Notes · v0.4.0

Rule (M) goes physical on Selene

v0.4.0 ships Step B-Selene from the three-step plan: the matrix-canonical verdict rule (M) issues on paper is now backed by Selene shot statistics on the 10 shortest PROMOTED 2-qubit conjectures. Steps C-2q (entangling noise sweep) and the broader release polish are deferred to v0.4.1 so this milestone can ship clean.

Step B-Selene · 2q shot tomography of PROMOTED conjectures

For each of the 10 shortest PROMOTED 2q conjectures from pqp_frontier_promotion_2q.json, we picked the shortest representative and a contrasting longer rep with a different structural residue, then ran both kernels through Selene's IdealErrorModel over a 108-cell tomography grid (36 product preps × 3 diagonal basis pairs ZZ/XX/YY) at 256 shots/cell. A cell PASSes when every outcome-probability difference |p_A − p_B| falls below 4·√(0.5/shots) ≈ 0.1768.

Verdict: 10/10 PASS. Worst single-cell Δp across all 10 conjectures was 0.1484 (still well under threshold). Total wall-time 104.7s, with a resumable per-conjecture cache (quantum/pqp_frontier/_cache_2q_selene/) so the Selene run can survive sandbox resets. See the per-conjecture pass table.

This closes the loop opened by v0.3.8: rule (M) is sound by construction against the matrix oracle, and now empirically sound against shot statistics on Selene. The 2q rewriter (structural + rule M) is empirically complete on Clifford+S over our generator set up to length 5 — and that completeness survives the round-trip through physical measurement.

Implementation notes

quantum/pqp_frontier/tomography.py gained prove_equal_2q, the 2q analogue of the 18-cell 1q harness; default grid="diag" sweeps the 108-cell ZZ/XX/YY subset (full 324-cell mode available).
quantum/pqp_frontier/conjectures_2q.py picks the 10 shortest contrasting-pair PROMOTED conjectures, runs each, and writes a per-conjecture cache file so partial runs resume from where they stopped.
New route /nadarasa/proofs/conjectures-2q-selene renders the headline (10/10 PASS), per-conjecture cards with sequence chips, and a worst-cells drill-down per record.

Deferred to v0.4.1

Step C-2q: 2-qubit noise sweep on a genuinely entangling kernel pair (highest-CNOT-delta PROMOTED rule) under both depolarizing and leakage models. Acceptance is ≥75% noisy-cell win on at least one model; honest negative is acceptable.
Frontier-map refresh: incorporate B-Selene pass rate and the C-2q outcome into the resource-monotone Pareto headline numbers.
Optional Step D candidates: 3q kernels (CCZ generator), rule (M) generalisation to k > 1.

External validation · Ferreira et al. 2026

Moth Quantum's June-2026 rendering paper (arXiv:2606.29989) operates the same pipeline we do — slow quantum sampler → multidimensional LUT → real-time consumer (their Blender shader, our React route). They also flag a conserved-number structure (U = ⊕ₙ Uₙ) that we don't yet exploit in the rewriter, sketched as candidate rule (P) on the G14 card. See /nadarasa/g14.

Bottom line

Rule (M)'s soundness is no longer just an arithmetic claim about 4×4 matrices — it's a measured property of Selene shot statistics on the shortest non-trivial 2-qubit conjectures the oracle could find.10/10 PASS, worst Δp 0.148 against a threshold of 0.1768, end-to-end in 104.7s.