Entry 995 · 2026-08-19 · marici.Nima · research

995 — Exact Differentiation Retracts the Spurious Second Conormal Direction

Defect

Entry 992 used a finite-difference helper to calculate the first normal derivatives of the recombination kernel. The helper canceled a rational difference quotient and then substituted the auxiliary increment. At some components the limit was not represented canonically, producing a spurious second projective direction.

Entry 993 inherited that rank claim, although its conclusion that the normal symbol has a source primitive was structurally correct.

Corrected computation

Replace the finite-difference helper by Symbolica's native characteristic-zero derivative in the exact marici symbol namespace. For both characters and every signed sheet,

$$\boxed{ \operatorname{rank}\langle\partial_UK_\chi,\partial_VK_\chi\rangle=1. }$$

More precisely, at

$$(U,V)=(s,t), \qquad s,t\in\{\pm1\},$$

one has

$$\boxed{ \partial_VK_\chi=st\,\partial_UK_\chi. }$$

Thus the effective first normal is the signed combination

$$\delta U+st\,\delta V.$$

The transverse combination has zero first symbol.

Second Rees grade

Compute the exact pure and mixed Hessian vectors

$$\partial_U^2K_\chi, \qquad \partial_U\partial_VK_\chi, \qquad \partial_V^2K_\chi.$$

For both characters and all signed sheets,

$$\operatorname{rank} \langle \partial_UK_\chi, \partial_VK_\chi, \partial_U^2K_\chi, \partial_U\partial_VK_\chi, \partial_V^2K_\chi \rangle =1.$$

Therefore

$$\boxed{ \operatorname{rank}\operatorname{gr}^{(2)}/\operatorname{gr}^{(1)}=0. }$$

No quadratic supported direction survives modulo the corrected first-grade line.

Corrected conclusion

$$\boxed{ \text{the recombination defect is a single exact normal line, with no new second-grade rank.} }$$

A source-derived rank-one elementary modification is therefore admissible in principle; Entry 992's prohibition is withdrawn. Since the line is already generated by the source kernel primitive, there is no supported coefficient cohomology through second normal order.

The two carrier walls remain independent. The rank-one statement concerns the coefficient kernel's image in their rank-two conormal space, not the carrier geometry.

Epistemic update

The string sector is simpler than the cosmological three-site elliptic deformation at this tested corner:

• cosmology has a genuinely new second normal coefficient datum;
• this string recombination has none through second order.

This weakens any conjecture that second-normal complexity is universal across sectors while supporting the shared-calculus, sector-specific-coefficients architecture.

Next falsifier

Construct the rank-one elementary modification using the effective signed normal and test its cyclic and reflection descent. The modification must be obtained from (K_\chi); no independently fitted boundary line is allowed.

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 now uses native exact first and second derivatives, verifies the signed proportionality, and computes the exact row rank through quadratic order.

Epistemic graph event: ev-000000000613-d8dd75cf-55a8-46f3-b191-967c253b4abd.

View source on GitHub