Exponent-lattice formulation of square-completed ancestry shadows

Geometry

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Status: proved exact criterion and coordinate-budget corollary

Date: 2026-08-15

Depends on: ES-SQUARE-TRAP-SIGNED-BOX-IDENTITY.md, ES-SQUARE-MULTIPLICATIVE-SHADOW-IFF.md, ES-SQUARE-SQUAREFREE-FACTOR-LIFT.md

Claim boundary: gives an exact finite-lattice reformulation of direct shadow along any modulus-ancestry edge, together with a strong sufficient box-budget test. It does not solve the resulting lattice-containment problem in full generality.

1. Ancestor exponent lattice

Fix an earlier layer

\boxed{j=\prod_{i=1}^d p_i^{E_i}}

and put

\boxed{m=4j-1.}

Define the exponent homomorphism

\boxed{ \phi_j:\mathbb Z^d \longrightarrow (\mathbb Z/m\mathbb Z)^\times, \qquad (z_1,\ldots,z_d) \longmapsto \prod_{i=1}^d p_i^{z_i}.}

Let

\boxed{L_j=\ker\phi_j.}

Thus two exponent vectors give the same residue modulo m exactly when they differ by an element of L_j.

The centered exponent box of the completed layer is

\boxed{ B_j = \prod_{i=1}^d[-E_i,E_i]_{\mathbb Z}.}

By the signed-box identity,

\boxed{ \mathcal R_m(j)=\phi_j(B_j).}

2. A later ancestry layer

Let

\boxed{k=\prod_{\nu=1}^t r_\nu^{F_\nu}}

and assume the modulus-ancestry relation

\boxed{m_j\mid m_k.}

The later completed signed box reduced modulo the ancestor modulus is

\boxed{ R_{k\to j} = \left\{ \prod_{\nu=1}^t r_\nu^{z_\nu}\pmod m: -F_\nu\le z_\nu\le F_\nu \right\}.}

For direct shadow by j, one needs

\boxed{R_{k\to j}\subseteq\phi_j(B_j).}

3. Immediate subgroup obstruction

If some prime factor r_\nu of k does not lie in the subgroup

\operatorname{im}\phi_j = \langle p_1,\ldots,p_d\rangle,

then direct shadow is impossible.

Indeed the later box contains the residue r_\nu itself, while the ancestor box is contained in im phi_j.

Therefore a necessary condition is

\boxed{ r_\nu\in\operatorname{im}\phi_j \quad\text{for every }\nu.}

Assume this from now on.

Choose arbitrary exponent lifts

\boxed{v_\nu\in\mathbb Z^d}

satisfying

\boxed{\phi_j(v_\nu)=r_\nu.}

Different choices differ by vectors in L_j and therefore do not change the criterion below.

4. The later signed box is a discrete zonotope

Define the discrete zonotope

\boxed{ Z(k\to j) = \left\{ \sum_{\nu=1}^t z_\nu v_\nu: -F_\nu\le z_\nu\le F_\nu \right\} \subseteq\mathbb Z^d.}

Then

\boxed{ R_{k\to j} = \phi_j(Z(k\to j)).}

The ancestor completed box is

\phi_j(B_j).

Therefore an element of the later box lies in the ancestor box exactly when its exponent vector can be shifted by a relation vector into B_j.

5. Exact lattice-cover criterion

Theorem — ancestry shadow as zonotope containment modulo the relation lattice

Along the ancestry edge j<k, the completed layer k is directly shadowed by j if and only if

\boxed{ Z(k\to j) \subseteq B_j+L_j.}

Equivalently, for every coefficient vector

(z_\nu), \qquad -F_\nu\le z_\nu\le F_\nu,

there exists a relation vector

\lambda\in L_j

such that

\boxed{ \sum_\nu z_\nu v_\nu-\lambda \in B_j.}

This criterion is independent of the chosen lifts v_\nu.

6. Coordinate-budget sufficient theorem

A simple strong sufficient condition avoids relation-lattice wrapping entirely.

Suppose lifts can be chosen so that for every ancestor coordinate i,

\boxed{ \sum_{\nu=1}^t F_\nu\,|(v_\nu)_i| \le E_i.}

Then for every allowed signed coefficient vector,

\left| \sum_\nu z_\nu(v_\nu)_i \right| \le \sum_\nu F_\nu|(v_\nu)_i| \le E_i.

Hence the whole zonotope already lies in B_j:

Z(k\to j)\subseteq B_j.

Therefore:

Corollary — coordinate-budget shadow

If there exist exponent lifts satisfying

\boxed{ \sum_\nu F_\nu|(v_\nu)_i|\le E_i \quad\text{for every }i,}

then

\boxed{S_k\bmod m_j\subseteq S_j.}

This is a finite weighted L^1 packing test.

7. Squarefree factor lift is a disjoint-support budget

In the squarefree factor-lift theorem, the earlier index factors as

j=A_1\cdots A_t

and each later prime r_\nu is congruent to A_\nu modulo m.

Choose v_\nu to be the exponent vector of A_\nu.

When the factorization distributes the ancestor prime-power exponents among the A_\nu, the coordinate budgets add exactly to the midpoint exponents E_i.

Thus

\sum_\nu |(v_\nu)_i|=E_i

and the coordinate-budget theorem recovers the factor-lift shadow.

The all-quotient Dirichlet families are therefore explicit exact packings of the ancestor exponent box.

8. Multiplicative stabilizer extensions use relation-lattice wrapping

The multiplicative-shadow iff theorem can behave differently.

A stabilizer direction need not admit a small representative inside the raw coordinate budget. Its products may leave B_j and return through the relation lattice L_j.

Thus the exact criterion

Z\subseteq B_j+L_j

strictly generalizes the no-wrap coordinate budget.

This separates two mechanisms cleanly:

geometric packing: Z subset B_j;
periodic packing: Z subset B_j+L_j only after relation-lattice reduction.

9. Canonical relation for even ancestors

j=2^e\prod_{i=1}^s p_i^{E_i}

is even, then

4j =2^{e+2}\prod_i p_i^{E_i} \equiv1\pmod{4j-1}.

Therefore the exponent vector

\boxed{ (e+2,E_1,\ldots,E_s) \in L_j.}

This canonical relation is responsible for many effective-dimension collapses.

For j=2^e p with p prime, it yields the one-dimensional interval theorem

\mathcal R(j)=\{2^z:-2e-2\le z\le2e+2\}.

For larger even support it supplies a distinguished lattice direction along which the ancestor box can be folded.

10. Effective dimension

The raw support dimension of the completed layer is

\omega(j)=d.

But the actual multiplicative geometry is governed by the quotient lattice

\boxed{\mathbb Z^d/L_j.}

and by the image of the bounded box B_j inside that quotient.

Thus a more useful invariant is the effective signed-box dimension, meaning the minimal rank of a lattice model needed to represent phi_j(B_j) after quotienting by explicit relations.

Examples:

prime powers: effective dimension one;
2^e p: raw dimension two but effective cyclic interval dimension one;
general multi-prime indices may retain higher effective dimension.

11. Research consequence

The remaining nonmultiplicative ancestry problem is now an exact finite geometry problem:

\boxed{ \text{Does a later discrete zonotope fit inside the ancestor exponent box modulo }L_j?}

This suggests both theorem and computation paths:

derive canonical generators for L_j from the modulus identity and factor relations;
reduce B_j to a low-dimensional fundamental-domain model;
classify few-prime later zonotopes by coordinate budgets;
use Smith-normal-form or finite quotient methods for exact automated shadow certification;
search for infinite ancestry families corresponding to fixed lattice-packing templates.