Follow the published construction into its registered claims, then explore reviewed research questions, attempts and the connections that remain open.
Future quotient and positive GNS reconstruction
What is determined by complete future behavior, and what additional positive word-kernel source is needed to reconstruct a cyclic operator realization?
Construction applies only with complete coherent K values and stated domain conditions; K from the original TFPT source has not been derived.
Read the exact result, assumptions and evidence
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.
Read the exact result, assumptions and evidence
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.
Original principles supply a concrete joint answer/K functional and experiment alphabet, positivity, reference, and null stability reproducing the intended physical source process.
Next investigation: Derive K and its operation alphabet from P1/P2 and original source structure; no GNS theorem replaces this selection.
Builds on: Future quotient from complete r; Conditional GNS reconstruction from supplied K
Future classes and the GNS representation are determined given complete r or a supplied positive K. This does not select the original answer alphabet or K; their common source selection remains open in the target problem.
Original historical attempts linked to this question
These are original research records. Their reported verdicts are distinct from the reviewed results above.
universalraum primitive source audit 20260915
Primitive-source audit — 15 September 2026
Recorded verdict: PASS
Historical inventory; not independently reviewed by this export.