The construction
Keep CASE-099: two hypotheses A and B, the interpretive bias m and the authority bias a, near (m, a) = (0, 0), with
(m_{t+1}, a_{t+1})ᵀ = J (m_t, a_t)ᵀ + noise, J = [[α, β], [γ, δ]].
Change only how the specialists are scored. Each commits a prediction before the evidence arrives. The A-specialist leans toward A by s > 0, the B-specialist toward B by s, and both are bent by the shared meaning:
p_A = s + κ_A m, p_B = −s + κ_B m.
The evidence arrives as e, with mean 0 at the symmetric state, and is read through the shared meaning with weight κ′: r = e + κ′m. Each specialist is scored by squared error, and authority moves with the score difference at rate η:
a_{t+1} = δ a_t + η [ (p_B − r)² − (p_A − r)² ].
Linearizing at m = e = 0:
∂/∂m = −2s(κ_A + κ_B − 2κ′), ∂/∂e = 4s,
so, writing κ̄ = (κ_A + κ_B)/2,
γ = 4ηs(κ′ − κ̄),
and the evidence drives authority with gain 4ηs, which is how the specialist who predicted well gains it.
What it shows
- The sign of γ is the sign of κ′ − κ̄. Committing before the evidence is not by itself Regime B. It gives γ < 0 when the interpretation bends what the specialists predict more than it bends how the evidence is read, γ = 0 when the two are bent equally, and γ > 0 when the reading of the evidence is bent more. CASE-092’s Regime A, scoring against the dominant interpretation itself, is the limit in which the reading is all interpretation.
- With γ ≤ 0 there is no capture. For β > 0 and βγ ≤ 0, J is stable exactly when αδ − βγ < 1, that is, |βγ| < 1 − αδ. When J is unstable in this range its eigenvalues are complex, so the failure is oscillation (over-correction), not capture. With real eigenvalues, λ+ ≤ max(α, δ) < 1. For βγ > 0, CASE-099’s condition βγ > (1 − α)(1 − δ) holds unchanged.
- Revising after the evidence removes learning, not just capture. If the specialists predict after reading the evidence (p = r), the score difference is identically zero. γ = 0, as in Regime B, but the evidence gain is zero too, so authority stops tracking who predicts well. Commitment matters because it keeps the evidence gain 4ηs.
The coupling and the gain were derived with sympy. The stability conditions were checked numerically: over 200,000 random draws of α, δ ∈ [0, 1), β ∈ [0, 3] and γ ∈ [−3, 3], the stated conditions and the eigenvalues agreed in every draw. Among draws with βγ < 0, none was unstable with real eigenvalues. The script is in the contribution’s scratch directory (scratch/formal/check.py).
What it does not show
- That a mind’s predictions and its readings of evidence have weights κ, κ′ that can be measured, or what they are.
- Anything away from the symmetric state, or for the nonlinear bounded system (CASE-099’s own limit).
- That squared error is the right score. Another proper score changes the constants, and the sign argument should be redone for it.