Reciprocity defect quotient for square-lift signature shadows

Geometry

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Status: proved structural theorem plus finite proof-mining signal

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this note does not prove QDSC, universal Direct-Shadow Completeness, universal López Type A/B coverage, or the Erdős-Straus conjecture. It isolates exactly where a square lift can create new local quadratic-signature information beyond its ancestor.

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1. Ancestor signature spaces

Let

d=4a-1

be squarefree. Let

V_d=\mathbb F_2^{\omega(d)}

be the local Legendre-sign space at the distinct primes dividing d.

Let

V_a =\operatorname{span} \{\lambda_d(\ell):\ell\mid a,\ \ell\text{ prime}\} \subseteq V_d

be the ancestor divisor-signature space.

Let

J_d:V_d\to\mathbb F_2

be the Jacobi functional, obtained by summing the local sign bits with the odd-exponent weights of d. Since d is squarefree, this is simply the parity of the local negative signs.

The Type A/B divisor-Jacobi theorem gives

V_a\subseteq\ker J_d.

Define the quadratic quotient dimension

\kappa(a)=\dim V_d-\dim V_a.

Because J_d is nonzero and annihilates V_a, kappa(a)>=1.

2. The defect quotient

Define the reciprocity defect quotient of the ancestor by

\boxed{ \mathcal R_a = \ker J_d/V_a. }

Its dimension is

\boxed{ \dim\mathcal R_a=\kappa(a)-1. }

Thus R_a measures exactly the local quadratic information that remains after removing:

the ordinary Jacobi bit;
the signatures already generated by divisors of the ancestor depth a.

When kappa(a)=1, this quotient is zero and SQUARE-LIFT-SIGNATURE.md gives automatic signature shadowing for every square lift.

3. Prime defect classes

For any integer q coprime to d with

(q/d)=+1,

define its defect class

\boxed{ \delta_a(q) =[\lambda_d(q)] \in\mathcal R_a. }

The class is zero exactly when

\lambda_d(q)\in V_a.

Now let

4j-1=d s^2

be a square lift of the ancestor.

Square-lift reciprocity proves

(\ell/d)=+1

for every prime ell|j, so every prime divisor of j has a well-defined defect class

\delta_a(\ell)\in\mathcal R_a.

4. Signature defect of the lift

Let

W_{j\to a} = \operatorname{span} \{\lambda_d(\ell):\ell\mid j\}

be the projected divisor-signature space of the lift.

Its image in the defect quotient is

\boxed{ \Delta_{j\to a} = (W_{j\to a}+V_a)/V_a \le\mathcal R_a. }

Equivalently,

\Delta_{j\to a} = \operatorname{span} \{\delta_a(\ell):\ell\mid j\}.

Theorem

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

\boxed{ \Delta_{j\to a}=0. }

This is the quotient form of the criterion W_{j->a} subset V_a.

Thus every square-lift signature exception is precisely a nonzero subspace of the ancestor's reciprocity defect quotient.

5. Conservation law for square-lift defects

Factor

j=\prod_i\ell_i^{e_i}.

Because

4j\equiv1\pmod d,

we have

j\equiv4^{-1}\pmod d.

But 4 is a square modulo every prime divisor of d. Therefore

\boxed{ \lambda_d(j)=0. }

By multiplicativity of the signature map,

\sum_i(e_i\bmod2)\lambda_d(\ell_i)=0.

Passing to the defect quotient gives the exact conservation law

\boxed{ \sum_i(e_i\bmod2)\delta_a(\ell_i)=0 \quad\text{in }\mathcal R_a. }

Consequence

New square-lift signature information cannot appear as an unbalanced single defect.

It must occur in a parity-cancelling configuration of prime-factor defect classes.

This is a structural restriction on every possible square-lift signature exception.

6. One-dimensional defect quotient

Suppose

\kappa(a)=2.

Then

\dim\mathcal R_a=1.

There is only one nonzero defect class.

Therefore a square lift has nonzero signature defect exactly when at least one prime divisor has the nonzero defect class, while the conservation law forces the total parity of nonzero-defect prime exponents to be even.

So every exception in this case is built from one of two patterns:

at least two odd-exponent prime factors carrying the same nonzero defect;
a prime carrying nonzero defect to an even positive exponent.

The full lift may contain additional zero-defect prime factors.

This converts a higher-dimensional signature phenomenon into a parity law.

7. Finite k <= 1200 exception structure

Among the 224 non-squarefree layers through j<=1200, only 17 escape their squarefree ancestor at full quadratic-signature resolution.

Of those:

16 have kappa(a)=2, so dim R_a=1
 1 has kappa(a)=3, so dim R_a=2

The one-dimensional cases exhibit the conservation law transparently. Examples include:

j=115  over a=13: prime factors 5 and 23 carry the same nonzero defect
j=205  over a=23: prime factors 5 and 41 carry the same nonzero defect
j=259  over a=29: prime factors 7 and 37 carry the same nonzero defect
j=319  over a=13: prime factors 11 and 29 carry the same nonzero defect
j=529  over a=59: the nonzero-defect prime 23 occurs with exponent 2
j=625  over a=13: the nonzero-defect prime 5 occurs with exponent 4
j=961  over a=107: the nonzero-defect prime 31 occurs with exponent 2

The unique k<=1200 exception with a two-dimensional defect quotient is

j=979
m_j=3915
ancestor a=109
ancestor modulus d=435
kappa(a)=3
j=11*89

and the two odd prime factors again carry equal defect classes, so their sum vanishes as required.

These are finite proof-mining observations. The defect quotient and conservation law are universal theorems.

8. Reciprocity-matrix interpretation

Let the rows be the primes dividing d and the columns the primes dividing the relevant depth. Put

A_{p,\ell} = \ell_p(\ell) \in\mathbb F_2,

where the entry is the local Legendre-sign bit.

Then:

the ancestor space V_a is the column space of the ancestor matrix;
the Jacobi functional is a canonical nonzero left-kernel vector;
R_a is the remaining left-quotient after removing that Jacobi direction;
square-lift prime factors contribute columns whose classes live in R_a;
the exponent-parity vector of j gives a right-kernel relation because lambda_d(j)=0.

So the signature exception problem is a finite binary reciprocity-matrix problem with both a canonical left constraint and a canonical right dependency.

9. Why this matters for QDSC

The full quadratic-signature shield showed no collective obstruction without a direct signature residual through k<=1200.

The square-lift part of that phenomenon now has a sharper architecture:

\boxed{ \kappa(a)=1 \Rightarrow \text{automatic ancestor signature shadow}, }

while for kappa(a)>1, all genuinely new lift information is confined to

\boxed{ \mathcal R_a \text{ of dimension } \kappa(a)-1, }

and must satisfy the defect conservation law.

Thus a potentially large local-signature system reduces to a small quotient with a built-in parity dependency.

The next target is to classify which defect configurations can actually occur from the norm-form depths

j=(d s^2+1)/4

and prove that their induced signature constraints are already shadowed by lower-codimension layers.

10. Novelty boundary

Quadratic reciprocity, Legendre symbols, quotient vector spaces, and parity relations are classical. The candidate contribution is the reciprocity defect quotient and conservation law specialized to square-lift López Type A/B shadowing inside the minimal-depth framework.

Publication priority remains subject to broader literature review and external mathematical review.