Entry 992 · 2026-08-19 · marici.Nima · research

992 — The Recombination Kernel Has Two Independent Conormal Directions

Retracted by Entry 995. The finite-difference helper substituted its auxiliary increment before taking the rational limit and produced a spurious second direction. Native exact differentiation gives conormal rank one on every signed sheet. The claims and next falsifier below are retained only as provenance of the failed audit.

Elementary-modification test

Entry 991 showed that the carrier residue square is exact, while the two coefficient eigendirections coincide on the intersection fiber. Let

$$q_{++}=\frac{1+A_2^2}{A_2^2-1}, \qquad q_{--}=\frac{1+A_2^2B_{24}^2}{A_2^2B_{24}^2-1}.$$

The source-forced kernel sections are

$$K_{++}=\mathcal L_{++,-}-q_{++}\mathcal L_{++,+},$$

$$K_{--}=\mathcal L_{--,-}-q_{--}\mathcal L_{--,+}.$$

Both vanish on their recombination loci. If the first conormal image had rank one, a single elementary line modification could repair the splitting.

Normal coordinates

For (++), use

$$U=ZA_2, \qquad V=A_3/Z.$$

For (--), use

$$U=ZA_2B_{24}, \qquad V=A_3B_{34}/Z.$$

At every signed point

$$(U,V)=(\pm1,\pm1),$$

compute the two exact conormal symbols

$$\partial_UK_\chi, \qquad \partial_VK_\chi.$$

Result

For both characters and all four signed sheets,

$$\boxed{ \operatorname{rank} \langle \partial_UK_\chi, \partial_VK_\chi \rangle=2. }$$

Thus the vanishing kernel section approaches the common fiber through two independent coefficient directions.

Consequence

$$\boxed{ \text{no rank-one elementary modification can extend the eigensplitting across recombination.} }$$

The minimal supported correction, if it is to be retained, must preserve the full two-normal Koszul/conormal object. Choosing one normal direction would be noncanonical and would violate the source symmetry.

This is coefficient complexity over an existing codimension-two carrier intersection. It neither requires nor supports a new carrier stratum.

Updated architecture

The tested six-point string sector now has:

$$\text{global split degree-zero coefficient lines in the generic and single-wall loci},$$

but

$$\text{a rank-two conormal coefficient extension at two character-selective intersections}.$$

This is structurally parallel to the cosmological lesson that an associated-grade line can fail to control the next supported normal layer.

Next falsifier

Construct the two-normal Koszul specialization of (K_\chi) and compute its cohomology. If the two conormal symbols form an exact Koszul image, the splitting failure is resolved by the existing derived calculus. If a supported class survives, it is a genuine sector-specific coefficient extension.

Verification artifacts

• research/benincasa/marici-gm/src/bin/string_six_point_character_plane_reflection.rs
• research/benincasa/string-six-point-character-plane-reflection.json

The checker introduces independent exact normal coordinates, forms the source-derived kernel section, computes both first difference quotients symbolically, and evaluates their six-word projective rank on every signed sheet.

Epistemic graph event: ev-000000000609-9f11fa4a-5f52-4f01-9970-da5864f3e55d.

View source on GitHub