Nadarasa · Proofs · Selene/Guppy · v0.3.7
Conjectures, proved + promoted
Two passes on every flagged conjecture: (1) Selene shot-tomography confirms the equality is physical (PASS = the TS oracle is honest), and (2) the new rule (N) Euler-normalisation pass on the bastard-rewriter collapses every matrix-equivalent representative to a single canonical residue. A PROMOTED badge means the rewriter now proves the equality itself — the conjecture is closed.
→ Track B: 2-qubit oracle synthesis lifts the same enumeration to {H⊗I, I⊗H, CZ, S⊗I, I⊗S} over 4×4 unitaries.
Rule (N) promotion — Headline
11/11 conjectures promoted · 0 partially reduced · 0 unchanged. Across 363 sequences at maxLen = 5, every matrix-equivalent class now shares one residue — the rewriter is empirically complete on the 1-qubit Clifford+T fragment up to this length.
Selene PASS
11/11
shots/cell
384
wall-time
183s
(1) matrix class
[[0.71+0.00i, 0.71+0.00i], [0.71+0.00i, -0.71+0.00i]]
H ≡ SHSHS
residues (base): ∅ ↔ X(π/2) · Z(π/2) · Z(π/2)
spiders: 0→0 vs 3→3
+ Euler (N): 2 → 1 distinct residues · canonical = X(π/2) · Z(π/2) · Z(π/2)
| prep \ basis | Z | X | Y |
|---|---|---|---|
| |0⟩ | 0.50/0.50 | 0.00/0.00 | 0.49/0.52 |
| |1⟩ | 0.49/0.55 | 1.00/1.00 | 0.50/0.47 |
| |+⟩ | 0.00/0.00 | 0.47/0.51 | 0.52/0.46 |
| |-⟩ | 1.00/1.00 | 0.52/0.49 | 0.50/0.52 |
| |+i⟩ | 0.51/0.52 | 0.52/0.45 | 1.00/1.00 |
| |-i⟩ | 0.47/0.52 | 0.52/0.49 | 0.00/0.00 |
worst tv = 0.0677 · threshold (4σ) = 0.1443 · 16.4s
(2) matrix class
[[0.71+0.00i, 0.00+0.71i], [0.00+0.71i, 0.71+0.00i]]
SHS ≡ HSSSH
residues (base): Z(π/2) · Z(π/2) ↔ X(3π/2)
spiders: 2→2 vs 3→1
+ Euler (N): 2 → 1 distinct residues · canonical = X(π/2) · Z(π) · Z(π)
| prep \ basis | Z | X | Y |
|---|---|---|---|
| |0⟩ | 0.52/0.48 | 0.50/0.50 | 0.00/0.00 |
| |1⟩ | 0.51/0.46 | 0.47/0.51 | 1.00/1.00 |
| |+⟩ | 0.54/0.50 | 0.00/0.00 | 0.51/0.53 |
| |-⟩ | 0.53/0.54 | 1.00/1.00 | 0.49/0.48 |
| |+i⟩ | 1.00/1.00 | 0.47/0.47 | 0.51/0.49 |
| |-i⟩ | 0.00/0.00 | 0.54/0.51 | 0.54/0.51 |
worst tv = 0.0521 · threshold (4σ) = 0.1443 · 16.8s
(3) matrix class
[[1.00+0.00i, 0.00+0.00i], [0.00+0.00i, 0.00-1.00i]]
SSS ≡ HSHSH
residues (base): Z(3π/2) ↔ X(π/2) · Z(π/2)
spiders: 3→1 vs 2→2
+ Euler (N): 2 → 1 distinct residues · canonical = Z(3π/2)
| prep \ basis | Z | X | Y |
|---|---|---|---|
| |0⟩ | 0.00/0.00 | 0.52/0.51 | 0.53/0.46 |
| |1⟩ | 1.00/1.00 | 0.53/0.51 | 0.49/0.48 |
| |+⟩ | 0.49/0.52 | 0.47/0.46 | 1.00/1.00 |
| |-⟩ | 0.51/0.52 | 0.56/0.49 | 0.00/0.00 |
| |+i⟩ | 0.46/0.54 | 0.00/0.00 | 0.49/0.51 |
| |-i⟩ | 0.49/0.45 | 1.00/1.00 | 0.48/0.51 |
worst tv = 0.0781 · threshold (4σ) = 0.1443 · 16.1s
(4) matrix class
[[0.50+0.50i, 0.50-0.50i], [0.50+0.50i, -0.50+0.50i]]
