nucleation

CASE-152: Absent attempt or unresponsive receiver: with Y ∼ N(ab, 1), a the strength of an attempted influence and b the receiver's responsiveness, the settings (a, b) = (0, 1) and (1, 0) give the same response distribution; K(a, b) = a²b²/2, the Fisher matrix has rank at most one and vanishes at the origin, and the real log canonical threshold is ½ with multiplicity 2, not the regular d/2 = 1, while changing b with a held fixed separates the two

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Properties10
StatusActive
Standingstipulated
Kindconstruction
Tagsmathematical-statistics, pragmatics, information-geometry
Linepragmatic-transport
SupportsCLAIM-450 — Three standards for a communicative distinction come apart: statistical distinguishability of an interpreter's response profiles, measured locally by the Fisher metric and never raised by a parameter-independent garbling; usability by a bounded receiver, which a transformation can raise; and pragmatic efficacy in the act; and where response models are singular, as Watanabe's theory treats, a missing response can come from an absent attempt or an unresponsive receiver alike, and only intervention or later dynamics separate them
IllustratesCLAIM-450 — Three standards for a communicative distinction come apart: statistical distinguishability of an interpreter's response profiles, measured locally by the Fisher metric and never raised by a parameter-independent garbling; usability by a bounded receiver, which a transformation can raise; and pragmatic efficacy in the act; and where response models are singular, as Watanabe's theory treats, a missing response can come from an absent attempt or an unresponsive receiver alike, and only intervention or later dynamics separate them, CLAIM-407 — A coarse-graining exactly informationally closed under a process's ordinary dynamics need not preserve its responses to an intervention on the mechanism: for a declared intervention distribution ρ, independent of the state, the deficiency Γ = I(S; Y^(I) | T(S), I) can be positive, as in the hidden governance bit, where it is ρ(h = 1) bits; and Γ = 0 exactly when, for every intervention ρ charges, the response kernels are constant on T's fibres over the states of positive probability
VariantsCASE-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
Cited as “grounds” byCLAIM-305 — Resolving a singularity by blow-up reparametrizes the states and adds no observation, so it cannot make indistinguishable states distinguishable: every point over a singular point gives the same distribution, and the Fisher information can stay degenerate there; what the resolution buys is a normal form for the divergence, and so the learning coefficient
Cited as “illustrated by” byCLAIM-188 — Where the model with a candidate structure is singular at the parameters that switch the candidate off, with real log canonical threshold λ below d/2 and multiplicity m, the reification rule's Gaussian Occam term overcharges keeping the candidate by (d/2 − λ) log n + (m − 1) log log n to leading order, so near the threshold the rule prunes candidates the Bayes factor keeps, and its derivation holds only where that model stays regular

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The case

A stipulated toy, not a theory of communication. An observed response Y ∼ N(ab, 1), where a is the strength of an attempted communicative influence and b the responsiveness of the receiving process. The distribution depends only on the product ab.

  • Same distribution. (a, b) = (0, 1), no attempt and a responsive receiver, and (1, 0), an attempt and an unresponsive receiver, both give N(0, 1), and so does every point on the two axes.
  • Divergence. Against the reference N(0, 1), K(a, b) = D_KL = ½a²b².
  • Fisher matrix. The score is (y − ab)(b, a), so G(a, b) = [[b², ab], [ab, a²]]: determinant zero everywhere, rank at most one, zero at the origin. The regular asymptotics, which need G invertible (THEORY-082), fail on the axes.
  • Learning coefficient. For a prior smooth and positive near the origin, ∫ exp(−n a²b²/2) π(a, b) da db ≍ (log n)/√n, so λ = ½ with multiplicity m = 2. The free energy grows as ½ log n − log log n, not as the regular (d/2) log n = log n (THEORY-087).
  • Intervention. Change b with a fixed. At (0, 1) the response stays N(0, 1); at (1, 0) it moves to N(b′, 1). An intervention on the receiver separates what no number of ordinary observations does.

What it can show

That a missing response is compatible with an absent attempt and with an unresponsive receiver, that the nominal parameter count overstates the model’s complexity there, and that the distinction is recoverable by intervention (CLAIM-407).

What it cannot show

That the two settings are different communicative acts. Whether “an attempt that was not taken up” is an act at all is a question for the account of communicative kinds (TERM-040), not for the likelihood.

Note of 2026-10-31: the same λ, different multiplicity

A3 of the exchange “Analyze nucleation concepts” rebuilt this case as gradient flow on L = ½(uv − 1)² and added a contrast, rechecked. With a nonzero target the true set ab = c ≠ 0 is a smooth curve, with λ = ½ and m = 1. At the target 0 the true set is the crossing ab = 0, with λ = ½ and m = 2. So λ alone does not identify the geometry of the optimal set; the multiplicity separates them. A14 gave a variant for a three-factor probe Y ∼ N(q²c₁c₂, σ²) on a box. When the true response is zero, λ = ¼ and m = 1, from the q⁴ term. When it is positive, λ = ½. A15 gave the Bernoulli version, σ(uv), with the same pair of values as this case. The singularity is the probe’s and the parameterization’s, as this case already says (“What it cannot show”).

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