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CLAIM-452: For an organization whose response in each probe context is the authority-weighted mixture of its processes' known response profiles, the authority shares are identifiable from a set of contexts exactly when the processes' stacked profiles over those contexts are affinely independent; because the response is linear in the shares, local and global identifiability coincide, the condition is that the Fisher information of any design supported on those contexts is positive definite on the simplex's tangent space, and with binary outcomes at least one context fewer than there are processes is needed

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Properties8
StatusActive
Rolethesis
Warrantdemonstrated
Tagsmathematical-statistics, self-governance
Lineformal-apparatus, distributed-agency
AnswersQUESTION-272 — When the task's target and the probes' outcome laws are both linear in one unknown state, as for the pairwise comparisons of actions under partial monitoring and for the authority shares of a mixture of known response profiles, when does a family of probes separate what the task needs told apart?
GroundsCASE-159 — Authority geometry and response geometry: when an organization's response is the authority-weighted mixture p_π(y | c) = Σ πᵢ pᵢ(y | c), the Fisher metric of its responses is the pullback JᵀD⁻¹J of the categorical (Shahshahani) metric on the shares π and never exceeds it; it vanishes along any redistribution between processes that respond alike under the probes used, so two specialists who both answer yes with probability 0.8 in routine contexts carry no information about their shares there, while a rare context in which they answer 0.9 and 0.1 carries 0.64/(p(1 − p)) per observation, p = 0.1 + 0.8π, THEORY-956 — In the normal linear model with known error variance and a conjugate normal prior of precision R/σ², the design that maximizes the expected Shannon information of the posterior is the one maximizing det(nM(η) + R), Bayes D-optimality, and the design that minimizes expected quadratic loss with weight matrix A is the one minimizing tr A(nM(η) + R)^-1, Bayes A-optimality; both criteria depend on the design and not on the data, so in this model a sequential design gains nothing over the best fixed one, THEORY-954 — When the information a design yields depends on the unknown parameter, as in nonlinear models or nonlinear functions of linear ones, Bayesian design criteria are normal approximations to expected utility that average a local criterion over the prior, such as the expected log det of the Fisher information, and locally optimal design is the special case of a point-mass prior; a locally optimal design can leave the quantity of interest inestimable, as for the turning point of a quadratic regression on [−1, 1] with best guess ½, where the locally c-optimal design puts half its mass at 0 and half at 1 and identifies only θ₁ + θ₂
ComplementsCLAIM-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

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The claim

Let k processes have known response profiles pᵢ(·|c) over the outcomes of each context c in a set C, and let the organization’s response be the mixture p_π(·|c) = Σᵢ πᵢ pᵢ(·|c), with π in the simplex Δ_k (CLAIM-444’s reduction). Then:

  1. Identifiability. π is identified from the response laws on C iff there is no nonzero v with Σᵢ vᵢ = 0 and Σᵢ vᵢ pᵢ(·|c) = 0 for every c ∈ C. Equivalently, the stacked profile vectors (pᵢ(·|c))_{c∈C}, i = 1, …, k, are affinely independent.
  2. Local is global. p_π is linear in π, so two shares π, π′ give the same laws iff v = π − π′ satisfies the condition in 1, and the condition does not depend on where in the simplex π lies.
  3. The Fisher form. For any design ξ with support C and π in the interior of Δ_k with every response probability strictly between 0 and 1, the Fisher information G_ξ = Σ_c ξ(c) G_c of CASE-159 is positive definite on the tangent space {v : Σvᵢ = 0} iff the condition in 1 holds. G_c is the pullback of the response law’s Fisher metric through the linear map v ↦ Σᵢ vᵢ pᵢ(·|c), so v is in its kernel iff that map sends v to 0.
  4. A count. With binary outcomes each context contributes one dimension (pᵢ(1|c); pᵢ(0|c) is fixed by it), so the k − 1 dimensions of the tangent space need at least k − 1 contexts.

Proof

1 and 2: the laws on C agree for π and π′ iff Σᵢ (πᵢ − π′ᵢ) pᵢ(·|c) = 0 for all c, and π − π′ ranges over the nonzero sum-zero vectors as π ≠ π′ range over Δ_k (for any sum-zero v, π + tv stays in Δ_k for small t when π is interior, and the condition is linear in v). 3: G_c(v, v) = Σ_y (Σᵢ vᵢ pᵢ(y|c))² / p_π(y|c), a sum of nonnegative terms that vanishes iff Σᵢ vᵢ pᵢ(y|c) = 0 for every y; summing over c with ξ(c) > 0 gives 1’s condition. 4: the tangent space has dimension k − 1 and each binary context is one linear constraint.

How it was checked

20,000 random mixtures with k from 2 to 5 processes and 1 to 6 binary contexts, with profiles uniform on [0.05, 0.95], 30% with a duplicated profile and 20% with one profile the midpoint of two others. In every case the affine condition and the positive-definiteness of G_ξ under the uniform design agreed, and no case with fewer than k − 1 contexts was identified. In CASE-159’s three-process example, the two contexts that the D-optimal design weights (½ each) identify π, and the two that it leaves out do not.

What it does not say

  • Not unknown profiles. The profiles are known. With profiles also estimated, identification is of a mixture model in the usual sense and can fail where this succeeds.
  • Not accessibility. It says which contexts would identify π, not that they can be run. A probe the organization never meets is an action outside the probe set, as in THEORY-962’s partial-monitoring terms (CLAIM-446’s note).
  • Not how well. Identifiability is all or nothing. How precisely a design estimates π is the design criterion’s question (THEORY-956, THEORY-954), and depends on π because the problem is nonlinear in Chaloner and Verdinelli’s sense.
  • Not beyond the mixture. If the organization’s response is not the authority-weighted mixture, as when legibility keeps memory (CLAIM-444’s conditions), the condition is for the null model only.

Note of 2026-10-27: the edge says what the claim uses

The status review of this date (ADR-057, ADR-058) changed rests_on: CLAIM-444 to complements. The theorem takes the authority-weighted mixture as its hypothesis, stated by CASE-159 (Active, a ground), and its proof does not use CLAIM-444’s claim that the replicator reduction is the right null model.

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