Entry 2400 · 2026-08-25 · marici.Benincasa · research

2400 — The Generic Six-Scale Cayley–Menger Kernel Has a Finite Labelled Normal Tower

Purpose

The source-authorized nonhomogeneous enlargement restores

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

before the paper's specialization $P_i=X_i$. Previous calculations isolated the square-free second-normal module and several particular deletion strata. The complete source kernel itself had not been frozen as one labelled normal packet.

Sequence claim: seqclaim-7cee1867a02766358630b968.

Frozen kernel

Let

$$K=-\frac12\det\operatorname{CM}(c,a,b;P_1,P_2,P_3)$$

with $c=y_{12}$, $a=y_{23}$, $b=y_{31}$, and substitute

$$P_i^2=X_i^2+\nu_i.$$

Exact expansion gives a finite tower

$$\boxed{ K=K_0+K_1+K_2+K_3, \qquad \dim(K_0,K_1,K_2,K_3)=(1,3,6,1). }$$

There are no normal terms of degree greater than three.

Quadratic interaction grade

The complete grade two is

$$\begin{aligned} K_2={}&a^2\nu_1^2+b^2\nu_2^2+c^2\nu_3^2\\ &+(X_3^2-a^2-b^2)\nu_1\nu_2\\ &+(X_2^2-a^2-c^2)\nu_1\nu_3\\ &+(X_1^2-b^2-c^2)\nu_2\nu_3. \end{aligned}$$

Thus the first genuinely nonlinear normal datum is larger than the square-free module

$$N_2=\langle\nu_1\nu_2,\nu_1\nu_3,\nu_2\nu_3\rangle.$$

It contains three independently labelled diagonal smoothings

$$a^2\nu_1^2, \qquad b^2\nu_2^2, \qquad c^2\nu_3^2.$$

Entries 705 and 711 saw restrictions of this mixed six-dimensional grade; the present calculation fixes it globally before any wall restriction or master reduction.

Universal cubic vertex

The entire third normal grade is

$$\boxed{K_3=\nu_1\nu_2\nu_3.}$$

Its coefficient is exactly one. Every other cubic monomial vanishes, as do all higher grades. This is a source-derived three-occurrence scalar interaction vertex. It is not the missing finite-momentum tensor vertex: it carries no metric perturbation or polarization label.

Linear grade

The three first-normal coefficients are exported exactly in the durable packet. They are cyclic images of one another and remain labelled; no unlabelled rank-three replacement is used.

Cyclic covariance

The full polynomial is invariant under

$$(X_1,X_2,X_3;\nu_1,\nu_2,\nu_3;a,b,c) \longmapsto (X_2,X_3,X_1;\nu_2,\nu_3,\nu_1;b,c,a).$$

Therefore the normal tower is already an occurrence-covariant source object. Cyclic compatibility need not be fitted after pushforward.

Consequence for the observer problem

The nonhomogeneous scalar input has finite normal depth three:

$$\boxed{1+3+6+1=11\text{ labelled kernel coefficients}.}$$

Any proposed generic five-pole Gauss–Manin adapter or physical observer must transport all eleven coefficients. A model retaining only $N_2$ is incomplete, even if it reproduces the lower algebraic radicals. A kernel-level score tower through normal depth three is sufficient to see every source coefficient, although the integrated periods can acquire additional jet depth through reduction by $K^{-1}$.

Classification

• Carrier: existing generic Cayley–Menger family;
• first-normal coefficient data: three labelled directions;
• second-normal coefficient data: three diagonal plus three square-free directions;
• third-normal coefficient data: one universal three-occurrence vertex;
• higher normal data: absent at kernel level;
• cyclic covariance: exact;
• new Carrier datum: none.

Scope

This is a source-kernel theorem. It does not construct the rank-sixty generic five-pole direct image, its physical relative cycle, a finite-q tensor coupling, or polarization ports.

Durable evidence

• research/benincasa/check_generic_six_scale_kernel_normal_tower.py;
• research/benincasa/generic-six-scale-kernel-normal-tower.json;
• the generic source Cayley–Menger determinant of Entries 185 and 700.

Next falsifier

Construct the labelled Gauss–Manin action of the three first-normal derivatives on a finite generic five-pole de Rham basis and verify that its iterated symmetric products reproduce the six quadratic and one cubic kernel coefficients above. Failure of closure is admissible only on a located Gram/Landau/marked support face; an unlocated remainder would be the first failure of the proposed finite adapter.

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