A registered result in its source context

Original ledger classification: Construction [E]/[O] (admissible-sector OS measure)

Recorded statement: CONSTRUCTING THE AMBIENT MEASURE ON THE GAP-DECOUPLED ADMISSIBLE SECTOR (2026-06-22) as a bona fide Osterwalder-Schrader object, with the honest fence (the constructive step v275's roadmap calls for). [E] REFLECTION POSITIVITY: the seam…

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Paper context: Origin Theory

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QGAMB.MEASURE.01

Original ledger classification: Construction [E]/[O] (admissible-sector OS measure)

Original claim text

CONSTRUCTING THE AMBIENT MEASURE ON THE GAP-DECOUPLED ADMISSIBLE SECTOR (2026-06-22) as a bona fide Osterwalder-Schrader object, with the honest fence (the constructive step v275's roadmap calls for). [E] REFLECTION POSITIVITY: the seam transfer T (v221/v311; spectrum {1,(2/3)^6,(1/3)^6}) is symmetric PSD, so the finite-volume transfer-matrix measures are RP (OS-0). [E] EXPONENTIAL CLUSTERING: the connected 2-pt C(n)/C(n-1) -> lambda_2=(2/3)^6 (corr. length xi=1/Delta, Delta=6 ln(3/2)). [E] TIGHTNESS => PROJECTIVE LIMIT: chi=sum_n C(n)=729/665 is FINITE, so Var(S_Lambda)/Lambda -> chi converges => the finite-volume measures are uniformly tight (Prokhorov) and the projective limit mu=lim mu_Lambda EXISTS on the admissible sector. [E] OS RECONSTRUCTION: RP + clustering + positive transfer give H_OS=-log T>=0 with unique vacuum and gap 6 ln(3/2) (v240) -- mu is a genuine OS measure on the admissible sector. [E] NEG: an ungapped transfer (lambda_2->1) sends xi,chi->inf and Var/Lambda->inf, tightness FAILS -- the construction is the gap. [O] THE FULL AMBIENT MEASURE: the non-admissible (metric/dressed) sector is NOT built; the full 4D QG.AMB.01 needs the metric-direction measure + its coupling. By the gap-decoupling margin Delta-31/(4 pi^2)~1.648>0 (v76/v275) the physical readouts do NOT need it, but QG.AMB.01 stays open.

Original classification detail

[E] the admissible-sector measure is CONSTRUCTED (RP + clustering + tightness => projective limit + OS reconstruction); [O] the full ambient (metric-sector) measure stays open, gap-decoupled. A genuine PARTIAL construction of QG.AMB.01 on the decoupled sector, NOT a closure.

Dependencies as recorded

QGAMB.ROADMAP.01; v275; v311; v240; v221; v76

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Origin Theory
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Sections that explicitly cite this claim

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2026-06-22 (frontier round: the OS step discharged at the many-body level, the ambient measure built on the gap-decoupled sector, and the carrier Cartan determinants grounded in Lean: v329, v330, FORM.CARTAN.DET.01)Research changelog; TFPT — Introduction and reading guide

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The closure ledger: what is done, and what genuinely remainsTFPT — Introduction and reading guide
Honest statusinput 4 origin theory
2026-06-22 (keystone unification + status reframe: ONE open theorem, ONE decoupled foreign problem: v335, FORM.QGEO.BW.01 qgeoSymIsCorollary)Research changelog; TFPT — Introduction and reading guide
2026-06-23 (Directions 3 \& 7 — the two hard functional-analysis residuals ADVANCED: the QG measure as one-loop fluctuations around the parameter-free saddle ( v365), and the seam collar verified IN MMST's free-fermion class so the scaling limit is [formula in source] ( v366))Research changelog; TFPT — Introduction and reading guide
Front matter / supportResearch contracts
Honest statusorigin theory
2026-06-22 (endgame round: the necessity of [formula in source] reduced to the order-4 CM selection, the QG metric sector characterized as the conformal-mode problem, [formula in source] and the [formula in source] commutation grounded in Lean: v331, v332, FORM.MU4COMM.01)Research changelog; TFPT — Introduction and reading guide
Honest statusOrigin Theory

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Registered verification modules

v330_qgamb_admissible_measure.pyRegistered verification module · frontier
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Constructing the ambient measure on the gap-decoupled admissible sector as a bona fide Osterwalder-Schrader object (with the honest fence). [E] REFLECTION POSITIVITY: the seam transfer T (v221/v311; spectrum {1,(2/3)^6,(1/3)^6}) is symmetric PSD => the finite-volume measures are RP. [E] EXPONENTIAL CLUSTERING: C(n)/C(n-1) -> lambda_2=(2/3)^6 (xi=1/Delta, Delta=6 ln(3/2)). [E] TIGHTNESS => PROJECTIVE LIMIT: chi=sum_n C(n)=729/665 finite, so Var(S_L)/L -> chi => uniform tightness (Prokhorov) => the projective limit mu=lim mu_Lambda EXISTS on the admissible sector. [E] OS RECONSTRUCTION: H_OS=-log T>=0, unique vacuum, gap 6 ln(3/2) (v240) -- mu is a genuine OS measure on the admissible sector. [E] NEG: an ungapped transfer (lambda_2->1) sends chi->inf, tightness FAILS. [O] THE FULL AMBIENT MEASURE: the non-admissible (metric) sector is NOT built; by the gap margin Delta-31/(4pi^2)~1.648>0 (v76/v275) physical readouts do not need it, but QG.AMB.01 stays open. A genuine PARTIAL construction, NOT a closure.

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