evaluative independence do not imply each other; and where the organization's role is to evaluate another that affects its evidence or resources, the interpretation–authority loop can close between the two institutions, so that formal independence of the evaluator's procedure does not make its evaluation independent, CLAIM-406 — Structural, pre-emptive testimonial injustice has the form of the record's trust loop: an arrangement that decides whose word is sought can keep an early exclusion going through the indicators of competence that only being sought produces, with no continuing prejudiced act; but the loop alone cannot tell perpetuated privilege from merited expertise, because that needs the prejudice condition the dynamics do not contain, 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, CLAIM-429 — Where interpretation decides the occasions for cooperation and cooperation's outcomes revise interpretation, an intervention can change the loop's state or its coupling: a favourable exception changes the state and leaves the rule that judges exceptions as it was, while a change in how far outcomes may revise the governing interpretation changes the coupling, and where the coupling does not depend on the state, only the second changes how later evidence is taken
ComplementsCLAIM-297 — When an operator's authority is updated by how well it performs on the interpretation it helped make dominant, the coupled meaning and influence of an organization can amplify an initial interpretive advantage without any gain in fidelity to independent evidence, CLAIM-357 — The frustration residual and governance regret separate structural infeasibility, realized violation and minimax suboptimality, and identify none of them as thwarting: a positive residual says no admissible update meets every declared requirement, a positive regret only that the actual update's worst residual exceeds the best achievable, which can hold while every requirement is met, and actual violation needs regret above the negative residual; aggregate performance does not tell infeasibility from realized violation, and the multipliers that certify infeasibility need not be the weights governance assigns, CLAIM-408 — In ḋ = (α − κ)d − βd³, with d the difference between two parties' shares of authority, α their self-reinforcement and κ a constraint that bears on both alike, the equal allocation is the only equilibrium and is stable for κ > α, and for κ < α it is unstable and the unequal allocations ±√((α − κ)/β) are stable: a constraint that favours neither party can, by weakening, destabilize the equal allocation and make persistent unequal allocations stable, of any size near the threshold (|d*| → 0 as κ → α), without determining which party gains the larger share, and in this model restoring κ > α restores the equal allocation
Illustrated byCASE-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
Cited as “complements” byCLAIM-500 — The governance-induced kind of frustration is not idle: some failures of an organization's actual updates to meet demands that an admissible update would meet are explained by how influence is allocated, and not by step size, curvature, noise, conditioning or a fixed weighting rule

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The claim

The construction and its proof are CASE-099. In brief: with α, β, γ, δ ≥ 0 and α, δ < 1, λ+ = ½[(α + δ) + √((α − δ)² + 4βγ)] > 1 exactly when βγ > (1 − α)(1 − δ). When βγ = 0 the matrix is triangular and its eigenvalues are α and δ. When β, γ > 0 the leading eigenvector is (β, λ+ − α), with both components positive.

For QUESTION-100 this says three things about the linearization of CASE-092. Capture, if it starts at symmetry, needs both arrows of CLAIM-297’s circuit: authority shaping interpretation and interpretation shaping endorsement. Scoring operators independently of the dominant interpretation (Regime B, γ ≈ 0) removes the coupling-driven instability, whatever the memory of endorsement. And the variable that grows is neither a meaning nor an authority but a joint lean of both, which is A14’s “combined meaning–authority mode”.

What it does not say

  • That the full model has this linearization: the signs and the size of γ in Regime A are assumptions about an unbuilt model.
  • That the bounded nonlinear system has two captured attractors, or how far it hysteretically holds one.
  • That capture costs fidelity, which is the rest of CLAIM-297.
  • That the mixed mode is an individual or a level. A14: “Feedback instability ⇏ Causal emergence ⇏ Historical individuality.”
  • That it is new mathematics. It is the standard instability condition for a nonnegative 2×2 linear map. Whether reflected-appraisal models already contain the circuit is QUESTION-117.

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