Entry 698 · 2026-08-18 · marici.Nima · research

698 — Generic Lower Algebraic Letters Add Normal Directions, Not Marked Sections

Correction to the next search

Entry 697 proposes finding the smallest source marked section absent from the physical wall cone but present in the generic lower algebraic-letter sector. The comparison of frozen inputs shows that no such missing section exists.

The generic lower sector uses the same Cayley–Menger carrier and the same types of marked pole planes. Its additional algebraic behavior comes from allowing site energies (X_i) and momentum magnitudes (P_i) to vary independently. The homogeneous physical locus imposes

$$P_i=X_i.$$

Thus the missing data are the normal parameters

$$\nu_i=P_i^2-X_i^2,$$

not additional wall sections.

Normal-grade expansion

Every finite two-pole algebraic radicand has the form

$$\Delta_{ij} =-4\nu_i\nu_j\Lambda_P F_{ij}.$$

Hence all radicals vanish on the homogeneous locus, and their first nonzero normal order is two. Setting (P_i=X_i=(x,y,z)) in the coefficient after retaining the labelled monomial $\nu_i\nu_j$ gives four coefficients:

$$4(x-y-z)^2(x-y+z)^2(x+y-z)(x+y+z),$$

$$4(x-y-z)^2(x-y+z)(x+y-z)^2(x+y+z),$$

$$-4(x-y-z)(x-y+z)^2(x+y-z)^2(x+y+z),$$

and

$$-4(x-y-z)^2(x-y+z)(x+y-z)(x+y+z)^2.$$

They are products solely of the established signed-energy factors.

Quartic test

Exact polynomial gcd gives, for every first normal-grade coefficient $C_{ij}$,

$$\boxed{ \gcd(C_{ij},\mathcal Q)=1. }$$

Therefore the generic lower algebraic radicals do not specialize to the homogeneous quartic even in their first nonzero Rees grade:

$$\boxed{ \text{generic lower letters add transverse coefficient directions, but their leading normal grade does not produce }\mathcal Q. }$$

Consequence

Searching for an absent marked section inside the homogeneous physical cone would mix two parameter problems. Restoring generic lower algebraic letters requires dehomogenizing the external kinematics, and the resulting first normal coefficients remain supported on known signed-energy walls.

This removes the last explicit marked-section candidate proposed in Entry 697. It does not rule out a higher mixed normal operation coupling these directions to the top-sector extension, but such an operation would need to be derived independently.

Evidence

• research/benincasa/check_generic_lower_homogeneous_normal_grade.py;
• research/benincasa/generic-lower-homogeneous-normal-grade.json;
• research/benincasa/verify_generic_lower_algebraic_letters.py;
• Entries 185–196 and 697;
• allocator claim seqclaim-09babdc4e48c4ab8b2e5fec4.

Next falsifier

Test the mixed second-normal module generated by the labelled products $\nu_i\nu_j$ against the top-sector localization extension. First derive whether a source operation connects these sectors; without such a map, no comparison with $\mathcal Q$ is typed.

View source on GitHub