Square-lift reciprocity and infinite ancestor-shadow families

Geometry

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Status: proved theorem family

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this note does not prove universal Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture. It gives an exact infinite family of shadow reductions inside the squarefree-lift core.

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1. Square-lift setup

Let

d=4a-1

be squarefree, and let s be any positive odd integer. Define

\boxed{ 4j-1=d s^2, \qquad j=\frac{d s^2+1}{4}. }

Then the squarefree ancestor of the layer j is exactly a.

The exact Type A/B traps are

T_j=\{-e,-4e\pmod{4j-1}:e\mid j\}.

We study their projection modulo the ancestor modulus d.

2. Reciprocity theorem for divisors of j

Theorem

For every divisor e|j,

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

Consequently,

\boxed{ \left(\frac{-e}{d}\right) = \left(\frac{-4e}{d}\right) =-1. }

Thus

\boxed{ T_j\bmod d \subseteq \{u\in(\mathbb Z/d\mathbb Z)^\times:(u/d)=-1\}. }

Proof

It is enough to prove the first claim for every prime divisor ell|j.

Because gcd(j,4j-1)=1, the prime ell is coprime to d s.

From

d s^2=4j-1

and ell|j, we obtain

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

Hence

-d\equiv s^{-2}\pmod\ell,

so -d is a quadratic residue modulo ell.

Odd ell

Because d is squarefree and d=3 mod 4, quadratic reciprocity for the Jacobi symbol gives

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

The right side is +1 by the congruence above. Therefore

(\ell/d)=+1.

ell = 2

If 2|j, then

4j-1\equiv7\pmod8.

Since every odd square is 1 mod 8, the identity 4j-1=d s^2 gives

d\equiv7\pmod8.

Therefore

(2/d)=+1.

By multiplicativity, every divisor e|j satisfies (e/d)=+1.

Finally d=3 mod 4, so (-1/d)=-1, while 4 is a square modulo d. Hence

(-e/d)=(-4e/d)=-1.

QED.

3. Interpretation

The square-lift operation cannot project Type A/B traps arbitrarily into the ancestor unit group.

It is confined to the ancestor's Jacobi-negative half.

So the projection excess from SQUAREFREE-LIFT-CORE.md,

E_j=(T_j\bmod d)\setminus T_a,

always satisfies

\boxed{ E_j \subseteq \{u:(u/d)=-1\}\setminus T_a. }

The only possible new projected residues are therefore nontrap quadratic nonresidues of the squarefree ancestor.

This identifies the exact location where a square-lift can create genuinely new residue information.

4. Saturated-ancestor shadow theorem

Call the ancestor a Jacobi-saturated when

\boxed{ T_a = \{u\in(\mathbb Z/d\mathbb Z)^\times:(u/d)=-1\}. }

Theorem

If a is Jacobi-saturated, then every odd square-lift

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

satisfies

\boxed{ T_j\bmod(4a-1) \subseteq T_a. }

Hence

\boxed{E_j=\varnothing.}

Proof

The reciprocity theorem places the projected trap set inside the complete Jacobi-negative half of the ancestor. Saturation identifies that half with T_a. QED.

Thus every Jacobi-saturated ancestor generates an infinite square-lift direct-shadow family.

5. Three explicit infinite families

The first three ancestors are Jacobi-saturated.

a = 1

d=3, \qquad T_1=\{2\},

which is the complete Jacobi-negative unit set modulo 3.

Therefore for every odd s,

\boxed{ j=\frac{3s^2+1}{4}}

has

T_j\bmod3\subseteq T_1.

a = 2

d=7, \qquad T_2=\{3,5,6\},

which is the complete quadratic-nonresidue set modulo 7.

Therefore for every odd s,

\boxed{ j=\frac{7s^2+1}{4}}

has

T_j\bmod7\subseteq T_2.

a = 4

d=15, \qquad T_4=\{7,11,13,14\},

which is the complete Jacobi-negative unit set modulo 15.

Therefore for every odd s,

\boxed{ j=\frac{15s^2+1}{4}}

has

T_j\bmod15\subseteq T_4.

These are infinite exact shadow families, not finite empirical patterns.

6. Relation to the finite k <= 1200 core

The k<=1200 squarefree-lift replay found many non-squarefree moduli whose projection excess is empty. The theorem above explains an infinite subcollection immediately: every lift with squarefree ancestor modulus 3, 7, or 15 is forced to have zero excess.

It also explains why the exceptional projection layers cannot be arbitrary. They can only arise when the ancestor's Jacobi-negative half is strictly larger than its exact Type A/B trap set.

For example, the ancestor

a=3, \qquad d=11,

has only three Type A/B trap residues while the Jacobi-negative half has five. Square-lifts over 11 can therefore create projection excess, and the first such example occurs at

j=25, \qquad 4j-1=99=11\cdot3^2.

7. New classification problem

The projection-excess question now splits cleanly into two pieces.

First, classify the Jacobi-saturated ancestors:

\boxed{ T_a = \{u:(u/(4a-1))=-1\}. }

Second, for a nonsaturated ancestor, determine which square multipliers s^2 still satisfy

T_j\bmod(4a-1)\subseteq T_a.

The finite data shows that both outcomes occur for the same ancestor, so saturation is sufficient but not necessary for an individual lift to be shadowed.

8. Why this sharpens the proof search

The residual hierarchy is now

\boxed{ \text{fixed-negative character layer} \to \text{squarefree ancestor} \to \text{Jacobi-negative projection} \to \text{ancestor trap vs. nontrap split} \to \text{projection excess only}. }

A large half-space obstruction has collapsed to a small exact residue difference set.

That difference set, rather than the whole lifted trap layer, is the object that must be controlled in the next Direct-Shadow Completeness proof stage.

9. Novelty boundary

Quadratic reciprocity and Jacobi symbols are classical. The candidate contribution is the square-lift specialization to the López Type A/B trap system and the resulting infinite ancestor-shadow families inside the minimal-depth/shadow framework.

Publication priority remains subject to external review and a broader literature search.