Multiplicative semigroup shadow families from saturated Type A/B sources

Shadow

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Status: proved general shadow theorem with infinite composite-depth families

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this theorem gives exact structural gaps inside the López Type A/B minimal-depth spectrum. It does not prove universal Type A/B coverage or the Erdős-Straus conjecture. Literature priority remains under review.

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1. Saturated source

For a source depth j, put

m_j=4j-1

and

H_j=\langle\ell\bmod m_j:\ell\mid j,\ \ell\text{ prime}\rangle.

The general multiplicative theorem gives

T_j\subseteq-H_j.

Call j coset-saturated when

\boxed{T_j=-H_j.}

Every power-of-two depth is coset-saturated by the Mersenne theorem. The initial depth j=1 is also saturated, with

H_1=\{1\}\pmod3, \qquad T_1=\{2\}=-H_1.

2. Semigroup shadow theorem

Theorem

Let j be coset-saturated. Let k>j satisfy

\boxed{m_j\mid m_k}

and suppose every prime divisor p|k satisfies

\boxed{p\bmod m_j\in H_j.}

Then the complete Type A/B layer at depth k is directly shadowed by depth j:

\boxed{ T_k\bmod m_j\subseteq T_j. }

Proof

Every divisor e|k is a product of prime divisors of k. Since each such prime lies in the subgroup H_j,

e\bmod m_j\in H_j.

Also

4j\equiv1\pmod{m_j},

and j in H_j, so

4\equiv j^{-1}\pmod{m_j}

lies in H_j as well.

Therefore

e,4e\in H_j

for every e|k, and hence

-e,-4e\in-H_j=T_j.

Thus every target trap reduces into the source trap set. QED.

3. The depth-1 structural semigroup

Take

j=1.

Then

m_1=3, \qquad H_1=\{1\}, \qquad T_1=\{2\}.

Suppose every prime divisor of k>1 satisfies

\boxed{p\equiv1\pmod3.}

Then every divisor e|k is also 1 mod 3, and k=1 mod 3. Consequently

3\mid4k-1,

so the modulus-ancestry condition is automatic.

The semigroup shadow theorem gives:

Corollary

If k>1 has no prime divisor outside the residue class 1 mod 3, then

\boxed{T_k\bmod3=T_1=\{2\},}

and therefore

\boxed{C_{AB}(n)\ne k}

for every integer n for which C_AB(n) is defined.

So the multiplicative semigroup

\boxed{ \mathcal S_3 = \{k>1:p\mid k\Rightarrow p\equiv1\pmod3\} }

is an infinite family of global structural gaps.

Examples include

7, 13, 19, 31, 37, ...
7^2 = 49
7*13 = 91
13^2 = 169
7*19 = 133
7*13*19 = 1729

The composite examples are not consequences of the prime-depth dichotomy; they are new instances supplied by multiplicative source saturation.

4. Binary saturated sources

For every a>=1, take

j=2^a, \qquad m_j=2^{a+2}-1, \qquad H_j=\langle2\rangle.

The Mersenne theorem gives

T_j=-H_j.

Therefore any k>j satisfying

k\equiv j\pmod{m_j}

and

p\bmod m_j\in\langle2\rangle \qquad\text{for every }p\mid k

is directly shadowed by j.

This produces a hierarchy of multiplicatively defined structural-gap families attached to the saturated binary sources.

The original Mersenne shadow lattice is the special case where the later target itself is another power of two.

5. Why the semigroup condition is natural

The trap-coset theorem says the target divisor set matters multiplicatively.

If the source trap fills an entire coset -H_j, then exact shadowing no longer requires checking target divisors one by one. It is enough that the prime generators of the target divisor monoid map into H_j.

Thus coset saturation converts an exact residue problem into a prime-factor membership test.

This gives a new general mechanism:

\boxed{ \text{source coset saturation} + \text{target prime-factor containment} + \text{modulus ancestry} \Longrightarrow \text{complete direct shadow}. }

6. Counting consequence from the depth-1 semigroup

Even without using the full asymptotic theory of multiplicative sets, the family already strengthens the supply of explicit composite structural gaps.

For example, every product

k=pq

of primes

p\equiv q\equiv1\pmod3

lies in S_3 and is globally impossible as a minimal depth.

Classical multiplicative-set methods such as Wirsing or Selberg-Delange predict and prove much denser counting asymptotics for the full semigroup. A publication version should state the sharp asymptotic only after the constant and analytic hypotheses are written carefully.

The exact shadow theorem itself is elementary and independent of those counting refinements.

7. Next theorem targets

classify all coset-saturated depths j satisfying T_j=-H_j;
the finite search currently suggests j=1 and the powers of two may be the complete list, but this is not proved;
derive sharp counting asymptotics for the structural semigroup S_3;
classify intersections and unions of the saturated-source semigroup families;
determine whether their union has positive density among all depth values;
use saturated-source shadows as exact deletions before attacking the remaining composite-depth DSC-P problem.