Home › Cubature › X₂₃ and X₂₄ Locally hosted existence theorem · 3 September 2026 Positive H₃-invariant cubature on X₂₃ and X₂₄ X₂₃ admits strictly positive H₃-invariant cubature weights exact through degree 71, and X₂₄ through degree 75. No maximality assertion is included. The proof is exact linear algebra over ℚ(√5). This statement does not enlarge Paper VI v1.3, whose proved range remains 1 ≤ f ≤ 9. On this page Claim boundary Binary64 correction Theorem Invariant reduction Exact field Finite systems Certificate substitution Exact positivity Rank corollary Printed finite audit Quoted replay Publication status: six Zenodo preprints and one Zenodo report. Paper VII is version 1.6 at DOI 10.5281/zenodo.22233573. Not peer reviewed unless a specific work is later linked to a peer-reviewed publication. Computational reproducibility: exact verifiers and/or finite certificates are supplied where listed. Independent third-party reproduction: none recorded on this site unless a named reproduction and its public evidence are linked. Source: locally hosted theorem text X23_X24_positive_existence.md · 7685 bytes · SHA-256 d6b2ffb747a1b373a9dced0ff5089b395b226b80d0365367b97cb4f3f72317ea Copy . Not a Zenodo preprint. No DOI. Does not enlarge Paper VI v1.3. Companion dossier: OMEGA_NAUGHT_Typed_Network_Exact_Proof_Dossier_v1.0.pdf · 1105802 bytes · SHA-256 2e4c1f1afd3a6fa1b9b1076722d7589ba89cbea07be3a8ab258f5a44a142e3e8 Copy . Printed finite-audit extract: ClassI_Cubature_f23_f24_audit_extract_v1.1.json · SHA-256 354976dc65ddcb7194e8f46fe9651234b4bdaea51b25cc9f82fa0d0815bf4688 Copy . The 5 September 2026 typed-network dossier cites a complete f=23,24 coefficient certificate, exact verifier, and 109-residual ledger as LaTeX filecontents. Those filecontents are not extractable from the hosted PDF. This archive enumerates the printed finite audit data (grid sizes, free orbits, rank witnesses, minima, canonical weight-text hashes, cited payload hashes) and the PDF itself. The 117 exact a+b√5 coefficient pairs and the runnable verifier are not hosted as separate artifacts. Implementation conformance is not run. Independent third-party reproduction is currently recorded as none. A same-environment replay quoted in the dossier is not a clean-room reproduction. Neither argument proves failure or impossibility at any higher degree. Claim boundary This page states an existence theorem on the two fixed grids X 23 X 23 ​ and X 24 X 24 ​ . It is independent of Ω 0 Ω 0 ​ and of SRSD. It does not rewrite frozen CLASS_I_CUBATURE/1.0 or G6_RELEASE/1.1. CLASS_I_CUBATURE/1.1 remains the frequency-22 correction candidate. The 5 September 2026 typed-network dossier reprints the finite sizes, free-orbit rationals, rank witnesses, minima, and canonical weight-text hashes. The 117 exact a + b 5 a + b 5 ​ masses and the runnable verifier are cited as LaTeX filecontents* and are not extractable from the hosted PDF. Implementation conformance is not run. Independent third-party reproduction is currently