Original ledger status
HOR.CLOCK.08
HOR.CLOCK.08
THE MEASURE IS THE AREA (the per-mode S^{1/2} of the Born square is the zero-mode NORM, computed exactly; the R1 residue narrows to one determinant-ratio cancellation): (1) ZERO-MODE NORM = AREA [I]: on the sphere S^2(r) the L^2 norm^2 of each of the three normalised l = 1 harmonics is EXACTLY r^2 = A/(4pi) (all three modes integrated exactly) - the norm of every zero mode IS the horizon area, hence the entropy, up to the fixed 4pi. (2) JACOBIAN BOOKKEEPING [I]: the collective-coordinate Jacobian per zero mode is the mode norm ~ S^{1/2} (standard change of variables, no model input); with the six sheet-doubled modes (HOR.CLOCK.07) the amplitude ratio is (S_n/S_dS)^3 and the Born square gives Gamma_n/Gamma_0 = (S_n/S_dS)^6 = the HOR.CLOCK.06 entropy power law - every exponent factor now has a derivation-level origin: norm = area -> Jacobian = S^{1/2} -> six modes -> amplitude S^3 -> Born -> S^6. (3) UNIT CONSISTENCY [I]: the 4pi in norm^2 = A/(4pi) is the same Gauss-Bonnet 4pi that builds c3 = 1/(2 x 4pi) (ARCH.QUAD.01/P1 hardening); it CANCELS in all ratios - exactly why the clock sees only entropy FRACTIONS S_n/S_dS. (4) THE LAST STEP [P] (recorded, not claimed): show the non-zero-mode fluctuation determinants cancel in the ratio, det'_n/det'_dS = 1 + O(deficit) - plausible (same local near-Nariai geometry) but the one remaining computation; R1 = norm (exact, done) + Jacobian (standard) + Born (rule) + det-ratio (open)
Recorded status: Identity [I] (exact sphere integrals + bookkeeping) + Physical [P] (det-ratio)
Dependencies, status detail and source
Canonical status: Identity: ||Y_1m||^2 = A/(4pi) exactly (all three modes); Jacobian = S^(1/2) per mode; 4pi cancels in ratios; R1 residue = det-ratio cancellation only
Dependencies: HOR.CLOCK.07; HOR.CLOCK.06; HOR.NARIAI.01; ARCH.QUAD.01 (c3 primitive)
Scripts: v131_measure_is_area.py
Location: tfpt_horizon_readouts
Original CSV, line 216