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Original ledger classification: Identity [I] (entropy-level + power-law + triple-coupling identities) + Physical [P] (derivation)

Recorded statement: THE CLOCK IS AN ENTROPY POWER LAW (the transfer eigenvalues are the Nariai entropy fractions to the hexagon power; the ring coupling is the fractional entropy deficit): (1) THE THREE ENTROPY LEVELS [I]: the established Nariai bookkeeping…

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HOR.CLOCK.06

Original ledger classification: Identity [I] (entropy-level + power-law + triple-coupling identities) + Physical [P] (derivation)

Original claim text

THE CLOCK IS AN ENTROPY POWER LAW (the transfer eigenvalues are the Nariai entropy fractions to the hexagon power; the ring coupling is the fractional entropy deficit): (1) THE THREE ENTROPY LEVELS [I]: the established Nariai bookkeeping (HOR.NARIAI.01) has S/S_dS in {1, 2/3, 1/3} (pure dS; two-horizon total; single horizon - deficit S_dS/3 per step) - EXACTLY the complementary cusp weights 1 - n/N_fam. (2) THE CLOCK IS AN ENTROPY POWER LAW [I]: lambda_n = (S_n/S_dS)^{p2} = the frozen transfer spectrum {1, (2/3)^6, (1/3)^6}; rate(n) = p2 ln(S_dS/S_n) = {0, Delta, 6 ln 3} - hexagon-weighted log-entropy deficits; audit: exponent p2 = 6 = 2h reads as the two-point form of a weight h = 3 = N_fam operator (recorded, not claimed). (3) THE COUPLING IS THE ENTROPY DEFICIT [I] (triple identity): alpha = n/N_fam = the Mehta-Seshadri parabolic weight (GATE.UWALL.07/HOR.CLOCK.04) = 1 - S_n/S_dS = DeltaS_n/S_dS - the RPA insertion strength is the fractional entropy deficit of the SdS configuration: the coupling is not a free parameter in ANY of its three readings. (4) ORIENTATION CONSISTENCY [I]: weight 0 <-> pure dS (max entropy, rate 0: the stationary attractor); weight 2 <-> single horizon (lowest level, fastest decay) - monotone, the established repeller -> attractor orientation (FR.KOIDE.05/HOR.NARIAI.01/HOR.ORIENT.01). (5) R1 THERMODYNAMIC FORM [P] (recorded, not claimed): derive Gamma_n ~ (S_n/S_dS)^{p2} from the near-Nariai one-loop - a Gibbons-Hawking-type entropy power law with exponent = the hexagon size; the exponent (weight h = N_fam) is the remaining quantum content

Original classification detail

Identity: lambda = (S/S_dS)^p2; rate = p2 ln(S_dS/S); alpha = entropy-deficit fraction = parabolic weight; orientation consistent; R1 = Gibbons-Hawking power law

Dependencies as recorded

HOR.CLOCK.05; HOR.CLOCK.04; HOR.NARIAI.01; HOR.ORIENT.01; GATE.UWALL.07

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Registered verification modules

  • v129_entropy_power_law.pyRegistered verification module · registry
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    THE CLOCK IS AN ENTROPY POWER LAW (what the resummation argument 1 − n/N_fam IS physically): THE THREE ENTROPY LEVELS [I] — the established Nariai bookkeeping (v101) has S/S_dS ∈ {1, 2/3, 1/3} (pure dS; two-horizon total; single horizon — deficit S_dS/3 per step): EXACTLY the complementary cusp weights; THE CLOCK IS AN ENTROPY POWER LAW [I]: λₙ = (Sₙ/S_dS)^{p₂} = the frozen transfer spectrum, rate(n) = p₂ ln(S_dS/Sₙ) — hexagon-weighted log-entropy deficits (audit: exponent p₂ = 6 = 2h reads as the two-point form of a weight h = 3 = N_fam operator); THE COUPLING IS THE ENTROPY DEFICIT [I] (triple identity): α = n/N_fam = MS parabolic weight (v115/v126) = 1 − Sₙ/S_dS = ΔSₙ/S_dS — the RPA insertion strength is the fractional entropy deficit, not a free parameter in ANY of its three readings; ORIENTATION CONSISTENT [I]: weight 0 ↔ pure dS (max entropy, stationary attractor), weight 2 ↔ single horizon (fastest decay) — the established repeller → attractor flow; R1 THERMODYNAMIC FORM [P]: derive Γₙ ∝ (Sₙ/S_dS)^{p₂} from the near-Nariai one-loop — a Gibbons–Hawking-type entropy power law with exponent = the hexagon size

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