Coordinate-sweep contraction constants: selected source audit

The selected trace solution has an existential universal sweep threshold, but this review has not extracted a useful numerical value. The hashed reading record distinguishes these selected module excerpts from a complete imported-proof review. The source commit remains fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. No Lean, Comparator or independent kernel ran.

TraceModel.lean imports Harmonic.Main, inside OAI.ThorpNine.Harmonic. The parallel Casimir module contains similar names but is not the selected import for this path. TraceDecay.uniform_nonconstant_decay takes positive level/dimension contraction coefficients a,b and chooses m>=max(14/a,2/b), then u=2m. The regular trace conclusion uses P=2u. This chain provides a proof witness for existence; the preceding finite four-card calibration does not choose the universal witness.

In Harmonic/Main lines 415–478, the eventual level coefficient combines a spectral coefficient and a dimension coefficient. An eventual threshold D is obtained, and the finite part takes a positive lower bound of f(d)=-log(blockGap(d))/2^d over d<D. The resulting all-d coefficient is the minimum of the eventual and finite bounds. In lines 1087–1138, the dimension contraction similarly uses f(d)=-log(blockGap(d))/(1+log((2^d)!)) and a finite positive minimum. Neither numerical D nor the underlying spectral coefficients were selected in this review.

Harmonic/AverageMap lines 271–292 defines a finite convolution minorant. Its exponent is classically chosen from existence of a positive convolution power; its mass is the full-group size times the minimum mass of that selected power. Lines 463–480 give blockGap=1-mass/(2*2^exponent), and lines 687–694 instantiate it for the palindrome shuffle. This is more information than merely saying the gap is positive, but evaluating a literal full-group minimum is a separate finite construction with factorial-sized state space. An explicit sharper minorant or spectral estimate could be an engineering bridge; it is not supplied by these references.

The product decision remains conditional: do not advertise a source-certified sweep schedule from the tiny fixtures or a hand-picked P. Next inspect the spectral-regime numerical dependencies, and consider an exact four-card palindrome minorant as a finite calibration only. A changed proof with quantitative witnesses, a validated independent-bit backend and correspondence to an actual implementation remain separate obligations. Framework dossier.

Exact two/four-slot calibration

The new finite utility implements the selected recursive butterfly model, whose child action precedes the root switch. InsertionPerm's physical-sweep lemma identifies the sweep average with the inverse butterfly average; BlockProduct expresses the palindrome as butterfly times its adjoint. Selected excerpts are now included in the reading record, but those lemmas' proof/import closure was not executed. An independent actual-card swap oracle checks all sixteen four-slot coin assignments, including this orientation, before comparing the inverse law with the earlier physical-sweep law.

For four slots the two independent butterfly assignments account for 256 equally weighted coin strings. All 24 permutations occur with counts 8 or 16. The minimum probability is 1/32, so exponent zero and mass 3/4 give an explicit uniform minorant and gap-formula bound 5/8. This constructed minorant is a valid finite reference; it is not an evaluation of the source's classically selected palindromeMinorant value.

The exact four-slot witness provides the centered 24-by-24 regular operator B=A-J, in a declared permutation order. Rational arithmetic verifies symmetry, zero constant direction and B^2=(1/4)B. Its scaled-projection rank is two, and a displayed nonzero column is a zero-sum eigenvector of eigenvalue 1/4. Thus the palindrome's nonconstant norm is exactly 1/4, and the corresponding physical sweep's nonconstant squared norm is 1/4. The two-slot palindrome is uniform and its centered operator is zero. No floating eigensolver is used.

For a point mass on S_n, the conservative regular norm inequality gives TV^2<=((n!-1)/4)c^v when c bounds the sweep's nonconstant squared norm. At n=4, c=1/4 reaches the source target squared 1/(4n^10) at 13 sweeps; using the weaker explicit-minorant c=5/8 reaches it at 37. Both preceding steps fail their respective bound. These are sufficient bounds for the four-slot reference, not its optimal mixing time: direct exact-law enumeration already passes at six sweeps. At n=2 the exact and minorant calibrations are one and ten sweeps respectively.

50 controls reconstruct the law/transition matrix through separate physical swaps and direct group transitions, verify exact mass/witness/bound arithmetic, and reject dimensions beyond two. This is a usable finite calibration, with a factorial-sized regular matrix and explicit domain cap. It neither computes all small dimensions below an unknown threshold D nor supplies the eventual spectral constants, universal P, fair-bit backend or formal program/proof acceptance. A numerical bridge should expose those remaining witnesses instead of extrapolating the four-slot result.