Full quadratic-signature coset for Type A/B traps

Geometry

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Status: proved structural theorem with finite proof-mining evidence

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this note does not prove universal Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture. López 2024 records the Type B Jacobi-nonresidue property. The vector-valued local-signature packaging below is being treated as a novelty candidate within the minimal-depth/shadow program pending broader literature review.

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1. Local signature map

Fix k>=1 and write

m=4k-1=\prod_{i=1}^r p_i^{a_i}.

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

\boxed{ \lambda_m(x)= \left( \ell_{p_1}(x),\ldots,\ell_{p_r}(x) \right) \in\mathbb F_2^r, }

where

\left(\frac{x}{p_i}\right)=(-1)^{\ell_{p_i}(x)}.

Thus each coordinate records the Legendre sign at one distinct prime divisor of m.

Let

\eta_m=\lambda_m(-1).

Its p_i coordinate is 1 exactly when p_i=3 mod 4.

2. Divisor-signature space

Let P(k) denote the distinct prime divisors of k, including 2 when appropriate. Define

\boxed{ V_k=\operatorname{span}_{\mathbb F_2} \{\lambda_m(\ell):\ell\in P(k)\}. }

Every divisor e|k is obtained by choosing exponents of these primes. Since quadratic signatures only remember exponent parity, and a divisor may independently include or omit one copy of each prime divisor, the set of signatures attained by divisors of k is exactly

\boxed{ \{\lambda_m(e):e\mid k\}=V_k. }

3. Quadratic-signature trap theorem

Recall

T_k=\{-e,-4e\pmod m:e\mid k\}.

Theorem

The set of local quadratic signatures attained by the complete Type A/B trap set is the affine subspace

\boxed{ \lambda_m(T_k)=\eta_m+V_k. }

Proof

For e|k, multiplicativity of Legendre symbols gives

\lambda_m(-e)=\eta_m+\lambda_m(e).

Because 4 is a square modulo every odd prime,

\lambda_m(4)=0,

and therefore

\lambda_m(-4e)=\eta_m+\lambda_m(e).

As e ranges over divisors of k, lambda_m(e) ranges over exactly V_k. Hence

\lambda_m(T_k)=\eta_m+V_k.

QED.

4. Quadratic quotient dimension

Define

\boxed{ \kappa(k) =r-\dim V_k. }

Then the trap signatures occupy exactly a fraction

\boxed{2^{-\kappa(k)}}

of all possible local Legendre-sign vectors.

The Jacobi theorem is the weakest universal projection of this statement. The product of the local Legendre signs is the Jacobi symbol, so the previous Jacobi=-1 trap signature is one nonzero linear functional vanishing on V_k.

Therefore

\boxed{\kappa(k)\ge1}

for every k.

When kappa(k)>1, the vector-valued signature shield excludes strictly more unit classes than the single Jacobi bit can see.

5. Candidate restriction

For a target candidate at depth K, write

x\equiv r\pmod L, \qquad L=\operatorname{lcm}(840,4K-1).

At an earlier layer j<K, some prime-sign coordinates of lambda_{m_j}(x) are fixed by x=r mod L; the remaining coordinates are free choices obtainable by CRT.

Intersect the affine trap-signature space

\eta_{m_j}+V_j

with the fixed signs.

The resulting forbidden set on the free sign coordinates is again an affine subspace. It therefore falls into four exact cases:

empty: the earlier layer is automatically safe at quadratic-signature resolution;
full: every free signature lands in the trap-signature envelope, giving one direct signature obstruction;
codimension one: safety is equivalent to one linear XOR equation;
higher codimension: only a smaller affine subset of free signatures is forbidden.

This converts the earlier residue system into a finite affine-subspace avoidance problem over F_2.

6. Signature shield theorem

If one can choose the free local Legendre signs so that, for every earlier j<K, the resulting signature lies outside

\eta_{m_j}+V_j,

then the target candidate has a reduced avoiding arithmetic progression and therefore infinitely many primes of exact Type A/B depth K.

The proof is the same CRT/Dirichlet construction used for the scalar character shield, but with prescribed local Legendre signs at the relevant primes. A Type A/B trap would force its full local signature into the forbidden affine coset, contradicting the construction.

This criterion is sufficient, not necessary: an integer can share a trap's quadratic signature while still avoiding the much smaller exact residue set T_j.

7. Relation to the multiplicative trap coset

Let

H_k=\langle\ell\bmod m_k:\ell\mid k,\ \ell\text{ prime}\rangle.

The multiplicative theorem gives

T_k\subseteq-H_k.

Applying the local Legendre-sign map gives

\lambda_m(-H_k)=\eta_m+V_k.

Thus the quadratic-signature theorem is exactly the maximal elementary-2 quotient visible through the individual Legendre characters of the full multiplicative trap coset.

The hierarchy is

\boxed{ T_k \subseteq -H_k \longrightarrow \eta_m+V_k \longrightarrow \text{Jacobi }-1. }

Each arrow discards information.

8. Finite replay through k <= 1200

A proof-mining replay of the frozen 41,470 directly novel candidates through k<=1200 used this full local quadratic-signature system without consulting the stored exact avoiding witness.

It found:

full quadratic-signature shield solved: 30,786
direct signature residual:              10,684
collective linear inconsistency found:       0
unresolved non-direct signature systems:     0

Every one of the 30,786 non-direct-signature candidates was solved by the deterministic base solution of the codimension-one XOR system; no randomized repair trial was needed in that replay.

This is a finite diagnostic, not a universal theorem that collective quadratic-signature obstruction never occurs.

The key conceptual observation is that the richer local signature geometry independently resolves candidates that the scalar Jacobi shield leaves in its negative core.

9. Next theorem target

The strongest finite pattern now asks for a signature-level analogue of Direct-Shadow Completeness:

If no single earlier layer is a full quadratic-signature obstruction, can the collection of earlier affine signature traps ever cover all globally compatible sign assignments?

The k<=1200 replay found no such collective obstruction.

A proof would remove the entire elementary-2 quotient from the exact DSC-P problem and leave only:

direct signature residual layers;
higher-order multiplicative quotient information;
prime-power lifting inside a fixed signature;
the final exact divisor-generated residue sets.

That is a substantially smaller target than the original global covering system.