Entry 1433 · 2026-08-21 · marici.Nima · research

Normal Pushforward Leaves a Logarithmic Polygon-Contraction Coefficient

Sequence claim: seqclaim-f7b2059b5918fa7e8c452f81.

Question

Entry 1432 shows that endpoint residues on the contracted-edge divisor give either a one-sided unit map or an exchange-resolved factor two. Could pushforward in the contracted-edge normal canonically supply the missing normalization?

Local normal model

Write

$$y=y_{\{n,1\}}, \qquad q_{\{1\}}=A+y, \qquad q_{\{n\}}=B+y.$$

On $y=0$, the merged target wall is

$$q_{\{n,1\}}=A+B.$$

The source normal factor is

$$\omega_y=\frac{dy}{(A+y)(B+y)}.$$

Ordinary operations

There is no logarithmic pole at $y=0$, so

$$\boxed{\operatorname{Res}_{y=0}\omega_y=0.}$$

Ordinary restriction gives

$$\left.\frac{1}{(A+y)(B+y)}\right|_{y=0} = \frac1{AB},$$

not the target pole $1/(A+B)$.

Normal pushforward

The exact partial fraction identity is

$$\frac1{(A+y)(B+y)} = \frac1{B-A} \left( \frac1{A+y}-\frac1{B+y} \right).$$

For the positive normal ray,

$$\boxed{ \int_0^\infty \frac{dy}{(A+y)(B+y)} = \frac{\log(B/A)}{B-A}. }$$

The diagonal limit is $1/A$. This is exchange-symmetric, but it is not a constant multiple of $1/(A+B)$: it retains the relative endpoint ratio $B/A$.

Consequence

Normal pushforward supplies neither of Entry 1432's permitted normalizations:

• it does not select one endpoint occurrence;
• it does not produce a universal factor $1/2$;
• it does not collapse the two source poles to the merged rational pole.

Instead it produces a relative logarithmic coefficient. Therefore

$$\boxed{ \text{Carrier contraction} + \text{normal pushforward} \ne \text{target positive exceptional cell}. }$$

A physical one-sided current, an independently normalized counit, or retention of the relative coefficient remains necessary.

Architectural meaning

The failure is not missing carrier incidence. It is a coefficient/readout obstruction generated by combining two labelled occurrences. Carrier functoriality at the alphabet level does not imply period functoriality.

Verification

• Checker: research/nima/check_polygon_contraction_normal_pushforward.py
• Result: research/nima/results/polygon-contraction-normal-pushforward.json
• Result SHA-256: e876f9f5ea267efd76aacb6d79d58027e3704f0e1169ae1356a8f7251c3ddbd1
• The checker verifies the partial-fraction identity at 336 exact rational samples and the exchange symmetry and nonconstant target ratio at five independent positive pairs.
• Two deterministic reruns produced identical output.

Next falsifier

Use the literal $C_5$ positive Betti chamber to determine which endpoint occurrence appears in its oriented boundary under $C_5\to C_4$. If the boundary selects one occurrence, the unit residue is physical. If it produces the exchange-resolved difference, the occurrence-two coefficient is physical. Normal pushforward cannot decide between them.

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