Classification of Jacobi-saturated Type A/B layers

Status: proved theorem

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this theorem does not prove universal Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture. It classifies exactly when one Type A/B trap layer fills the complete Jacobi-negative half of its unit group.

Read with:

• QUADRATIC-TRAP-SIGNATURE.md
• MULTIPLICATIVE-TRAP-COSET.md
• DYADIC-TRAP-LATTICE.md
• SQUARE-LIFT-RECIPROCITY.md

1. Definition

For

let

Every trap has Jacobi symbol -1 modulo m_k. Call depth k Jacobi-saturated if the converse also holds:

2. The classification theorem

Theorem

A Type A/B layer is Jacobi-saturated if and only if

Equivalently, the only Jacobi-saturated moduli 4k-1 are

Proof

Let

and let H_k be the subgroup generated by the prime divisors of k, as in MULTIPLICATIVE-TRAP-COSET.md.

That theorem gives

If T_k is the entire Jacobi-negative half, both inclusions must be equalities. In particular,

The full saturation theorem in DYADIC-TRAP-LATTICE.md proves

Hence write

The case r=0, namely k=1, is exceptional because the two trap families coincide modulo 3; it is checked directly below. Assume first that r>=1, and put

Then

The dyadic trap theorem gives

and

because the order of 2 mod (2^n-1) is exactly n.

If the layer is Jacobi-saturated, the Jacobi-negative half has size phi(m_k)/2. Therefore, for r>=1, saturation forces

We show this can occur only for n=3,4.

An elementary phi bound

For every odd prime power p^a,

For a=1, this is p-1>=sqrt(p), true for every odd prime p>=3; larger a only strengthens the inequality.

By multiplicativity,

for every odd positive integer N.

For n>=9,

The strict inequality holds at n=9 and remains true thereafter.

The remaining values 5<=n<=8 are explicit:

n=5:  2^n-1=31,  phi=30  >10
n=6:  2^n-1=63,  phi=36  >12
n=7:  2^n-1=127, phi=126 >14
n=8:  2^n-1=255, phi=128 >16

Thus phi(2^n-1)=2n is impossible for every n>=5.

For the remaining dyadic depths:

k=1 (r=0): |T_1|=1, phi(3)/2=1
k=2 (r=1, n=3): |T_2|=3, phi(7)/2=3
k=4 (r=2, n=4): |T_4|=4, phi(15)/2=4

Direct enumeration shows that each of these trap sets is exactly the Jacobi-negative unit set.

Therefore the only Jacobi-saturated depths are

QED.

3. Explicit saturated trap sets

They are:

and

Each is exactly the complete Jacobi-negative unit set of its modulus.

4. Consequence for square-lift shadowing

SQUARE-LIFT-RECIPROCITY.md proves that if

with 4a-1 squarefree, then

lies entirely in the ancestor's Jacobi-negative half.

The classification above therefore gives exactly three ancestor depths for which this character information alone forces complete projection shadowing for every odd square multiplier s:

Thus the universal square-lift families

are the complete families arising solely from Jacobi saturation of the squarefree ancestor.

Other ancestors can still shadow particular square-lifts, but that requires finer exact divisor-residue structure beyond the scalar Jacobi character.

5. Why this matters

This separates two mechanisms cleanly:

1. character-forced lift shadowing, completely classified by the three ancestors 1,2,4;
2. exact-residue lift shadowing, which occurs for additional individual square-lifts and is the genuinely finer problem.

The next classification target is therefore not the saturated case. It is the non-saturated projection-excess condition

for specific square multipliers s.

6. Novelty boundary

Euler phi bounds, multiplicative groups, Jacobi symbols, and Mersenne moduli are classical. The candidate contribution is the exact classification of Jacobi saturation inside the López Type A/B trap system and its role in the minimal-depth square-lift shadow architecture.

Publication priority remains subject to external review.