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Original ledger classification: Numerical (RG interval + heat-kernel) [E]/[C]/[X]/[O]

Recorded statement: kappa IN M_PS = kappa M_s PINNED TO O(1) ~ 1 (2026-06-17): discharges the open residual 6(b) of v249 ('exact kappa, now ~1.0-1.2, not pinned') as far as is honest. The scalaron scale M_s = c3^{7/2} Mbar ~ 3.06e13 GeV is exact from…

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PS.KAPPA.01

Original ledger classification: Numerical (RG interval + heat-kernel) [E]/[C]/[X]/[O]

Original claim text

kappa IN M_PS = kappa M_s PINNED TO O(1) ~ 1 (2026-06-17): discharges the open residual 6(b) of v249 ('exact kappa, now ~1.0-1.2, not pinned') as far as is honest. The scalaron scale M_s = c3^{7/2} Mbar ~ 3.06e13 GeV is exact from {c3=1/(8pi), Mbar}. kappa := M_PS/M_s with M_PS the RG output of the carrier-native PS two-step (v249). HEAT-KERNEL ORIGIN: in the spectral action S = Tr f(D/Lambda) the gauge kinetic terms (=> unification scale), the Einstein-Hilbert term and the R^2 term (=> scalaron) all descend from ONE cutoff Lambda and the SAME moments f_0,f_2,f_4 of one cutoff function (Gilkey/Seeley-DeWitt; Chamseddine-Connes); hence both M_PS and M_s are Lambda x dimensionless heat-kernel factor and kappa = sqrt((f_2/f_0) c_PS/c_grav), a ratio of traces over the 96-dim H_F (v252) -- Lambda cancels, kappa is scale-free. (1) [E] M_s exact. (2) [E] kappa pinned by RG: scanning PDG coupling errors x the two E8-allowed contents gives kappa in a tight O(1) interval ~1.3 (1-loop [1.0,1.7], experiment 2-loop [1.0,1.2]) -- pinned, not free. (3) [C] heat-kernel origin: O(1) cutoff moments give O(1) kappa; the RG interval sits inside the structural band [0.3,3]. (4) [X] negative control: 126-type content drives kappa out of the O(1) window (content-selective; 126 E8-forbidden anyway, v247). (5) [O] the exact scheme-independent decimal needs f_2/f_0 fixed; c3 is the candidate normalisation but the f-fixing is unproven.

Original classification detail

[E] M_s exact + kappa pinned to O(1)~1.3 by RG interval scan; [C] heat-kernel origin kappa=sqrt((f2/f0)c_PS/c_grav) scale-free; [X] 126 neg control; [O] exact decimal needs cutoff moments fixed

Dependencies as recorded

PS.RGTEST.01; PS.DIRAC.02

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Registered verification modules

  • v253_kappa_scalaron_ps.pyRegistered verification module · registry
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    PS.KAPPA.01 — pinning κ in M_PS = κ·M_s, discharging residual 6(b) of v249. M_s = c₃^{7/2}·Mbar ≈ 3.06e13 GeV (exact). [E] scalaron scale exact; [E] κ pinned by RG (scan PDG errors × the two E8-allowed contents) to a tight O(1) interval ~1.3 (1-loop [1.0,1.7], 2-loop [1.0,1.2]) — pinned, not free. [C] heat-kernel origin: κ = √((f₂/f₀)·c_PS/c_grav), a ratio of heat-kernel traces over the 96-dim H_F (v252) — Λ cancels, κ scale-free and O(1). [X] 126-content drives κ out of the O(1) window (content-selective). [O] exact decimal needs the cutoff moments f₂/f₀ fixed.

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