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A registered result in its source context

Original ledger classification: Formal/Identity [E] (exact tensorial factors, scalaron mass, positivity witness) + [C] (verdict/surviving route) + [O] (AP2 + R^2 gate unchanged)

Recorded statement: THE R^2 KILL TEST OF THE ENTROPIC-ACTION BRIDGE, EXECUTED (2026-07-13; work package 5 of GRAV.ENTROPIC.ACTION.01/v473) + THE EXPLICIT LORENTZIAN-POSITIVITY WITNESS -- decides v473's interpretation 2 negatively in its naive form; closes NO…

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GRAV.ENTROPIC.SCALARON.01

Original ledger classification: Formal/Identity [E] (exact tensorial factors, scalaron mass, positivity witness) + [C] (verdict/surviving route) + [O] (AP2 + R^2 gate unchanged)

Original claim text

THE R^2 KILL TEST OF THE ENTROPIC-ACTION BRIDGE, EXECUTED (2026-07-13; work package 5 of GRAV.ENTROPIC.ACTION.01/v473) + THE EXPLICIT LORENTZIAN-POSITIVITY WITNESS -- decides v473's interpretation 2 negatively in its naive form; closes NO gate. [E] MAX-SYMMETRIC EIGENVALUES (the Appendix-B flattening of Bianconi PRD 111, 066001 (2025) verified): with R_munurhosigma = (R/12)(gg - gg) on a general diagonal 4d metric the relative curvature operator R~ g~^-1 has eigenvalues EXACTLY {R (scalar); R/4 x4 (Ricci); R/6 x6 (Riemann on flattened 2-forms)}, computed symbolically including the 6x6 two-form flattening. [E] EXACT VACUUM f(R): L_B(R) = -Tr_F ln(I - beta R~ g~^-1) = -[ln(1-bR) + 4 ln(1-bR/4) + 6 ln(1-bR/6)] with exact series 3 beta R + (17/24) beta^2 R^2 + O(R^3) -- the linear 3 re-derives her Eq. (45) (= the v473 coefficient pin) and 17/24 = (1/2)(1 + 1/4 + 1/6) is the EXACT tensorial R^2 factor (v473 quoted the raw (c3/6)^2 'before tensorial factors'). [E] THE RAW ENTROPIC SCALARON IS TRANS-PLANCKIAN: with the pinned beta' = c3/6 the EH normalisation matches identically and BOTH routes (Starobinsky matching and f'/(3 f'')) give m_raw^2 = 4608 pi^2 Mbar^2/17, i.e. m_raw = pi sqrt(4608/17) Mbar ~ 51.7 Mbar ~ 1.3e20 GeV -- HEAVY, not light; the TFPT scalaron is c3^{7/2} Mbar = 3.06e13 GeV, a factor ~4.1e6 below; the mechanism interpretation 2 requires must supply EXACTLY m_raw^2/m_TFPT^2 = (72/17)(8 pi)^9 ~ 1.69e13 in mass^2 -- the kill-test number now WITH tensorial factors. [E] DOMAIN: positivity of I - beta R~ g~^-1 holds iff R < 1/beta = 384 pi^2 Mbar^2 (first log pole; then 4/beta, 6/beta) -- a trans-Planckian curvature bound, so the raw action is well-defined at all sub-Planckian curvatures (the obstruction is the mass, not the domain). [E] LORENTZIAN-POSITIVITY WITNESS (the v473 [C] caveat made explicit): in her warm-up sector G = g + alpha dphi x dphi on Minkowski, a TIMELIKE gradient gives G g^-1 eigenvalues {1 - alpha v^2, 1,1,1} -- nonpositive for v^2 >= 1/alpha, so ln(G g^-1) is UNDEFINED on an open field region; spacelike and Euclidean stay positive for all v -- Lorentzian positivity is a genuine restriction, TFPT's OS/reflection-positive route is the sturdier construction. [C] VERDICT: of v473's three interpretations, (1) SURVIVES (the entropic action covers the Einstein + Lambda level: beta' pinned v473, Lambda channel closed v473/v474), (2) is DEAD in naive form (this module), (3) KMS-spectral renormalisation of the log action remains the ONLY route to the R^2 sector. [O] UNCHANGED: the AP2 compression and the R^2 gate itself stay open; no status marker moves anywhere

Original classification detail

R^2 kill test executed [E]/[C]/[O]: exact max-symmetric eigenvalues {R, R/4, R/6}, exact vacuum f(R) = 3 beta R + (17/24) beta^2 R^2, raw entropic scalaron m^2 = 4608 pi^2/17 Mbar^2 (~51.7 Mbar, trans-Planckian) => v473 interpretation 2 dead in naive form, required enhancement exactly (72/17)(8 pi)^9 ~ 1.69e13; Lorentzian timelike witness 1 - alpha v^2 <= 0; KMS renormalisation the only surviving R^2 route; nothing closes

Dependencies as recorded

GRAV.ENTROPIC.ACTION.01; GRAV.ENTROPIC.HODGE.01; v36; v60; v359; v473; v474

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  • v475_entropic_scalaron_gate.pyRegistered verification module · frontier
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    GRAV.ENTROPIC.SCALARON.01: the R^2 kill test of the entropic-action bridge EXECUTED (work package 5 of v473) + the explicit Lorentzian-positivity witness. [E] max-symmetric eigenvalues (her Appendix-B flattening verified): R~ g~^-1 has eigenvalues exactly {R; R/4 x4; R/6 x6} on a general diagonal 4d metric, incl. the 6x6 two-form flattening. [E] exact vacuum f(R): L_B(R) = -[ln(1-bR) + 4 ln(1-bR/4) + 6 ln(1-bR/6)] = 3 beta R + (17/24) beta^2 R^2 + O(R^3) -- the 3 re-derives her Eq. (45) (the v473 pin), 17/24 the exact tensorial R^2 factor. [E] raw scalaron TRANS-PLANCKIAN: with beta' = c3/6 both routes give m_raw^2 = 4608 pi^2/17 Mbar^2, m_raw ~ 51.7 Mbar ~ 1.3e20 GeV -- heavy, not light (TFPT: c3^{7/2} Mbar = 3.06e13 GeV); required enhancement EXACTLY (72/17)(8 pi)^9 ~ 1.69e13 in mass^2 -- v473's interpretation 2 (light G-trace mode) KILLED in naive form. [E] domain: first log pole at R = 384 pi^2 Mbar^2 (trans-Planckian) -- the obstruction is the mass, not the domain. [E] Lorentzian witness: timelike gradient gives eigenvalue 1 - alpha v^2 <= 0 for v^2 >= 1/alpha (ln undefined); spacelike/Euclidean positive -- the v473 caveat exhibited, the OS route sturdier. [C] verdict: interpretation 1 survives (Einstein + Lambda level), 3 (KMS-spectral renormalisation) is the only surviving R^2 route. [O] unchanged: AP2 compression + the R^2 gate; no marker moves. Exact (sympy, Wolfram-mirrored).

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