HSHS ≡ SSSH
residues (base): X(π/2) · Z(π/2) ↔ Z(3π/2)
spiders: 2→2 vs 3→1
+ Euler (N): 2 → 1 distinct residues · canonical = X(π/2) · Z(π/2)
| prep \ basis | Z | X | Y |
|---|---|---|---|
| |0⟩ | 0.55/0.48 | 0.00/0.00 | 0.51/0.51 |
| |1⟩ | 0.49/0.49 | 1.00/1.00 | 0.52/0.50 |
| |+⟩ | 0.51/0.52 | 0.48/0.49 | 0.00/0.00 |
| |-⟩ | 0.51/0.50 | 0.50/0.50 | 1.00/1.00 |
| |+i⟩ | 0.00/0.00 | 0.47/0.55 | 0.53/0.49 |
| |-i⟩ | 1.00/1.00 | 0.51/0.50 | 0.47/0.47 |
worst tv = 0.0807 · threshold (4σ) = 0.1443 · 16.6s
(5) matrix class
[[0.71+0.00i, 0.71+0.00i], [0.00-0.71i, 0.00+0.71i]]
HSSS ≡ SHSH
residues (base): Z(3π/2) ↔ X(π/2) · Z(π/2)
spiders: 3→1 vs 2→2
+ Euler (N): 2 → 1 distinct residues · canonical = X(π/2) · Z(π/2)
| prep \ basis | Z | X | Y |
|---|---|---|---|
| |0⟩ | 0.48/0.44 | 0.48/0.47 | 1.00/1.00 |
| |1⟩ | 0.51/0.53 | 0.54/0.48 | 0.00/0.00 |
| |+⟩ | 0.00/0.00 | 0.48/0.47 | 0.48/0.51 |
| |-⟩ | 1.00/1.00 | 0.51/0.49 | 0.50/0.45 |
| |+i⟩ | 0.48/0.55 | 1.00/1.00 | 0.51/0.52 |
| |-i⟩ | 0.49/0.52 | 0.00/0.00 | 0.48/0.47 |
worst tv = 0.0625 · threshold (4σ) = 0.1443 · 16.1s
(6) matrix class
[[0.50+0.50i, 0.50-0.50i], [-0.50+0.50i, -0.50-0.50i]]
HSHSS ≡ SSSHS
residues (base): X(π/2) · Z(π) ↔ Z(3π/2) · Z(π/2)
spiders: 3→2 vs 4→2
+ Euler (N): 2 → 1 distinct residues · canonical = X(π/2) · Z(π)
| prep \ basis | Z | X | Y |
|---|---|---|---|
| |0⟩ | 0.52/0.50 | 0.46/0.49 | 0.00/0.00 |
| |1⟩ | 0.50/0.51 | 0.53/0.48 | 1.00/1.00 |
| |+⟩ | 0.50/0.49 | 1.00/1.00 | 0.48/0.51 |
| |-⟩ | 0.55/0.49 | 0.00/0.00 | 0.47/0.51 |
| |+i⟩ | 0.00/0.00 | 0.49/0.51 | 0.44/0.47 |
| |-i⟩ | 1.00/1.00 | 0.47/0.47 | 0.53/0.52 |
worst tv = 0.0573 · threshold (4σ) = 0.1443 · 16.3s
(7) matrix class
[[0.50+0.50i, 0.50-0.50i], [0.00+0.71i, -0.71+0.00i]]
HSHST ≡ SSSHT
residues (base): X(π/2) · Z(3π/4) ↔ Z(3π/2) · Z(π/4)
spiders: 3→2 vs 4→2
+ Euler (N): 2 → 1 distinct residues · canonical = X(π/2) · Z(3π/4)
| prep \ basis | Z | X | Y |
|---|---|---|---|
| |0⟩ | 0.51/0.46 | 0.15/0.18 | 0.17/0.14 |
| |1⟩ | 0.51/0.50 | 0.88/0.86 | 0.87/0.84 |
| |+⟩ | 0.51/0.50 | 0.84/0.84 | 0.12/0.17 |
| |-⟩ | 0.48/0.48 | 0.15/0.17 | 0.84/0.84 |
| |+i⟩ | 0.00/0.00 | 0.46/0.50 | 0.52/0.53 |
| |-i⟩ | 1.00/1.00 | 0.48/0.48 | 0.50/0.48 |
worst tv = 0.0495 · threshold (4σ) = 0.1443 · 16.4s
(8) matrix class
[[0.71+0.00i, 0.71+0.00i], [0.50-0.50i, -0.50+0.50i]]
HSSST ≡ SHSHT
residues (base): Z(7π/4) ↔ X(π/2) · Z(π/2) · Z(π/4)
spiders: 4→1 vs 3→3
+ Euler (N): 2 → 1 distinct residues · canonical = X(π/2) · Z(π/2) · Z(π/4)
| prep \ basis | Z | X | Y |
|---|---|---|---|
| |0⟩ | 0.51/0.49 | 0.15/0.14 | 0.89/0.86 |
| |1⟩ | 0.52/0.48 | 0.86/0.85 | 0.17/0.17 |
| |+⟩ | 0.00/0.00 | 0.49/0.51 | 0.54/0.51 |
| |-⟩ | 1.00/1.00 | 0.54/0.50 | 0.48/0.50 |
