Square-lift quadratic-signature shadow theorem

Geometry

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Status: proved theorem family plus finite proof-mining signal

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this note does not prove universal Direct-Shadow Completeness, universal López Type A/B coverage, or the Erdős-Straus conjecture. It proves an infinite shadow family at the full local quadratic-signature resolution.

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1. Setup

Let

d=4a-1

be squarefree and let s be positive odd. Put

4j-1=d s^2.

Thus a is the squarefree ancestor depth of j.

Let the distinct prime divisors of d be

p_1,\ldots,p_r.

For every unit u mod d, write

\lambda_d(u) = \left( \left(\frac{u}{p_1}\right),\ldots, \left(\frac{u}{p_r}\right) \right) \in\mathbb F_2^r,

with +1 encoded by 0 and -1 by 1.

Let

V_a = \operatorname{span}_{\mathbb F_2} \{\lambda_d(\ell):\ell\mid a,\ \ell\text{ prime}\}.

Then the ancestor trap-signature theorem gives

\lambda_d(T_a)=\eta_d+V_a,

where

\eta_d=\lambda_d(-1).

2. Projected divisor-signature space

Define

\boxed{ W_{j\to a} = \operatorname{span}_{\mathbb F_2} \{\lambda_d(\ell):\ell\mid j,\ \ell\text{ prime}\}. }

Because signatures are multiplicative, as e ranges over the divisors of j, the set of signatures lambda_d(e) is exactly W_{j->a}.

Since 4 is a square modulo every odd prime divisor of d, both Type A and Type B projected traps have the same signature shift by eta_d.

Theorem

The complete projected quadratic-signature image of the lifted trap set is

\boxed{ \lambda_d(T_j\bmod d) = \eta_d+W_{j\to a}. }

Proof

For every divisor e|j,

\lambda_d(-e)=\eta_d+\lambda_d(e)

and

\lambda_d(-4e)=\eta_d+\lambda_d(e).

The divisor signatures fill exactly the span W_{j->a}. QED.

3. Exact signature-shadow criterion

The ancestor trap signatures are

\eta_d+V_a.

The lifted projected trap signatures are

\eta_d+W_{j\to a}.

Therefore:

Theorem

The square-lift layer j is completely shadowed by its ancestor a at full local quadratic-signature resolution if and only if

\boxed{ W_{j\to a}\subseteq V_a. }

Equivalently, it is enough to test the prime generators:

\boxed{ \lambda_d(\ell)\in V_a \quad\text{for every prime }\ell\mid j. }

This is an exact finite-dimensional linear-algebra criterion. Unlike exact residue containment, there is no need to enumerate every divisor of j once the prime generators are known.

4. Reciprocity places every lift inside the Jacobi kernel

SQUARE-LIFT-RECIPROCITY.md proves that for every divisor e|j,

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

At the vector level this says every element of W_{j->a} lies in the kernel of the Jacobi functional

J_d:\mathbb F_2^r\to\mathbb F_2.

Hence

\boxed{ W_{j\to a}\subseteq\ker J_d. }

The ancestor divisor-signature space also lies in this kernel:

V_a\subseteq\ker J_d.

5. Quotient-dimension-one automatic shadow theorem

Let

\kappa(a)=r-\dim V_a

be the quadratic quotient dimension from QUADRATIC-SIGNATURE-QUOTIENT.md.

The Jacobi functional is nonzero and annihilates V_a, so kappa(a)>=1.

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

then V_a is a codimension-one subspace contained in the codimension-one Jacobi kernel. Therefore

\boxed{V_a=\ker J_d.}

Combining this with square-lift reciprocity gives:

Theorem

If the squarefree ancestor a has

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

then every odd square-lift

4j-1=(4a-1)s^2

satisfies

\boxed{ \lambda_d(T_j\bmod d) \subseteq \lambda_d(T_a). }

Thus every such lifted layer is automatically shadowed by its squarefree ancestor at full local quadratic-signature resolution.

QED.

6. Relation to Jacobi saturation

JACOBI-SATURATION.md classified the much stronger exact-residue condition in which the ancestor trap set fills the entire Jacobi-negative half. That happens only for

a=1,2,4.

The present theorem is broader because it asks only for equality after passing to local Legendre-sign vectors.

Whenever kappa(a)=1, the ancestor trap signature coset fills the entire Jacobi-negative signature hyperplane even though the exact trap set is usually much smaller.

Thus:

\boxed{ \text{exact Jacobi saturation} \Longrightarrow \text{signature saturation}, }

but not conversely.

7. Finite signal through k <= 1200

An exact replay of the non-squarefree layers through j<=1200 gives:

non-squarefree layer moduli:              224
square-lifts whose projected signatures
are NOT contained in the ancestor coset:  17
signature-shadowed square-lifts:          207

Every one of the 17 finite exceptions has a squarefree ancestor with

\kappa(a)>1.

Sixteen have kappa(a)=2; one has kappa(a)=3.

The first exceptions are:

j=115,  m=459,  ancestor a=13,  d=51
j=205,  m=819,  ancestor a=23,  d=91
j=259,  m=1035, ancestor a=29,  d=115
j=319,  m=1275, ancestor a=13,  d=51
j=520,  m=2079, ancestor a=58,  d=231

These counts are finite proof-mining data. The automatic-shadow theorem for kappa(a)=1 is universal.

8. Why this matters for QDSC

The full quadratic-signature shield through k<=1200 found that collective higher-codimension signature constraints repeatedly collapse onto lower-codimension shadows.

The theorem above supplies one infinite arithmetic mechanism for that collapse:

\boxed{ \text{square lift} + \kappa(a)=1 \Longrightarrow \text{ancestor signature shadow}. }

So higher-order local sign information created by square factors is often not new at all. It is inherited from the squarefree ancestor.

Only square lifts over ancestors with a genuinely higher-dimensional quotient

\kappa(a)>1

can create new quadratic-signature projection information.

This localizes the square-lift part of the QDSC proof problem to the higher-quotient ancestor set.

9. Next theorem target

For kappa(a)>1, classify the subspace

W_{j\to a}

inside the Jacobi kernel and determine exactly when it escapes V_a.

Because both spaces are generated by prime-factor signature vectors, this becomes a reciprocity-matrix problem rather than an exact residue enumeration problem.

That is the next natural bridge from square-lift ancestry to the quadratic-signature direct-shadow theorem.

10. Novelty boundary

Local Legendre symbols, quadratic reciprocity, vector spaces over F_2, and affine signature cosets are classical. The candidate contribution is the exact square-lift projection theorem and the automatic ancestor-shadow criterion inside the López Type A/B minimal-depth/shadow framework.

Publication priority remains subject to broader literature review and independent scrutiny.