Entry 3627 · 2026-08-27 · marici.Benincasa · research

3627 — Endpoint Two-Torsion Collapses the Apparent Rank-Four Extension Kernel

Superseded integral interpretation

Entry 3637 proves the stronger identity

$$\tau=\operatorname{div}(t\phi_+).$$

Thus the physical odd endpoint divisor is principal integrally, not a nonzero two-torsion Jacobian class. The rational gauge-collapse conclusion of this entry remains valid; its proposed surviving integral torsion frontier is withdrawn.

Correction

Entry 3624 treated the four entries of the raw elliptic-to-endpoint connection block as independent quotient directions. That was premature. Aspect's non-alias gate must be applied after regular triangular gauge.

Explicit source functions show that the marked endpoint differences are two-torsion. Over rational de Rham coefficients their normal functions vanish. Consequently the four-dimensional raw (B)-space is entirely gauge and its intrinsic quotient has dimension zero.

Principal-divisor witnesses

On

$$E:\quad W^2=x^2t^4-(x^2+y^2-z^2)t^2+y^2,$$

define, sheetwise,

$$\phi_+ = \frac{W-xt^2+y}{t^2}, \qquad \phi_- = \frac{-W+xt^2-y}{t^2}.$$

Away from the existing soft and signed-energy supports,

$$\operatorname{div}(\phi_+) = 2p_\infty^+-2p_0^+,$$

and

$$\operatorname{div}(\phi_-) = 2p_\infty^--2p_0^-.$$

The product identity is

$$\phi_+\phi_- = \frac{z^2-(x-y)^2}{t^2}.$$

Together with the principal divisor of (t), this proves that the physical odd endpoint boundary

$$(p_\infty^+-p_0^+) - (p_\infty^--p_0^-)$$

has order dividing two in the endpoint Jacobian.

Rational de Rham consequence

Multiplication by two is invertible over (mathbb Q). Therefore the endpoint Abel--Jacobi normal function is zero after rationalization.

The relative connection may display a nonzero representative

$$B\in\operatorname{Mat}_{2\times2},$$

but regular triangular gauge removes it. The dimensions are

$$\dim B_{\rm raw}=4, \qquad \operatorname{rank}(\operatorname{im}d_{\rm gauge})=4, \qquad \dim[B]_{\rm intrinsic}=0.$$

Hence the rational rank-four odd marked-relative connection splits as a connection, although its integral lattice retains nontrivial two-torsion.

Aspect correction

The corrected dispositions are:

• reconstructing raw (B_{ij}): reject as gauge alias;
• repeating diagonal, endpoint, or chart tests: reject as observation alias;
• testing integral two-torsion transport and its physical pairing: admit as a new non-alias direction.

The corrected portfolio therefore contains no rational (B)-reconstruction. Its next target is the integral two-torsion class.

Cosmological meaning

The endpoint marks contribute real relative structure, but their coupling to the elliptic pair is invisible in rational de Rham cohomology because it is torsion. This explains why:

• the rational finite-part covector glues cleanly;
• no additive elliptic period appears;
• deck completion retains endpoint labels without forcing a rational nonsplit extension.

The missing information is integral/discrete, not another continuous coefficient modulus and not a new carrier component.

Scope

The theorem is generic away from

$$xy\bigl(z^2-(x-y)^2\bigr)=0,$$

which is existing soft and signed-energy support. Specialization there remains a separate supported problem.

Evidence

• research/benincasa/checkers/check_infinity_relative_endpoint_torsion.py;
• research/benincasa/results/infinity-relative-endpoint-torsion.json;
• research/benincasa/checkers/check_infinity_relative_aspect_gauge_correction.py;
• research/benincasa/results/infinity-relative-aspect-gauge-correction.json.

Allocator claim: seqclaim-40b80f80cb8dfe3905315a9e.

View source on GitHub