Nadarasa · Track 2 · Synesthete's PQP Frontier
Bastard-spider rewriter
A 5-rule ZX rewriter — spider fusion (F), identity removal (I), Hadamard cancellation (HH), colour change (CC), and bastard absorption (B) — applied to fixpoint against the phase-atlas generators for each Nadarasa kernel. Rules are anchored in PQP Thm 8.34 / Cor 8.35 / Cor 9.21 / Eq. 9.20 / Thm 8.72; rule (B) is the v0.3.3 bastard formulation of the modal projector.
G1 · QFT cphase ladder
spiders 3 → 2(−1) · H 4 → 0(−4)
HH · 1F · 1CC · 1
- Residual spiders
- X(3π/2) · B(0)
- Atlas |G| (Thm 11.12)
- 4
Show trace · 3 steps
- HHannihilate H–H pair #3–#4
- Ffuse Z-spiders #5 → #2, sum phases
- CCH–Z(α)–H → X(α) at #2
spiders 4 → 2(−2) · H 6 → 0(−6)
HH · 2F · 2CC · 1
- Residual spiders
- X(7π/4) · B(0)
- Atlas |G| (Thm 11.12)
- 8
Show trace · 5 steps
- HHannihilate H–H pair #3–#4
- Ffuse Z-spiders #5 → #2, sum phases
- HHannihilate H–H pair #6–#7
- Ffuse Z-spiders #8 → #2, sum phases
- CCH–Z(α)–H → X(α) at #2
spiders 5 → 2(−3) · H 8 → 0(−8)
HH · 3F · 3CC · 1
- Residual spiders
- X(15π/8) · B(0)
- Atlas |G| (Thm 11.12)
- 16
Show trace · 7 steps
- HHannihilate H–H pair #3–#4
- Ffuse Z-spiders #5 → #2, sum phases
- HHannihilate H–H pair #6–#7
- Ffuse Z-spiders #8 → #2, sum phases
- HHannihilate H–H pair #9–#10
- Ffuse Z-spiders #11 → #2, sum phases
- CCH–Z(α)–H → X(α) at #2
spiders 6 → 2(−4) · H 10 → 0(−10)
HH · 4F · 4CC · 1
- Residual spiders
- X(31π/16) · B(0)
- Atlas |G| (Thm 11.12)
- 32
Show trace · 9 steps
- HHannihilate H–H pair #3–#4
- Ffuse Z-spiders #5 → #2, sum phases
- HHannihilate H–H pair #6–#7
- Ffuse Z-spiders #8 → #2, sum phases
- HHannihilate H–H pair #9–#10
- Ffuse Z-spiders #11 → #2, sum phases
- HHannihilate H–H pair #12–#13
- Ffuse Z-spiders #14 → #2, sum phases
- CCH–Z(α)–H → X(α) at #2
G2-real · two ladders + glue
spiders 4 → 2(−2) · H 6 → 0(−6)
HH · 2F · 2CC · 1
- Residual spiders
- X(7π/4) · B(0)
- Atlas |G| (Thm 11.12)
- 8
Show trace · 5 steps
- HHannihilate H–H pair #3–#4
- Ffuse Z-spiders #5 → #2, sum phases
- HHannihilate H–H pair #6–#7
- Ffuse Z-spiders #8 → #2, sum phases
- CCH–Z(α)–H → X(α) at #2
spiders 5 → 2(−3) · H 8 → 0(−8)
HH · 3F · 3CC · 1
- Residual spiders
- X(15π/8) · B(0)
- Atlas |G| (Thm 11.12)
- 16
Show trace · 7 steps
- HHannihilate H–H pair #3–#4
- Ffuse Z-spiders #5 → #2, sum phases
- HHannihilate H–H pair #6–#7
- Ffuse Z-spiders #8 → #2, sum phases
- HHannihilate H–H pair #9–#10
- Ffuse Z-spiders #11 → #2, sum phases
- CCH–Z(α)–H → X(α) at #2
spiders 6 → 2(−4) · H 10 → 0(−10)
HH · 4F · 4CC · 1
- Residual spiders
- X(31π/16) · B(0)
- Atlas |G| (Thm 11.12)
- 32
Show trace · 9 steps
- HHannihilate H–H pair #3–#4
- Ffuse Z-spiders #5 → #2, sum phases
- HHannihilate H–H pair #6–#7
