Home › Cubature Papers IV and VI Cubature audit No property of Ω₀, SRSD, or the rational floor map is required for the results on this page. The claims below concern finite radial Class-I icosahedral grids and exact linear algebra over ℚ(√5). The only surviving SRSD statement is generic lossless external representation. On this page Class-I construction Frequency-22 correction Harmonic obstruction Positive cubature 1 ≤ f ≤ 9 10 ≤ f ≤ 21 Frequency 22 X₂₃ and X₂₄ Open from f ≥ 25 Source works: Paper IV, 10.5281/zenodo.22033917 ; Paper VI, 10.5281/zenodo.22068873 . Typed compatibility of Paper VI with Papers I–V is an interoperability statement; it does not make the cubature theorem logically depend on Ω₀, SRSD, or the rational floor map. This page supplies. Class-I cubature ranges, certificates, and remaining public packaging status. Hypotheses / scope. No property of Ω₀ or SRSD is required. Ranges 1–9, 10–21, 22, 23–24, and ≥25 stay separate. Evidence. Written proof · Exact certificates for 1–9 and 22 · Existence statement for 23–24 without the 117 masses · 10–21 unpackaged · ≥25 open. Publication: not peer reviewed. Independent third-party reproduction: none recorded. Full status record . 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 Not peer reviewed unless a specific work is later linked to a peer-reviewed publication.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. How a Class-I rule is built A radial Class-I grid is generated at frequency f. The icosahedral group H₃ partitions nodes into orbits. Orbit mass is the total weight of an orbit; per-node weight is that mass divided by orbit size. Polynomial exactness means spherical moments through a stated degree are reproduced exactly. Invariant moment equations reduce the linear system. Exact arithmetic over ℚ(√5) decides positivity or obstruction. Class-I grid construction and orbit masses Schematic of a radial Class-I icosahedral grid: nodes grouped into symmetry orbits, orbit mass distinct from per-node weight, polynomial exactness as matching spherical moments. Class-I nodes by H₃ orbits Grid: radial Class-I frequency f Orbit mass ≠ per-node weight Exactness: spherical moments through degree t Invariant equations over ℚ(√5) Exact arithmetic: positivity or obstruction Independent of Ω₀ and of SRSD Schematic only. Frequencies 1–9 have published certificates; 10–21 remain unpackaged; 22 has a positive degree-69 certificate and a degree-71 obstruction; 23–24 have a stated existence theorem whose 117 exact coefficient pairs are not extractable from the hosted dossier PDF; f ≥ 25 is open. Frequency-22 correction Correction (August 27, 2026). The earlier summary selected the minimum positive degree-69 orbit after binary64 conversion and incorrectly named (22, 0, 0). Exact comparison in ℚ(√5) gives (11, 11, 0) with mass 0.003874805888537519508839054528…. The Julia verifier also contained a stale assertion and a scalar-transpose runtime error; both are corrected in this release. The corrected Python and Julia verifiers pass separately and emit the unchanged canonical degree-69 and degree-71 solution digests. Neither run is independent third-party reproduction. The 24 August 2026 research-addendum PDF is superseded by v1.1, which reports the corrected orbit, both CERTIFICATE PASS runtimes, and the unchanged theorem. The central result is unchanged: degree 69 is strictly positive, while the unique forced degree-71 solution has negative mass at (21, 1, 0). The theorem and the two canonical solution digests are unchanged. The corrected minimum positive degree-69 orbit is ( 11 , 11 , 0 ) ( 11 , 11 , 0 ) with orientation mass 0.003874805888537519508839054528… . The unique forced negative degree-71 mass remains at ( 21 , 1 , 0 ) ( 21 , 1 , 0 ) with orientation -0.006609660200017031631251308016577647 . Open the frequency-22 audit Claim-status table Proved ranges, separately certified cases, hosted existence statements, and open cases are listed as separate rows. Do not compress them into a single all-frequency theorem. Cubature claims with separate mathematical, evidence, and review axes. Claim Scope Mathematical status Evidence Machine-checkable? Publication/review Equal-weight harmonic obstruction Paper IV: every complete radial Class-I grid (five-design); limiting equal-weight measure mu_rad (degree-six obstruction) ; radial Class-I grids and their weak limit mu_rad theorem claim paper proof; exact invariant evaluation where supplied Zenodo preprint ; not peer reviewed Positive cubature exactness for 1 ≤ f ≤ 9 1 to 9 ; fixed radial Class-I grids theorem claim paper proof; exact moment system; exact weights yes where artifact supplied Zenodo preprint ; not peer reviewed Positivity of orbit masses for 1 ≤ f ≤ 9 1 to 9 ; fixed radial Class-I grids theorem claim exact inequalities; exact weights yes Zenodo preprint ; not peer reviewed Target-degree exactness for 10 ≤ f ≤ 21 10 to 21 ; fixed radial Class-I grids unpackaged author result; not part of the fully reproducible release corpus CLASS_I_CUBATURE/1.1 records the range as metadata only; weight files, moment-system