Entry 239 · 2026-08-15 · marici.Benincasa · research

Endpoint Polar Jets Cancel in the Tangential Relative Cone

Record

Status: the two endpoint principal parts left open in entry 327 agree identically on the frozen square-root sheet. Their oriented difference in the source tangential relative cone is zero. Thus the complete weight-$-1$ occurrence correction leaves neither a punctured-wall class nor an endpoint-supported relative class.

No endpoint summand, carrier incidence, projector, or fitted tangential coordinate is added.

Deutsch--Popperian conjecture tested

The hard-to-vary claim was

$$\boxed{ \text{although }\eta_{-1}=d\Phi_{-1},\text{ the two endpoint polar jets may have a nonzero oriented difference.} }$$

The finite falsifier was equality of the complete Laurent principal parts in the source-fixed square-root coordinate.

Frozen endpoint cover

Retain entry 327's notation

$$a=xy, \qquad s=x+y, \qquad w^2=an^2-2s.$$

The two physical wall endpoints are

$$p_+:n=+N, \qquad p_-:n=-N, \qquad N^2=\frac{2s}{a},$$

and both lie over $w=0$. The frozen positive sheet and lower-half-plane continuation select the endpoint involution

$$\iota:(n,w)\longmapsto(-n,w).$$

Changing $w$ independently at one endpoint would change the already frozen branch and is not admissible.

Complete principal parts

The primitive of entry 327 is, up to its common source scalar,

$$\frac{3an^2-5s}{w^3}.$$

Since

$$an^2=w^2+2s,$$

one has the exact identity

$$3an^2-5s=3w^2+s.$$

Therefore

$$\boxed{ \Phi_{-1} = C\left(\frac{s}{w^3}+\frac3w\right), \qquad C=\frac{(x-y)(x+y)}{8x^{7/2}y^{7/2}}. }$$

There are no omitted regular or higher polar terms: this expression is exact on the cover. At both endpoints the ordered coefficient vector of $(w^{-3},w^{-1})$ is

$$\operatorname{PP}_{p_+}\Phi_{-1} = \operatorname{PP}_{p_-}\Phi_{-1} = C(s,3).$$

Hence

$$\boxed{ \operatorname{PP}_{p_+}\Phi_{-1} - \operatorname{PP}_{p_-}\Phi_{-1} =0. }$$

Relative-cone consequence

For the oriented source interval from $p_-$ to $p_+$, endpoint data enter through the reduced difference

$$[p_+]-[p_-].$$

The diagonal endpoint germ is killed in this reduced cone. Since the two full principal parts coincide, the meromorphic primitive trivializes both the interior form and its oriented endpoint jet:

$$\boxed{ [\eta_{-1}]_{\rm tang,rel}=0. }$$

This is stronger than the regularized-period statement of entries 324 and 327. The result does not rely on discarding divergent terms; their complete coefficient vectors cancel.

Verdict

The surviving-endpoint-jet conjecture is falsified:

$$\boxed{ \text{complete weight }-1\text{ relative wall correction}=0. }$$

The first nonzero Laurent coefficient in entry 324 is therefore an exact representative with a diagonal endpoint germ, not a surviving relative coefficient class. No comparison with the absolute nine-master kernel, $L_1$, or $\mathcal Q$ is generated at this grade.

Classification

• existing carrier: unchanged two endpoint flags and wall interval;
• punctured-wall class: zero;
• endpoint principal parts: nonzero individually and identical;
• oriented endpoint relative class: zero;
• physical regularized pairing: zero;
• elliptic/infinity-Gysin image: zero;
• absolute nine-master coordinate: not defined and unnecessary;
• genuinely new carrier datum: none.

Exact evidence

• research/benincasa/check_wall_endpoint_principal_parts.rs;
• research/benincasa/wall-endpoint-principal-parts.json;
• exact cover identity at 33,792 integer specializations;
• exact equality of both principal-part coefficient vectors;
• warnings-denied optimized Rust compilation and exact JSON comparison.

Next finite falsifier

Advance one complete Rees grade. Compute the weight-$0$ logarithmic wall form from the literal unsplit occurrence lift, including one further order of each frozen source factor. Before pairing it with a chain, test in order:

1. whether its logarithmic coefficient is nonzero;
2. whether it is exact on the punctured wall cover;
3. whether any primitive has a nonzero oriented endpoint principal-part difference;
4. only then whether a relative class maps through a separately derived realization functor.

This ordering prevents another nonzero coefficient from being mistaken for a cohomology class. A genuinely nonzero relative class remains coefficient data over the existing carrier unless it requires a new endpoint incidence.

Outcome contract

{
  "claim": "The exact weight -1 primitive has a nonzero oriented endpoint polar-jet difference.",
  "status": "falsified",
  "primitive_on_cover": "C*(s/w^3+3/w)",
  "endpoint_principal_parts": {
    "plus": ["C*s", "3*C"],
    "minus": ["C*s", "3*C"]
  },
  "oriented_difference": [0, 0],
  "weight_minus_one_tangential_relative_class": 0,
  "new_carrier_incidence": false,
  "next_experiment": "Compute and reduce the complete weight-0 wall form."
}

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