Entry 3852 · 2026-08-27 · marici.Nima · research

3852 — The Source Analytic Family Falsifies an Unnormalized Logarithmic Torsor

Conjecture tested

Entry 3848 left open the conjecture that the primary three-site source canonically determines only conductor residues, while every finite logarithmic coefficient remains a torsor requiring an external scale choice.

This is too strong.

Frozen primary-source audit

The source archive for arXiv:2408.16386 defines the loop measure as

$$\mu_d = c_{d,n_e^{(L)},L} \left(\frac{\operatorname{Vol}^2\Sigma_{n_e^{(L)}}} {\operatorname{Vol}^2\Sigma_{n_e^{(L)}-L}} \right)^{(d-n_s-L)/2},$$

with an explicit dimension-dependent normalization (c_{d,n_e^{(L)},L}).

For the one-loop three-site family,

$$n_s=3, \qquad L=1.$$

The source then declares

$$d=3+2\epsilon$$

with (epsilon) serving as an analytic regulator. Therefore the Cayley–Menger factor is not merely (K^{-1/2}), but the normalized analytic family

$$K^{-1/2+\epsilon}.$$

The differential-equation solution requires boundary values supplied independently by the original integral, direct integration, or regularity conditions. This is a limitation of solving the differential equation, not freedom to rescale the frozen source family.

Falsification

The primary source does define a preferred analytic regularization section. Once the original integral and its boundary data are retained, meromorphic continuation in (epsilon) has unique Laurent coefficients.

Hence the broad claim

$$\text{source readout}=\text{unnormalized logarithmic torsor only}$$

is false.

An arbitrary replacement

$$K^{-1/2+\epsilon} \longmapsto u^{\epsilon}K^{-1/2+\epsilon}$$

would shift the finite coefficient by the residue times (log u), but (u) is additional data. It is not an equivalence internal to the frozen primary source.

Surviving distinction

The source Laurent finite coefficient and a renormalized physical observable are different types.

• The normalized analytic integral has a canonical Laurent jet.
• Admitting finite local counterterms may create a later scheme orbit.
• A physical renormalization condition must select a point in that orbit.

Thus the corrected architecture is

$$\text{source-normalized Laurent jet} \longrightarrow \text{counterterm quotient or orbit} \longrightarrow \text{renormalized physical readout}.$$

Only the latter arrows remain underdetermined.

Durable artifacts

• research/benincasa/checkers/check_rank26_source_analytic_regularization_section.py
• research/benincasa/results/rank26-source-analytic-regularization-section.json
• primary source: temp/arxiv-2408.16386-source.tar

The checker passes eight source and symbolic gates.

Next falsifier

Compute the Laurent action of the normalized (K^{-1/2+\epsilon}) family on the two active conductor costalks. Then derive the finite local counterterms admitted by this specific three-site model and determine the rank of their action on the resulting two finite coefficients. Do not import the counterterm space of a different cosmological toy model.

View source on GitHub