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A registered result in its source context

Original ledger classification: Identity [I] (model mechanism exact) + [A] (the theorem proper stays open)

Recorded statement: THE EH FORM FROM THE REPLICA VARIATION, GAPPED-MODEL LEVEL (the first movement on R3/SEAM.THEOREM.01; the gate narrows, does NOT close): (1) CONICAL DEFICIT EXACT [I] - on C/Z_N the heat-trace deficit is the classical image sum (1/4N) sum…

Statement, assumptions and sources

Paper context: Origin Theory

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SEAM.EHMODEL.01

Original ledger classification: Identity [I] (model mechanism exact) + [A] (the theorem proper stays open)

Original claim text

THE EH FORM FROM THE REPLICA VARIATION, GAPPED-MODEL LEVEL (the first movement on R3/SEAM.THEOREM.01; the gate narrows, does NOT close): (1) CONICAL DEFICIT EXACT [I] - on C/Z_N the heat-trace deficit is the classical image sum (1/4N) sum csc^2(pi k/N) = (N^2-1)/(12N) = C(2pi/N), C(gamma) = (1/12)(2pi/gamma - gamma/2pi), t-INDEPENDENT (exact N=2..6, 40-digit N=2..40); (2) GAP => CUTOFF-INDEPENDENT [I]: for -Delta + m^2 the deficit zeta-term is the exact Mellin transform Delta zeta(s) = C m^{-2s} => Delta log det' = 2 C(gamma) ln m - FINITE, no UV cutoff (the massless case would give the familiar log-epsilon; the gap supplies the scale); (3) EH LINEARISATION [I]: C = (2pi-gamma)/(12pi) + O(def^2), Int sqrt(g)R_cone = 2(2pi-gamma) => Delta log det' = (ln m/12pi) Int sqrt(g) R + O(def^2) - the replica variation of a gapped 2d determinant IS of Einstein-Hilbert form with cutoff-independent coefficient; (4) TARGET EQUATION [I, audit]: k_model = c3/2 <=> ln m = 12pi c3/2 = 3/4 = q(A3) - the A3 glue norm as the required seam gap parameter, recorded NOT asserted. HONEST SCOPE [P]: 2d bulk-scalar model; SEAM.THEOREM.01 stays open [A] - remaining: (i) the Calderon (boundary) version, (ii) the q(A3) normalisation; established: the MECHANISM (gap => cutoff-independent EH coefficient under replica) is exact at model level

Original classification detail

Identity [I]: EH mechanism exact at gapped-model level; gate narrowed, open [A]

Dependencies as recorded

SEAM.THEOREM.01; SEAM.AREACOEFF.03; SEAM.KHALF.01; v59; v73; v90

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Registered verification modules

  • v150_replica_eh_model.pyRegistered verification module · registry
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    EH form from replica (R3, model level): conical deficit exact + t-independent (Cheeger image sum (N²−1)/(12N)); gap ⇒ Δlogdet′ = 2C(γ)ln m FINITE, cutoff-independent; linearisation = (ln m/12π)·∫√gR — the EH form; target equation k = c₃/2 ⟺ ln m = 3/4 = q(A₃); Calderón transfer + normalisation stay open (Calderón transfer since answered: the DtN kernel is conically clean, v151)

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