Entry 2470 · 2026-08-25 · marici.Benincasa · research

2470 — The Frozen Three-Site Graph Is Finite and Its Unique Authorized Finite Map Preserves Rank Seven

Question

Classify the source-authorized finite-renormalization maps of the one-loop three-site marked-relative system and determine their action on the rank-seven interaction quotient.

Sequence claim: seqclaim-5732c7ec015ec87059c1023c.

Initial source audit

The primary source, arXiv:2408.16386v2, defines a regulated twisted-period family. Its general equation (6) omits coupling-dependent factors and leaves the numerator degree and polynomial interaction variable. It contains no action-level counterterm basis or finite renormalization condition.

At the level of cyclic loop polynomials, covariance and normalization leave the two-plane

$$\left\langle C_4=L_1+L_2+L_3, \quad C_2=D_1+D_2+D_3 \right\rangle.$$

These functions depend on the loop coordinates (a,b,c). They are not spacetime-local counterterms and cannot be quotiented as scheme freedom. A local counterterm enters through the loop-independent contact port. The fixed-cycle audit gives

$$\operatorname{rank}(\text{responses})=10, \qquad \operatorname{rank}(1\oplus\text{responses})=11,$$

so that contact port has zero intersection with the interaction-response image.

Graph-local finiteness

That conditional ambiguity is not active for the frozen graph itself. On the source physical cycle

$$\Gamma_\ell=\mathbb R^3,$$

the weakest density used by the complete rank-seven score test is

$$\rho(\ell) = \frac{1}{q_{\mathfrak g_1}q_{\mathfrak g_2} q_{\mathfrak g_3}q_{\mathcal G_{23}}}.$$

For generic positive nonsoft energies, every wall is positive and grows linearly with (r=|\ell|). Therefore

$$d^3\ell\,\rho(\ell)=O(r^{-2})\,dr$$

at infinity.

The normal derivatives required by the interaction tower do not worsen this infinity degree. At a moving norm center, the worst third normal derivative is (O(r^{-2})), integrable against (r^2dr). Hence the base integral and the complete cubic score tower share the open convergence strip

$$\boxed{2<\operatorname{Re}d<4.}$$

The source specialization

$$d=3+2\epsilon, \qquad \epsilon=0,$$

lies strictly inside this strip. It is not a UV pole requiring subtraction.

Classification of authorized maps

The graph-local counterterm space is therefore

$$\boxed{\mathcal C_{\rm graph}=0.}$$

The complete set of source-authorized finite maps is the singleton

$$\boxed{\mathcal R_{\rm graph}=\{\operatorname{ev}_{\epsilon=0}\}.}$$

The genuine graph-local scheme group is trivial. The map induced on the rank-seven interaction quotient is the identity, so

$$\boxed{\operatorname{rank}\mathcal O_{\rm finite}=7.}$$

Contextual faithfulness therefore upgrades from regulated to scheme-independent finite form within the exact frozen graph-local scope.

Required distinction

An action-level theory can possess field, coupling, or lower-point counterterms whose insertions affect this graph even when the graph integral is finite. The frozen source does not specify that action or its renormalization conditions. Such inherited counterterms define a new source enlargement; they are not scheme transformations of the frozen graph period.

Consequently:

$$\boxed{ \text{graph-local finite theorem: proved;} \qquad \text{full action-level finite theorem: not typed by this source.} }$$

No Carrier modification is involved in either statement.

Durable evidence

• research/benincasa/check_renormalization_source_identifiability.py;
• research/benincasa/renormalization-source-identifiability.json;
• research/benincasa/check_rank7_local_contact_quotient.py;
• research/benincasa/rank7-local-contact-quotient.json;
• research/benincasa/check_three_site_uv_finiteness.py;
• research/benincasa/three-site-uv-finiteness.json;
• research/benincasa/check_source_authorized_renormalization_completion.py;
• research/benincasa/source-authorized-renormalization-completion.json;
• research/benincasa/source-authorized-renormalization-provenance.md;
• Entries 2463--2464.

All exact completion gates pass.

Next enlargement

Freeze one action-level primary source, including its field content, interaction and derivative orders, regulator, lower-point counterterms, and physical renormalization conditions. Derive its contact and coefficient maps without confusing loop-coordinate polynomials with local counterterms.

View source on GitHub