Internal audit: actual one-sided Nahm vacuum

8 September 2026. This is a self-audit, not an independent agent audit or external specialist review.

Algebraic checks

• Both actual edges are included. Their collapsed rows are opposite, but their scalars are 1 and A^4. The resulting characters are exactly inverse; dropping the scalar of the second edge would change the target.
• The source meridian and canonical longitude are raw combinatorial rows. The source’s SL2 eigenvalue lift is not silently substituted for the native scalar B^(prime sum). The declared orientation gives Omega(a,d)=-2 and the complement is therefore qr-pt=1.
• The actual unsaturated image is Z h + Z h_s. Its even intersection is derived before introducing graph coordinates. For p divisible by four, Gamma is {x in 2Z^2 : x2-x1 in 2q Z}, not the full integer lattice.
• The cone computation is symbolic in p and q. Strict copositivity, not positive definiteness, is the required condition. At p/q=-4 an axis is null; below it an axis is negative. The q=0 vertical direction is kept as a detector obstruction.
• The native phase condition is proved for the full even image. Neither the moment character nor the longitude scalar is altered to make a sign argument work.
• The opposite-phase lemma compares coefficient signs, not valuations alone. The longitude-square support can differ from the unit support. Every supported longitude term has the same negative coefficient phase, so equality of supports is unnecessary.
• The generic coefficient-field embedding into Laurent series is injective. A nonzero value in that field extension proves the original k-vector-space class is nonzero. This is not a specialization at a single numerical A.
• The sector is a quotient of the native six-root object. Nonzero image of the specific vacuum detects that vacuum upstairs; no reverse lifting of arbitrary sector theorems or injectivity of sector projection is asserted.

Series and original relation sides

The verifier derives the +4 sum from raw full-image Weyl multiplication independently of the two displayed Nahm formulas. It checks S f_j, both (c_e W(h_e)+A^2)S relations, and the filling relation. A supported input with odd w coordinates is included; vanishing of unsupported odd z inputs is not mistaken for absence of all odd-root support.

For a fixed supported input, a=(a1,a2)>=0 gives the exact exponent

D=2(a1^2+a22-a1*a2)+L.a+c,

with L=(2+2alpha2-4beta1,2alpha1-4beta2) and c=2(alpha.beta-alpha1*alpha2)+2alpha2. The inequality D >= (a1^2+a22)/2 - ||L||^2/2 + c supplies every finite coefficient cutoff. No numerical convergence heuristic is used.

The old-index ray is -2h+h_s=(-4,0;0,-2), with character exactly one. Both ordinary-index factors have constant coefficient one for every positive ray multiple. The failure is unordered summability, irrespective of hypothetical cancellation after a prohibited regrouping. The new seed excludes that ray by support.

Scope checks and useful failures

• The datum is the single published two-tetrahedron figure-eight triangulation. The infinitely many slopes are handled symbolically; no topology or slope census is performed.
• The family is sufficient, not a classification of surviving fillings. Failure of its phase or cone certificate does not imply a zero target or a zero filled vacuum.
• The +4 case lacks strict filling angles and defeats the original-character ordinary-index average. We have not shown that it lies outside all prior regular-intersection or clean-excess criteria.
• The named-trace corollary depends on the retained generic local-collar comparison and vacuum reconstruction. The source’s conflicting wording in the earlier CK comparison remains disclosed in that publication; this note does not erase the caveat or re-audit that whole publication.
• No triangulation independence or new surgery invariant is proved. Panitch–Romo Conjecture 7.1 is universal over ideally triangulated oriented manifolds and fillings; this one-datum family does not resolve it.
• No priority search establishes novelty of this application. This is an ordinary research note, with no qualifying author outreach.

Next decisive obligation

Remove the unit-phase obstruction through a genuine alternative local solution or a controlled combination with the same exact moment character and proved convergence. Do not simply modify native signs. Alternatively, identify the sign certificate intrinsically in actual raw lattice/peripheral parity data without enumerating triangulations.