Skip to main content

A registered result in its source context

Original ledger classification: Numerical (finite-block) [E]/[O]

Recorded statement: GNS / OS RECONSTRUCTION (2026-06-17, finite toy): from the discrete seam data (a *-algebra M + the state omega) one reconstructs a Hilbert space, a cyclic vector, the modular flow (v238/v239) AND a POSITIVE Hamiltonian -- so two QFT…

Statement, assumptions and sources

Paper context: Research contracts

The exact identifier, original classification and full claim are available below.

The source wording and classification below are preserved as recorded. A paper link, shared script or authored topic relation does not upgrade the claim or report a new proof run. Read later reviewed findings separately.

Read the exact registered claim and its dependencies

DYN.GNS.01

Original ledger classification: Numerical (finite-block) [E]/[O]

Original claim text

GNS / OS RECONSTRUCTION (2026-06-17, finite toy): from the discrete seam data (a *-algebra M + the state omega) one reconstructs a Hilbert space, a cyclic vector, the modular flow (v238/v239) AND a POSITIVE Hamiltonian -- so two QFT primitives (the Hilbert space and a positive Hamiltonian) are OUTPUTS of the seam data, not extra inputs. (1) [E] GNS: <A,B>=Tr(rho A^dag B) is Hermitian and positive-definite (rho>0), dim n^2; the identity is cyclic (pi(A)1 = A spans M) and separating. (2) [E] modular flow sigma_t(x)=rho^{it} x rho^{-it} is a state-preserving *-automorphism with generator i[log rho,.] = Lambda_Sigma (v238); the modular operator Delta:X|->rho X rho^{-1} is positive. (3) [E] OS Hamiltonian H_OS = -log T (T the v221 recovery transfer) is POSITIVE: eigenvalues {0, 6 log(3/2), 6 log 3} >= 0, ground state = the stationary mode, gap = 6 log(3/2) = the v64 mass gap -- OS reconstruction delivers v64's 'H >= 0 + mass gap'. (4) [E] H_OS = -L: the OS Hamiltonian is MINUS the v238 dissipative generator -- the recovery semigroup e^{L tau} IS the Euclidean evolution e^{-H_OS tau}; two Hamiltonians from one seam (modular thermal-time + OS positive-energy). (5) [O] residual: the geometric/operator-level identification = QGEO.SYM.01/BW and the full reconstruction = the standard OS theorem under the named RP/gap hypotheses (Layer 2) -- a target, not a closure

Original classification detail

[E] GNS Hilbert space + cyclic/separating vector from (M,omega); modular flow = v238; OS Hamiltonian H_OS = -log T >= 0 with gap = v64 mass gap, and H_OS = -L. [O] the geometric/operator-level OS step = QGEO.SYM.01 + Layer-2 RP/gap -- a target, not a closure

Dependencies as recorded

DYN.SEMIGROUP.01; QEC.SEAM.01; QG.GAPDEC.01

Open the original ledger source, row 326

Location recorded in the ledger

These document locations are recorded directly on the ledger row. They are document level navigation, rather than a newly inferred section citation.

Sections that explicitly cite this claim

A literal claim identifier is present in these paper sections. The section and its premises determine what the citation supports.

Navigation through a shared verification script

These sections cite a script associated with the claim. This is a source navigation join, rather than an explicit citation of the claim or a theorem equivalence.

All 9 linked sections

Registered verification modules

  • v240_gns_os_reconstruction.pyRegistered verification module · registry
    Read the description recorded in the module registry

    DYN.GNS.01 — GNS / OS reconstruction (finite toy): the Hilbert space and a POSITIVE Hamiltonian are OUTPUTS of the seam data (M, ω), not inputs. [E] GNS: ⟨A,B⟩=Tr(ρ A†B) Hermitian positive-definite (ρ>0), dim n², identity cyclic+separating. [E] modular flow σ_t(x)=ρ^{it} x ρ^{−it} is a state-preserving *-automorphism, generator i[log ρ,·]=Λ_Σ (v238). [E] OS Hamiltonian H_OS=−log T (the v221 transfer) is POSITIVE — eigenvalues {0, 6 log(3/2), 6 log 3}≥0, gap = 6 log(3/2) = the v64 mass gap — so OS reconstruction delivers v64's 'H≥0 + mass gap'. [E] H_OS=−L: the OS Hamiltonian is minus the v238 dissipative generator (recovery semigroup = Euclidean time); two Hamiltonians from one seam. [O] the geometric/operator-level step = QGEO.SYM.01/BW + the standard OS theorem under RP/gap (Layer 2) — a target, not a closure.

A registered module is a source reference. Its presence does not assert that the module was executed in this snapshot.

Search later reviewed research for this identifierRead the coverage boundaries