Entry 301 · 2026-08-16 · marici.Benincasa · research

Four Enhanced-Point Rees Classes Have Smith Type 1-2-2

Result

The four source-generated enhanced points of the total-energy split model carry a canonical occurrence/deck-orbit higher-Rees lattice

$$\mathcal H_{\rm exc} = \mathbb Z\langle h_{++},h_{+-},h_{-+},h_{--}\rangle.$$

Their source-normalized leading functionals do not give four independent directions in the nine-master algebraic kernel. After stripping the common physical Leray scalar and the invertible kinematic monomials, the realization map is

$$\boxed{ \Phi_{\rm exc} = \begin{pmatrix} 1&-1&1&-1\\ 1&1&-1&-1\\ 1&-1&-1&1 \end{pmatrix}, }$$

in point order

$$(++,+-,-+,--)$$

and target order

$$(y e_3,\;x e_5,\;e_6).$$

It has

$$\boxed{ \ker\Phi_{\rm exc} = \mathbb Z(1,1,1,1), }$$

and Smith normal form

$$\boxed{\operatorname{SNF}(\Phi_{\rm exc})=(1,2,2).}$$

Therefore

$$\boxed{ \operatorname{rank}\operatorname{im}\Phi_{\rm exc}=3, \qquad \operatorname{coker}\Phi_{\rm exc} \simeq(\mathbb Z/2)^2. }$$

The same two occurrence parities already govern the integral conductor extension of Entry 280. Thus the four enhanced germs add a rank-three algebraic higher-Rees layer plus the diagonal relation; their only saturation defect is the existing pair of occurrence-identification factors of two. No new carrier stratum or new torsion prime appears.

Frozen sign-resolved germs

Index the four enhanced points by

$$P_{\epsilon\delta}: \qquad (a,b)=(\epsilon y,\delta x), \qquad \epsilon,\delta\in\{\pm1\}.$$

Use the sign-resolved weighted coordinates

$$E=\tau^2, \qquad a=\epsilon\bigl(y+\tau^2r\bigr), \qquad b=\delta\bigl(x-\tau^2r+\tau^3n\bigr).$$

The tangent relation is

$$\epsilon(a-\epsilon y)+ \delta(b-\delta x)=O(\tau^3).$$

Because the Cayley--Menger family and the source double-pole numerator depend on $a^2$ and $b^2$, their weighted leading terms are independent of $(\epsilon,\delta)$:

$$K = \tau^6 \left[ 4x^2y^2n^2+ 8xy(x+y)(r^2-1) \right] +O(\tau^7),$$

$$K_1 = 16xy(x+y)\tau^4+O(\tau^5).$$

No sign-dependent normalization has been introduced.

Orientation produces the three characters

The global source orientation is $da\wedge db$. In the displayed chart,

$$da\wedge db = \epsilon\delta\, \tau^5\,dr\wedge dn.$$

The three source double-pole masters have numerators proportional to

$$(aK_1,\;bK_1,\;K_1).$$

At $P_{\epsilon\delta}$, multiplication by the Jacobian sign gives

$$\epsilon\delta(\epsilon y,\delta x,1) = (\delta y,\epsilon x,\epsilon\delta).$$

Entry 226 fixes the common physical Leray coefficient at the positive chart. Transport through the frozen deck/occurrence action therefore yields

$$\boxed{ \Lambda_{\epsilon\delta} = -\frac{2\pi^2}{xy} \left( \delta y\,e_3+ \epsilon x\,e_5+ \epsilon\delta\,e_6 \right), }$$

in the equation-(58) de Rham normalization. The overall common wavefunction prefactor remains outside the source normalization exactly as in Entry 226.

For $(\epsilon,\delta)=(+,+)$, this reduces to

$$-\frac{2\pi^2}{xy}(y e_3+x e_5+e_6),$$

the previously computed physical-corner functional.

The three target rows are the nontrivial sign characters

$$\delta,\qquad\epsilon,\qquad\epsilon\delta.$$

The trivial character is absent.

