Complete classification of the depth-1 Type A/B shadow component

Shadow

Read with COSET-SOURCE-SEMIGROUP-SHADOW.md and STRUCTURAL-GAP-ASYMPTOTIC.md.

Status: proved exact classification

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this theorem classifies global direct shadowing by the first Type A/B layer. It does not classify all structural gaps and does not prove universal López Type A/B coverage or the Erdős-Straus conjecture.

Read with COSET-SOURCE-SEMIGROUP-SHADOW.md and STRUCTURAL-GAP-ASYMPTOTIC.md.

1. First layer

At depth

j=1,

we have

m_1=3

and

T_1 = \{-1,-4\}\pmod3 = \{2\}.

Call a later layer k>1 globally shadowed by depth 1 when every integer satisfying a Type A/B trap congruence at depth k is automatically in the depth-1 trap modulo 3.

2. Classification theorem

Theorem

For every k>1, the following are equivalent:

the complete Type A/B layer at depth k is globally directly shadowed by depth 1;
every prime divisor p|k satisfies

p\equiv1\pmod3;

every positive divisor e|k satisfies

e\equiv1\pmod3.

Thus the depth-1 shadow component is exactly the multiplicative semigroup

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

Proof

(2) iff (3) is immediate from unique factorization.

(2) implies (1)

If every prime divisor of k is 1 mod 3, then every divisor e|k is 1 mod 3, and in particular

k\equiv1\pmod3.

Hence

3\mid4k-1=m_k.

For every target trap residue,

-e\equiv-1\equiv2\pmod3

and, because 4=1 mod 3,

-4e\equiv-e\equiv2\pmod3.

Therefore every target trap is contained in

T_1=\{2\}\pmod3.

So depth 1 globally shadows depth k.

(1) implies (2)

Assume depth 1 globally shadows depth k.

First, 3 must divide m_k=4k-1. If not, then

\gcd(m_k,3)=1.

Take the target trap residue

x\equiv-1\pmod{m_k}.

By CRT this progression contains integers in every residue class modulo 3, including residues not equal to 2. Such integers hit layer k but avoid T_1, contradicting global shadowing.

Therefore

3\mid4k-1,

k\equiv1\pmod3.

Now let p|k be any prime divisor. Since p itself is a divisor of k, the target trap includes

x\equiv-p\pmod{m_k}.

Because 3|m_k, every integer in this progression has residue

x\equiv-p\pmod3.

Global shadowing by depth 1 requires this residue to equal the unique trap residue 2 mod 3. Thus

-p\equiv2\pmod3,

and hence

p\equiv1\pmod3.

This holds for every prime divisor p|k. QED.

3. Spectrum consequence

For every

k\in\mathcal S_3,

we have

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

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

So S_3 is not merely an explicit subset of the gap set. It is exactly the portion of the global structural-gap set explained by the first Type A/B layer alone.

4. Counting consequence

The Selberg-Delange analysis in STRUCTURAL-GAP-ASYMPTOTIC.md gives

|\mathcal S_3\cap[1,X]| \sim C_3\frac{X}{\sqrt{\log X}}

with explicit C_3>0.

Therefore the single first layer already explains

\boxed{ (C_3+o(1))\frac{X}{\sqrt{\log X}} }

global impossible depths up to X.

This gives the current strongest proved lower bound for the full structural-gap counting function.

5. Why this matters

The first trap layer is extremely small:

T_1=\{2\}\pmod3.

Yet its exact shadow basin in depth space has size of order

X/\sqrt{\log X}.

This illustrates the central geometry of the project: tiny early congruence layers can cast very large multiplicative shadows over the later minimal-depth spectrum.

6. Next classification targets

The natural analogues are:

classify global direct shadowing by depth 2, where

m_2=7, \qquad T_2=-\langle2\rangle=\{3,5,6\};

classify shadowing by depth 4, where

m_4=15

and the source is again coset-saturated;

determine whether the union of the exact depth-1, depth-2, and depth-4 shadow components has a sharper asymptotic than any one component;
extend exact source-basin classifications to every coset-saturated power-of-two source.