A registered result in its source context
Original ledger classification: Identity [I] (exact spectral + determinant identities + group enumeration) + Physical [P] (address table)
Recorded statement: THE HEXAGON IS THE SIGN-TWISTED FAMILY SPECTRUM (first exact piece of the H2 dictionary; the v20 lepton coefficients are resolvent determinants of the exact W(A3) monodromy GATE.UWALL.09): (1) SIGN-TWIST LEMMA [I]: -M0 has order 6 ((-M0)^3…
Statement, assumptions and sources
Paper context: Standard Model and flavor structure
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FLAV.H2.02
Original ledger classification: Identity [I] (exact spectral + determinant identities + group enumeration) + Physical [P] (address table)
Original claim text
THE HEXAGON IS THE SIGN-TWISTED FAMILY SPECTRUM (first exact piece of the H2 dictionary; the v20 lepton coefficients are resolvent determinants of the exact W(A3) monodromy GATE.UWALL.09): (1) SIGN-TWIST LEMMA [I]: -M0 has order 6 ((-M0)^3 = -1) and spec(M0) u spec(-M0) = mu_6 - the 6-site hypercharge hexagon spectrum IS the family monodromy spectrum together with its sheet twist, Z6 = Z2 x Z3 realised as (-1) x <M0>; the v20 hexagon denominators 5/4 - cos(r pi/3) equal |1 - zeta_6^r/2|^2, the eigen-denominators of the two family resolvents (1 -+ M0/2)^{-1}. (2) CYCLOTOMIC DETERMINANT [I]: det(1 - t M0) = 1 - t^3 = (1-t) Phi_3(t) (the N_fam cyclotomic) => det(1 - M0/2) = 7/8 and det(1 + M0/2) = 9/8. (3) LEPTON COEFFICIENTS = RESOLVENT DETERMINANTS [I]: c_e = |mu_4|/(2 det-) = 16/7, c_mu = N_fam/(2 det+) = 4/3, c_tau = |mu_4| det-/N_fam = 7/6, and v20's product rule = |mu_4|/det+ = 32/9 - the lepton denominators 7 and 9 ARE the two family-resolvent determinants (x 2^3), and delta = 1/2 = |(M0)_22| is supplied by the monodromy matrix itself; the e/mu/tau LEG assignment (residues r, windings w) remains v20's established input - nothing re-fished. (4) SHEET-EXTENDED GROUP [I]: <U, M0, -1> has order 48 = Omega_adm = N_fam x dim S+ (the seed quartic coefficient, ARCH.QUAD.01) = |S4 x Z2| (exact enumeration; audit cross-link). (5) RESIDUE [P] (recorded): the dictionary's VALUES are exact; its ADDRESS TABLE (which hexagon site each fermion occupies; the quark-sector composition) remains open
Original classification detail
Identity: hexagon = spec(M0) u spec(-M0); lepton 7 and 9 = det(1 -+ M0/2) x 8; product rule = mu4/det+; H2 values exact - address table open
Dependencies as recorded
GATE.UWALL.09; FLAV.H2.01; v20 (leg assignment); ARCH.QUAD.01 (Omega_adm); E8.GLU.01
Open the original ledger source, row 202Location recorded in the ledger
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- Standard Model and flavor structure
- Research contracts
Navigation through a shared verification script
These sections cite a script associated with the claim. This is a source navigation join, rather than an explicit citation of the claim or a theorem equivalence.
- Charged leptons: completely closed in [formula in source]tfpt 2 standard model
- What this changes, and what stays openE8 audit and bootstrap
- The closure ledger: what is done, and what genuinely remainsTFPT — Introduction and reading guide
- Front matter / supportResearch contracts
- The Sheet Diamond and the Winding Line: masses and transport from one surfacetfpt 2 standard model
- 2026-06-12 (external review integrated: [formula in source] typing, marker migration, website parity)Research changelog; TFPT — Introduction and reading guide
- Status summary and open audit pointsArchitecture, E8 and the electromagnetic fixed point
- Research Contract 1 — [formula in source]tfpt research contracts
All 21 linked sections
- Nachweise der erneuten Prüfung
- Status summary and open audit points
- Front matter / support
- 2026-06-11
- Front matter / support
- What this changes, and what stays open
- Front matter / support
- What this changes, and what stays open
- Status summary and open audit points
- Research Contract 1 — [formula in source]
- Charged leptons: completely closed in [formula in source]
- Massen und Yukawa-Kopplungen aus derselben markierten Struktur
- Ausführbare Nachweise dieser Fortsetzung
Registered verification modules
- v118_hexagon_family_dictionary.pyRegistered verification module · registry
Read the description recorded in the module registry
THE HEXAGON IS THE SIGN-TWISTED FAMILY SPECTRUM (first exact piece of the H2 dictionary; the v20 lepton coefficients are resolvent determinants of the exact W(A₃) monodromy): SIGN-TWIST LEMMA [I] — −M̃₀ has order 6 ((−M̃₀)³ = −1) and spec(M̃₀) ∪ spec(−M̃₀) = μ₆: the 6-site hypercharge hexagon spectrum IS the family monodromy spectrum plus its sheet twist (ℤ₆ = ℤ₂×ℤ₃ = (−1)×⟨M̃₀⟩); the v20 denominators 5/4 − cos(rπ/3) = |1 − ζ₆ʳ/2|² are the eigen-denominators of the two family resolvents (1 ∓ M̃₀/2)⁻¹; CYCLOTOMIC DETERMINANT [I]: det(1 − tM̃₀) = 1 − t³ = (1−t)Φ₃(t) ⇒ det(1 ∓ M̃₀/2) = (7/8, 9/8); LEPTON COEFFICIENTS = RESOLVENT DETERMINANTS [I]: c_e = |μ₄|/(2det₋) = 16/7, c_μ = N_fam/(2det₊) = 4/3, c_τ = |μ₄|det₋/N_fam = 7/6, product rule = |μ₄|/det₊ = 32/9 — the lepton 7 and 9 ARE the family-resolvent determinants (×8), and δ = ½ = |(M̃₀)₂₂| is supplied by the matrix itself; the e/μ/τ leg assignment stays v20's established input (nothing re-fished); SHEET-EXTENDED GROUP [I]: ⟨U, M̃₀, −1⟩ has order 48 = Ω_adm = N_fam·dim S⁺ (the seed quartic coefficient); residue [P]: the dictionary's VALUES are exact, its ADDRESS TABLE (site per fermion, quark composition) remains open
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