A registered result in its source context
Original ledger classification: Reduction [E]/[O] (OS gap = transfer gap + single premise)
Recorded statement: CLOSING THE OS STEP: THE BULK GAP IS THE TRANSFER GAP (2026-06-22, second quantization) + the single sufficient premise. The v308 chain left ONE analytic input -- the OS identification 'OS-reconstructed bulk gap = transfer gap'. [E] OS GAP…
Statement, assumptions and sources
Paper context: Origin Theory
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QGEO.OSGAP.01
Original ledger classification: Reduction [E]/[O] (OS gap = transfer gap + single premise)
Original claim text
CLOSING THE OS STEP: THE BULK GAP IS THE TRANSFER GAP (2026-06-22, second quantization) + the single sufficient premise. The v308 chain left ONE analytic input -- the OS identification 'OS-reconstructed bulk gap = transfer gap'. [E] OS GAP = TRANSFER GAP: the OS/Euclidean many-body Hamiltonian is dGamma(-log T) (T the v221/v302 recovery transfer); its spectrum is the set of sums of single-mode energies, so the many-body (bulk) gap = the smallest positive single-mode energy = 6 ln(3/2), INDEPENDENT of the number of modes -- discharges the v308 OS input at the many-body level. [E] H => omega o rho = omega: under the premise H the covariance is rotation-covariant C=f(L), so K=log((1-C)/C)=g(L) commutes with rho=exp(i(pi/2)L) ([rho,K]=0 = QGEO.SYM.01 + geometric flow, v201/v309/v323). [E] H => SRE => holomorphic => (E8)_1: |det Cartan(E8)|=1 vs D8=4 (v237/v308). [E] SINGLE SUFFICIENT PREMISE: both bedrock IDs (QGEO.SYM.01 + SEAM.EQUIV.01) follow from the ONE premise H = 'raw collar = rotation-invariant flat tau=i pillowcase state', Lean-pinned (FORM.SEAMEQUIV.01 + FORM.QGEO.BW.01). [O] RESIDUAL: the NECESSITY of H on the raw (continuum) collar (NEG: a non-rotation-invariant collar breaks mark-locality). Sharpens v240/v308/v323/v325; does NOT close the bedrock.
Original classification detail
[E] the many-body OS-gap=transfer-gap identity (second quantization, size-independent) + the sufficiency chain (rotation-invariance => QGEO.SYM.01 + SRE => (E8)_1); [O] the necessity of H on the raw collar. Discharges the v308 OS input at the finite level and assembles H as the single sufficient premise; does NOT close the bedrock.
Dependencies as recorded
SEAM.EQUIV.CHAIN.01; QGEO.BW.01; QGEO.KEYSTONE.01; v240; v308; v323; v325; v237
Open the original ledger source, row 429Location recorded in the ledger
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- Origin Theory
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
Navigation through a shared verification script
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- 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
- 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
- Research Contract 2 — [formula in source]tfpt research contracts
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
Registered verification modules
- v329_os_gap_reduction.pyRegistered verification module · frontier
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Closing the OS step: the bulk gap IS the transfer gap (second quantization), and assembling the single sufficient premise of the whole bedrock. [E] OS GAP = TRANSFER GAP: the OS/Euclidean many-body Hamiltonian dGamma(-log T) has spectrum = sums of single-mode energies, so the many-body (bulk) gap = the smallest positive single-mode energy = 6 ln(3/2), INDEPENDENT of system size -- exactly 'OS bulk gap = transfer gap' (the v308 open input), discharged at the many-body level. [E] H => omega o rho = omega: under H the covariance is rotation-covariant C=f(L), K=g(L) commutes with rho=exp(i(pi/2)L) (v201/v309/v323). [E] H => SRE => (E8)_1: det Cartan(E8)=1 vs D8=4 (v237/v308). [E] both bedrock IDs follow from the ONE premise H = 'raw collar = rotation-invariant flat tau=i pillowcase state' (Lean-pinned FORM.SEAMEQUIV.01 + FORM.QGEO.BW.01). [O] residual: the NECESSITY of H on the raw collar. Sharpens v240/v308/v323/v325; does NOT close the bedrock.
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