Skip to main content

A registered result in its source context

Original ledger classification: Identity [I] (log-det + ring ladder + coupling arithmetic) + Physical [P] (derivation)

Recorded statement: 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…

Statement, assumptions and sources

Paper context: Horizon readouts

The exact identifier, original classification and full claim are available below.

The source wording and classification below are preserved as recorded. A paper link, shared script or authored topic relation does not upgrade the claim or report a new proof run. Read later reviewed findings separately.

Read the exact registered claim and its dependencies

HOR.CLOCK.05

Original ledger classification: Identity [I] (log-det + ring ladder + coupling arithmetic) + Physical [P] (derivation)

Original claim text

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

Original classification detail

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 as recorded

HOR.CLOCK.04; HOR.CLOCK.03; HOR.CLOCK.02; FLAV.H2.02 (hexagon); SYNTH.INVENTORY.02 (R1)

Open the original ledger source, row 212

Location recorded in the ledger

These document locations are recorded directly on the ledger row. They are document level navigation, rather than a newly inferred section citation.

Navigation through a shared verification script

These sections cite a script associated with the claim. This is a source navigation join, rather than an explicit citation of the claim or a theorem equivalence.

Registered verification modules

  • v127_ring_resummation.pyRegistered verification module · registry
    Read the description recorded in the module registry

    THE CLOCK IS A LOG-DETERMINANT (the geometric tail is the standard ring/RPA resummation; the 'O(1) resummation' demanded by v107 is now TYPED): LOG-DET IDENTITY [I] — for a rank-1 insertion P (tr Pᵏ = 1): −ln det(1 − αP) = −ln(1 − α) = Σₖ αᵏ/k, so the resummed clock is EXACTLY rate = −p₂ Tr ln(1 − αP): p₂ = 6 identical one-loop ring towers, ONE PER HEXAGON SITE (= positive root of A₃), coupling α = the Mehta–Seshadri parabolic weight; RING LADDER [I]: the k = 1 ring is the CLASSICAL term (6α = |ℤ₂|n = the v104/v126 entropy rate), the k = 2 ring is the first quantum correction (1/3 at weight 1); partial sums converge to {Δ, 6 ln 3} monotonically from below; COUPLING CROSS-LINK [I, audit]: κ_seam = 8/(24π) = 1/(3π) = α₁/π — the seam coupling is the first parabolic weight over π; THE WALL = RING DIVERGENCE [I]: radius of convergence |α| < 1, divergence at α → 1 (n → N_fam) — the v124 wall is the standard RPA instability; R1 FINAL FORM [P]: 'the seam clock is the RPA determinant of one mode on six sites' — resummation type identified, derivation open

A registered module is a source reference. Its presence does not assert that the module was executed in this snapshot.

Search later reviewed research for this identifierRead the coverage boundaries