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A registered result in its source context

Original ledger classification: Identity [I] (exact spectral bridge + coefficient identities) + Physical [P] (re-target)

Recorded statement: THE CLOCK AND THE WALL SHARE A SPECTRUM (the v124 resummed-clock weights ARE the Mehta-Seshadri parabolic weights; R1's first checkpoint passes classically): (1) THE WEIGHTS ARE THE PARABOLIC WEIGHTS [I]: the arguments n/N_fam = {0, 1/3,…

Statement, assumptions and sources

Paper context: Horizon readouts

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HOR.CLOCK.04

Original ledger classification: Identity [I] (exact spectral bridge + coefficient identities) + Physical [P] (re-target)

Original claim text

THE CLOCK AND THE WALL SHARE A SPECTRUM (the v124 resummed-clock weights ARE the Mehta-Seshadri parabolic weights; R1's first checkpoint passes classically): (1) THE WEIGHTS ARE THE PARABOLIC WEIGHTS [I]: the arguments n/N_fam = {0, 1/3, 2/3} of the resummed logarithm are EXACTLY spec A0* (the exact U_wall residue, GATE.UWALL.07); the spectral bridge lambda_n = (1 - alpha_n)^{p2} over alpha in spec A0* reproduces the frozen transfer spectrum {1, (2/3)^6, (1/3)^6} - the horizon clock and the flavor wall share their spectrum through the complement map alpha -> 1 - alpha: the transfer eigenvalues are the hexagon-power of the COMPLEMENTARY parabolic weight. (2) CHECKPOINT 1 PASSES CLASSICALLY [I]: the resummed clock's linear coefficient p2/N_fam = 2 = |Z2| equals the established classical entropy-deviation rate 2H = |Z2| x Hubble (HOR.CLOCK.01) - the slope-2 linear response demanded by HOR.CLOCK.03 is already the classical entropy rate; the genuinely QUANTUM content of R1 isolates to the geometric tail sum_{k>=2} alpha^k/k. (3) THE DETERMINANT IS THE ENTROPY CURVATURE [I]: det(1 - A0*) = (1)(2/3)(1/3) = 2/9 = |Z2|/N_fam^2 - exactly the established gravity entropy curvature at the anchor (HOR.ORIENT.01), a fourth independent appearance; tr(1 - A0*) = 2 = |Z2|; det A0* = 0 (the parabolic degree-0 marker). (4) R1 RE-TARGETED [P] (recorded, not claimed): derive 'rate = -p2 ln(1 - alpha)' with alpha the parabolic weight - first order is classical (HOR.CLOCK.01's entropy rate), the tail is the quantum part; ONE SPECTRUM, TWO GATES, ONE OBJECT A0*

Original classification detail

Identity: clock weights = parabolic weights (spec A0*); lambda = (1-alpha)^6; linear term = classical entropy rate |Z2|H; det(1-A0*) = entropy curvature 2/9; quantum content = geometric tail

Dependencies as recorded

HOR.CLOCK.03; HOR.CLOCK.01; HOR.ORIENT.01; GATE.UWALL.07 (A0*); SYNTH.INVENTORY.02 (R1)

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Registered verification modules

  • v126_clock_wall_bridge.pyRegistered verification module · registry
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    THE CLOCK AND THE WALL SHARE A SPECTRUM (the v124 resummed-clock weights ARE the Mehta–Seshadri parabolic weights; R1's first checkpoint passes classically): THE WEIGHTS ARE THE PARABOLIC WEIGHTS [I] — the arguments n/N_fam = {0, 1/3, 2/3} of the resummed logarithm are EXACTLY spec A₀* (the exact U_wall residue, v115); spectral bridge λₙ = (1 − αₙ)^{p₂} over α ∈ spec A₀* reproduces the frozen transfer spectrum {1, (2/3)⁶, (1/3)⁶} — the horizon clock and the flavor wall share their spectrum through α ↦ 1−α; CHECKPOINT 1 PASSES CLASSICALLY [I]: the linear coefficient p₂/N_fam = 2 = |ℤ₂| equals the established classical entropy-deviation rate 2H = |ℤ₂|×Hubble (v104) — the slope-2 linear response is already the classical entropy rate, so the genuinely QUANTUM content of R1 isolates to the geometric tail Σ_{k≥2} αᵏ/k; THE DETERMINANT IS THE ENTROPY CURVATURE [I]: det(1 − A₀*) = 2/9 = |ℤ₂|/N_fam² — exactly the v102 gravity entropy curvature at the anchor (fourth independent appearance); tr(1 − A₀*) = 2 = |ℤ₂|, det A₀* = 0; R1 re-targeted [P]: derive 'rate = −p₂ ln(1 − α)' with α the parabolic weight — first order classical, tail quantum: ONE SPECTRUM, TWO GATES, ONE OBJECT A₀*

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