2463 — The Frozen Three-Site Source Authorizes a Regulated-Period Theorem but No Finite Renormalization Map
Scope question
Entry 2461 proves faithfulness of the regulated physical score Gram form. Must the present objective also select and test a UV-renormalized finite representative?
Sequence claim: seqclaim-4706aef0d6eaa164a22c6526.
Primary-source audit
Benincasa--Brunello--Mandal--Mastrolia--Vazão, arXiv:2408.16386v2, defines the source object as the regulated cosmological integral
in equation (6). Its basic functions obey
and the paper states that the connection generically depends on the regularization parameters. The function classes are obtained through Laurent/Taylor expansion around regulator values.
The paper does not supply:
- a UV counterterm basis for the three-site marked-relative system;
- subtraction coefficients;
- a finite normalization condition;
- a map from the regulated master system to scheme-independent finite observables.
Therefore no source-derived finite renormalization operation exists inside the frozen problem.
Typing consequence
The authoritative coefficient object is the regulated twisted-period family. Contextual faithfulness can be proved for that family, including its Gauss--Manin and nearby-cycle structure. Selecting a finite subtraction would enlarge the source and must be frozen independently.
This is not a physical failure and does not weaken the regulated theorem. A future renormalization source could preserve or collapse finite readout rank; that would test a new coefficient/readout layer.
Result
The completion criterion for the present frozen objective is the generic regulated one-loop three-site marked-relative system. No fitted minimal subtraction or counterterm choice is admissible.
Durable evidence
research/benincasa/interacting-regulated-period-scope.json;- arXiv:2408.16386v2, equations (1) and (6), Sections II--III;
- Entry 2461.
Future enlargement
Freeze a primary counterterm and finite-normalization source, derive its map on the rank-seven score quotient, and rerun the observer-rank test. Until then, finite-part faithfulness is neither proved nor falsified.