lean qubit_h_m7_stabRankP_le_nine in LeanProofs.QubitHM7StabRank, machine-checked by Lean
Implied per-copy exponent γ ≤ 0.4528 against a published 0.3962, no improvement.
This bound is backed by a machine-checked theorem: qubit_h_m7_stabRankP_le_nine in LeanProofs.QubitHM7StabRank. The recorded lake build confirms it compiles.
1 run; 0.01 CPU-hours; 0.01 h wall-clock; on Apple silicon laptop, one core; direct construction, no search.
χ(|H⟩⊗7) ≤ 9 from the partial decomposition of Kissinger, van de Wetering, and Vilmart (arXiv:2202.09202, Section 4.2): |T⟩⊗5 = √ 2 (I ⊗ ⟨T|)|cat6⟩ has three terms, each a Clifford image of (stabilizer ⊗ |T⟩), so four T states are removed at a cost of three terms and χ(T^t) ≤ 3 χ(T^{t-4}). At t = 7 the leftover |T⟩ of each term joins the remaining |T⟩⊗2, and |T⟩⊗3 has rank 3, giving 3 x 3 = 9. Improves the 12 that QPG's table lists at seven copies. Witness built here by research/constructions/kvv_cat.py from the three cat_6 terms and the board's m=3 witness, and verified in exact arithmetic; exponent log_2(9)/7 = 0.4528. Now at the lean tier: qubit_h_m7_stabRankP_le_nine in LeanProofs.QubitHM7StabRank proves stabRankP 2 |qubitH⟩⊗7 ≤ 9 against IsStabP 2 from this witness, the pointwise identity being decided by the kernel on integer coordinate vectors (see lean_proofs/LeanProofs/Stabilizer/Reflect.lean).
Kissinger, van de Wetering, and Vilmart · arXiv:2202.09202 · literature
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"notes": "chi(|H>^7) <= 9 from the partial decomposition of Kissinger, van de Wetering, and Vilmart (arXiv:2202.09202, Section 4.2): |T>^5 = sqrt 2 (I (x) <T|)|cat_6> has three terms, each a Clifford image of (stabilizer (x) |T>), so four T states are removed at a cost of three terms and chi(T^t) <= 3 chi(T^{t-4}). At t = 7 the leftover |T> of each term joins the remaining |T>^2, and |T>^3 has rank 3, giving 3 x 3 = 9. Improves the 12 that QPG's table lists at seven copies. Witness built here by research/constructions/kvv_cat.py from the three cat_6 terms and the board's m=3 witness, and verified in exact arithmetic; exponent log_2(9)/7 = 0.4528. Now at the lean tier: qubit_h_m7_stabRankP_le_nine in LeanProofs.QubitHM7StabRank proves stabRankP 2 |qubit_H>^7 <= 9 against IsStabP 2 from this witness, the pointwise identity being decided by the kernel on integer coordinate vectors (see lean_proofs/LeanProofs/Stabilizer/Reflect.lean).",
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