Character-shield obstruction completeness

Status: proved theorem inside the Type A/B minimal-depth program

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this theorem does not prove Direct-Shadow Completeness, universal López Type A/B coverage, or the Erdős-Straus conjecture. The underlying quadratic-nonresidue fact for Type B is prior art in López 2024; see QUADRATIC-PRIOR-ART-NOTE.md.

This note strengthens QUADRATIC-TRAP-SIGNATURE.md and supersedes the possibility of a genuinely collective obstruction inside the quadratic character shield itself.

The result is exact and universal: at the squareclass/character level, global inconsistency occurs if and only if one earlier layer is already immutable and character-negative by itself.

1. Setup

Let

Work in the F_2 squareclass vector space V generated by odd primes. For an odd integer n, define

Let

Let F_k be the coordinate subspace generated by the odd primes dividing L, equivalently

Finally define the fixed-only row span

Thus a fixed-only earlier row has no odd-prime squareclass coordinate outside the target progression modulus L.

2. Squareclass saturation theorem

Theorem

For every k>=1,

In words: every squareclass combination of earlier moduli that is supported entirely on the fixed primes is already generated by earlier rows that are individually supported entirely on those fixed primes.

There are no genuinely collective new fixed squareclasses.

Proof

The inclusion

is immediate.

For the reverse inclusion, it suffices to prove the following statement:

For every fixed odd prime p whose coordinate occurs in at least one vector of W_k, the basis vector e_p lies in U_k.

Indeed, every vector in W_k cap F_k is a sum of fixed-prime basis vectors whose coordinates occur in W_k. If each such basis vector belongs to U_k, then the whole intersection lies in U_k.

We treat the possible fixed primes.

The prime 3

If the 3 coordinate occurs and k>1, the earlier row

is present. Hence

For k=1, W_k=0 and there is nothing to prove.

The prime 7

If the 7 coordinate occurs in an earlier row, necessarily M>7, and

is itself an earlier fixed-only row. Thus e_7 in U_k.

The prime 5

A squareclass coordinate 5 can first occur in an integer congruent to 3 mod 4 at 15=3*5. Thus if the 5 coordinate occurs in W_k, then M>15, so both earlier rows 3 and 15 are available. Therefore

and e_5 in U_k.

A prime p dividing M with p = 3 mod 4

If its coordinate occurs in an earlier row, then p<M. Since

the prime p itself is one of the earlier moduli:

It is fixed-only, so

A prime p dividing M with p = 1 mod 4

Write

Because M=3 mod 4 and p=1 mod 4,

Suppose the p coordinate occurs in an earlier row. Then some odd multiple of p strictly below M is congruent to 3 mod 4.

If q=3, the only positive odd multiple of p below M=3p is p itself, but

so it cannot be an earlier modulus m_j=3 mod 4. Therefore occurrence of the p coordinate forces

Since q=3 mod 4, this gives q>=7, and hence

Both 3 and 3p are therefore earlier moduli, and both are fixed-only. Since

we obtain

This covers every fixed prime coordinate that can occur in W_k, proving

Therefore

QED.

3. Character-shield completeness theorem

Fix an admissible target candidate

The Legendre signs at primes dividing L are fixed by r. They define a linear functional

where sign +1 is bit 0 and sign -1 is bit 1.

The free prime signs may be chosen arbitrarily by CRT. The goal of the quadratic character shield is to extend epsilon to the free coordinates so that every earlier modulus has Jacobi sign +1, equivalently so that the resulting character functional vanishes on W_k.

A linear extension vanishing on W_k exists if and only if

By the squareclass saturation theorem,

Therefore we obtain:

Theorem

The simultaneous quadratic character shield is solvable if and only if every fixed-only earlier layer already has positive Jacobi sign:

Equivalently,

Thus collective quadratic-character inconsistency adds no obstruction beyond one fixed-only earlier layer.

QED.

4. Why this matters

The original character-shield formulation appeared to require solving a potentially large linear system over F_2 for every candidate.

The theorem collapses that global problem to a local test.

There can be many linear dependencies among the earlier squareclass rows, but none can create a new character obstruction unless some individually fixed-only earlier modulus is already Jacobi-negative.

This is an exact character-level analogue of the broader Direct-Shadow Completeness phenomenon:

It does not prove exact Direct-Shadow Completeness because a fixed-only earlier layer with Jacobi sign -1 need not be an exact Type A/B trap hit. The Jacobi-negative half of the unit group is much larger than T_j.

Therefore the unresolved arithmetic has been localized further:

only fixed-only, Jacobi-negative earlier layers require the finer exact trap geometry.

5. Finite regression evidence

Before the proof above was identified, an exact recomputation of the frozen k<=1200 candidate bundle gave:

directly novel candidates:             41,470
character-shield solvable:              30,414
character-shield inconsistent:          11,056
candidates with a direct fixed-only
Jacobi-negative earlier layer:          11,056
inconsistent candidates without such
a direct character obstruction:              0

Thus the finite data had already been exhibiting the theorem exactly.

The proof shows that this collapse is not a k<=1200 coincidence.

6. Computational consequence

The quadratic analyzer no longer needs Gaussian elimination to decide character-shield solvability.

It may instead:

1. determine which earlier squareclasses use only primes dividing L;
2. evaluate the Jacobi sign of r on those fixed-only moduli;
3. declare the shield solvable exactly when all such signs are +1.

Gaussian elimination remains useful as a regression check and for studying the dependency/cycle space, but it is no longer necessary for the existence decision.

7. New exact residual core

Define the fixed-negative character core of a candidate by

If this set is empty, the character shield independently proves a reduced avoiding progression and infinitely many exact-depth primes.

If it is nonempty, the only reason the character shield fails lies inside these fixed-negative layers. Exact trap avoidance must distinguish the candidate residue from the much smaller sets T_j inside their Jacobi-negative regions.

This set is now a primary object for the next proof stage.

8. Relation to the larger diamond

The theorem sharpens the active architecture to

The large F_2 character system has disappeared from the obstruction side. What remains is a much smaller exact-residue problem inside a directly identifiable family of earlier layers.

9. Novelty boundary

The squareclass argument uses elementary unique factorization, congruences modulo 4, CRT, and linear algebra over F_2. López 2024 already records the Type B Jacobi-nonresidue property and Mordell's quadratic-residue restriction is classical.

The candidate novelty is the minimal-depth/shadow-specific saturation theorem and resulting character-shield obstruction completeness reduction. Publication priority remains subject to broader literature and external mathematical review.