Sharp one-qubit full-Clifford tensor threshold: width 5

research/full-clifford-one-qubit/.gitattributes

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+# Preserve the frozen Windows verification output byte for byte.
+RESULT.json -text whitespace=cr-at-eol

research/full-clifford-one-qubit/README.md

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+# Sharp one-qubit full-Clifford tensor threshold
+
+This scoped algebraic result gives pointwise tensor independence from width **5**
+for the full one-qubit Gaussian-dyadic Clifford group modulo its four scalar
+phases. The group has 96 exact lifts and 24 projective classes. The exact Gram
+ranks at widths 1 through 9 are `4,10,20,23,24,24,24,24,24`.
+
+This sharpens the n=1 sufficient pointwise threshold 8 in Ali Romman's
+[PR #5526](https://github.com/CrocSwap/integer-mult-bounds/pull/5526), at exact head
+`45f7de73b168721cdf7ec8c1769b1b594493716a`, especially
+[`PROOF.md` Sections 1 and 5](https://github.com/aliromman/integer-mult-bounds/blob/45f7de73b168721cdf7ec8c1769b1b594493716a/research/clifford-pointwise-faithfulness/PROOF.md).
+The scope refines the formal-interface question to this smallest full-Clifford
+alphabet. It does not change the selected result or supply a new kappa,
+physical implementation, paid invoice or multiplication theorem.
+
+## Argument and evidence
+
+The exact generators are `S=diag(1,i)` and
+`V=((1+i,1-i),(1-i,1+i))/2`. Both normalize the Pauli group. The enumerated
+24 projective classes exhaust the one-qubit Clifford group: X has six signed
+Pauli images, Z has four anticommuting signed Pauli images, and this action
+determines a projective matrix. The exact scalar kernel in `Z[i,1/2]` consists
+of the four Gaussian units, as in the predecessor.
+
+For the enumerated representatives define
+`H_f(g,h)=(tr(U_g* U_h)/2)^f`, the normalized tensor Gram matrix.
+Exact Gaussian rational elimination gives `rank(H_4)=23` and `rank(H_5)=24`.
+The latter Gram matrix is positive definite. Since
+`H_(f+1)=H_f Hadamard H_1`, the Schur product theorem propagates this:
+`H_f >= lambda I` implies `H_(f+1) >= lambda (I Hadamard H_1)=lambda I`.
+Here `H_1` is positive semidefinite with diagonal one. Thus every width at
+least 5 works; the explicit width-4 null vector in [RESULT.json](RESULT.json)
+proves sharpness. The verifier computes unnormalized trace powers, which
+differ from `H_f` by the nonzero scalar `2^f` and have identical ranks/kernels.
+
+For one fixed residual in the 96-element group algebra, the predecessor's
+four-residue Fourier argument therefore makes widths **5,6,7,8** sufficient
+to decide vanishing at every positive width. The width-4 null vector times
+the central residue-zero filter vanishes at widths 1 through 7 but not 8.
+The saved witness is checked at both widths 4 and 8. Consequently no universal
+test using only positive widths below 8 suffices. Fixed residual coefficients
+are essential to this four-width statement.
+
+## Reproduction and limits
+
+Python 3.12 and the standard library suffice. From this directory:
+
+```sh
+python clifford-experiment.py
+python -I -O -W error clifford-experiment.py
+```
+
+The research author ran both commands under CPython 3.12.10 on Windows; their
+outputs were byte-identical, SHA256
+`2163c2f6f4bbc77e126a3cc96e778fcded03512be6784b898afd21875389ce7d`.
+The saved output includes every rank, null vectors and the representatives
+stored as `2U`. The standalone verifier imports no predecessor executable.
+It checks unitarity of all 96 elements, independently constructs 1,152 literal
+tensor Gram entries at widths 4 and 5, and checks the same-residue witness.
+Checks do not use removable Python assertions. Its SHA256 is
+`400844199eb130c02215cd4affea736f3f80a08a7ee1f60572619bcd855b204a`.
+
+This is author-checked exact evidence with a brief AI sanity review, not
+independent human or proof-kernel acceptance. No selected cold replay,
+physical supplier verification, full repository package wrapper, registered
+CI replay or full theorem audit is claimed. This small package uses the
+standalone commands above and does not opt into the generic replay manifest.
+Canonical base: `32890bc5de034ab8fc93ac88ead7279a6a416d04`, priority revision 7.
+The predecessor is a mathematical source; executing this verifier does not
+require its checkout or an unmerged Git object.
+
+## Open next questions
+
+What are the exact pointwise thresholds for larger full-Clifford groups?
+They could sharpen sufficient algebraic identity tests. A first step is a
+bounded exact Gram computation retaining the central residue. Group size and
+exact arithmetic cost are the immediate blockers; no larger cutoff follows
+from this one-qubit calculation.
+
+## Attribution
+
+Ali Romman retains credit for PR #5526's general theorem, phase conventions
+and Fourier implication. Credit also the monomial results in
+[#5521](https://github.com/CrocSwap/integer-mult-bounds/pull/5521),
+[#5522](https://github.com/CrocSwap/integer-mult-bounds/pull/5522),
+[#5529](https://github.com/CrocSwap/integer-mult-bounds/pull/5529) and
+[#5542](https://github.com/CrocSwap/integer-mult-bounds/pull/5542), and DLamdata's
+general-theorem audit [#5547](https://github.com/CrocSwap/integer-mult-bounds/pull/5547).
+This result concerns the larger full-Clifford alphabet and does not repeat
+those audits. Schur products, Pauli normalizers and Fourier inversion are
+classical. No global literature-priority claim is made.
+
+Prepared for MatShep with substantial OpenAI Codex assistance (GPT-6.1 Sol
+research; GPT-6 Astra brief validation and publication). Apache-2.0.