Original ledger status
HOR.CLOCK.05
HOR.CLOCK.05
THE CLOCK IS A LOG-DETERMINANT (the geometric tail of HOR.CLOCK.03/04 is the standard ring/RPA resummation; the 'O(1) resummation' demanded by HOR.CLOCK.02 is now TYPED): (1) LOG-DET IDENTITY [I]: for a rank-1 insertion P (tr P^k = 1), -ln det(1 - alpha P) = -ln(1 - alpha) = sum_k alpha^k/k - so the resummed clock is EXACTLY rate = -p2 Tr ln(1 - alpha P): p2 = 6 identical one-loop ring towers, ONE PER HEXAGON SITE (= positive root of A3), with coupling alpha = the Mehta-Seshadri parabolic weight. (2) RING LADDER [I]: the k = 1 ring is the CLASSICAL term (6 alpha = |Z2| n = the HOR.CLOCK.01/04 entropy rate); the k = 2 ring is the first quantum correction (1/3 at weight 1); the partial sums converge to the exact rates {Delta, 6 ln 3} monotonically from below. (3) COUPLING CROSS-LINK [I, audit]: the established seam coupling kappa = (c/24pi)(Lambda/Mbar^2) at Lambda = Mbar^2 with c = 8 (HOR.CLOCK.02) is 8/(24pi) = 1/(3pi) = alpha_1/pi - the seam coupling is the first parabolic weight over pi. (4) THE WALL = RING DIVERGENCE [I]: radius of convergence |alpha| < 1; at alpha -> 1 (n -> N_fam) the sum diverges - the HOR.CLOCK.03 wall at the family count is the standard RPA instability. (5) R1 FINAL FORM [P] (recorded, not claimed): show that the near-Nariai one-loop effective action of the seam mode, with insertion strength = parabolic weight and multiplicity p2, is the ring sum - 'the seam clock is the RPA determinant of one mode on six sites'; the resummation type is identified, the derivation itself remains open
Recorded status: Identity [I] (log-det + ring ladder + coupling arithmetic) + Physical [P] (derivation)
Dependencies, status detail and source
Canonical status: Identity: clock = -p2 Tr ln(1 - alpha P) (ring/RPA determinant; one tower per hexagon site); k=1 ring = classical entropy rate; wall = RPA instability; kappa = alpha_1/pi
Dependencies: HOR.CLOCK.04; HOR.CLOCK.03; HOR.CLOCK.02; FLAV.H2.02 (hexagon); SYNTH.INVENTORY.02 (R1)
Scripts: v127_ring_resummation.py
Location: tfpt_horizon_readouts
Original CSV, line 212