Original ledger status
HOR.CLOCK.03
HOR.CLOCK.03
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
Recorded status: Identity [I] (exact spectral reading + series) + Physical [P] (semiclassical derivation)
Dependencies, status detail and source
Canonical status: 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: HOR.CLOCK.02; HOR.CLOCK.01; FR.KOIDE.06 (transfer spectrum); SYNTH.INVENTORY.02 (R1)
Scripts: v124_resummed_clock.py
Location: tfpt_horizon_readouts
Original CSV, line 209