nucleation

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

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Properties12
StatusActive
Version3
Rolethesis
Warrantdemonstrated
Tagscomplex-systems, self-governance
Linedistributed-agency, formal-apparatus
AnswersQUESTION-253 — In the coupled meaning–authority model, is authority's effect on what becomes legible as evidence a third channel beside β and γ, a slower variable with an instability condition of its own, or is it already the commitment regime's κ′, how far the prevailing interpretation bends the reading of the evidence?
Rests onCLAIM-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
GroundsCASE-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-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-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
UsesTERM-064 — legibility, of a distinction to an observer for a task: whether the observer, with the rules, categories and resources it actually has, can recover the distinction from what it is given and bring it to bear on the task; a relation among a representation, an observer's rule class, its resources and a task, bounded above by what the source makes possible for an ideal receiver, and directed, since what is legible to a governing institution and what of its governing is legible to others are different relations
Illustrated byCASE-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-156 — A legibility filter as the ν of the meaning–authority–legibility model: in a safe world where 15% of raw cues are alarms, an admission rule P(admit | y, r) = σ(c + χry) with c = 0 and χr = 2 admits 88.1% of alarms and 11.9% of reassurances, so 56.6% of the admitted record is alarms with the world and the sensors unchanged; at symmetry the mean admitted evidence is χσ′(c)r to first order, so ν = η_m χ σ′(c); and r_t = λᵗr₀ + ε Σ λ^(t−1−j) a_j is an exponentially weighted history of authority

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The claim

The system is CASE-124:

J₃ = | α  β  ν |
     | γ  δ  μ |
     | 0  ε  λ |

with all entries nonnegative and 0 ≤ α, δ, λ < 1.

  1. The threshold. det(I − J₃) = (1 − λ)[(1 − α)(1 − δ) − βγ] − ε[νγ + μ(1 − α)]. For a nonnegative matrix, the spectral radius is below one exactly when I − J₃ is a nonsingular M-matrix, that is, when its leading principal minors are positive. Where the two-variable loop is itself stable (βγ < (1 − α)(1 − δ)), instability therefore comes to det(I − J₃) < 0. With μ = 0 this is the review’s form, γ(β + νε/(1 − λ)) > (1 − α)(1 − δ).
  2. The reduction. Dividing by 1 − λ, det(I − J₃) / (1 − λ) = (1 − α)(1 − δ_eff) − β_eff γ, with β_eff = β + νε/(1 − λ) and δ_eff = δ + με/(1 − λ). That is CLAIM-330’s expression for a two-variable system with those couplings, and J₃ is unstable exactly when that two-variable system is, whether or not the original loop was stable. The regime’s contribution is its steady-state gain: authority’s effect on r, times r’s effect, divided by how fast r forgets.
  3. The mode. Away from the threshold, J₃ has three eigenvalues and the projection on (m, a) is not a closed first-order two-variable system. In CASE-124’s second witness, the impulse response of m has Hankel rank 3, and a lean in r keeps m and a off symmetry long after the two-variable system would have returned.

How it was checked

  • The determinant identity and the reduction, symbolically (sympy).
  • The equivalence of “spectral radius > 1” with the reduced two-variable condition, on 200,000 random draws with α, δ, λ uniform on [0, 0.99) and β, γ, ν, μ, ε uniform on [0, 2): no mismatch.
  • Both witnesses numerically.

What it answers

QUESTION-253 asked for “a showing either that the condition reduces to CLAIM-330’s with κ′ replaced by a function of r (the layer is κ′ renamed) or that it allows an instability, or a slow mode, that no choice of the two-variable parameters gives”. The threshold reduces, with the regime’s steady-state gain absorbed into β (the ν arrow) or δ (the μ arrow, a direct effect of the regime on scoring). The question’s own reading, r setting κ′, is not the μ arrow: since γ = 4ηs(κ′ − κ̄) (CLAIM-340), a regime that only varies κ′ adds a term in r·m to the authority update, second order at symmetry, so it moves no linear threshold at all. In that reading the layer is not even κ′ renamed at linear order; it matters only away from symmetry. The slow mode does not reduce. So whether legibility is “renamed” depends on what is asked of the model: for whether symmetry breaks, it is; for how long a lean lasts and how the system returns, it is not.

What it does not say

  • That legibility must be modelled as a state. If the regime is an instantaneous function of m and a, it folds into the couplings and the two-variable model suffices. Whether an organization’s legibility regime has dynamics of its own is empirical, and the record holds no case of it (QUESTION-253’s “What it does not ask”).
  • Anything outside the linearization at symmetry: not lock-in, not bistability, not the size an advantage reaches.
  • That a slower regime is worse. It adds more at the threshold, and holds a lean longer, whether the lean tracks a real difference or not.
  • That the third arrow is the only way to get a third mode. Any third state coupled into the loop would do; the construction’s content is that a legibility regime is one natural reading of such a state.

Note of 2026-10-25: when the slow regime can be reduced

From A29–A30 of the exchange of 2026-10-24, continued. Treating the legibility lean r as slow, dπ/dt = F(π, r) and dr/dt = εG(π, r), suggests the reduced flow dr/dt ≈ εG(π*(r), r). THEORY-771 licenses that when the fast authority dynamics contract to an equilibrium. A30: “the fast authority dynamics may cycle rather than converge to a unique equilibrium. In such cases we cannot automatically replace π by a function π*(ρ).” A valid slow reduction may then need averaging over an invariant distribution, or several attracting regimes. The harmonic part of the scores (CLAIM-444, THEORY-894) and Sato and Crutchfield’s cycling learners (THEORY-936) are where that happens. This claim’s result is a linearization at a stable symmetric state and stands as it is. Borkar’s two-timescale stochastic approximation was named with no work cited, and is not registered.

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