nucleation

CLAIM-154: A candidate structure is worth naming as a unit when its effect-to-noise ratio outruns the Occam cost of keeping it: reify when d'² exceeds the log prior-to-posterior variance ratio, where the model with the candidate stays regular at the parameters that switch it off

4 min read

Properties11
StatusActive
Version3
Rolethesis
Warrantdemonstrated
Tagsmodel-comparison, complex-systems, individuation
Linedistributed-agency, formal-apparatus
GroundsTHEORY-081 — The evidence of any model formed from a full model by changing only its prior is the full evidence times the full posterior expectation of the prior ratio, so the Savage–Dickey ratio is its point-mass case and choosing which parameters to keep is evidence maximisation over the prior, THEORY-087 — The Bayesian complexity penalty grows as (d/2) log n in a regular model, where the curvature at the optimum sets the Occam factor, and as λ log n in a singular one, where λ is the real log canonical threshold; with the prior positive on the optimal set, λ falls below d/2 when that set is infinite or has a point of degenerate curvature, and equals d/2 when it is finitely many points of non-degenerate curvature
SupportsCLAIM-157 — Relative to a stated target, a combined form is one system while its joint dynamics are nearly decomposable with the form as a block whose gap beats the Occam factor, and it lasts as long as that block persists; the unit is the gap-isolated block, not any member, CLAIM-174 — A candidate factor is an agent at a level when it is one system there, when a goal-directed description beats a device description by more than the Occam cost, and when its policy would adapt if its actions' consequences changed; it is a willed agent when, further, a will-organization over several regimes describes it better than none, and what passes that is the jurisdiction of a will-organization, CLAIM-332 — A loosely organized population can change what an outside institution does through its members' separately motivated acts of making, selecting and passing things on, without forming a collective intention or judgment, when the organized pattern of those acts, and not only the acts of identifiable individuals, earns its place in explaining the change
Objected 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
ComplementsCLAIM-164 — What a spectral gap makes definite is the subspace of the cluster it isolates, not a single direction: the gap picks the object, and the object is a vector only when its own gap is wide, CLAIM-307 — Retrieval pressure and evidence are different criteria for carving a category, and they move in opposite directions with copy count: evidence that a distinction exists grows with its copies, the search a bin saves shrinks with them, so the bins a deadline forces are often the ones the data least supports
UsesTERM-046 — notation for the threshold, drive and unity criteria

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The claim

From MOM-CLM-160. For one candidate direction w, with a Gaussian prior of variance s² and a Laplace posterior with mean μ and variance σ², pruning w is a nested model. Its log Bayes factor is the Savage–Dickey ratio:

ln BF(keep) = ½ μ²/σ² − ½ ln(s²/σ²) = ½ (d’² − Occam), in the notation of TERM-046.

So the rule is: keep (reify) w if and only if d’² > Occam, that is δ/ε > τ with τ² = ln(s²/σ²). The threshold is not a taste parameter. It is the prior-to-posterior information gain about w. In the conjugate Gaussian case the rule is the Bayes factor exactly, not an approximation of it.

Where it holds. The derivation needs a Laplace posterior for w. That needs the model with the candidate to be regular where w = 0: its Fisher information non-singular there, so that no parameter attached to the candidate goes unidentified when the candidate is switched off, and a prior smooth and positive there. Outside the conjugate case the rule is then Laplace’s approximation of the same Bayes factor (THEORY-087’s regular expansion). Two candidates fail the condition: a coupling whose effect is a strength times a responsiveness, and a hidden unit behind an output weight (THEORY-082). There the penalty is λ log n − (m − 1) log log n, and the rule prunes near the threshold what the Bayes factor keeps (CLAIM-188, checked on CASE-152’s model).

Why this record holds it

The re-check of 2026-10-11 used y ~ N(θ, 1) and θ ~ N(0, s₀²). The exact log Bayes factor was computed from the marginal likelihoods of the mean under each model, independently of MOM’s derivation. It agreed with ½(d’² − Occam) to 9 × 10⁻¹⁶ in every setting, with the same keep and prune decisions as MOM’s table.

What it gives this record is the answer to a question that was left as a matter of taste: how much stronger interaction inside a candidate system must be than across its edge before the system earns a name. The owner’s claim that a dancing couple “merits some privileged nomenclature” (CLAIM-175) becomes a decidable question once the coupling is measured. Simon’s near-decomposability (LIT-905) says what to compare, CLAIM-164 says what the named object is, and this claim says how big the gap must be. That bearing is this record’s. MOM makes the same join for its own spectral sectors.

What it does not say

  • That the criterion is free of choices. The prior variance s² and the candidate direction are inputs. The rule prices a candidate; it does not find one. The target-relativity of CLAIM-152 applies.
  • That the magnitude is trustworthy. Away from the threshold the Laplace form can understate the evidence badly (by ten nats in MOM’s logistic case). It is a decision rule.
  • That any couple, firm or person has been measured. Nothing here applies it to a real system.
  • That the threshold survives misspecification. When every candidate model is misspecified, which a block or coupling model of a real system is, Bayes and MDL can prefer the wrong model for large n, and the remedy (Safe Bayes, η < 1) changes the effective Occam cost (LIT-238 §6.3, NOTE-760). The review cites that result and does not prove it. CLAIM-188’s singularity is a limit now written into the title; this is a separate caution.
  • That the Occam term is a description length. Under a modeller’s prior, the evidence penalty matches the description-length penalty only for hypotheses that are random under that prior (THEORY-259, LIT-225, NOTE-801). A hypothesis the prior treats as special is where the two rankings diverge.

Note of 2026-10-12: evidence now linked

THEORY-081 now grounds this claim. The Savage–Dickey ratio is the point-mass case of a general identity for models that differ only in their prior. Its caveat, that Laplace forms are only as good as the full posterior, sits with CLAIM-188.

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