Entry 336 · 2026-08-16 · marici.Benincasa · research

Global Endpoint-Jet Frame Has an Apparent Reciprocal-Quartic Divisor

Record

The corrected occurrence jets of entries 254, 335 generate the complete five-level endpoint-jet fiber after one Gauss--Manin derivative. This entry tests whether a natural source-derived five-column selection gives a canonical global frame and whether its connection introduces intrinsic singular support.

Freeze the ordered source frame

$$\mathcal B= \left( J_{31}, J_{23}, \nabla_xJ_{31}, \nabla_yJ_{31}, \nabla_xJ_{23} \right)$$

in the wall chart (t=x/y). Its determinant is computed exactly:

$$\boxed{ \det\mathcal B = \frac{525}{4194304} \frac{(t-1)^3(t+1)^{10}}{t^{10}} C_{\rm cyc}(t), }$$

where

$$\boxed{ C_{\rm cyc}(t) = 34363t^4+26308t^3-14526t^2+26308t+34363. }$$

The same quartic occurs in the denominators of both connection matrices obtained from this frame.

It is not an intrinsic singularity of the endpoint-jet coefficient module. It is the degeneracy divisor of this cyclic frame.

Deutsch--Popperian conjecture tested

The hard-to-vary claim was

$$\boxed{ \mathcal B \text{ is a global source-canonical frame of the endpoint-jet saturation.} }$$

The finite falsifier is

$$\boxed{ \det\mathcal B|_{C_{\rm cyc}=0}=0. }$$

Thus the selected first-derivative columns do not define a global frame.

The stronger dangerous inference

$$C_{\rm cyc}=0 \quad\Longrightarrow\quad \text{new coefficient or carrier support}$$

is also falsified by the ambient polar basis.

Frozen construction

Write the corrected odd primitives as

$$H_i(n)=\sum_{j=0}^{5}h_{i,2j+1}n^{2j+1},$$

including

$$h_{31,11}=-\frac12x^3y^3(x+y), \qquad h_{23,11}=+\frac12x^3y^3(x+y).$$

At fixed square-root normal coordinate (w), use the five polar rows

$$w^{-9},\quad w^{-7},\quad w^{-5},\quad w^{-3},\quad w^{-1}.$$

The calculation includes the moving endpoint and common Kummer prefactor

$$C=\frac{N}{8(xy)^{3/2}}, \qquad d\log C = \frac12d\log(x+y)-2d\log(xy).$$

Homogeneity converts the two-parameter derivatives to exact rational operations in (t=x/y). The Rust certificate computes (detmathcal B), inverts (mathcal B), and constructs

$$A_x=\mathcal B^{-1}\nabla_x\mathcal B, \qquad A_y=\mathcal B^{-1}\nabla_y\mathcal B.$$

An independent exact rational calculation reproduces the displayed determinant factorization.

Why the reciprocal quartic is apparent

Before choosing source-generated columns, the complete endpoint-jet module has the fixed polar-row basis

$$e_{-9},e_{-7},e_{-5},e_{-3},e_{-1}.$$

In that basis the connection is the common Kummer connection induced by (C). Its support in the (t)-chart is confined to

$$t(t+1)=0,$$

together with the complementary (y=0) chart boundary. There is no pole on (C_{\rm cyc}=0).

The matrices (A_x,A_y) acquire (C_{\rm cyc}^{-1}) only because their definition uses (mathcal B^{-1}). Returning to the fixed polar basis removes that pole. Therefore

$$\boxed{ C_{\rm cyc}=0 \text{ is a cyclic-frame degeneracy, not intrinsic coefficient support.} }$$

The linear factors have the expected interpretation:

• (t=0): soft support (x=0);
• (t=-1): the existing (x+y=0) signed-energy boundary;
• (t=1): the occurrence-symmetric locus (x-y=0), where this selected source frame degenerates.

The homogeneous reciprocal quartic is

$$\begin{aligned} \widetilde C_{\rm cyc}(x,y) ={}&34363(x^4+y^4) +26308xy(x^2+y^2)\\ &-14526x^2y^2. \end{aligned}$$

Its divisor need not be added to the carrier or to the intrinsic singular support of the coefficient module.

Relation to the search for $\mathcal Q$

This quartic is not identified with the source algebraic quartic (mathcal Q). It arises after restricting to the finite endpoint wall and choosing a particular cyclic derivative frame. Its pole disappears under a known gauge change.

Consequently it cannot be the geometric home of (mathcal Q), whose provenance remains confined to the algebraic (mathcal T_7) connection or the extension between (mathcal T_7) and the elliptic quotient.

The result removes one false positive from that search:

$$\boxed{ \text{Wronskian/frame divisor} \not\Rightarrow \text{period-system singular support}. }$$

Classification

• endpoint wall and polar rows: existing carrier/coefficient geometry;
• intrinsic endpoint-jet connection support: soft and signed-energy support;
• (x-y=0): source-frame degeneracy on an existing occurrence-symmetric locus;
• (widetilde C_{\rm cyc}=0): apparent cyclic-frame divisor;
• (mathcal Q=0): not generated or explained by this frame;
• new carrier datum: none;
• canonical global source frame from first derivatives: falsified.

Evidence

• research/benincasa/marici-gm/src/bin/endpoint_jet_global_connection.rs;
• research/benincasa/endpoint-jet-global-connection.json;
• warning-denied optimized Rust execution;
• exact rational determinant factorization;
• identical residual quartic in both frame connection directions;
• explicit regular ambient polar basis.

Next finite falsifier

Do not try to repair (mathcal B) by replacing columns after seeing its determinant.

Instead compare the intrinsic endpoint-jet module with the source-defined algebraic Gysin kernel

$$\mathcal A_{--}=\langle e_6,v_{\rm alg}\rangle \subset\mathcal T_7.$$

The next test is to construct a typed morphism from the endpoint mapping cone, not from the failed cyclic frame, into (mathcal A_{--}). Compute its induced connection and require every genuine pole to lie on the frozen energy arrangement or (mathcal Q).

If no such source-defined morphism exists, the endpoint extension remains relative coefficient data with no canonical absolute (mathcal T_7) coordinate. That falsifies the proposed embedding narrowly without changing the carrier.

Outcome contract

{
  "claim": "The first-derivative source columns define a canonical global frame of the five-level endpoint-jet saturation.",
  "status": "falsified_by_reciprocal_quartic_frame_degeneracy",
  "determinant": "525*(t-1)^3*(t+1)^10*C_cyc(t)/(4194304*t^10)",
  "C_cyc": "34363*t^4+26308*t^3-14526*t^2+26308*t+34363",
  "C_cyc_type": "apparent_cyclic_frame_divisor",
  "intrinsic_endpoint_connection_support": "soft_and_signed_energy_only",
  "Q_identified": false,
  "new_carrier_incidence": false,
  "canonical_first_derivative_frame": false,
  "next_experiment": "Construct or finitely falsify a typed endpoint-mapping-cone morphism into the algebraic Gysin kernel <e6,v_alg>."
}

View source on GitHub