A registered result in its source context
Original ledger classification: Identity [I] (uniformization, normal form, curvature, slope) + Physical [P] (clock bridge)
Recorded statement: TRISECTION NORMAL FORM (the canonical coordinate of the SdS moduli line; answers the HOR.ORIENT.01 curvature-matching question at the geometric level): the horizon cubic is uniformized by ANGLE TRISECTION - r = 2cos(theta) gives r^3-3r =…
Statement, assumptions and sources
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HOR.TRISECT.01
Original ledger classification: Identity [I] (uniformization, normal form, curvature, slope) + Physical [P] (clock bridge)
Original claim text
TRISECTION NORMAL FORM (the canonical coordinate of the SdS moduli line; answers the HOR.ORIENT.01 curvature-matching question at the geometric level): the horizon cubic is uniformized by ANGLE TRISECTION - r = 2cos(theta) gives r^3-3r = 2cos(3theta) exactly, cubic <=> cos(3theta) = -3m; centered angle psi: m = cos(psi)/N_fam, Nariai psi=0, pure dS psi=pi/2; deck involution psi->-psi (= the x->1/x deck of HOR.NARIAI.01), Z3 trisection rotation = the triality deck (GATE.QGEOM.03/v72 cross-ref). NORMAL FORM [I]: sigma(psi) = S_tot/S_dS = 4/3 - (2/3)cos(2psi/3) - mean |mu4|/N_fam, amplitude |Z2|/N_fam, frequency |Z2|/N_fam: one cosine, every constant a glue atom. CANONICAL CURVATURE [I]: sigma''(0) = 8/27 = (2/3)^3 = (|Z2|/N_fam)^N_fam - the Koide constant to the family power. INVARIANT BASE SLOPE [I]: dsigma/dm|_Nariai = -8/9 = -rank(E8)/N_fam^2 (coordinate-invariant, cross-checked in both coordinates; generally -(4/3)sin(2psi/3)/sin(psi)); dimensionful dS_tot/dM|_N = -r_N/(N_fam c3). BRIDGE [P]: flavor multiplier (2/3)^{2N_fam} is a RATE, gravity curvature (2/3)^{N_fam} is a CURVATURE - same base 2/3 = |Z2|/N_fam, exponent ratio |Z2|; the full identification needs the gravity-side CLOCK (near-Nariai evaporation generator), the same missing-object class as P2's continuous transfer generator: one clock, two known geometries. No open gate closed; figure trisection.pdf
Original classification detail
Identity: one cosine of glue atoms; canonical curvature (2/3)^N_fam; clock [P]
Dependencies as recorded
HOR.NARIAI.01; HOR.ORIENT.01; FR.KOIDE.04 (rate); GATE.QGEOM.02 (triality)
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- 2026-06-14 (operator origin of the Koide 53/54 factor: the missing Sheet-Diamond corner, v183)Research changelog; TFPT — Introduction and reading guide
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- 2026-06-11Research changelog; TFPT — Introduction and reading guide
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
- Front matter / supportHorizon readouts
- The maximal black hole is the anchor: SdS in seam unitsHorizon readouts
Registered verification modules
- v103_trisection_normal_form.pyRegistered verification module · registry
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Trisection normal form (the canonical coordinate exists): the SdS horizon cubic is uniformized by ANGLE TRISECTION — r = 2cos θ gives r³−3r = 2cos 3θ exactly, so cubic ⇔ cos 3θ = −3m (ℤ₃ trisection deck = the triality of coker Q = ℤ/N_fam); centered angle ψ: m = cos(ψ)/N_fam and S_tot/S_dS = 4/3 − (2/3)cos(2ψ/3) — ONE cosine whose mean (|μ₄|/N_fam), amplitude and frequency (|ℤ₂|/N_fam) are glue atoms; canonical curvature at the anchor = (2/3)³ = (|ℤ₂|/N_fam)^N_fam — the Koide constant to the family power; invariant base slope dσ/dm = −8/9 = −rank E₈/N_fam² (cross-checked in both coordinates); dimensionful dS/dM = −r_N/(N_fam·c₃); bridge: flavor rate (2/3)^{2N_fam} vs gravity curvature (2/3)^{N_fam} — same base, exponent ratio |ℤ₂|; the missing piece is ONE clock (near-Nariai evaporation generator) [I] + [P]
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