specification, verifier, and transcript are not enumerated on this site unpackaged; not part of the fully reproducible release corpus Zenodo preprint-associated artifact ; not peer reviewed Positive degree-69 rule on the fixed frequency-22 grid 22 ; fixed radial Class-I grid certificate-backed theorem claim exact moment system; rank certificate; exact positive weights yes Zenodo preprint-associated artifact ; not peer reviewed Degree-70/71 obstruction on the fixed frequency-22 grid 22 ; fixed radial Class-I grid obstruction forced negative weight; exact dual certificate yes Zenodo preprint-associated artifact ; not peer reviewed Strictly positive H₃-invariant rules on X₂₃ and X₂₄ 23 and 24 ; fixed radial Class-I grids X_23 and X_24 theorem claim locally hosted existence proof; 5 September 2026 typed-network dossier (printed finite audit data); H_3-invariant reduction to U_6^r U_10^s; exact linear system over Q(sqrt(5)); certificate substitution and exact positivity as stated in the proof proof and printed finite-audit extract hosted; 117 exact Q(sqrt(5)) masses, runnable verifier, and 109-residual ledger file are cited as LaTeX filecontents and are not extractable from the hosted PDF locally hosted theorem statement ; not peer reviewed Rank and affine dimension four on the f=23 and f=24 moment systems 23 and 24 ; H_3-orbit moment systems on X_23 and X_24 corollary two-sided modular determinants of chosen minors, as stated in the hosted proof and reprinted in the 5 September 2026 dossier modular determinant argument hosted; chosen minors are not separately hashed on this site locally hosted theorem statement ; not peer reviewed Existence of strictly positive H_3-orbit-constant rules of target degree t_f for every f ≥ 25 25 and above ; fixed radial Class-I grids conjecture none supplied no Zenodo preprint-associated artifact ; not peer reviewed Cross-language replay Named implementation and release only (G6_RELEASE/1.1, f ≤ 9) Implementation property Reproduction of frozen outputs Yes Not theorem verification Equal-weight harmonic obstruction Every complete radial Class-I grid is a spherical five-design. The limiting equal-weight sampling measure differs from uniform spherical area first at harmonic degree six. Paper IV also gives finite-frequency computations and an asymptotic expansion for the degree-six moment. The displayed exact invariant evaluation is a paper theorem, not a site-wide certificate. Paper IV DOI Positive cubature, 1 ≤ f ≤ 9 Paper VI v1.3 proves a strictly positive H 3 H 3 ​ -orbit-constant rule of degree t f t f ​ on each frozen grid X f X f ​ for 1 ≤ f ≤ 9 1 ≤ f ≤ 9 . Target degrees are t f = { 3 f + 2 , f odd , 3 f + 3 , f even. t f ​ = { 3 f + 2 , 3 f + 3 , ​ f odd , f even. ​ Exact weights: orbit_masses_f1_f9.csv . Frequency-9 affine family: f9_affine_family.csv . Frozen contract: CLASS_I_CUBATURE/1.0 . Replay of those frozen files is an implementation property of G6_RELEASE/1.1. It does not enlarge Paper VI beyond f = 9. Reproduce these certificates Target-degree range 10 ≤ f ≤ 21 A post-publication computation is reported for a strictly positive target-degree rule on each fixed Class-I grid with 10 ≤ f ≤ 21 10 ≤ f ≤ 21 . Maximality is not claimed. This is not Paper VI v1.3, and it is not the f = 22 obstruction. Provenance gap: the proposed additive contract CLASS_I_CUBATURE/1.1 currently records only the metadata positive_target_degree_rule on that range. It does not enumerate frozen weight files, a moment-system specification, a verifier, SHA-256 pins, or an expected transcript for f = 10, …, 21. Those objects are therefore not exposed here at the same standard as the frequency-22 package. This range is an unpackaged author result, not part of the fully reproducible release corpus. Frozen CLASS_I_CUBATURE/1.0 and G6_RELEASE/1.1 bytes are unchanged. Frequency 22 on the fixed grid X₂₂ On the frozen node set X 22 X 22 ​ a strictly positive orbit-constant rule exists through degree 69, and no nonnegative weights on those same nodes reach degree 70. See the dedicated audit. Audit this result X₂₃ and X₂₄ A locally hosted existence theorem states that X 23 X 23 ​ admits strictly positive H 3 H 3 ​ -invariant cubature weights exact through degree 71, and X 24 X 24 ​ through degree 75. No maximality assertion is included. The argument is exact linear algebra over Q ( 5 ) Q ( 5 ​ ) . It does not enlarge Paper VI v1.3. The 5 September 2026 typed-network dossier reprints the finite sizes, free orbits, minima ( 21 , 1 , 1 ) ( 21 , 1 , 1 ) and ( 23 , 1 , 0 ) ( 23 , 1 , 0 ) , rank witnesses, and canonical weight-text hashes. The 117 exact a + b 5 a + b 5 ​ masses and the runnable verifier are not extractable from the hosted PDF. Implementation conformance is not run. Read the existence theorem · Typed network dossier · X23_X24_positive_existence.md Open from f ≥ 25 Existence of strictly positive H 3 H 3 ​ -orbit-constant rules of target degree t f t f ​ for every f ≥ 25 f ≥ 25 remains an open conjecture. Neither the X 23 X 23 ​ nor the X 24 X 24 ​ argument proves failure or impossibility at any higher degree.
