PROOF_AUDIT.md

# Internal proof audit · QÆNTHRIX 3.0.0

This audit covers nine core theorems and 29 supporting propositions. It is an
internal review, not an independent referee report or proof-assistant certificate.

## Core theorem checks

1. **Spectral-gap repair.** Constant rank and compactness provide a uniform
   positive gap. Inverse square roots act on the support and vanish on its
   complement. The repair formula is basis-independent and continuous.
   The square-root bound comes from a Sylvester integral, so no commutativity
   of the two Gramians is assumed. Kernel leakage contributes sqrt(epsilon)
   to the total correction estimate. Gap equality can permit complete failure.
2. **Grassmannian minimax.** For m<r use rank-nullity. For r<=m<n restrict to
   E0 plus a varying line in E0-perp. The complement output bundle has rank
   m-r, but Whitney multiplication forces its degree-(m-r+1) Chern class to
   be nonzero. The projective base dimension n-r is enough for that class.
   This forces a kernel inside the target support, giving error lambda,
   including when the background eigenvalue mu is positive.
3. **Labelled costs.** Each positive output label is a separate orthogonal
   space. Apply the global lower bound to each block and to the pooled sum.
   Tolerances are per-label operator norms. Complete final attenuation is
   allowed; earlier attenuation denominators remain positive.
4. **Reference radius.** Target and reference ranks agree, with 1<=r<n.
   A blind reference has a null domain vector and a missed codomain vector.
   The explicit principal-angle replacement yields a rank-r projector at
   distance exactly sigma_min(P W). Zero and unit margins are handled
   separately. The r=n case has no blind locus and is deliberately excluded.
5. **Frame stability.** The metric is Frobenius norm in the polar inequality.
   Hermitian real parts of contractions give positive trace terms. The
   operator-norm input estimate incurs sqrt(r). The explicit phase family
   proves inverse-margin order, not an asserted optimal constant for the
   full projector-restricted Frobenius problem.
6. **Minimum atlas.** The lower bound uses the complex Pluecker projective
   variety and Krull's height theorem, both classical. Each fixed reference
   contributes one linear equation in exterior coordinates. The upper
   bound uses independent polynomials, distinct leading degrees, a nonzero
   Vandermonde coefficient, and degree <=r(n-r). It therefore works at any
   r(n-r)+1 distinct real nodes. Characteristic zero is essential.
7. **Coordinate conditioning.** Cauchy-Binet supplies the determinant sum.
   Every compression singular value is <=1, so the product is <=the
   smallest value. This yields a uniform certificate, not a general
   optimality assertion for its numerical constant.
8. **Transport.** Every edge has invertible support overlap. The inverse
   square root is support-restricted. Closed products act unitarily on the
   starting subspace, and a frame change conjugates their matrices. The
   spanning-tree argument gives necessity and sufficiency of identity on
   every cycle. In the stated U(2) example the commutator norm is exactly
   162/3481, so scalar transition phases do not suffice.
9. **Decoding.** The repaired ledger is an isometry. Before repair the total
   Gram error is bounded by the sum of label errors. A left decoder exists
   for total error <1; its noise bound follows from the least singular
   value. The one-dimensional example attains that bound. This does not
   remove the lower bounds on whether the assumed factors can exist.

## Supporting assumptions

The chronological word is fixed and known and reserves start empty. Prefix
compression acts on the reachable subspace, not arbitrary preloaded memories.
Gramians use full earlier visible products; raw label normals do not determine
label ranks. Multiple-label excess uses the kernel of the addition map, not
just pairwise intersections. Nonunitary coordinate charts carry their metric.
A path factor carries an initial generator and history. Relational matrix
states live in dimension n^2 and do not recover an original common vector phase.
Nonlinear scalar energy readouts do not preserve complex amplitude information.

## Counterexamples and boundaries checked

- Equal event multisets can have different chronological separated ranks.
- A fixed unrestricted no-feedback unitary cannot implement genuine transfer.
- At the positive-gap error threshold a target direction may be discarded.
- Four real jet references at -2,-1,1,2 miss one explicit two-plane in C^4.
- A largest-margin chart selector can be discontinuous at chart ties.
- Reference conditioning diverges with inverse margin near the blind locus.
- Two rank-two comparison cycles returning to the same subspace can fail to commute.
- Changing rank, discontinuous parameter choices, and altered input
  representations are outside the hypotheses of the core obstruction.

## Verification evidence

`VERIFICATION.json` records the complete executed audit. Rational Wronskian
checks are independent determinant-polynomial computations. Random complex
checks cover every rank in dimensions two through eight. Numerical residuals
are finite observations, not formal proof substitutes or uniform bounds.