recorded as none. No maximality assertion is included. Neither argument proves failure or impossibility at any higher degree. Binary64 minimum-orbit correction Correction (certificate format 1.1). Candidate v1.0 evaluated a+b√5 in binary64. Cancellation produced incorrect printed minima and used floating comparison to select the minimum orbit. Exact algebraic comparison names (21,1,1) at f=23 and (23,1,0) at f=24. Exact weight vectors, moment identities, positivity, rank witnesses, and canonical weight-text hashes are stated to be unchanged. This is the same defect class as the hosted frequency-22 correction that had named ( 22 , 0 , 0 ) ( 22 , 0 , 0 ) instead of ( 11 , 11 , 0 ) ( 11 , 11 , 0 ) . The two format families are distinct: CLASS_I_CUBATURE/1.1 remains the frequency-22 candidate; CLASS_I_CUBATURE_F23_F24_EXACT/1.1 is the later f=23,24 wrapper. Theorem Let X f ⊂ S 2 X f ​ ⊂ S 2 be the fixed radially projected Class-I icosahedral grid. Then X 23 admits strictly positive H 3 -invariant cubature weights exact through degree 71 , X 23 ​ admits strictly positive H 3 ​ -invariant cubature weights exact through degree 71 , and X 24 admits strictly positive H 3 -invariant cubature weights exact through degree 75. X 24 ​ admits strictly positive H 3 ​ -invariant cubature weights exact through degree 75. No maximality assertion is included. The numerical targets coincide with t 23 = 3 ⋅ 23 + 2 = 71 t 23 ​ = 3 ⋅ 23 + 2 = 71 and t 24 = 3 ⋅ 24 + 3 = 75 t 24 ​ = 3 ⋅ 24 + 3 = 75 from the Paper VI formula. That coincidence does not import the Paper VI theorem onto these frequencies. 1. Reduction to invariant polynomials Let H 3 H 3 ​ be the full icosahedral reflection group. For an orbit O λ ⊂ X f O λ ​ ⊂ X f ​ , let W λ W λ ​ denote its total mass and assign each node x ∈ O λ x ∈ O λ ​ the weight w x = W λ / ∣ O λ ∣ w x ​ = W λ ​ /∣ O λ ​ ∣ . Define Q f [ p ] = ∑ x ∈ X f w x p ( x ) . Q f ​ [ p ] = x ∈ X f ​ ∑ ​ w x ​ p ( x ) . Both Q f Q f ​ and normalized spherical integration I [ p ] = ∫ S 2 p d σ I [ p ] = ∫ S 2 ​ p d σ are H 3 H 3 ​ -invariant. For any polynomial p p , form its Reynolds average R p = 1 ∣ H 3 ∣ ∑ g ∈ H 3 p ∘ g . R p = ∣ H 3 ​ ∣ 1 ​ g ∈ H 3 ​ ∑ ​ p ∘ g . Then Q f [ p ] = Q f [ R p ] Q f ​ [ p ] = Q f ​ [ R p ] and I [ p ] = I [ R p ] I [ p ] = I [ R p ] . Consequently, Q f [ p ] = I [ p ] Q f ​ [ p ] = I [ p ] for every polynomial of degree at most t t if it holds for every H 3 H 3 ​ -invariant polynomial of degree at most t t . The invariant ring of the reflection group H 3 H 3 ​ is a polynomial algebra on homogeneous generators of degrees 2 , 6 , 10 2 , 6 , 10 . On S 2 S 2 , the degree-two generator x 2 + y 2 + z 2 x 2 + y 2 + z 2 equals 1. Hence invariant polynomial restrictions are generated by two functions of degrees 6 and 10. Let V V be the twelve unnormalized icosahedral vertices and R 2 = φ + 2 R 2 = φ + 2 . Define S m ( x ) = ∑ v ∈ V ( x ⋅ v R ) m . S m ​ ( x ) = v ∈ V ∑ ​ ( R x ⋅ v ​ ) m . The verifier uses U 6 = S 6 U 6 ​ = S 6 ​ and U 10 = S 10 U 10 ​ = S 10 ​ . These raw power sums are an equivalent generator basis. If H 6 , H 10 H 6 ​ , H 10 ​ denote the corresponding harmonic generators obtained from the Legendre orbit sums, direct expansion on S 2 S 2 gives H 6 = 231 S 6 − 396 16 , H 6 ​ = 16 231 S 6 ​ − 396 ​ , H 10 = 46189 S 10 − 114114 S 6 + 145236 256 . H 10 ​ = 256 46189 S 10 ​ − 114114 S 6 ​ + 145236 ​ . The leading coefficients are nonzero, so this is an invertible triangular change of generators. Therefore every invariant restriction of degree at most t t is a linear combination of U 6 r U 10 s U 6 r ​ U 10 s ​ with 6 r + 10 s ≤ t 6 r + 10 s ≤ t . Thus it is sufficient to verify ∑ λ W λ U 6 ( x λ ) r U 10 ( x λ ) s = ∫ S 2 U 6 r U 10 s d σ (1) λ ∑ ​ W λ ​ U 6 ​ ( x λ ​ ) r U 10 ​ ( x λ ​ ) s = ∫ S 2 ​ U 6 r ​ U 10 s ​ d σ ( 1 ) for all such ( r , s ) ( r , s ) . 2. Exact field of computation Put K = Q ( 5 ) K = Q ( 5 ​ ) . For the orbit indexed by a ≥ b ≥ c ≥ 0 a ≥ b ≥ c ≥ 0 , a + b + c = f a + b + c = f , take p a , b , c = a v 1 + b v 2 + c v 3 p a , b , c ​ = a v 1 ​ + b v 2 ​ + c v 3 ​ with v 1 = ( 0 , 1 , φ ) , v 2 = ( − 1 , φ , 0 ) , v 3 = ( 1 , φ , 0 ) . v 1 ​ = ( 0 , 1 , φ ) , v 2 ​ = ( − 1 , φ , 0 ) , v 3 ​ = ( 1 , φ , 0 ) . All coordinates of p a , b , c p a , b , c ​ , its squared norm, and its scalar products with the vertices lie in K K . Because the degrees 6 and 10 are even, U 6 ( x a , b , c ) ∈ K U 6 ​ ( x a , b , c ​ ) ∈ K and U 10 ( x a , b , c ) ∈ K U 10 ​ ( x a , b , c ​ ) ∈ K . The spherical monomial identity ∫ S 2 x 2 i y 2 j z 2 k d σ = ( 2 i − 1 ) ! ! ( 2 j − 1 ) ! ! ( 2 k − 1 ) ! ! ( 2 i + 2 j + 2 k + 1 ) ! ! ∫ S 2 ​ x 2 i y 2 j z 2 k d σ = ( 2 i + 2 j + 2 k + 1 )!! ( 2 i − 1 )!! ( 2 j − 1 )!! ( 2 k − 1 )!! ​ and the vanishing of monomials with an odd exponent show that every right-hand side in (1) also lies in K K . Hence the entire moment system A f W f = b f A f ​ W f ​ = b f ​ is an exact linear system over Q ( 5 ) Q ( 5 ​ ) . 3. The finite systems The Class-I triangulation has 20 f 2 20 f 2 faces and 30 f 2 30 f 2 edges, so Euler’s formula gives ∣ X f ∣ = 10 f 2 + 2 ∣ X f ​ ∣ = 10 f 2 + 2 . The H 3 H 3 ​ -orbits are indexed by partitions of f f into at most three parts. For f = 23 f = 23 , ∣ X 23 ∣ = 5292 ∣ X 23 ​ ∣ = 5292 and there are 56 orbits. The number of invariant rows is N ( 71 ) = ∑ s = 0 7 ( ⌊ 71 − 10 s 6 ⌋ + 1 ) = 12 + 11 + 9 + 7 + 6 + 4 + 2 + 1 = 52. N ( 71 ) = s = 0 ∑ 7 ​ ( ⌊ 6 71 − 10 s ​ ⌋ + 1 ) = 12 + 11 + 9 + 7 + 6 + 4 + 2 + 1 = 52. Therefore A 23 ∈ K 52 × 56 A 23 ​ ∈ K 52 × 56 . For f = 24 f = 24 , ∣ X 24 ∣ = 5762 ∣ X 24 ​ ∣ = 5762 and there are 61 orbits, and N ( 75 ) = ∑ s = 0 7 ( ⌊ 75 − 10 s 6 ⌋ + 1 ) = 13 + 11 + 10 + 8 + 6 + 5 + 3 + 1 = 57. N ( 75 ) = s = 0 ∑ 7 ​ ( ⌊ 6 75 − 10 s ​ ⌋ + 1 ) = 13 + 11 + 10 + 8 + 6 + 5 + 3 + 1 = 57. Thus A 24 ∈ K 57 × 61 A 24 ​ ∈ K 57 × 61 . 