Universal square-lift shadow classification — Free Computation Foundation

Geometry

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Status: proved classification theorem inside the Type A/B minimal-depth program

Date: 2026-08-15

Project: Free Computation Foundation / CENTL

Claim boundary: this does not prove universal Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture. It completely classifies one natural infinite shadow mechanism: full shadowing of every odd square lift of a fixed Type A/B modulus.

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1. Universal square-lift shadow property

Fix a Type A/B depth j and put

m=4j-1.

For every positive odd integer c, define

K_c=\frac{mc^2+1}{4}, \qquad 4K_c-1=mc^2.

Say that j has the universal square-lift shadow property if

\boxed{ T_{K_c}\bmod m\subseteq T_j \quad\text{for every positive odd }c. }

For c>1, this says every later layer in the entire square-lift tower is directly shadowed by the base.

2. Classification theorem

Theorem

For a Type A/B depth j, the following are equivalent:

j has the universal square-lift shadow property;
the base trap set is the complete Jacobi-negative half of the unit group,

T_j=\left\{u:\left(\frac u{4j-1}\right)=-1\right\};

j is one of

\boxed{1,2,4.}

Therefore

\boxed{ \text{universal odd square-lift shadow bases} =\{1,2,4\}. }

3. Proof that Jacobi saturation implies universal shadowing

This direction is the reciprocity tower theorem already proved in RECIPROCITY-TOWER-SHADOWS.md.

If the base is Jacobi-saturated, then for every divisor e|K_c, quadratic reciprocity gives

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

Hence both lifted trap residues

-e, \qquad -4e

have Jacobi sign -1 mod m and therefore belong to the saturated base trap set.

Thus

T_{K_c}\bmod m\subseteq T_j

for every odd c.

4. Converse: a non-saturated base always has an escaping square lift

Now assume the base is not Jacobi-saturated.

The quadratic trap theorem still gives

T_j\subseteq N_m^-,

where

N_m^-= \left\{u:\left(\frac u m\right)=-1\right\}.

Since the inclusion is strict, choose

\boxed{v\in N_m^-\setminus T_j.}

Set

\boxed{u=-v\pmod m.}

Because

\left(\frac{-1}{m}\right)=-1,

we have

\left(\frac u m\right) = \left(\frac{-1}{m}\right) \left(\frac v m\right) =( -1)( -1)=+1.

Thus u is a Jacobi-positive unit modulo m.

Choose a prime in the positive class

By Dirichlet's theorem, there exist infinitely many primes ell satisfying

\boxed{\ell\equiv u\pmod m.}

Choose such an odd prime.

Then

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

Since m=3 mod 4, quadratic reciprocity gives

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

Therefore -m is a quadratic residue modulo ell. Equivalently, there exists c_0 such that

\boxed{ c_0^2\equiv-m^{-1}\pmod\ell.}

Choose an odd positive integer c in this residue class. This is always possible because ell is odd: adding ell flips parity without changing the residue class.

Now

mc^2+1\equiv0\pmod\ell,

\ell\mid K_c=\frac{mc^2+1}{4}.

Therefore ell is a divisor of the lifted depth and

-\ell\in T_{K_c}.

Reducing to the base modulus gives

-\ell \equiv -u \equiv v \pmod m.

But v was chosen outside T_j. Hence

T_{K_c}\bmod m\not\subseteq T_j.

So the square lift K_c escapes the base shadow.

This proves:

\boxed{ \text{universal square-lift shadow} \Longrightarrow \text{Jacobi saturation}. }

5. Finish the classification

RECIPROCITY-TOWER-SHADOWS.md proves the exact Jacobi-saturation classification

\[ \boxed{ T_j=N_{4j-1}^- \iff j\in\{1,2,4\}. }</div>

Combining both directions gives

\boxed{ \forall\text{ odd }c, \ T_{((4j-1)c^2+1)/4}\bmod(4j-1)\subseteq T_j \iff j\in\{1,2,4\}. }

QED.

6. Constructive counter-lift certificate for every other base

The proof is constructive once one chooses a missing Jacobi-negative residue v.

For every

j\notin\{1,2,4\},

one can generate a finite certificate of failure of universal square-lift shadowing:

find

v\in N_m^-\setminus T_j;

set u=-v mod m;
find a prime

\ell\equiv u\pmod m;

solve

c^2\equiv-m^{-1}\pmod\ell;

choose c odd;
form

K=(mc^2+1)/4;

verify

\ell\mid K;

verify the lifted trap residue

-\ell\pmod m

lies outside T_j.

This gives an explicit later layer in the base's square-lift tower that is not shadowed by the base.

7. Why the classification matters

Before the converse, the three towers based at 1, 2, and 4 were infinite sufficient shadow families.

The converse upgrades the statement:

these are the only bases whose entire odd square-lift tower can collapse under one base layer.

Therefore any universal theorem for the many other square-lift shadows visible in the finite shadow graph must use additional information beyond the scalar Jacobi character.

That pushes the theory naturally toward:

full local quadratic signatures;
multiplicative quotient classes;
exact two-box divisor geometry;
higher p-adic structure.

The hierarchy is not optional. The classification proves the scalar quadratic mechanism has been exhausted completely.

8. Relationship to the observed finite shadow map

The finite direct-shadow graph contains many square-lift ancestry edges where the base is not 1, 2, or 4.

This theorem says those cannot arise from a base that shadows every odd square lift merely because of Jacobi saturation.

They must be selective:

\boxed{ \text{specific lift arithmetic} + \text{finer quotient/residue structure} }

is responsible.

This isolates the next classification problem very sharply.

9. New theorem target

Classify, for a general base j, the set

\boxed{ \mathcal C_j = \left\{ c\text{ odd}: T_{K_c}\bmod(4j-1)\subseteq T_j \right\}. }

The present theorem gives the complete universal case:

\boxed{ \mathcal C_j=\{1,3,5,\ldots\} \iff j\in\{1,2,4\}. }

For every other base, C_j is a proper subset of the odd integers.

The next diamond question is whether C_j itself has a finite congruence, multiplicative, or automata-like description controlled by the quotient Gamma_j and the two-box trap geometry.

10. Novelty boundary

Dirichlet's theorem and quadratic reciprocity are classical. López Type A/B congruences are prior art.

The candidate contribution is the complete classification of universal Type-A/B square-lift shadow bases inside the minimal-depth/shadow framework, together with the constructive counter-lift mechanism for every non-saturated base.

Targeted arXiv searches on 2026-08-15 did not locate this exact formulation. That negative search does not establish publication priority.