| |+i⟩ | 0.48/0.47 | 0.86/0.86 | 0.88/0.85 |
| |-i⟩ | 0.46/0.52 | 0.16/0.13 | 0.13/0.15 |
worst tv = 0.0573 · threshold (4σ) = 0.1443 · 16.8s
(9) matrix class
[[0.71+0.00i, 0.00+0.71i], [0.00-0.71i, -0.71+0.00i]]
SHSSS ≡ SSHSH
residues (base): Z(3π/2) · Z(π/2) ↔ X(π/2) · Z(π)
spiders: 4→2 vs 3→2
+ Euler (N): 2 → 1 distinct residues · canonical = X(π/2) · Z(π)
| prep \ basis | Z | X | Y |
|---|---|---|---|
| |0⟩ | 0.46/0.46 | 0.55/0.52 | 1.00/1.00 |
| |1⟩ | 0.51/0.50 | 0.49/0.50 | 0.00/0.00 |
| |+⟩ | 0.50/0.53 | 1.00/1.00 | 0.50/0.48 |
| |-⟩ | 0.52/0.54 | 0.00/0.00 | 0.53/0.45 |
| |+i⟩ | 1.00/1.00 | 0.49/0.48 | 0.51/0.49 |
| |-i⟩ | 0.00/0.00 | 0.46/0.50 | 0.49/0.51 |
worst tv = 0.0781 · threshold (4σ) = 0.1443 · 17.2s
(10) matrix class
[[0.71+0.00i, 0.50-0.50i], [0.71+0.00i, -0.50+0.50i]]
SSSTH ≡ THSHS
residues (base): Z(7π/4) ↔ X(π/2) · Z(π/2) · Z(π/4)
spiders: 4→1 vs 3→3
+ Euler (N): 2 → 1 distinct residues · canonical = X(π/2) · Z(π/2) · Z(π/4)
| prep \ basis | Z | X | Y |
|---|---|---|---|
| |0⟩ | 0.55/0.51 | 0.00/0.00 | 0.50/0.52 |
| |1⟩ | 0.48/0.48 | 1.00/1.00 | 0.49/0.54 |
| |+⟩ | 0.15/0.17 | 0.51/0.46 | 0.14/0.13 |
| |-⟩ | 0.84/0.86 | 0.48/0.47 | 0.88/0.85 |
| |+i⟩ | 0.14/0.16 | 0.48/0.53 | 0.85/0.87 |
| |-i⟩ | 0.84/0.84 | 0.49/0.52 | 0.12/0.16 |
worst tv = 0.0521 · threshold (4σ) = 0.1443 · 16.6s
(11) matrix class
[[0.50+0.50i, 0.00+0.71i], [0.50-0.50i, -0.71+0.00i]]
STHSH ≡ THSSS
residues (base): X(π/2) · Z(3π/4) ↔ Z(3π/2) · Z(π/4)
spiders: 3→2 vs 4→2
+ Euler (N): 2 → 1 distinct residues · canonical = X(π/2) · Z(3π/4)
| prep \ basis | Z | X | Y |
|---|---|---|---|
| |0⟩ | 0.49/0.54 | 0.48/0.47 | 1.00/1.00 |
| |1⟩ | 0.48/0.51 | 0.49/0.52 | 0.00/0.00 |
| |+⟩ | 0.17/0.13 | 0.86/0.88 | 0.48/0.47 |
| |-⟩ | 0.85/0.86 | 0.13/0.14 | 0.52/0.44 |
| |+i⟩ | 0.89/0.86 | 0.85/0.83 | 0.50/0.53 |
| |-i⟩ | 0.14/0.15 | 0.14/0.13 | 0.48/0.54 |
worst tv = 0.0833 · threshold (4σ) = 0.1443 · 17.4s
Each conjecture is an equality the TS-side matrix oracle asserts but the bastard rewriter cannot prove. We confirm the oracle on Selene shots: for the shortest representative vs a contrasting longer one, the 18-cell tomography grid must agree within shot noise. PASS = the conjecture is physically real, the rewriter has a genuine completeness gap. FAIL = the TS oracle has a bug.
v0.3.7 Track A: added rule (N) Euler normalisation to the bastard-rewriter. A single-qubit wire's interior is replaced by the canonical Z(γ)·X(β)·Z(α) form via direct 2×2 matrix decomposition. Sound by construction (drops only global phase, which rule B already discards). Conjectures whose representatives now share a single residue are promoted from open candidates to proved-equal-by-the-rewriter.