4. Exact certificate substitution Let W 23 W 23 ​ and W 24 W 24 ​ be the explicit vectors encoded in a hash-pinned exact certificate. Every entry is stored as W λ = a λ + b λ 5 W λ ​ = a λ ​ + b λ ​ 5 ​ with a λ , b λ ∈ Q a λ ​ , b λ ​ ∈ Q . The exact verifier reconstructs from the definitions: all orbit representatives; every U 6 ( x λ ) U 6 ​ ( x λ ​ ) and U 10 ( x λ ) U 10 ​ ( x λ ​ ) ; every spherical moment b r , s b r , s ​ ; the complete matrices A 23 A 23 ​ and A 24 A 24 ​ . It then performs arithmetic on rational pairs representing a + b 5 a + b 5 ​ . Direct substitution gives A 23 W 23 − b 23 = 0 A 23 ​ W 23 ​ − b 23 ​ = 0 in all 52 coordinates and A 24 W 24 − b 24 = 0 A 24 ​ W 24 ​ − b 24 ​ = 0 in all 57 coordinates. Every residual is the exact pair ( 0 , 0 ) ( 0 , 0 ) , not a floating-point approximation. The complete 109 residuals are recorded in the verification ledger. The normalization identities are ∑ λ = 1 56 W 23 , λ = 1 , ∑ λ = 1 61 W 24 , λ = 1. λ = 1 ∑ 56 ​ W 23 , λ ​ = 1 , λ = 1 ∑ 61 ​ W 24 , λ ​ = 1. Those pinned vectors, verifier, and ledger are cited as LaTeX filecontents in the 5 September 2026 dossier. They are not extractable from the hosted PDF and are not hosted as separate artifacts. This section is the substitution argument of the proof, not a replayable certificate package on this site. 5. Exact positivity For z = a + b 5 ∈ K z = a + b 5 ​ ∈ K , its real sign is decidable using rational arithmetic. When a a and b b have opposite signs, comparison with zero reduces to comparing a 2 a 2 and 5 b 2 5 b 2 . Because a , b ∈ Q a , b ∈ Q , equality a 2 = 5 b 2 a 2 = 5 b 2 is impossible unless a = b = 0 a = b = 0 . Therefore the comparison is exact. Applying this test gives W 23 , λ > 0 W 23 , λ ​ > 0 for all 56 orbits and W 24 , λ > 0 W 24 , λ ​ > 0 for all 61 orbits. Hence every per-node weight w x = W λ / ∣ O λ ∣ w x ​ = W λ ​ /∣ O λ ​ ∣ is strictly positive. Combining positivity, normalization, the 109 identities (1), and the invariant-reduction argument proves that X 23 X 23 ​ is exact through degree 71 and X 24 X 24 ​ is exact through degree 75. This proves the existence theorem. Rank and affine-dimension corollary Take p = 1000000000000000031 p = 1000000000000000031 . This integer is prime. The two numbers r + = 968101896520624317 , r − = 31898103479375714 r + ​ = 968101896520624317 , r − ​ = 31898103479375714 satisfy r ± 2 ≡ 5 ( m o d p ) r ± 2 ​ ≡ 5 ( mod p ) and r + + r − ≡ 0 ( m o d p ) r + ​ + r − ​ ≡ 0 ( mod p ) . Thus they define the two reductions of Q ( 5 ) Q ( 5 ​ ) into F p F p ​ . All denominators appearing in the chosen minors are invertible modulo p p . For f = 23 f = 23 , deleting the four declared free columns gives a 52 × 52 52 × 52 minor M 23 M 23 ​ , with det ⁡ M 23 ( + ) ≡ 650097745717062046 ≢ 0 ( m o d p ) , det M 23 ( + ) ​ ≡ 650097745717062046  ≡ 0 ( mod p ) , det ⁡ M 23 ( − ) ≡ 139778200766197651 ≢ 0 ( m o d p ) . det M 23 ( − ) ​ ≡ 139778200766197651  ≡ 0 ( mod p ) . If det ⁡ M 23 = 0 det M 23 ​ = 0 in K K , every defined modular image would be zero. Therefore det ⁡ M 23 ≠ 0 det M 23 ​  = 0 and rank ⁡ A 23 = 52 rank A 23 ​ = 52 . Hence its solution space has