A registered result in its source context
Original ledger classification: Identity [I] (exact spectral reading + series) + Physical [P] (semiclassical derivation)
Recorded statement: THE RESUMMED CLOCK (R1's bend acquires a closed form; the HOR.CLOCK.02 job description transforms): (1) THE SPECTRUM READS (1 - n/N)^{p2} [I]: the frozen transfer eigenvalues {1, (2/3)^6, (1/3)^6} (v54/v69/v95) are EXACTLY lambda_n = (1 -…
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HOR.CLOCK.03
Original ledger classification: Identity [I] (exact spectral reading + series) + Physical [P] (semiclassical derivation)
Original claim text
THE RESUMMED CLOCK (R1's bend acquires a closed form; the HOR.CLOCK.02 job description transforms): (1) THE SPECTRUM READS (1 - n/N)^{p2} [I]: the frozen transfer eigenvalues {1, (2/3)^6, (1/3)^6} (v54/v69/v95) are EXACTLY lambda_n = (1 - n/N_fam)^{p2} for n = 0, 1, 2, with p2 = 6 = |R+(A3)| the hexagon size. (2) CLOSED-FORM CLOCK [I]: rate(n) = -p2 ln(1 - n/N_fam) reproduces {0, Delta, 6 ln 3} exactly; the HOR.CLOCK.02 bend rate(2)/rate(1) = log_{3/2}3 becomes a one-line consequence. (3) THE LINEARISATION CARRIES THE SHEET [I]: rate(n) = |Z2| n + n^2/3 + 2n^3/27 + ... - the linear (one-loop) term has slope exactly |Z2| = 2 relative to the integer classical Ginsparg-Perry clock {0,-1,-2} (HOR.CLOCK.01): the quantum clock is the SHEET-DOUBLED classical clock plus the geometric tail sum_k (n/N)^k/k. (4) THE WALL AT N_fam [I]: rate -> infinity as n -> N_fam (pole of the resummed logarithm at the family count) => the cusp ladder ends at n = N_fam - 1 = 2: the three-level transfer spectrum is forced by the clock itself. (5) R1 RE-SHARPENED [P] (recorded, not claimed): closed-form semiclassical target - derive rate(n) = -p2 ln(1 - n/N_fam) from the near-Nariai one-loop at kappa = 1/(3pi); the linear response must give slope |Z2| x classical and the resummation must exponentiate the geometric series; all spectra frozen in v54/v69/v95/v104/v107 - nothing fitted
Original classification detail
Identity: transfer spectrum = (1 - n/N)^6; rate = -6 ln(1 - n/3); linearisation slope |Z2|; pole at N_fam forces three levels; R1 target now closed-form
Dependencies as recorded
HOR.CLOCK.02; HOR.CLOCK.01; FR.KOIDE.06 (transfer spectrum); SYNTH.INVENTORY.02 (R1)
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- How to read this documenttfpt 1 architecture e8
- The whole integer skeleton from one pair [formula in source]input 4 origin theory
- Front matter / supportArchitecture, E8 and the electromagnetic fixed point
- The closure ledger: what is done, and what genuinely remainsTFPT — Introduction and reading guide
- Honest statusinput 4 origin theory
- The whole integer skeleton from one pair [formula in source]input 4 origin theory
- 2026-07-15 (II) ( v487 — the lazy split FORCED by the resummed clock: the walk's one-step spectrum is the complete v124 clock ladder below the wall, deck parity [formula in source] rung index, and the two comb tones are one bend apart — the v486 \ ``split selection'' closes onto established structure; no gate closes, no marker moves)Research changelog; TFPT — Introduction and reading guide
- The whole integer skeleton from one pair [formula in source]Origin Theory
All 19 linked sections
- Honest status
- Front matter / support
- The hard reduction: the live residual as typed interfaces
- How to read this document
- Honest status
- 2026-06-12
- Front matter / support
- Honest status
- 2026-06-22 (the translation clock: the discrete [formula in source] dynamic bridge is the order- [formula in source] Coxeter element, a clock with two coprime hands [formula in source] , read law-inclusive ( [formula in source] ) or live-only ( [formula in source] ): v319)
- The whole integer skeleton from one pair [formula in source]
- Front matter / support
Registered verification modules
- v124_resummed_clock.pyRegistered verification module · registry
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THE RESUMMED CLOCK (R1's bend acquires a closed form — the transfer spectrum IS the geometric resummation of a weight-linear clock): SPECTRUM READING [I] — the frozen transfer eigenvalues {1, (2/3)⁶, (1/3)⁶} are EXACTLY λₙ = (1 − n/N_fam)^{p₂} for n = 0,1,2 (p₂ = 6 = the hexagon size); CLOSED-FORM CLOCK [I]: rate(n) = −6 ln(1 − n/3) reproduces {0, Δ, 6 ln 3} and the v107 bend log₃∕₂3 becomes a one-line consequence; THE LINEARISATION CARRIES THE SHEET [I]: rate(n) = 2n + n²/3 + 2n³/27 + … — the one-loop term has slope exactly |ℤ₂| = 2 relative to the integer classical Ginsparg–Perry clock {0,−1,−2}: quantum clock = sheet-doubled classical clock + geometric tail; THE WALL AT N_fam [I]: rate → ∞ as n → 3 (pole at the family count) ⇒ the cusp ladder ends at n = 2 — the three-level transfer spectrum is forced by the clock itself; R1 re-sharpened [P]: derive −p₂ ln(1 − n/N_fam) from the near-Nariai one-loop at κ = 1/(3π), first checkpoint = linear-response slope |ℤ₂|; all spectra frozen (v54/v69/v95/v104/v107)
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