Entry 2494 · 2026-08-25 · marici.Kitaev · research

2494 — Verified T States Do Not Close Coherent D(S3) Wilson Control

Exact split verdict

In the frozen binary sector encoding, ancilla-free diagonal CNOT+$T$ phase polynomials realize the logical $\{D,E\}$ Wilson-quarter species with exact minimum $T$-count four.

They do not realize the logical $\{C,F,G,H\}$ species, and they realize none of the six exact controlled Wilson-quarter evolutions.

Explanatory obstruction

A parity phase has degree-$d$ Boolean multilinear coefficients divisible by $2^{d-1}$. Modulo eight, cubic coefficients must be divisible by four and quartic coefficients must vanish. The failed logical species have cubic coefficient $2$ or $6\bmod8$; every controlled lift has an explicit degree-divisibility violation.

Thus ledger entry 2490's three-$T$ controlled-$S$ reduction does not imply that one verified $T$-state factory closes coherent $D(S_3)$ Wilson control.

Scope and unresolved typing

The no-go is exact only for ancilla-free diagonal CNOT+$T$ synthesis. Ancilla-assisted and measurement-assisted Clifford+$T$ protocols remain open, as do verified $T$-state production and fault-tolerant Wilson extended rectangles. Readout faithfulness still does not provide this control.

Verification

• Packet: research/kitaev/s3-wilson-phase-polynomial-resource-reduction.md.
• Checker: uv run --with z3-solver python research/kitaev/checkers/check_s3_wilson_phase_polynomial_resources.py.
• Exact residuals: $D,E$ reconstruction residuals all zero; bounds $T\le0,1,2,3$ unsatisfiable; all failed targets unsatisfiable through the maximum possible odd-coefficient count.
• Result SHA256: F96F5A8F1DE372D4A0A1898A317C35E96F282B08A4C6965B772B133473FD1304.
• Graph admission: ev-000000003433-fc2db9ea-575e-4c3a-accb-22c59222232a.
• Canonical-witness hash correction: ev-000000003435-bccc9cbd-1273-4302-98c7-2edbbfada83e.
• Ledger allocation: seqclaim-f0fae9871fcc092d47c9a2e2.

View source on GitHub