What could actually be measured

Layer II, continued — the critical sharpness, the criterion, and the exact place where universality stops

The mechanics page ended with a potential whose shape is forced but whose coefficient is not. This page collects what follows from that. It is the last of the three theory pages, and it is where the formal layer meets the possibility of measurement.

Where the two prices balance Machine-Checked

The potential has a unique minimum, and at that minimum the two prices are equal:

κ^{*} = α C_{F} / c_{0}, c_{0} κ^{*} = \frac{α C _{F}}{κ ^{*}}, V^{*} = 2 c_{0} κ^{*}

The boundary rests where the price of maintaining it equals the price of the uncertainty it excludes — the balance point between knowing and paying.

The criterion, and three levels that must not be conflated Machine-Checked

With the tension written as σ = σ₀·κ², the value at the least-action boundary is definite. But three distinct statements live here, and collapsing them is the easiest way to overclaim:

Level	Statement	Status
General law	`σ* = α·C_F / C₀`	Machine-Checked
Dimensionless criterion	`σ* = π²α / (3C₀)` in the logistic core	Machine-Checked
Gauge value	`σ* = 1 ⟺ π²α = 3C₀`	Gauge condition

The third line is a normalisation, not a derived fact — which is exactly why the coefficient c₀ stays free on the mechanics page. In the biological anchor, the criterion is what a cortex satisfies at β ≈ 1.

Other structure that survives machine checking

The critical locus branches Machine-Checked

For the Landau form V(φ;r) = −rφ² + φ⁴ the critical locus decomposes, and its projection is a branched cover:

Crit = {φ = 0} \cup {φ^{2} = r /2}

Fibre over $r$ : a singleton for $r \leq 0$ ; three points for $r > 0$ ; the unique branch point at $r = 0$ .

An earlier draft called this "a splitting of limits". No diagram was specified, and without a diagram there is no limit to preserve. The phrase is retained only as a metaphor; the theorem above is the statement.

A conserved charge Machine-Checked Interpretation attached

The generalised force is the negative gradient of the potential, and the energy bracket vanishes identically along the equation of motion. The algebraic core is machine-checked; the reading of it is flagged as interpretation:

Self-continuity — "I am the same I" — is a conserved charge of the boundary dynamics, not an assumption. The content of the self may drift while its being a self is constrained. This is the standard Noether statement for a time-translation symmetry. It is a candidate formal model of boundary continuity, not a proof of philosophical personal identity. Interpretation

An earlier draft dressed this in categorical language it did not need. That wrapper is neither needed nor given, and it is not claimed.

An attractor, with a rate Machine-Checked

For A(σ) = σ/(1+ε) + D with ε > 0 there is a unique fixed point and the error contracts exactly:

σ_{\infty} = D (1 + \frac{1}{ε}), σ_{n} - σ_{\infty} = \frac{σ _{0} - σ _{\infty}}{( 1 + ε ) ^{n}}

What formalisation added. The earlier drafts mentioned only an "assembly efficiency ε". Writing it down forced the sign hypothesis ε > 0, since ε = −1 makes the denominator vanish. The drafts also asserted convergence without a rate; the exact geometric law supplies the rate.

Where universality stops Machine-Checked

This is the sharpest honest statement in the theory, and it is worth stating plainly rather than burying:

Layer	Grade	Holds?
State / intervention	G4 — free / 2-universal	Yes — `Caus(V)` is the free category on `V`; every model factors uniquely through it
Kernel	G4 — free / 2-universal	No — there is no single universal kernel

The failure is machine-checked twice over: once by a separation carrying no numerical premise, and once by the invariant I(F) = C_F·F′(0). This is strictly more informative than an earlier draft's "structural universality principle", which claimed G2 while delivering G0.

What is not claimed here. Nothing on this page describes a measured system. The criterion is a formal statement about the model's own quantities; whether any real agent realises it is the question the experiment exists to settle, and it has not been run. Empirical prediction