Full local quadratic signature quotient of Type A/B traps

Status: exact theorem note and active proof direction

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this does not prove Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture. López 2024 already contains the Type B Jacobi-nonresidue statement; see QUADRATIC-PRIOR-ART-NOTE.md.

The Jacobi character records only one bit from a composite modulus. This note keeps the complete vector of local Legendre signs at the distinct prime factors of m_k=4k-1 and shows that the Type A/B trap signatures form one exact affine subspace.

1. Local signature map

Let

For every unit u mod m, define its local quadratic signature

Encode +1 as 0 and -1 as 1, so signatures lie in

The map is multiplicative, hence linear in this bit representation.

2. Divisor-signature subgroup

Let the distinct prime factors of k be

Define

Lemma

As e ranges over the positive divisors of k, the set of signatures chi_m(e) is exactly H_k.

Proof

Every divisor

has signature

so all divisor signatures lie in H_k.

Conversely, every binary choice of the generators is realized by the divisor obtained by using exponent 1 for the selected primes and exponent 0 for the others. QED.

3. Trap-signature coset theorem

Recall

Since 4 is a square modulo every odd prime,

Therefore

and

Using the divisor-signature lemma gives:

Theorem

The image of the complete Type A/B trap set under the local quadratic signature map is exactly the affine coset

Thus all exact trap residues occupy one distinguished point in the quotient

Every unit residue whose quotient signature differs from

is automatically outside T_k.

QED.

4. Relation to the Jacobi theorem

Because

for every p_i|m, the integer k is a quadratic residue modulo every prime factor of m:

This imposes at least the parity relation corresponding to the Jacobi character. Equivalently, the divisor-signature subgroup lies in the kernel of the Jacobi functional, recovering the fact that every trap has global Jacobi sign -1.

The quotient Q_k can be strictly larger than one bit. When

there are independent quadratic characters that distinguish trap signatures more finely than the Jacobi symbol alone.

5. Annihilator formulation

Let

be the annihilator of H_k.

Every character psi in H_k^perp is constant on the trap-signature coset:

Therefore a unit x is quadratically certified safe whenever there exists

such that

The ordinary Jacobi symbol corresponds to one particular nonzero element of this annihilator. When dim Q_k>1, additional independent separator characters exist.

6. Important special cases

Prime target modulus

If m=4k-1 is prime, V_m has dimension one and the quotient has dimension one. The local signature theorem reduces to the ordinary quadratic-residue/Jacobi distinction.

Prime k

If k itself is prime, then

for every p_i|m, so chi_m(k)=0. Since k has only one prime generator,

Hence the trap set occupies one single local Legendre-sign vector even when m is composite.

More generally, an odd prime power k=ell^a with odd a also forces chi_m(ell)=0, so again H_k=0.

7. Finite structural signal through k=3000

An exact enumeration of the layer invariant through all 1<=k<=3000, including the base layer k=1, gives quotient dimensions:

dim Q_k = 1: 2026 layers
dim Q_k = 2:  785 layers
dim Q_k = 3:  173 layers
dim Q_k = 4:   16 layers

The counts sum to all 3000 layers. An earlier draft listed 2025 in the first row and therefore omitted the base layer k=1; this has been corrected.

Thus roughly one third of the tested layers carry more quadratic information than the Jacobi bit alone.

These counts are finite proof-mining data, not an asymptotic theorem.

8. Quadratic signature shield

For an exact-depth target candidate, Legendre-sign variables at primes outside the fixed target modulus may be selected by CRT.

For each earlier layer j, the unsafe local sign patterns form the affine subspace

Therefore simultaneous quadratic-signature avoidance becomes a finite Boolean problem:

choose the free local Legendre bits so that, for every earlier layer, the induced local signature does not lie in its trap-signature affine subspace.

The Jacobi character shield is the codimension-one projection of this stronger system.

9. Why this matters for the residual core

CHARACTER-SHIELD-COMPLETENESS.md proves that the Jacobi shield has no genuinely collective obstruction beyond a fixed-only Jacobi-negative earlier layer.

The present quotient gives a way to split that residual negative half further. A fixed-negative layer may still be quadratically safe because its complete local signature can lie outside the much smaller coset chi(-1)+H_j.

Thus the new hierarchy is

but when Jacobi is -1, one may still have

Only residues landing in the distinguished trap-signature quotient class require still finer exact residue analysis.

10. Next theorem target

The immediate computational/theoretical question is whether the simultaneous full-signature shield has its own direct-obstruction completeness property:

A positive answer would compress the exact shadow problem another major step before exact residue geometry is needed.

11. Novelty boundary

Local Legendre symbols, character groups, affine subspaces and annihilators are classical. The candidate contribution is the Type-A/B-specific identification of the entire trap-signature image as one quotient point and its integration into the minimal-depth/shadow/fiber theorem program.

Publication priority remains subject to broader literature and external review.