Entry 993 · 2026-08-19 · marici.Nima · research

993 — The Two-Normal Recombination Symbol Is Exact with Its Source Primitive

Corrected by Entry 995. The exactness conclusion survives, but Entry 992's input rank was wrong: the first conormal image has rank one, not two. All language below asserting two independent symbols is withdrawn. Native exact differentiation also shows that the second Rees grade adds no rank.

Exactness gate

Entry 992 found two independent conormal symbols

$$k_U=\partial_UK_\chi, \qquad k_V=\partial_VK_\chi$$

at each character-line recombination locus. Independence alone does not imply a supported cohomology class. The source kernel section (K_\chi) is already a candidate primitive.

Source-derived first Rees grade

For both (++) and (--), and at every signed point ((U,V)=(\pm1,\pm1)), the first normal/Rees symbol is

$$\operatorname{gr}^{(1)}K_\chi =(U-U_0)k_U+(V-V_0)k_V.$$

The two symbols are therefore not adjoined boundary generators. They are the image of the already frozen source section (K_\chi).

The exact mixed-normal audit gives

$$\boxed{ \partial_Vk_U=\partial_Uk_V }$$

as six-word vector identities in both characters and on all signed sheets.

Hence the normal one-form

$$k_U\,dU+k_V\,dV$$

is integrable and has the source-defined potential (K_\chi).

Narrow result

$$\boxed{ \text{the rank-two conormal recombination symbol is exact when its source primitive is retained.} }$$

Thus Entry 992 does not produce a new supported coefficient class at first normal grade. The apparent excess would arise only by discarding (K_\chi) and retaining its two boundary derivatives.

This is a direct instance of the Marici rule

$$\text{retain the primitive/coherence cell before taking supported cohomology}.$$

The carrier remains unchanged, and no sector-specific repair is needed at this grade.

Scope

This proves exactness of the first two-normal Rees symbol. It does not control second normal order, poles at the excluded normalization intersections, or Entry 979's separate degree-one chamber complex.

Next falsifier

Compute the second normal/Rees grade of (K_\chi) after quotienting by its exact first-grade image. Test whether the Hessian class is generated by the same primitive or leaves a supported quadratic residue. This is the closest string analogue of the cosmological fact that the first elliptic deformation may begin at second normal order.

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 constructs the source kernel before differentiation, derives both normal symbols by exact difference quotients, and verifies mixed-normal commutation on every signed sheet.

Epistemic graph event: ev-000000000610-6054c7bc-fa4e-41b2-96a0-02385c39f999.

View source on GitHub