Quadratic cycle core for the Type A/B character shield

Geometry

This note refines QUADRATIC-TRAP-SIGNATURE.md. It explains exactly when the quadratic character shield can fail and turns each failure into a finite squareclass-cycle certificate.

Source in the repository

Status: exact linear-algebra theorem note and active proof direction

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this does not prove universal Direct-Shadow Completeness, universal López Type A/B coverage, or the Erdős-Straus conjecture.

This note refines QUADRATIC-TRAP-SIGNATURE.md. It explains exactly when the quadratic character shield can fail and turns each failure into a finite squareclass-cycle certificate.

Read together with QUADRATIC-PRIOR-ART-NOTE.md: the Type B quadratic-nonresidue property itself is already present in López 2024. The present object is the simultaneous exact-depth character system and its obstruction geometry.

1. Squareclass vectors

For each earlier layer j<k, write

m_j=4j-1=\prod_p p^{a_{j,p}}.

Only primes occurring to odd exponent affect the Jacobi symbol. Define the squareclass vector

v_j=(a_{j,p}\bmod2)_p

in the F_2 vector space generated by odd primes.

For a fixed target candidate x=r mod L, split the prime coordinates into:

fixed primes p|L, whose Legendre signs are already determined by r;
free primes p∤L, whose Legendre signs may be chosen by CRT.

Accordingly write

v_j=(f_j,g_j),

where f_j is the free-prime part and g_j the fixed-prime part.

Let epsilon be the vector of fixed Legendre-sign bits, so the character-shield equation for layer j is

\boxed{f_j\cdot z=b_j}, \qquad b_j=g_j\cdot\epsilon.

2. Exact cycle criterion

Theorem

The simultaneous character shield is solvable if and only if every F_2 dependency among the free squareclass rows is compatible with the fixed signs.

Equivalently, for every coefficient vector alpha=(alpha_j) satisfying

\sum_j\alpha_j f_j=0,

we must also have

\boxed{\sum_j\alpha_j b_j=0.}

Proof

This is the standard solvability criterion for a linear system over a field. The equations f_j·z=b_j are inconsistent exactly when a linear combination of their left sides vanishes while the same linear combination of their right sides equals 1. QED.

3. Multiplicative interpretation

A row dependency

\sum_j\alpha_j f_j=0

means that in the product

M_\alpha=\prod_{j:\alpha_j=1}m_j,

every free prime occurs with even total exponent.

Thus the free part of M_alpha is a square. Only primes already fixed by the target modulus L can carry an odd squareclass contribution.

The incompatible case

\sum_j\alpha_jb_j=1

means that the product of the corresponding fixed Legendre signs is -1.

Therefore a failed character shield has a concrete certificate:

\boxed{ \text{a subset of earlier moduli whose free squareclasses multiply to a square but whose fixed character product is }-1. }

We call such a subset a quadratic cycle obstruction.

It obstructs the character shield only. It does not imply union shadowing or failure of exact trap avoidance.

4. Immediate corollaries

Free squareclass independence

If the vectors f_j are linearly independent, the character shield is automatically solvable for every fixed right-hand side.

Hence character difficulty can occur only after squareclass dependencies appear.

New-prime triangularity

If the earlier layers can be ordered so that each row contains a free prime to odd exponent that appears in no preceding row, then the system is triangular and solvable.

Thus a large part of the Type A/B history is automatically harmless whenever new odd prime factors keep entering the squarefree kernels of 4j-1.

Cycle-space localization

Let F be the matrix with rows f_j. Then the possible character obstructions live entirely in the row-dependency space

\ker(F^T).

The dimension

\boxed{c=(\#\text{rows})-\operatorname{rank}(F)}

is the quadratic cycle rank. When c=0, no character obstruction exists.

This gives another compression of the proof problem: hundreds of earlier layers may reduce to a very small dependency space.

5. Why this may matter for DSC-P

The exact trap system has already shown strong overlap. The quadratic character view now says that the coarse half-space obstruction is itself governed only by squareclass cycles.

The proof architecture becomes

\boxed{ \text{direct novelty} \to \text{fiber peel} \to \text{character shield} \to \text{quadratic cycle core if inconsistent} \to \text{exact trap geometry only inside that core}. }

A universal DSC-P proof would follow from proving that every quadratic cycle obstruction left after peeling is still escapable by the finer exact residues.

An even stronger route would be to prove that direct novelty itself forbids incompatible squareclass cycles in the relevant reduced system, but current evidence does not justify that statement yet.

6. Certificate mining

The companion analyzer quadratic_cycle_core_analyzer.py performs Gaussian elimination with provenance bitsets.

For each candidate it records:

row count and free-squareclass rank;
quadratic cycle rank;
whether the character system is solvable;
when inconsistent, an explicit subset of earlier layers whose free squareclasses cancel and whose fixed right-hand sides sum to 1;
the size and layer profile of the first obstruction cycle.

The analyzer does not use the stored avoiding witness.

7. Falsifiable theorem targets

The next questions are:

Are obstruction cycles uniformly small after fiber peeling?
Do only a few cycle shapes recur?
Are obstruction cycles forced to use a bounded set of small primes?
Does every obstruction cycle correspond to a residual fiber-kernel signature already observed computationally?
Can one exact trap residue always escape each character cycle unless a direct shadow exists?

A positive structural answer to the last question would connect the character core directly to DSC-P.

8. Novelty boundary

Linear-system solvability and dependency certificates are standard algebra. The candidate contribution is their specialization to simultaneous avoidance of López Type A/B minimal-depth traps, integrated with shadowing and fiber elimination.

This should remain a potentially novel framework claim pending publication-grade prior-art review.