Matthew Alan Newman · 23 August 2026 · additive site update

OMEGA NAUGHT Site Update Data

Papers I–VI / G₆ additive extension. The existing OMEGA_NAUGHT/1.0 contract is retained unchanged. Paper VI is added as a distinct positive-cubature component.

1. Frozen contract

contract=OMEGA_NAUGHT/1.0
algebra=Q(T)[lambda]/(lambda^2-T*lambda-T^2)
bits=15
Q=339949771344778
S=1000000000000000
L60=339949771344778929576702809321279193551009905014259416689444
U60=339949771344778929576702809321279193551009905014259416689445

Do not rename, rewrite, or reinterpret this contract.

2. Paper VI

Newman, M. (2026). Exact Positive Icosahedral Cubature on Fixed Radial Class-I Grids: Quadratic-Field Interoperability, Exact Replay, and Typed Compatibility with Papers I–V. Zenodo. doi:10.5281/zenodo.22068873.

3. Replace G₅ by G₆

G₆ = (S_SRSD, A_Ω, D_quant, D_rat, H_ico, C_fin, C_cub; J₆)

C_cub is the positive fixed-Class-I cubature component. SRSD remains the sole frozen foundation. The tuple is a typed assembly; it does not assert that every component is derived from every other.

4. Cubature data

|X_f| = 10f² + 2. Strictly positive H₃-orbit-constant probability cubature is certified for 1 ≤ f ≤ 9. Exact degrees 5, 9, 11, 15, 17, 21, 23, 27, 29. Equivalently t_f = 3f+2 (f odd) or 3f+3 (f even). The published all-frequency continuation is open for f ≥ 10.

5. Frequency nine

Rank 11 with 12 orbit variables: W(t) = W^part + t W^null, with W₃₃₃(t) = t. Complete strict-positivity domain t_L < t < t_U, where t_L = −W^part₉₀₀ / W^null₉₀₀ and t_U = −W^part₈₁₀ / W^null₈₁₀. Orientation decimals: t_L ≈ 0.0261933635545922987, t_U ≈ 0.0380455174948779754. Witness t = 1/27. Separate exact affine-family members maximize the minimum total orbit mass and the minimum per-node weight.

6. Coefficient map

a + b√5 ↦ (a − b) + (2b/T) λ

Native field Q(√5). Inside K_Ω, √5 = 2λ/T − 1. Dependency: Class-I geometry → invariant moments → Q(√5) cubature coefficients → OMEGA representation. Do not reverse this into a claim that OMEGA generates or proves the cubature theorem.

7. Same nodes, distinct measures

supp μ_f = supp ν_f = X_f, but generally μ_f ≠ ν_f. Paper VI reweights the same fixed Class-I nodes; it does not alter the node set.

8. SRSD compatibility

D_s E_s = id. The cubature measure may be transported losslessly by ν̂_f,s = (E_s)# ν_f. This is transport of an already-established rule, not a derivation of the cubature theorem.

9. CSV hashes

orbit_masses_f1_f9.csv
eed325c55fe183b36b46b591f758935bcf748a91c2427a09a965337895354b98

f9_affine_family.csv
851a5ff03452e21f7839285cd63423385c936907568e852a174056197c20a95f

10. Nested contracts

CLASS_I_CUBATURE/1.0
NEWMAN_GLOBAL_OBJECT/1.0

Implementation identifiers, not new publication titles or DOI records.

11. Status language

After exact Paper-VI certificate replay and global cross-runtime checks pass:

OMEGA_NAUGHT/1.0 ................. PASS
CLASS_I_CUBATURE/1.0 ............. PASS
NEWMAN_GLOBAL_OBJECT/1.0 ......... PASS

GLOBAL OBJECT CONFORMANCE = PASS

12. Full cubature runner checklist

CSV hashes; |X_f|; invariant systems 1≤f≤9; stored coefficients; probability normalization; strict positivity; degree sequence; omitted residuals; rank M₉=11; affine family; positivity interval; t=1/27; orbit-mass max–min; per-node max–min; every Q(√5)→OMEGA translation. Certificate replay of the published CSVs is what the live runners certify. Native reconstruction of spherical moment matrices is not claimed.

13–14. Final rule

G₅ → G₆ = G₅ + C_cub + new typed interfaces, with OMEGA_NAUGHT/1.0 unchanged.

Live companion · Conformance spec · TeX