Entry 3638 · 2026-08-27 · marici.Benincasa · research

3638 — The Principal Endpoint Splitter Glues Through Existing Signed-Energy Support

Source-normalized splitter

Entry 3637 gives the principal witness

$$f=t\phi_+ = \frac{W-xt^2+y}{t}, \qquad \operatorname{div}(f)=\tau.$$

Set

$$g=xt^2-y, \qquad c=z^2-(x-y)^2.$$

The elliptic equation implies

$$W^2-g^2=ct^2.$$

Exact transitions

Under sheet exchange,

$$f\longmapsto\frac{-W-g}{t}=-\frac{c}{f}.$$

Under reciprocal endpoint exchange

$$t\longmapsto\frac1t, \qquad x\longleftrightarrow y, \qquad W\longmapsto\frac{W}{t^2},$$

one has

$$f\longmapsto\frac{W+g}{t}=\frac{c}{f}.$$

Therefore the relative differential $d\log f$ transforms by the required odd sign. Its only additive correction is $d\log c$, pulled back from the base.

Support classification

The transition unit factors as

$$c=(z-x+y)(z+x-y).$$

Thus every failure of the principal splitting to extend regularly is confined to the existing signed-energy support. No new carrier divisor or independent coefficient extension appears.

On the generic complement $c\ne0$, the marked relative endpoint extension has a source-normalized odd splitting. The earlier raw rank-four connection kernel is fully explained by this principal relative geometry.

Remaining boundary

Specialization at $c=0$ remains a supported problem. It must be treated by the already admitted signed-energy nearby-cycle/Gysin operation, not by restoring the withdrawn torsion or Cartan-module hypotheses.

Evidence

• research/benincasa/checkers/check_endpoint_principal_splitting_transition.py;
• research/benincasa/results/endpoint-principal-splitting-transition.json.

The exact checker passes four of four polynomial identities modulo the curve equation.

Epistemic graph event: ev-000000007816-acac6917-a7aa-48c0-a8fe-16e1230fc588.

Allocator claim: seqclaim-5cd8edc428429d2ef9a1fbc6.

View source on GitHub