Shadow semigroups from the three Jacobi-saturated bases

Shadow

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Status: proved theorem family 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 all modulus-ancestry shadows whose source is one of the three Jacobi-saturated layers 1,2,4.

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1. Setup

Let

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

and put

m=4j-1\in\{3,7,15\}.

These are exactly the Jacobi-saturated Type A/B layers:

\boxed{ T_j = \left\{ u\in(\mathbb Z/m\mathbb Z)^\times: \left(\frac u m\right)=-1 \right\}. }

Let K>j be any later depth satisfying the modulus-ancestry condition

\boxed{m\mid4K-1.}

Equivalently,

K\equiv j\pmod m.

2. Exact saturated-base ancestry criterion

Theorem

For j in {1,2,4} and any ancestry child K, the following are equivalent:

the complete later Type A/B trap layer is directly shadowed by j;
every divisor e|K has positive Jacobi symbol modulo m;
every prime divisor ell|K has positive Jacobi symbol modulo m.

In formulas,

\boxed{ T_K\bmod m\subseteq T_j \iff \left(\frac e m\right)=+1 \ \forall e\mid K \iff \left(\frac\ell m\right)=+1 \ \forall \ell\mid K. }

Proof

Every target trap residue is

-e

-4e

for some divisor e|K.

Since

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

and

\left(\frac4m\right)=+1,

we have

\left(\frac{-e}{m}\right) = \left(\frac{-4e}{m}\right) =-\left(\frac e m\right).

Because the base trap set is exactly the Jacobi-negative units, both target residues land in T_j exactly when

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

Thus full shadowing is equivalent to positivity for every divisor.

By multiplicativity of the Jacobi symbol, every divisor is positive iff every prime divisor of K is positive. QED.

3. Base j = 1: an exact multiplicative semigroup

Here

m=3, \qquad K\equiv1\pmod3.

For a prime ell !=3,

\left(\frac\ell3\right)=+1 \iff \ell\equiv1\pmod3.

Therefore:

Corollary

A later depth K>1 is completely shadowed by layer 1 iff

\boxed{ \text{every prime divisor of }K\text{ is }1\pmod3. }

The ancestry condition is then automatic because a product of such primes is itself 1 mod 3.

So layer 1 deletes the entire multiplicative semigroup

\boxed{ \mathcal S_3 = \left\{ K>1:\ell\mid K\Longrightarrow\ell\equiv1\pmod3 \right\}. }

Every K in S_3 is a global structural gap in the minimal-depth spectrum.

Examples include

7, 13, 19, 31, 37, 43, ...
49 = 7^2
91 = 7*13
133 = 7*19
169 = 13^2
247 = 13*19
...

These are not finite-search absences. Every hit at such a depth already hits layer 1.

4. Base j = 2: quadratic-residue semigroup modulo 7

Here

m=7, \qquad K\equiv2\pmod7.

The positive Jacobi classes are the quadratic residues

\boxed{1,2,4\pmod7.}

Therefore:

Corollary

An ancestry child K>2 is completely shadowed by layer 2 iff

\boxed{ \ell\bmod7\in\{1,2,4\} \quad\text{for every prime }\ell\mid K. }

Together with the ancestry condition

K\equiv2\pmod7,

this gives an exact multiplicative description of every full shadow sourced at layer 2.

For prime target depths this specializes immediately to

\boxed{ K\text{ prime},\quad K\equiv2\pmod7 \Longrightarrow 2\to K. }

5. Base j = 4: Jacobi-positive semigroup modulo 15

Here

m=15, \qquad K\equiv4\pmod{15}.

For every prime divisor ell of such a child,

\gcd(\ell,15)=1.

The exact criterion is

\boxed{ \left(\frac\ell{15}\right)=+1 \quad\forall\ell\mid K. }

Thus the complete source-4 shadow family is the intersection of the ancestry class

K\equiv4\pmod{15}

with the multiplicative semigroup generated by Jacobi-positive prime classes modulo 15.

For prime depth K, the ancestry residue 4 mod 15 is itself Jacobi-positive, so every prime

\boxed{K\equiv4\pmod{15}}

is automatically shadowed by layer 4.

6. Reciprocity towers are a special case

The reciprocity tower theorem considered

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

Quadratic reciprocity proved that every prime divisor of K_c is Jacobi-positive modulo the base m.

The saturated-base criterion now shows exactly why the tower theorem works:

\boxed{ \text{reciprocity forces the prime factors into the positive semigroup} \Longrightarrow \text{the saturated base shadows the whole target layer}. }

So the three universal square-lift towers are structured subfamilies of the much larger source-1, source-2, and source-4 shadow semigroups.

7. Prime-depth consequences

The source-1 semigroup alone proves:

\boxed{ K\text{ prime},\quad K\equiv1\pmod3 \Longrightarrow K\notin\mathcal D_\exists. }

Thus one half of the prime residue classes modulo 3 are explicit structural-gap depths.

The source-2 family independently gives

\boxed{ K\text{ prime},\quad K\equiv2\pmod7 \Longrightarrow K\notin\mathcal D_\exists. }

And source 4 gives

\boxed{ K\text{ prime},\quad K\equiv4\pmod{15} \Longrightarrow K\notin\mathcal D_\exists. }

These are all special cases of the complete prime-depth dichotomy, but here the actual shadow source is explicit.

8. Stronger structural-gap counting from source 1

Let

G(X)=\#\{k\le X:k\notin\mathcal D_\exists\}.

Every prime

K\equiv1\pmod3

is in the source-1 shadow semigroup.

Therefore

\boxed{ G(X) \ge \pi(X;3,1)-O(1). }

By the prime number theorem in arithmetic progressions,

\pi(X;3,1) \sim \frac12\frac{X}{\log X}.

Hence the currently immediate prime-only lower bound strengthens to

\boxed{ G(X) \ge \left(\frac12+o(1)\right) \frac{X}{\log X}. }

The full multiplicative semigroup S_3 contains many composite depths too, so this analytic lower bound is not expected to be sharp for the source-1 gap family itself.

9. Why this matters

The shadow graph is beginning to decompose into exact algebraic source families.

For the first three saturated layers, there is no remaining mystery:

\boxed{ \text{ancestry child} + \text{prime-factor character signs} \Longleftrightarrow \text{full direct shadow}. }

This turns three high-connectivity vertices of the shadow graph into fully classified multiplicative deletion operators.

The hard classification problem therefore moves to non-saturated bases, where the Jacobi sign no longer describes the exact trap set.

10. Next theorem target

For a general non-saturated base j, replace the scalar character condition by progressively finer structures:

\boxed{ \text{Jacobi sign} \to \text{full local quadratic signature} \to \text{multiplicative quotient }\Gamma_j \to \text{two-box exact divisor geometry}. }

The goal is to obtain an analogue of the saturated-base criterion at the first resolution fine enough to become exact.

That is now a natural route to classifying the remaining composite-child direct shadows.

11. Novelty boundary

Quadratic residues, Jacobi characters, and multiplicative semigroups are classical. López Type A/B congruences are prior art.

The candidate contribution is the exact source-wise shadow classification for the three Jacobi-saturated Type A/B layers inside the minimal-depth/shadow framework, and the resulting structural-gap semigroups.

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