Classification of universal square-lift signature shadowing

Geometry

Read with:

Status: proved if-and-only-if theorem

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this theorem is at the full local quadratic-signature resolution. It does not imply exact residue shadowing, universal Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture.

Read with:

1. Setup

Let

d=4a-1

be squarefree and let

V_a \subseteq \mathbb F_2^{\omega(d)}

be the local quadratic-signature space generated by the prime divisors of the ancestor depth a.

Let

\kappa(a) = \omega(d)-\dim V_a.

Let J_d be the Jacobi parity functional. Then

V_a\subseteq\ker J_d, \qquad \dim\ker J_d=\omega(d)-1.

For every positive odd s, define the square lift

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

4j_s-1=d s^2.

Let

W_s = \operatorname{span} \{\lambda_d(\ell):\ell\mid j_s,\ \ell\text{ prime}\}.

The projected trap-signature theorem gives

\lambda_d(T_{j_s}\bmod d)=\eta_d+W_s,

while the ancestor trap signatures are

\lambda_d(T_a)=\eta_d+V_a.

Hence the lift is ancestor-shadowed at signature resolution exactly when

W_s\subseteq V_a.

2. Universal-shadow classification

Theorem

The following are equivalent:

kappa(a)=1;
V_a=ker J_d;
every positive odd square lift satisfies

W_s\subseteq V_a;

every positive odd square lift is shadowed by the ancestor at full local quadratic-signature resolution.

Moreover, if

\boxed{\kappa(a)>1,}

then there are infinitely many positive odd s for which

\boxed{W_s\not\subseteq V_a.}

Thus

\boxed{ \kappa(a)=1 \iff \text{all square lifts are signature-shadowed}, }

and

\boxed{ \kappa(a)>1 \Longrightarrow \text{infinitely many square-lift signature exceptions}. }

3. Proof of the automatic direction

If kappa(a)=1, then V_a is a codimension-one subspace of the local sign space.

The Jacobi functional is nonzero and annihilates V_a, so

V_a=\ker J_d.

Square-lift reciprocity gives, for every prime ell|j_s,

\left(\frac\ell d\right)=+1.

Therefore

\lambda_d(\ell)\in\ker J_d=V_a.

Every generator of W_s lies in V_a, hence

W_s\subseteq V_a.

This proves 1 => 2 => 3 => 4.

4. Realizing every Jacobi-positive signature by split primes

To prove the converse strongly, we need one elementary realization lemma.

Lemma

For every vector

v\in\ker J_d,

there exist infinitely many odd primes ell such that

\boxed{ \lambda_d(\ell)=v }

and

\boxed{ \left(\frac{-d}{\ell}\right)=+1. }

Hence ell splits in

\mathbb Q(\sqrt{-d}).

Proof

For each prime p|d, choose a nonzero residue r_p mod p whose Legendre sign is the prescribed coordinate of v.

Use CRT to choose a reduced residue class R mod 4d satisfying

R\equiv1\pmod4, \qquad R\equiv r_p\pmod p \quad(p\mid d).

Dirichlet's theorem gives infinitely many primes

\ell\equiv R\pmod{4d}.

For each such prime,

\lambda_d(\ell)=v.

Because ell=1 mod 4, quadratic reciprocity gives

\left(\frac p\ell\right) = \left(\frac\ell p\right)

for every p|d, and

\left(\frac{-1}\ell\right)=+1.

Therefore

\left(\frac{-d}{\ell}\right) = \prod_{p\mid d} \left(\frac\ell p\right).

Since v in ker J_d, the product of its prescribed local signs is +1. Thus

(-d/\ell)=+1.

QED.

5. Constructing infinitely many exceptional square lifts

Assume now

\kappa(a)>1.

Then

V_a\subsetneq\ker J_d.

Choose

v\in\ker J_d\setminus V_a.

By the lemma, choose an odd split prime ell with

\lambda_d(\ell)=v.

Since (-d/ell)=+1, the congruence

d s^2\equiv-1\pmod\ell

has a solution s=s_0 mod ell.

Choose s_0 odd, replacing it by s_0+ell if necessary. Then every

\boxed{ s=s_0+2\ell n, \qquad n\ge0, }

is positive odd and satisfies the same congruence.

For

j_s=(1+d s^2)/4,

we therefore have

\ell\mid j_s.

Hence

v=\lambda_d(\ell)\in W_s.

But v notin V_a, so

W_s\not\subseteq V_a.

The values j_s grow strictly with n, giving infinitely many distinct square-lift depths that are not ancestor-shadowed at quadratic-signature resolution.

QED.

6. Defect-quotient formulation

The reciprocity defect quotient is

\mathcal R_a = \ker J_d/V_a, \qquad \dim\mathcal R_a=\kappa(a)-1.

The classification says:

\boxed{ \mathcal R_a=0 \iff \text{universal square-lift signature shadowing}. }

When R_a is nonzero, every nonzero defect class is represented by infinitely many split rational primes, and each such prime can be forced to divide infinitely many square-lift depths.

Thus R_a is not merely a formal quotient. Every one of its nonzero classes is arithmetically realizable in the square-lift family.

7. Relation to the finite k <= 1200 data

The finite replay found 207 ancestor-shadowed square lifts and 17 signature exceptions among the 224 non-squarefree moduli through 1200.

The theorem now explains the qualitative split exactly:

ancestors with kappa(a)=1 can never generate a signature exception, at any height;
ancestors with kappa(a)>1 genuinely possess defect directions and generate infinitely many exceptions, although the first one may occur above a finite search cutoff.

So absence of an exception at a finite range for a higher-quotient ancestor is a latency phenomenon, not a universal-shadow theorem.

This mirrors the earlier structural-gap versus finite-latency distinction for C_AB itself.

8. Why this matters for QDSC

The square-lift portion of the full quadratic-signature problem is now classified at the ancestor level.

There are exactly two regimes:

\boxed{ \begin{array}{ll} \kappa(a)=1:& \text{no new square-lift signature information can ever appear};\\[1mm] \kappa(a)>1:& \text{new defect information necessarily appears infinitely often}. \end{array}}

Therefore the hard part of QDSC cannot be spread uniformly across all squarefree ancestors. It is concentrated precisely on the higher-quotient ancestor set

\boxed{\{a:\kappa(a)>1\}.}

Within that set, the defect conservation law still forces the prime-factor defects of every lift to cancel in aggregate.

The next theorem problem is therefore sharper:

Given a nonzero defect configuration in R_a whose total parity sum is zero, prove that its induced unsafe affine signature constraint is already shadowed by an earlier codimension-one layer, or classify the exceptional configurations.

That is a finite-dimensional arithmetic problem for each ancestor quotient.

9. Novelty boundary

CRT, Dirichlet's theorem, quadratic reciprocity, split-prime criteria, and finite-dimensional vector spaces are classical. The candidate contribution is the exact if-and-only-if classification of universal square-lift Type A/B signature shadowing by the ancestor quotient dimension kappa(a), integrated with the minimal-depth/shadow framework.

Publication priority remains subject to broader literature and independent review.