Book 3 · The Mini-Beast · Chapter 19 of 44
Neural · Critical Curvature in Phase Space
When cognitive trajectory curvature crosses κ*, the brain commits.
κ_neural ≥ κ*
OrientationWhen the Brain Commits
A cognitive trajectory that curves sharply enough stops being reversible. That is the claim, and in this room it is unusually close to something people already report from the inside: the moment a decision stops being under consideration and starts being made.
The threshold is the same object as everywhere else, computed on the coherence manifold of the previous chapter.
Phase Transitions in Attention and Decision
Attention
Attentional selection has the shape of a fold: a graded competition resolves into a discrete winner, and the resolution is faster than the competition that preceded it. In dm³ terms the competition is the K phase and the resolution is F. The measurable prediction is that the curvature trajectory is monotone through the competition and crosses the band immediately before the resolution — not simultaneously with it.
Decision
Evidence-accumulation models already describe decisions as a boundary crossing, and they fit behavioural data well. The dm³ account is not in competition with them; it makes a different, additional claim — that the boundary is geometric, that its location is κ*, and that the same number governs immune clonal expansion. That is the part that can be wrong, and it is the part worth testing.
What would distinguish the accounts
An accumulation boundary is a property of the decision variable; κ* is a property of the manifold. If the threshold is genuinely geometric, it should be invariant under changes to the decision variable that leave the coherence structure alone — a different response modality, say. If it moves with the decision variable, it is an accumulation boundary with extra vocabulary.
The sharpest available test
- Curvature crosses 0.25–0.35 before the behavioural commitment, not at it. Simultaneity is uninformative; a crossing after commitment falsifies.
- κ* is invariant across response modalities for the same cognitive task.
- Seizure onset accelerates passage through κ* relative to ordinary state transitions.
A negative result here is worth as much as a positive one and is easier to obtain. The corpus’s standing practice is to publish it either way — see WP-24, which refuted an algebraic bridge in this author’s own material, in public, with the failing symbolic computation printed.
BridgesWhere This Connects
- WP-24 · Auditing the Criticality BridgeA bridge in this corpus that did not survive checking. The template for how to report a negative result here.
- Ch 4 · The Critical Curvature Threshold κ*Where the threshold is defined and the universality claim is made explicit.
- Ch 7 · The Four Biological OrbitsThe immune orbit crosses at κ* ≈ 0.15 — the comparison that makes the invariance claim testable.