Audit of the one-sided Nahm functional

8 September 2026. Self-audit of the argument; no independent or external review is claimed.

Accepted statements

1. The half-space seed (5) solves the plus recurrence at interior points and both boundary cases. Its minus-recurrence residual is nonzero. The target’s tetrahedron generator requires the plus recurrence only; no target relation was removed to use this seed.
2. Weyl multiplication gives the exact order (15). In particular the input’s first-coordinate pairing with the lattice’s second coordinate cancels. Tests use exact rational values and exact truncated Laurent series, not floating-point fits.
3. Strict positivity of Q on the recession cone suffices on every affine parity fiber. The proof uses compactness at infinity, so positivity only on integer points without a uniform cone statement is not substituted.
4. Necessity includes a rational-null-direction argument. For a nonvertical null ray, the unit might converge but a sufficiently shifted input does not. Vertical null rays contain both signs and fail even at the unit. These are obstructions to this unordered average, not target-vanishing theorems.
5. The projection is injective under the criterion, and isotropy makes its graph form symmetric. Strict copositivity is weaker than positive definiteness; the note does not claim an efficient general copositivity test.
6. Descent uses the full unsaturated image and original sides: H(a f_j)=0 and H((W(mu)-chi(mu))a)=0. It does not assume the quotient is an algebra. An odd-row check demonstrates that odd lattice translates cannot be discarded.
7. Vacuum positivity needs its separately declared exact phase character. A valuation check cannot replace that condition. The pure-w native longitude is a hypothesis, not a permitted renormalization or an arbitrary basis change.
8. Both the Weyl factor and seed shift contribute to the ratio c_l^2 q^(t.x). With positive scalar valuation, the longitude term has larger order than the positive unit series. Complete strict angles imply that scalar valuation is positive under the stated raw-longitude condition.
9. The abstract three-block example has the asserted raw rows, phases, complete angular data, skew pairing, no strict filling angles, and explicit nonzero vacuum. Its missing actual edge and normal-homology data are disclosed. It is not evidence that an additional actual filling has been covered.
10. The local sum reconstructing the ordinary index is a fixed-input identity. Swapping its extra summation with a global moment average would require a new summability proof and is not done.

Scope still open

No additional actual triangulation/filling has been certified. The joint actual realizability of strict cone positivity, the native phase condition, and the native pure-w longitude is the next gate. They may be restrictive or incompatible in some intended classes. A failure of these sufficient vacuum criteria does not invalidate the all-input convergence theorem or imply vacuum vanishing. The earlier odd peripheral functional is not evaluated here. Panitch–Romo Conjecture 7.1 remains open.

Verification boundary

The script checks local identities, exact phases, full-root parity behavior, original-sided global recurrences, image-lattice sensitivity, and the example through a cutoff justified by its exact quadratic degree. The general proof and actual topological applicability are not consequences of the finite checks. No triangulations, slopes, or normal-support configurations were enumerated.