research(proximity): add G139 Phi certificate probe

[coopertweak]

Summary

• add a stdlib-only exact probe for the G139 Phi_H weak-Sidon certificate
• certify diagonal selected cells n = 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576
• include two non-injective accident controls: G173 and n=512,p=138027521

Result

For an order-n subgroup H=<g>, the probe checks

Phi_H(r) = (g^r - 1)^n,  1 <= r <= n/2

Injectivity of this map certifies weak Sidon under unordered addition, hence the normalized G139 equation a+b=c+1 has only the lawful Mann families.

The first three diagonal cells are found by exact first-prime search below the deterministic Miller-Rabin bound used by the script. The larger cells are explicit diagonal-scale primes whose primality is checked by Pocklington from the included factorization of p-1.

diagonal-n8192:  injective=True, collision_count=0
diagonal-n16384: injective=True, collision_count=0
diagonal-n32768: injective=True, collision_count=0
diagonal-n65536: injective=True, collision_count=0
diagonal-n131072: injective=True, collision_count=0
diagonal-n262144: injective=True, collision_count=0
diagonal-n524288: injective=True, collision_count=0
diagonal-n1048576: injective=True, collision_count=0
G173-accident: non-injective, collision_count=3
n512-offdiag-accident: non-injective, collision_count=3

Scope

Finite exact-arithmetic evidence only. This is not a production n=2^30 closure, not a prize claim, and not a proof that the first prime after round(n^(158/30)) always works.

This builds on the G139 accident-census lane in PR #526 and Geoff's order-512 census comment on #466.

Verification

• python3 scripts/probes/probe_g139_phi_certificate.py
• python3 -m py_compile scripts/probes/probe_g139_phi_certificate.py
• git diff --check
• python3 scripts/forbidden_tokens.py