nucleation

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

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Properties10
StatusActive
Version2
Standingfictional
Kindconstruction
Tagscomplex-systems, self-governance
Linedistributed-agency
SupportsCLAIM-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-448 — An ordinal potential describes only the winning-move relation it is computed on; where an organization's legibility regime selects which contests are admitted or recorded, an acyclic record, even a persistent potential, can be the product of selective contestation, so observed coherence is not evidence that the organization's perspectives are integrated, and an intervention that improves its self-governance may at first increase the disagreement observed, CLAIM-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
IllustratesCLAIM-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-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-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

Graph View

The case

CASE-099’s linearization at the symmetric state, with a third coordinate:

m_{t+1} = α m_t + β a_t + ν r_t
a_{t+1} = γ m_t + δ a_t + μ r_t
r_{t+1} =         ε a_t + λ r_t
  • m is the meaning bias and a the authority bias, as in CASE-099, with β authority shaping interpretation and γ interpretation shaping endorsement (CLAIM-330).
  • r is the lean of the legibility regime toward one interpretation: which categories exist, what is recorded, which sources are channels. It is QUESTION-253’s “third, slower variable”.
  • ε ≥ 0: authority moves the regime.
  • ν ≥ 0: the regime moves what is interpreted (the route r → m, proposed in the owner’s review of 2026-10-21).
  • μ ≥ 0: the regime moves endorsement directly: what it records or scores counts toward authority whatever the reading. This is not the question’s reading of r as setting CLAIM-340’s κ′. There γ = 4ηs(κ′ − κ̄), so a regime that only varies κ′ (κ′ = κ′₀ + c r) adds 4ηs c r m to the authority update, a product that is second order at symmetry and enters no linearization. The μ arrow is the linear, additive effect a regime has when it changes the scoring itself.
  • 0 ≤ α, δ, λ < 1: each variable is stable alone, as in CLAIM-330. λ is the regime’s persistence; λ near 1 is a slow regime.

Witness 1: the loop stable, the system not

The review’s parameters, with μ = 0: α = δ = λ = 0.5, β = 0.1, γ = ν = ε = 0.5.

  • Two-variable system (m, a): spectral radius 0.7236 (= 0.5 + √0.05). βγ = 0.05 < (1 − α)(1 − δ) = 0.25, so it is stable.
  • Three-variable system: spectral radius 1.0333. Unstable.
  • The determinant of I − J₃ is (1 − λ)[(1 − α)(1 − δ) − βγ] − νεγ = 0.5 · 0.2 − 0.125 = −0.025 < 0.

Witness 2: stable, with a lean that outlasts

α = δ = 0.5, β = 0.1, γ = 0.5, ν = 0.02, μ = 0, ε = 0.1, λ = 0.95.

  • Eigenvalues 0.2798, 0.7139, 0.9563. Stable.
  • From (m, a, r) = (1, 1, 0), at t = 40 the state is (0.0056, 0.0062, 0.0977). Without r, the two-variable system decays at rate 0.7236, and (m, a) would be of order 10⁻⁶ by t = 40.
  • The impulse response of m to a shock in a has a 6×6 Hankel matrix of rank 3 (singular values 0.316, 0.0198, 0.0044, then 10⁻¹⁸). No linear two-state system has that response: the projection on (m, a) is not a closed first-order two-variable system.

Checked by symbolic determinant and numerics on 2026-10-21.

What it can show

That a regime which persists and is moved by authority behaves, at the instability threshold, exactly like stronger two-variable couplings, and away from the threshold like something the two-variable model cannot represent (CLAIM-400).

What it cannot show

  • That any institution’s categories have such dynamics, or that legibility must be modelled as a state. If the regime is an instantaneous function of m and a, it folds into β, γ, δ and the two-variable model suffices.
  • Anything beyond the linearization: no claim about nonlinear lock-in, bistability or the size of an advantage once it grows.
  • That the lean is wrong. r measures a lean toward one interpretation, not a distortion; tracking a real difference is a lean too.

Note of 2026-10-24: an observational semantics, and three neighbours

The owner: “i think we already have a three variable system somewhere, where the third component is described as legibility” (workbench entry of 2026-10-24, U18). The exchange made this case the core of its toy-model programme for protective lock-in, in place of a new two-state model (CASE-153). Additions and cautions:

  • A mechanism for ν. Selective admission of evidence gives ν = η_m χ σ′(c) and reads r as an exponentially weighted history of authority (CASE-156).
  • Two levels of legibility. TERM-064’s legibility is a relation of a representation, an observer’s capacities and a task. This case’s r is a signed lean of a maintained regime, not a score of how much information the organization has; a positive lean can help or harm. An observation kernel Z ∼ K_r(· | Y), with usable information measured as best risk for a bounded observer on a task fixed independently, would give r that semantics (CLAIM-199, THEORY-846). A regime can serve one task and disable another for the same observer (CASE-154).
  • An organizational baseline. March’s mutual learning of code and members (THEORY-142) already shows conformity destroying the diversity needed for later correction. The informative contrast is a minority that stays present and accurate but is not credited because evaluation runs through the prevailing regime (CASE-144’s note of 2026-10-24).
  • Performative feedback. A policy can change the world sampled as well as how it is read (CLAIM-402’s note of 2026-10-24).
  • An independent standard. Adaptation is judged against fixed reference demands and a feasible set (CLAIM-357, CLAIM-361), so a model cannot call its own persistence success.

Head-to-head results belong to QUESTION-100. A reported mean-field run is recorded, unchecked, in CASE-156.

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