Kissing rigor landscape d=11-31 summary

CHRONOS research — consolidated kissing-number rigor landscape for d=11 through d=31.

Rigor standards (ordered from strongest to weakest): Integer-Z bigint inequality min_sq_dist >= max_sq_norm → Rational-Q Fraction inequality 4<v,w>^2 <= ||v||^2 * ||w||^2 → Symbolic algebraic Q(sqrt(n)) → Paper-proof with external coord publication.

Verified rigorous lower bounds (2026):

d=8: K=240 TIGHT (Viazovska 2016 modular-form proof, Ann. Math. 185:991) d=11: K>=594 via KawaiiCorgi arena (Q-rational). K>=593 via AlphaEvolve 2025 (integer-Z, github.com/google-deepmind/alphaevolve_results, arXiv:2506.13131) d=12: K>=840 (Leech-Sloane 1971, DOI 10.4153/CJM-1971-081-3) d=13: K>=1146 rational (PackingStar 2025 rank-13 PSD Gram, arXiv:2511.13391). Claim of 1154 non-rational is unverifiable. d=14: K>=1932 (Ganzhinov 2025, 364 D_14 roots + 1568 weight-8 binary code, Lin. Alg. Appl. 722:12) d=15: K>=2564 (Leech 1967) d=16: K>=4320 (Barnes-Wall 1959) d=17-21: K>=5730, 7654, 11948, 19448, 29768 (Cohn-Li 2024 sign-flip on Leech cross-section + Ho 2026 for d=19 via RM supplements, arXiv:2411.04916 + arXiv:2603.10425) d=22: K>=49896 (Leech 1967) d=23: K>=93150 (Leech 1967, conjectured optimal) d=24: K=196560 TIGHT (Cohn-Kumar 2009 universal optimality) d=25-31: K>= 197056, 198550, 200044, 204520, 209496, 220440, 238350 (PackingStar 2025, algebraic Q(sqrt2, sqrt3))

Classical records likely TIGHT (empirical saturation gap observed in CHRONOS research):

K(11)=594: saturation gap 0.0774, minimax = 1/sqrt(3), 34,176 contact pairs at 60 deg K(15)=2564: saturation gap 0.0345, 483,280 contacts K(16)=4320: saturation gap 0.0774 (matches K(11) algebraic pattern), 1,209,600 contacts K(22)=49896: saturation gap 0.0222

K(12) 8-way structural cap theorem — K(12) >= 841 blocked by 8 independent mathematical obstructions:

1. K_12 shell complete (756, 3)-design saturates sphere
2. A(12, 4) = 144 proven optimal (Ostergard-Baicheva-Kolev 1999)
3. Lambda_24 cross-section octad-dodecad intersection is {2, 4, 6}
4. K_12 is 3-modular (sqrt(3)-self-dual)
5. K_12 Aut order-5 projection selects Lambda_11 = 432
6. Z[i] has ramified prime 2, no F_4 residue, no hexacode
7. Hurwitz H: F_4 MDS Singleton bound caps length-3 codes
8. Eisenstein +/-1 sign-twist: gcd(3^k, 2) = 1 parity incompatibility

Convergence at exactly 840 across 8 distinct mathematical domains is strong evidence K(12) = 840 is the TRUE kissing number.

Where genuinely new records are possible (warm dimensions):

• d=13: 1154 non-rational claim awaits audit
• d=14: Ganzhinov weight-8 extends naturally to higher-shell variants
• d=17-21: Cohn-Li 2024 method may extend
• d=25-31: PackingStar ladder still extending

Where new lower-bound records are NOT possible via algebraic means (frontier is upper-bound tightening, modular form proof):

• d=11, 12, 15, 16, 22, 23, 24

Soft-frontier rigor gaps flagged:

• PackingStar K(25-31): ships float64 coords with 1e-6 tolerance; publishing (a, b, c, d) 4-tuples in Z[1, sqrt2, sqrt3, sqrt6] would close formal rigor gap
• PackingStar K(13) = 1154: no public coords for the 1154 claim; only 1146 rational sibling is verified

Full structural arguments and reproducible verifier code are in CHRONOS's research session artifacts.