Complete classification of Type A/B multiplicative coset saturation

Theorem

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Status: proved universal classification theorem

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this classifies when the Type A/B trap set fills its divisor-generated multiplicative coset. It does not prove universal Type A/B coverage or the Erdős-Straus conjecture. Literature priority remains under review.

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1. Setup

For k>=1, let

m=4k-1, \qquad G_k=(\mathbb Z/m\mathbb Z)^\times,

and

H_k = \langle\ell\bmod m:\ell\mid k,\ \ell\text{ prime}\rangle.

The general multiplicative trap theorem gives

T_k\subseteq-H_k.

We call k coset-saturated when

\boxed{T_k=-H_k.}

2. Positive cases

k=1

Here

m=3, \qquad H_1=\{1\}, \qquad T_1=\{2\}=-H_1.

k=2^a

For every a>=1, the Mersenne shadow theorem proves

\boxed{ T_{2^a} =-\langle2\rangle =-H_{2^a}. }

Thus 1 and every positive power of 2 are coset-saturated.

3. Mixed even depths cannot saturate

Lemma

Let

k=2^b u, \qquad b\ge1, \qquad u>1\text{ odd}.

Then

\boxed{T_k\subsetneq-H_k.}

Proof

Because 2|k, the element 2 is a generator of H_k. Hence

2^{b+3}\in H_k.

Put

h=2^{b+3}.

Since u>=3,

h=2^{b+3} <2^{b+2}u-1 =4k-1=m.

Thus h is represented by its ordinary positive integer value modulo m.

If T_k=-H_k, then

h\in H_k=-T_k,

so h must be congruent modulo m to either

for some divisor e|k.

Since

1\le e\le k<m

and

4e\le4k=m+1,

there is no possible wrap producing h: the alternative 4e=h+m would exceed m+1 because h>1.

Therefore we would need either

e=h=2^{b+3}

e=h/4=2^{b+1}.

Neither can divide k=2^b u, because both require a power of 2 larger than the exact 2-adic exponent b of k.

Contradiction. QED.

4. Odd depths with multiple prime factors cannot saturate

Lemma

Let k be odd and have at least two distinct prime divisors. Then

\boxed{T_k\subsetneq-H_k.}

Proof

Let p be the smallest prime divisor of k, and write

k=p^a n, \qquad p\nmid n.

Because k has another distinct prime factor and p is smallest,

n\ge p.

Since p generates part of H_k,

h=p^{a+1}\in H_k.

Also

h=p^{a+1} <4p^a n-1 =4k-1=m,

because p<4n and in fact n>=p.

If T_k=-H_k, then h must be congruent to e or 4e for some divisor e|k.

Again 4e<=m+1, while h>1, so a wrapped equality 4e=h+m is impossible.

The equality

h=e

would require p^{a+1}|k, contradicting the exact exponent a.

The equality

h=4e

is impossible by parity because h is odd.

Contradiction. QED.

5. Odd prime powers cannot saturate

The only remaining odd depths are

k=p^a

with p an odd prime.

The prime-power trap dichotomy proves

\operatorname{ord}_{4p^a-1}(p)>2a+1

while

|T_{p^a}|=2a+1.

Since

H_{p^a}=\langle p\rangle,

we have

|H_{p^a}|>|T_{p^a}|,

\boxed{T_{p^a}\subsetneq-H_{p^a}.}

6. Classification theorem

The preceding cases exhaust all positive integers.

Theorem

For every k>=1,

\boxed{ T_k=-H_k \iff k=1\text{ or }k=2^a\text{ for some }a\ge1. }

Equivalently, the complete set of coset-saturated Type A/B depths is

\boxed{ \{1,2,4,8,16,32,\ldots\}. }

7. Meaning for the shadow program

The saturated-source mechanism in COSET-SOURCE-SEMIGROUP-SHADOW.md is now completely classified.

There are no hidden odd or mixed source families whose Type A/B traps fill the entire multiplicative coset.

Thus every exact semigroup shadow family generated by full source-coset saturation originates from one of the binary sources

\boxed{1,2,4,8,\ldots.}

This explains why the Mersenne structure kept reappearing in the global shadow graph: powers of two are not merely examples. They are the unique nontrivial saturated source family.

8. Relation to the exact divisor-sparsity factor

The quotient factorization gives

\frac{|T_k|}{\varphi(4k-1)} = 2^{-\kappa(k)}\Theta(k)^{-1} \frac{|T_k|}{|H_k|}.

The classification says

\boxed{ \frac{|T_k|}{|H_k|}=1 \iff k=1\text{ or }2^a. }

For every other depth,

\boxed{ \frac{|T_k|}{|H_k|}<1. }

So the exact divisor-sparsity factor is genuinely nontrivial everywhere outside the binary saturated spine.

9. Computational regression

Earlier finite search through k<=200000 had found exact saturation only at 1 and powers of two. That observation is now explained by the theorem and should be retained only as a regression test, not as evidence for an unproved classification.

10. Next targets

classify near-saturation, for example depths with bounded ratio |H_k|/|T_k|;
determine whether near-saturated source layers generate additional infinite shadow families;
quantify how the divisor-sparsity factor grows away from the binary spine;
combine the complete saturated-source classification with direct-shadow completeness and the composite-depth spectrum.