Kernel

Summing all four columns gives zero:

$$\sum_{\epsilon,\delta} \delta = \sum_{\epsilon,\delta} \epsilon = \sum_{\epsilon,\delta} \epsilon\delta =0.$$

Hence

$$h_{++}+h_{+-}+h_{-+}+h_{--} \in\ker\Phi_{\rm exc}.$$

Any three columns have determinant $\pm4$, so the matrix has rank three. The kernel is therefore exactly the primitive diagonal line:

$$\boxed{ \ker\Phi_{\rm exc} = \mathbb Z(h_{++}+h_{+-}+h_{-+}+h_{--}). }$$

This is an occurrence-orbit relation. It does not assert that four independent physical integration chains are present; the frozen physical Leray germ selects one chamber, while the other points are its occurrence/deck companions.

Smith calculation

The gcd of all entries is $1$. The gcd of all $2\times2$ minors is $2$. The four maximal minors are

$$(4,-4,4,-4),$$

so their gcd is $4$. Consequently the invariant factors are

$$d_1=1, \qquad d_1d_2=2, \qquad d_1d_2d_3=4,$$

and therefore

$$\boxed{(d_1,d_2,d_3)=(1,2,2).}$$

Equivalently, if the target coordinates are $(u,v,w)$, then

$$\operatorname{im}\Phi_{\rm exc} = \{(u,v,w)\in\mathbb Z^3: u\equiv v\equiv w\pmod2\}.$$

The two quotient parities may be represented by

$$u-w\pmod2, \qquad v-w\pmod2.$$

They are precisely the two independent occurrence axes inherited from the two frozen walls.

Relation to conductor gluing

Entry 280 found

$$0\to \mathcal K_{\Delta_{W_1}}\oplus \mathcal K_{\Delta_{W_2}} \to H^1(W) \to\mathbb Z_{\rm top} \to0$$

with extension class

$$(1,1)\in(\mathbb Z/2)^2.$$

The present Smith cokernel is not a new torsion group that merely happens to have the same order. Its two parity coordinates are induced by the same two sign flips $\epsilon$ and $\delta$. The top half-sum defect is nonzero on both parity axes, reproducing the $(1,1)$ pattern.

What is established is compatibility of the two integral occurrence lattices. A canonical identification of the full conductor extension with the higher-Rees realization still requires the physical relative-chain pairing; it is not inferred solely from isomorphic cokernels.

Nearby-cycle type

Each local exceptional class has, by Entry 226,

$$T_{\rm exc}=1, \qquad N_{\rm exc}=0.$$

Thus

$$\mathcal H_{\rm exc}/\ker\Phi_{\rm exc}$$

is a rank-three algebraic Tate higher-Rees grade. It is distinct from the rank-four logarithmic nilpotent image of Entry 300.

The current filtered picture at generic nonsoft total energy is therefore:

$$\begin{array}{c|c|c} \text{layer}&\text{rank}&\text{type}\\ \hline \operatorname{im}N_E^{(12)}&4& 1\text{ elliptic}+3\text{ algebraic logarithmic}\\ \operatorname{im}\Phi_{\rm exc}&3& \text{algebraic higher-Rees/Cut--nearby}\\ \ker\Phi_{\rm exc}&1& \text{diagonal occurrence relation}\\ \operatorname{coker}\Phi_{\rm exc}&(\mathbb Z/2)^2& \text{occurrence saturation defect} \end{array}$$

No splitting between the logarithmic and higher-Rees layers is claimed.

Classification

Deutsch--Popperian update M2.44

The hard-to-vary claim

$$\text{the four enhanced germs require four independent coefficient directions or a new global gluing generator}$$

is falsified.

The smaller surviving theorem is

$$\boxed{ \text{the four germs realize the augmentation-zero three-character lattice, with Smith type }(1,2,2)\text{ and no new carrier datum.} }$$

Scope boundary

This result does not yet compute:

1. the extension class between the higher-Rees image and the logarithmic nearby-cycle filtration;
2. the pairing with the global physical relative chain;
3. extension through soft support or collisions of enhanced points;
4. cyclic sewing among the three $q_{\mathcal G_{ij}}$ residue sectors;
5. a global integral basis of the complete rank-twelve variation.

Next hostile test

Construct the source intersection/duality pairing between

$$\operatorname{im}\Phi_{\rm exc} \quad\text{and}\quad \langle \Theta_{101}^{\rm fix}, \Theta_{110}^{\rm fix}, e_6/[8s] \rangle.$$

Determine whether its integral matrix is unimodular after adjoining exactly the two already forced half-sum corrections. A residual denominator, kernel, or support divisor not generated by the frozen walls and occurrence identifications is the next finite falsifier of global assembly.

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