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Original ledger classification: QG measure reduced to a one-loop Gaussian problem around the parameter-free saddle [E]/[C]/[O] (residual = the non-perturbative projective limit)

Recorded statement: Direction 3 (2026-06-23): the QG measure (gate C7/QG.AMB.01) as one-loop GAUSSIAN fluctuations around the PARAMETER-FREE saddle (v358/v359). Before v358 fluctuations around the gravitational saddle were not even dimensionlessly well-posed…

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QGAMB.SADDLE.01

Original ledger classification: QG measure reduced to a one-loop Gaussian problem around the parameter-free saddle [E]/[C]/[O] (residual = the non-perturbative projective limit)

Original claim text

Direction 3 (2026-06-23): the QG measure (gate C7/QG.AMB.01) as one-loop GAUSSIAN fluctuations around the PARAMETER-FREE saddle (v358/v359). Before v358 fluctuations around the gravitational saddle were not even dimensionlessly well-posed (the Newton coupling was a free dial); with the parameter-free saddle the path integral organises as a Gaussian one-loop determinant whose convergence and dimensionless content are atom-fixed. [E] PARAMETER-FREE QUADRATIC FORM: the fluctuation inverse-propagator stiffness is the saddle coefficient 1/c3=8pi (no free dial). [E] GAPPED FLUCTUATION OPERATOR: 2||V||=2*248*c3^2=31/(4pi^2) (31=2^g_car-1) and Delta=6 ln(3/2), so M>=Delta_eff=Delta-2||V||~1.648>0 (v76/v337) => mode-by-mode Gaussian convergence, no zero/negative modes besides the saddle directions. [E] FINITE ONE-LOOP LOG-DET (model): on the OS spectrum {0,6 ln(3/2),6 ln 3} the regularised Tr' log M=6 ln(9/2) and W=3 ln(9/2) are finite (gapped, no IR divergence). [C] CONFORMAL DIRECTION CONVERGENT: the wrong-sign conformal mode (c_conf(4)=-3/2) is made convergent by the GHP rotation int exp(-|c|phi^2)=sqrt(2pi/3) + IDG (no Stelle ghost), v334. [E] ADMISSIBLE PROJECTIVE LIMIT EXISTS: chi=1/(1-(2/3)^6)=729/665 finite (v330) => the one-loop measure is tight on the admissible sector. [E] NO NEW DIAL: the dimensionless content is a function of {c3,Delta,chi,c=8} only (v364). [O] RESIDUAL: the full non-perturbative projective limit (G6/QG.AMB.01) on the ambient sector remains OPEN. Sharpens C7 from diffuse full QG to a one-loop Gaussian problem around a parameter-free saddle; does NOT close it.

Original classification detail

[E] fixed quadratic form (1/c3=8pi) + gap-controlled convergence (Delta_eff~1.648>0) + finite model log-det (6 ln(9/2)) + tightness (chi=729/665) + 0 dials; [C] the GHP conformal contour + the one-loop Gaussian measure on the admissible sector; [O] the full non-perturbative projective limit. Sharpens C7; does NOT close it.

Dependencies as recorded

QG.AMB.01; v358; v359; v76; v337; v330; v334; v364; v59

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  • v365_qg_oneloop_saddle.pyRegistered verification module · core
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    Direction 3: the QG measure (gate C7/QG.AMB.01) as one-loop GAUSSIAN fluctuations around the PARAMETER-FREE saddle (v358/v359). [E] the fluctuation stiffness is 1/c3=8pi (no free dial), so the one-loop problem is well-posed (it was NOT before v358). [E] the fluctuation operator is gapped M>=Delta_eff=6 ln(3/2)-31/(4pi^2)~1.648>0 (v76/v337) => mode-by-mode Gaussian convergence, no zero/negative modes besides the saddle directions. [E] finite model log-det Tr' log M=6 ln(9/2) on the OS spectrum {0,6 ln(3/2),6 ln 3}. [C] the conformal mode converges on the GHP contour (int exp(-|c|phi^2)=sqrt(2pi/3), v334). [E] the admissible projective limit exists (chi=1/(1-(2/3)^6)=729/665, v330). [E] 0 new dimensionless dials (v364). [O] residual: the full non-perturbative projective limit (G6/QG.AMB.01) on the ambient sector. Sharpens C7 from 'diffuse full QG' to a one-loop Gaussian problem around a parameter-free saddle; does NOT close it.

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