Zero-product atom decomposition of square-lift multiplicative defects

Theorem

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Status: proved reduction using classical zero-sum theory

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: Davenport constants and zero-sum sequence theory in finite abelian groups are classical. This note applies them to the Type A/B square-lift multiplicative defect quotient. It does not prove exact Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture.

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1. Defect sequence

Let

d=4a-1

be a squarefree ancestor modulus and let

\mathcal M_a=K_a/D_a

be the multiplicative reciprocity defect quotient.

For a positive odd square lift

j=\frac{1+d s^2}{4},

factor

j=\prod_q q^{e_q}.

Each prime divisor has a defect class

\delta_a(q)\in\mathcal M_a.

Form the finite sequence S(j/a) containing e_q copies of delta_a(q) for every prime q|j.

Its length is

|S(j/a)|=\Omega(j),

where Omega counts prime factors with multiplicity.

The multiplicative conservation theorem gives

\boxed{ \prod_{g\in S(j/a)}g=1. }

Thus every square lift produces a zero-product sequence in the finite abelian group M_a.

2. Minimal zero-product atoms

A nonempty sequence in a finite abelian group is a minimal zero-product sequence when its total product is 1 but no nonempty proper subsequence has product 1.

Every finite zero-product sequence admits a factorization into minimal zero-product subsequences: repeatedly choose a minimal nonempty zero-product subsequence and remove it. The remainder remains zero-product.

Therefore:

Atom decomposition theorem

Every square-lift defect sequence admits a disjoint decomposition

\boxed{ S(j/a)=A_1\sqcup\cdots\sqcup A_r }

such that each A_i is a minimal zero-product sequence in M_a.

The decomposition need not be unique. Existence is enough for the structural reduction.

3. Davenport bound

Let

D(\mathcal M_a)

be the Davenport constant of the finite abelian group M_a: the maximum length of a minimal zero-sum / zero-product sequence, equivalently the least D such that every sequence of length D contains a nonempty zero-product subsequence under the standard convention adjusted by one as appropriate.

Using the convention that D(G) is the maximum atom length, every atom in the decomposition satisfies

\boxed{|A_i|\le D(\mathcal M_a).}

Hence the globally large prime-factor configuration of an arbitrary square lift decomposes into defect packets whose size is bounded solely by the ancestor quotient.

This bound is independent of the square multiplier s and the height of the lift.

4. Divisor interpretation

Each atom A_i corresponds to a divisor

e_i\mid j

obtained by multiplying the prime powers selected by that subsequence.

Because the atom has product defect 1,

\boxed{ e_i\bmod d\in D_a.}

Thus

-e_i,-4e_i\in-D_a.

So every atom determines a divisor whose projected Type A/B residues lie inside the ancestor's multiplicative trap coset.

If the whole defect sequence splits into more than one atom, the square lift contains multiple proper divisors with this property.

This is not yet exact ancestor trap membership because

T_a\subseteq-D_a

may be strict. The remaining difference is precisely the ancestor two-box / divisor-sparsity problem.

5. Proper neutral divisors above the Davenport threshold

Suppose

\Omega(j)>D(\mathcal M_a).

Take any D(M_a) terms of the defect sequence. By the defining zero-product property of the Davenport constant, they contain a nonempty zero-product subsequence.

Because at least one term of the full sequence lies outside those chosen terms, this zero-product subsequence is proper in the full factorization of j.

Hence:

Corollary

\boxed{ \Omega(j)>D(\mathcal M_a), }

then there exists a proper divisor

\boxed{1<e<j}

such that

\boxed{e\bmod d\in D_a.}

Since the total defect is also trivial, the complementary divisor

f=j/e

also satisfies

\boxed{f\bmod d\in D_a.}

Thus sufficiently long square-lift factorizations always split into two nontrivial multiplicatively ancestor-neutral pieces.

6. Atomic lifts are bounded-complexity objects

Call a square-lift defect configuration multiplicatively atomic when its nontrivial defect sequence is itself minimal zero-product.

Then necessarily

\boxed{ \Omega_{\rm defect}(j) \le D(\mathcal M_a), }

where Omega_defect counts only prime factors carrying nontrivial defect, with multiplicity.

Therefore any prospective genuinely primitive multiplicative exception at a fixed ancestor is forced into a bounded prime-factor complexity class.

Large lifts can still exist, but their defect content decomposes into bounded atoms.

7. Standard invariant-factor bounds

\mathcal M_a \cong C_{n_1}\oplus\cdots\oplus C_{n_r}, \qquad n_1\mid\cdots\mid n_r,

then the classical lower bound is

1+\sum_{i=1}^r(n_i-1) \le D(\mathcal M_a).

Equality is known for important classes including finite p-groups and groups of rank at most two. General upper bounds from zero-sum theory may be used when exact D(M_a) is unavailable.

For a cyclic defect quotient

\mathcal M_a\cong C_n,

we have simply

\boxed{D(\mathcal M_a)=n.}

For an elementary binary quotient

\mathcal M_a\cong C_2^r,

\boxed{D(\mathcal M_a)=r+1.}

8. Relation to the earlier binary conservation law

The reciprocity defect quotient

\mathcal R_a

records only the quadratic image of M_a.

Its conservation law sees prime exponents only modulo 2.

The present atom decomposition occurs in the full finite abelian group M_a, so it retains:

odd-order defect information;
higher prime-power orders;
full exponent multiplicities modulo the relevant group orders.

Thus the binary parity pairings observed in the k<=1200 data are the first visible special case of a general finite-abelian zero-product structure.

9. Why this is useful for exact shadowing

The unresolved exact problem inside an ancestor coset is the two-box set

-T_a = \phi_a(\mathcal B_{\mathbf b}) \cup \phi_a(\mathcal B_{\mathbf b}-\mathbf b) \subseteq D_a.

The atom theorem says that arbitrary square-lift factorizations need not be studied as one huge collection of primes. Their multiplicative defect content can be reduced to bounded-size neutral divisor packets.

This suggests a two-level proof strategy:

classify all possible zero-product atoms in M_a up to the Davenport bound;
for the corresponding neutral divisor residues in D_a, prove exact inclusion in the ancestor two-box trap or identify the finite exceptional residue types.

For ancestors with small M_a, this converts an unbounded factorization problem into a finite local classification.

10. Next computational and theorem targets

compute invariant factors of M_a for squarefree ancestors appearing through the current research range;
compute exact Davenport constants whenever the group class permits it;
classify observed square-lift defect sequences into zero-product atom types;
compare each atom's neutral divisor residue with T_a rather than merely D_a;
test whether every atom failing exact ancestor-trap membership is already shadowed by another earlier layer;
search for a uniform bound on the atom types needed in the directly novel candidate core.

The fifth target is a concrete route from classical zero-sum structure back to the exact Direct-Shadow Completeness theorem.