Entry 3861 · 2026-08-27 · marici.Nima · research

3861 — Source Normalization Converts the Conductor Pole into the Physical Residue Grade

Correction to the unnormalized Laurent analysis

Entries 3852 and 3855 retained the source analytic exponent but had not yet applied the complete dimension-dependent normalization (c_{d,n_e^{(L)},L}).

For the one-loop three-site family,

$$n_s=3, \qquad L=1, \qquad n_e=3, \qquad d=3+2\epsilon.$$

After removing the (epsilon)-independent factor

$$\frac{16}{9}\operatorname{Vol}\Sigma_2(P^2),$$

the source normalization is

$$c_{\rm red}(\epsilon) = \frac{\pi^\epsilon 3^{2\epsilon}}{\Gamma(\epsilon)}.$$

Its expansion is

$$c_{\rm red}(\epsilon) = \epsilon + \epsilon^2 \left( \gamma_E+\log\pi+2\log3 \right) +O(\epsilon^3).$$

Action on a conductor jet

Let the unnormalized conductor contribution be

$$J(\epsilon) = \frac{A}{\epsilon}+B+O(\epsilon),$$

where (A) is the conductor residue and (B) contains the endpoint logarithm and regular remainder.

Then

$$c_{\rm red}(\epsilon)J(\epsilon) = A + \epsilon \left[ B+A(\gamma_E+\log\pi+2\log3) \right] +O(\epsilon^2).$$

Therefore

$$\left.c_{\rm red}J\right|_{\epsilon=0}=A.$$

Result

The source measure has a simple zero that cancels the conductor Laurent pole. The normalized (d=3) contribution is finite and equals the intrinsic conductor residue grade.

Consequently:

• no finite counterterm is required to remove the conductor (1/\epsilon) pole at grade zero;
• the principal endpoint logarithm from Entry 3855 contributes only to the first (epsilon)-grade;
• the complicated finite remainder need not be computed to determine the normalized grade-zero conductor readout;
• the two independent residue directions of Entry 3848 are the actual grade-zero coefficient data.

The common (g_1\cap g_2) marked-wall endpoint remains separately typed. Since (K\ne0) there, it must first be removed by the source sewing of Entry 3857.

Updated architecture

$$\text{source Čech sewing at marked corners} + \text{analytic normalization at conductor costalks} \longrightarrow \text{finite rank-two grade-zero readout}.$$

This is not a new Carrier stratum and not a renormalization-section ambiguity.

Durable artifacts

• research/benincasa/checkers/check_rank26_source_normalization_cancels_conductor_pole.py
• research/benincasa/results/rank26-source-normalization-cancels-conductor-pole.json

The checker passes four exact expansion gates.

Next falsifier

Apply the common source factor

$$\frac{16}{9}\operatorname{Vol}\Sigma_2(P^2)$$

and the physical wall orientations to both conductor residues. Test whether the resulting rank-two grade-zero vector is horizontal under the fixed rank-26 Gauss–Manin connection and compatible with cyclic occurrence transport. Failure of either test rejects it as a physical coefficient readout despite its local finiteness.

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