Original ledger status
HOR.CLOCK.07
HOR.CLOCK.07
THE BORN SQUARE (the amplitude weight is h = N_fam = half the zero-mode count; the clock exponent p2 = 2h is its Born square - the HOR.CLOCK.05 multiplicity and the HOR.CLOCK.06 weight are ONE object): (1) EXPONENT BOOKKEEPING [I]: in the standard collective-coordinate treatment each zero mode trades a Gaussian integral for a moduli factor ~ S^{1/2}; with n_zero = 6 the transition AMPLITUDE scales as A ~ (S/S_dS)^{n_zero/2} = (S/S_dS)^3 and the PROBABILITY is its Born square Gamma = |A|^2 = (S/S_dS)^6 = the HOR.CLOCK.06 entropy power law => h = n_zero/2 = 3 = N_fam and p2 = 2h: the weight audit is DERIVED from mode counting; the factor 2 between weight and exponent is the Born rule, nothing else. (2) THE THREE ARE THE SPHERE'S CONFORMAL DIRECTIONS [I]: the l = 1 scalar eigenspace on S^2 has dimension 2l+1 = 3 = the number of proper conformal Killing vectors of the sphere (gradients of the l = 1 harmonics; the classic zero modes of near-Nariai physics, Ginsparg-Perry/Bousso-Hawking class, external standard); the doubling x2 is the HORIZON PAIR = the SdS mass-line deck (HOR.NARIAI.01): n_zero = |Z2| x 3 = 6 = p2 - every factor an established object. (3) CONSISTENCY [I]: multiplicity (E8.GRADE.01) = exponent (HOR.CLOCK.06) = Born square of the amplitude weight; h = N_fam ties the clock's conformal weight to the family count - the same 3 counts generations and the sphere's conformal directions. (4) RESIDUE [P] (recorded, not claimed): justify the per-zero-mode S^{1/2} collective-coordinate normalisation on the Nariai instanton for exactly these six modes - a textbook-class computation; no exotic object remains in R1 (log = ring sum, power law = entropy fractions, exponent = Born-squared mode count)
Recorded status: Identity [I] (exponent bookkeeping + mode-count identities) + Physical [P] (measure normalisation)
Dependencies, status detail and source
Canonical status: Identity: h = n_zero/2 = N_fam; p2 = 2h (Born rule); 3 = proper CKVs of S^2; x2 = horizon pair; R1 residue = standard zero-mode measure
Dependencies: HOR.CLOCK.06; HOR.CLOCK.05; E8.GRADE.01; HOR.NARIAI.01
Scripts: v130_born_square.py
Location: tfpt_horizon_readouts
Original CSV, line 215