Extremal-code geometry, exact mesh locality, and a one-photon falsifier
Passes 3973–3980 turn three tempting numerical patterns into explicit combinatorial objects, certify the current optical mesh against every single transposition, and separate photon capacity from propagation speed with a slope/intercept experiment.
The 57 becomes geometry
The support-intersection graph is explicitly reconstructed as
so 57 = C(10,2) + 2C(4,2). The MacWilliams family is one-parameter, but its elementary global bound is only A4 ≤ 245; 57 is not promoted as globally extremal.
Mesh neighborhood closed
Every single transposition of the 36-port order was replayed in exact multiquadratic arithmetic.
No radius-one neighbor improves the established mesh. Global optimality remains open.
Rank-48 tensor
The complete matrix-unit product table has 48 basis elements, center dimension seven, and 178 nonzero structure constants. The orbital-relation coordinate conversion is the remaining geometric step.
The photon experiment: separate intercept from slope
Use one fixed spectral envelope and randomly interleave mode alphabets M = 2, 4, 8, 16, 40 over two or more free-space lengths.
a(M) measures encoder, decoder, detector, and pulse-shape latency. Only a reproducible b(M) dependence that survives encoder swaps and length scaling challenges invariant-front propagation.
| M | Ideal bits/use | 1% symmetric error | 5% symmetric error |
|---|---|---|---|
| 2 | 1.0000 | 0.9192 | 0.7136 |
| 8 | 3.0000 | 2.8911 | 2.5732 |
| 40 | 5.3219 | 5.1883 | 4.7713 |
| 729 | 9.5098 | 9.3339 | 8.7480 |
Capacity trilemma
Direct sum: one photon in M alternatives gives at most log₂M ideal bits/use.
Tensor product: N log₂d needs N independent physical factors.
Serial dynamics: orthogonal state changes consume energy and elapsed time.
Time–bandwidth price
At half-bandwidth 20 GHz, 40 temporal modes require about 1 ns (0.300 m of vacuum flight), while 729 require about 18.225 ns (5.464 m). Capacity can grow by using more spacetime support without changing c.
Self-similar density
The address density is constant:
Self-similarity compresses routing description, not causal speed.
Evidence boundary
Exact: code reconstruction and line-graph identifications, one-parameter enumerator identities, all-630 mesh audit, Wedderburn multiplication tensor, and algebraic capacity/timing formulas.
Modeled: symmetric-channel information, illustrative timing sensitivity, and the Slepian time–bandwidth estimate.
Open: global A4 extremality, global mesh optimum, performed photon experiment, geometric orbital tensor, Monster words/class fusion, variable vacuum c, hardware, and laboratory validation.
Certificate: 769f2f1b29e832050fbf148f38ab64acd992766d5dc758a525b3f97e6e77d44e