affine dimension 56 − 52 = 4 56 − 52 = 4 . Likewise, for the selected 57 × 57 57 × 57 minor at f = 24 f = 24 , det ⁡ M 24 ( + ) ≡ 550949013298233594 ≢ 0 ( m o d p ) , det M 24 ( + ) ​ ≡ 550949013298233594  ≡ 0 ( mod p ) , det ⁡ M 24 ( − ) ≡ 966277299582556251 ≢ 0 ( m o d p ) . det M 24 ( − ) ​ ≡ 966277299582556251  ≡ 0 ( mod p ) . Consequently rank ⁡ A 24 = 57 rank A 24 ​ = 57 and dim ⁡ { W : A 24 W = b 24 } = 61 − 57 = 4 dim { W : A 24 ​ W = b 24 ​ } = 61 − 57 = 4 . Therefore the final proved statement is X 23 admits a strictly positive H 3 -invariant degree- 71 cubature rule, X 23 ​ admits a strictly positive H 3 ​ -invariant degree- 71 cubature rule, X 24 admits a strictly positive H 3 -invariant degree- 75 cubature rule. X 24 ​ admits a strictly positive H 3 ​ -invariant degree- 75 cubature rule. Neither argument proves failure or impossibility at any higher degree. Printed finite-audit data The following values are printed in the dossier and recorded in the extract. They do not replace the unhosted 117 coefficient pairs. Grid, rank, and certified minima printed in the 5 September 2026 dossier. f Nodes Orbits Rows / rank Affine dim. Degree Minimum Second-smallest 23 5292 56 52 / 52 4 71 ( 21 , 1 , 1 ) ( 21 , 1 , 1 ) ( 23 , 0 , 0 ) ( 23 , 0 , 0 ) 24 5762 61 57 / 57 4 75 ( 23 , 1 , 0 ) ( 23 , 1 , 0 ) ( 16 , 8 , 0 ) ( 16 , 8 , 0 ) Declared free-orbit masses, stored as reduced rationals in Q. f Orbit Weight 23 ( 17 , 5 , 1 ) ( 17 , 5 , 1 ) 82833 / 5496019 82833/5496019 23 ( 16 , 6 , 1 ) ( 16 , 6 , 1 ) 776496 / 85219693 776496/85219693 23 ( 13 , 9 , 1 ) ( 13 , 9 , 1 ) 67140 / 3095507 67140/3095507 23 ( 10 , 8 , 5 ) ( 10 , 8 , 5 ) 2643202 / 97050439 2643202/97050439 24 ( 15 , 8 , 1 ) ( 15 , 8 , 1 ) 687439 / 38273329 687439/38273329 24 ( 15 , 7 , 2 ) ( 15 , 7 , 2 ) 387392 / 39075271 387392/39075271 24 ( 13 , 7 , 4 ) ( 13 , 7 , 4 ) 2224115 / 77459599 2224115/77459599 24 ( 12 , 10 , 2 ) ( 12 , 10 , 2 ) 1469825 / 87141847 1469825/87141847 Canonical weight-text hashes, computed from the exact lines a,b,c:aNumerator/aDenominator:bNumerator/bDenominator in orbit order: f=23: f56551614718e7fd45de5a95cd01767a0060904b96ae2a42c13f99b165c0f9f7 Copy f=24: 86a96d2c91fffa5cbe2382299eb31353635d3d20396a61ae40a03bfd163f2d27 Copy Cited one-line certificate SHA-256 e886510ec03afe014ad143d6d0b45677fc2bb58a7df495374b5780cf79d5b5af Copy . That file is not hosted here. Quoted same-environment replay The dossier records a same-environment exact replay on 5 September 2026. It is not an external clean-room reproduction, not peer review, and not a verifier run executed by this archive. PASS f=23 tau=71 nodes=5292 orbits=56 rows=52 rank=52 free=4 min=(21, 1, 1) PASS f=24 tau=75 nodes=5762 orbits=61 rows=57 rank=57 free=4 min=(23, 1, 0) ALL PASS: 109 exact invariant moment identities; strict positivity and rank certified over Q(sqrt(5)). certificate_sha256 e886510ec03afe014ad143d6d0b45677fc2bb58a7df495374b5780cf79d5b5af Typed network dossier · Return to the cubature audit · Frequency-22 audit
