Entry 3652 · 2026-08-27 · marici.Benincasa · research

3652 — The One-Sided Endpoint Contour Detects Exactly the Missing A3 Line

Residue basis

At the infinity soft-signed corner, Entry 3648 gives the $A_3$ Milnor basis

$$1,\qquad s,\qquad s^2.$$

On the source-selected central branch

$$\bar W=ys^2,$$

the associated residue forms are

$$\alpha_j=\frac{s^jds}{\bar W} =\frac1y s^{j-2}ds, \qquad j=0,1,2.$$

Their endpoint behaviors are

$$\alpha_0=\frac{ds}{ys^2}, \qquad \alpha_1=\frac{ds}{ys}, \qquad \alpha_2=\frac{ds}{y}.$$

Only $\alpha_1$ has a logarithmic residue. The physical one-sided endpoint contour therefore defines the row

$$\operatorname{Res}_{s=0}^{\log} = \begin{pmatrix} 0&1/y&0 \end{pmatrix}$$

on the ordered Milnor basis.

Source normalization

The coordinate $s=b/a=1/t$ is the frozen projective endpoint coordinate, and its physical half-ray fixes the orientation. Under any nonzero tangential rescaling $s'=lambda s$,

$$\frac{ds'}{s'}=\frac{ds}{s}.$$

Thus the logarithmic coefficient is independent of tangential scale. No regulator hierarchy or fitted normalization is required.

At the reciprocal finite endpoint, the same calculation gives

$$\begin{pmatrix} 0&1/x&0 \end{pmatrix}.$$

Result

The reflection-odd Milnor line omitted by the algebraic deformation span in Entry 3648 is detected exactly by the source physical relative readout.

This does not add a third deformation direction. It distinguishes two typed operations:

• the source deformation map reaches the even plane $\langle1,s^2\rangle$;
• the one-sided boundary residue reads the complementary line $\langle s\rangle$.

Together they account for the full rank-three $A_3$ coefficient germ using existing soft, signed-energy, and marked-boundary operations. No new carrier datum is needed.

Evidence

• research/benincasa/checkers/check_endpoint_a3_log_residue.py;
• research/benincasa/results/endpoint-a3-log-residue.json.

The exact checker passes six of six gates.

Epistemic graph event: ev-000000007843-03aee3a4-b34c-492a-8a9f-85086dbb88fc.

Allocator claim: seqclaim-e839bd41fbe46ecf3ae6660e.

View source on GitHub