- Ffuse Z-spiders #8 → #2, sum phases
- HHannihilate H–H pair #9–#10
- Ffuse Z-spiders #11 → #2, sum phases
- HHannihilate H–H pair #12–#13
- Ffuse Z-spiders #14 → #2, sum phases
- CCH–Z(α)–H → X(α) at #2
spiders 7 → 2(−5) · H 12 → 0(−12)
HH · 5F · 5CC · 1
- Residual spiders
- X(63π/32) · B(0)
- Atlas |G| (Thm 11.12)
- 64
Show trace · 11 steps
- HHannihilate H–H pair #3–#4
- Ffuse Z-spiders #5 → #2, sum phases
- HHannihilate H–H pair #6–#7
- Ffuse Z-spiders #8 → #2, sum phases
- HHannihilate H–H pair #9–#10
- Ffuse Z-spiders #11 → #2, sum phases
- HHannihilate H–H pair #12–#13
- Ffuse Z-spiders #14 → #2, sum phases
- HHannihilate H–H pair #15–#16
- Ffuse Z-spiders #17 → #2, sum phases
- CCH–Z(α)–H → X(α) at #2
G12 · same skeleton as G1
spiders 3 → 2(−1) · H 4 → 0(−4)
HH · 1F · 1CC · 1
- Residual spiders
- X(3π/2) · B(0)
- Atlas |G| (Thm 11.12)
- 4
Show trace · 3 steps
- HHannihilate H–H pair #3–#4
- Ffuse Z-spiders #5 → #2, sum phases
- CCH–Z(α)–H → X(α) at #2
spiders 4 → 2(−2) · H 6 → 0(−6)
HH · 2F · 2CC · 1
- Residual spiders
- X(7π/4) · B(0)
- Atlas |G| (Thm 11.12)
- 8
Show trace · 5 steps
- HHannihilate H–H pair #3–#4
- Ffuse Z-spiders #5 → #2, sum phases
- HHannihilate H–H pair #6–#7
- Ffuse Z-spiders #8 → #2, sum phases
- CCH–Z(α)–H → X(α) at #2
spiders 5 → 2(−3) · H 8 → 0(−8)
HH · 3F · 3CC · 1
- Residual spiders
- X(15π/8) · B(0)
- Atlas |G| (Thm 11.12)
- 16
Show trace · 7 steps
- HHannihilate H–H pair #3–#4
- Ffuse Z-spiders #5 → #2, sum phases
- HHannihilate H–H pair #6–#7
- Ffuse Z-spiders #8 → #2, sum phases
- HHannihilate H–H pair #9–#10
- Ffuse Z-spiders #11 → #2, sum phases
- CCH–Z(α)–H → X(α) at #2
spiders 6 → 2(−4) · H 10 → 0(−10)
HH · 4F · 4CC · 1
- Residual spiders
- X(31π/16) · B(0)
- Atlas |G| (Thm 11.12)
- 32
Show trace · 9 steps
- HHannihilate H–H pair #3–#4
- Ffuse Z-spiders #5 → #2, sum phases
- HHannihilate H–H pair #6–#7
- Ffuse Z-spiders #8 → #2, sum phases
- HHannihilate H–H pair #9–#10
- Ffuse Z-spiders #11 → #2, sum phases
- HHannihilate H–H pair #12–#13
- Ffuse Z-spiders #14 → #2, sum phases
- CCH–Z(α)–H → X(α) at #2
What this proves (and what it doesn't)
- It proves the v0.3.3 "predictor-from-gates" rule is now executable: every kernel's diagram collapses to its bastard residue under the same 5 rules, and the residue's phase content matches the atlas group order.
- It does not prove full ZX completeness — we're working in the Clifford + dyadic-Rz fragment. JPV 2017 / NW 2017 would be needed to claim every equivalence is provable.
- Falsifier triggered? If any kernel above shows residual spiders whose phase denominators exceed the atlas |G|, the rewriter is unsound (or the atlas under-counts). Patch the bug before trusting Track 4.