Nadarasa · Track 1 · Synesthete's PQP Frontier
Phase-group atlas
Each point is a (kernel, register-size) pair plotted in phase-group coordinates: x = n qubits, y = log₂ |G| where G is the cyclic subgroup of (R/Z, +) generated by the kernel's spider phases. Colour encodes the order — cool for small cyclic groups, warm as the group grows. PQP Thm 11.12 says non-locality requires Z₄ ⊂ G; the atlas extends that litmus to a full coordinate system.
G1 · QFT cphase ladderG2-real · two ladders + glueG12 · same skeleton as G1
Atlas table
| Kernel | n | |G| | Group | distinct orders | Z₄ ⊂ G |
|---|---|---|---|---|---|
| G1 · QFT cphase ladder | 3 | 4 | Z₄ | 2 | yes |
| G1 · QFT cphase ladder | 4 | 8 | Z_{2^3} (= Z_8) | 3 | yes |
| G1 · QFT cphase ladder | 5 | 16 | Z_{2^4} (= Z_16) | 4 | yes |
| G1 · QFT cphase ladder | 6 | 32 | Z_{2^5} (= Z_32) | 5 | yes |
| G1 · QFT cphase ladder | 7 | 64 | Z_{2^6} (= Z_64) | 6 | yes |
| G1 · QFT cphase ladder | 8 | 128 | Z_{2^7} (= Z_128) | 7 | yes |
| G2-real · two ladders + glue | 3 | 8 | Z_{2^3} (= Z_8) | 3 | yes |
| G2-real · two ladders + glue | 4 | 16 | Z_{2^4} (= Z_16) | 4 | yes |
| G2-real · two ladders + glue | 5 | 32 | Z_{2^5} (= Z_32) | 5 | yes |
| G2-real · two ladders + glue | 6 | 64 | Z_{2^6} (= Z_64) | 6 | yes |
| G2-real · two ladders + glue | 7 | 128 | Z_{2^7} (= Z_128) | 7 | yes |
| G2-real · two ladders + glue | 8 | 256 | Z_{2^8} (= Z_256) | 8 | yes |
| G12 · same skeleton as G1 | 3 | 4 | Z₄ | 2 | yes |
| G12 · same skeleton as G1 | 4 | 8 | Z_{2^3} (= Z_8) | 3 | yes |
| G12 · same skeleton as G1 | 5 | 16 | Z_{2^4} (= Z_16) | 4 | yes |
| G12 · same skeleton as G1 | 6 | 32 | Z_{2^5} (= Z_32) | 5 | yes |
| G12 · same skeleton as G1 | 7 | 64 | Z_{2^6} (= Z_64) | 6 | yes |
| G12 · same skeleton as G1 | 8 | 128 | Z_{2^7} (= Z_128) | 7 | yes |
Reading the atlas
- QFT-skeleton kernels (G1, G12) land at
|G| = 2^(n-1): their phases are exactly the cphase ladder, so the group doubles with every extra qubit. - G2-real adds a slope cphase at angle
2π/N, pushing it one rung higher to|G| = 2^n. That is a structural separation visible at a glance — the same separation the resource monotones certify. - Falsifier check. All current kernels collapse onto a single power-of-two ray. That is informative: to diversify the program we need a generator whose phase denominator is NOT a power of 2 (e.g. a Toffoli-decomposed ancilla rotation, or a non-Clifford magic state). Until then, the atlas is one-dimensional.