Finite spatial volume and original neutral scalar/charge source; no spatial continuum or P1/P2 selection claim.
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Result
The original H is positive self-adjoint and has a faithful Gibbs state for every prescribed beta. The actual transfer reconstructs Hδ=−log Qδ/δ; removing the temporal regulator makes its unitary dynamics converge to the same H. Spatial lattice, couplings and beta remain declared inputs.
Required assumptions
Declared native charge/scalar source
Fixed finite periodic spatial lattice source of rounds20–26 with original G, cocycle, neutral Z8 charges, scalar field and H=Hdiag+Hhop; a,m,beta>0 and original couplings are supplied. Nontrivial memory requires J>0. The original grade k is additionally retained, without presuming its selection from P1/P2.
Applies to the cited operational seed category and its collar completion; it does not claim that no other original selector exists.
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Result
The original definition already assumes (A_loc, tau_t, Theta, omega, [u_Sigma], D_coll); its collar-completion theorem derives B_Sigma and spectral data from them, but does not select those inputs.
Required assumptions
P1/P2 remain declared starting axioms
This program checks which additional source, state, event, and observable structures follow from accepted TFPT principles; it does not revoke or replace P1/P2.
Do not identify source types
Observable algebra, OS reflection, state, collar operator, time evolution, carrier connection, family holonomy, and readout are distinct data until an original map relates them.
Construction applies only with complete coherent K values and stated domain conditions; K from the original TFPT source has not been derived.
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Result
Given a positive Gram kernel K with normalized reference, compatible composition/adjunction, and null-stable continuation, quotienting by null vectors and completing constructs H_K and the well-defined word action. Cyclic realizations of the same complete K values are unitarily equivalent on the common word domain; for unbounded operators, closures and self-adjointness remain additional requirements.
Required assumptions
Positive word kernel with null stability
The GNS/Gram reconstruction requires a Hermitian positive kernel K, normalized reference, declared composition and adjunction, and stability of null relations under every allowed continuation; these are conditions on a supplied K source, not TFPT axioms proved from bare future answers.
Applies exactly relative to the declared experiment alphabet and complete answer functional; no absolute ontology or physical TFPT source claim.
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Result
For a fixed experiment alphabet and complete answer functional r, u~v exactly when r(uw)=r(vw) for every allowed continuation w. The quotient is the minimal distinguishable process state relative to those answers; by itself it yields neither complex amplitudes, the Born rule, continuous time, a metric, nor E8.
Required assumptions
Declared complete answer functional
A fixed set of allowed experiments and the answer functional r for every finite experiment sequence are additional operational inputs; they define the future quotient only relative to that experiment alphabet.