Dyadic trap-coset and shadow lattice — Free Computation Foundation

Geometry

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Status: proved theorem family inside the Type A/B minimal-depth program

Date: 2026-08-14

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 gives an exact infinite subfamily of Type A/B trap layers and shadow relations.

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1. Dyadic depths

For

k=2^a, \qquad a\ge0,

the Type A/B modulus is

\boxed{m_a=4k-1=2^{a+2}-1.}

Thus the dyadic depth sequence is governed by Mersenne-type moduli.

The depth k=1=2^0 is a small exceptional endpoint because the divisor-generated subgroup D_1 is trivial. The cyclic description below begins at a>=1.

2. Exact dyadic trap-coset theorem

For a>=1, let

D_a=\langle2\rangle \le(\mathbb Z/m_a\mathbb Z)^\times.

This is exactly the multiplicative subgroup generated by the prime divisors of k=2^a.

Theorem

For every a>=1,

\boxed{T_{2^a}=-D_a.}

Moreover,

\boxed{|D_a|=a+2.}

Hence

\boxed{ T_{2^a} = \{-2^r\pmod{2^{a+2}-1}:0\le r<a+2\}. }

At the exceptional endpoint a=0,

k=1, \qquad m=3, \qquad D_1=\{1\}, \qquad T_1=\{2\}=-D_1.

Thus full multiplicative-coset saturation still holds at k=1, but not through the formula D_1=<2>.

Proof for a>=1

The divisors of 2^a are exactly

1,2,\ldots,2^a.

Therefore

T_{2^a} = \{-2^b,-2^{b+2}:0\le b\le a\}.

Modulo

m_a=2^{a+2}-1,

we have

2^{a+2}\equiv1.

The exponents appearing in the two trap families cover every residue class modulo a+2:

\{0,1,\ldots,a\} \cup \{2,3,\ldots,a+2\} \equiv \mathbb Z/(a+2)\mathbb Z.

It remains only to check that the order of 2 mod m_a is exactly a+2. If a smaller positive d<a+2 satisfied

2^d\equiv1\pmod{m_a},

then m_a would divide the strictly smaller positive integer 2^d-1, impossible because

0<2^d-1<2^{a+2}-1=m_a.

Thus ord_{m_a}(2)=a+2, and the trap set is exactly the negative coset of the cyclic subgroup generated by 2. QED.

3. Prime-power saturation classification

The dyadic family is the unique nontrivial prime-power family in which the multiplicative trap envelope is exact.

Theorem

Let

k=p^a

with p prime and a>=1. Then

T_k=-\langle p\rangle

if and only if

\boxed{p=2.}

Proof

The dyadic direction was proved above.

Now assume p is odd. The exact trap-cardinality formula gives

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

The normalized trap set inside \langle p\rangle is generated by exponent interval

-a,-a+1,\ldots,a,

because

4\equiv k^{-1}=p^{-a}\pmod{4p^a-1}.

Hence saturation would force

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

But then

p^{2a+1}\equiv1.

Using

p^a\equiv4^{-1},

we obtain

p^{2a+1}=p(p^a)^2\equiv p/16\equiv1 \pmod{4p^a-1},

p\equiv16\pmod{4p^a-1}.

For odd p>=3,

0<p<4p^a-1,

and therefore this congruence would force p=16, impossible. Thus no odd prime power saturates its multiplicative trap coset. QED.

4. Full saturation classification

Theorem

For every positive integer k,

\boxed{ T_k=-D_k \iff k\text{ is a power of }2, }

where D_k is the subgroup generated by the prime divisors of k.

Proof

For k=1, the equality was checked directly above. For k=2^a with a>=1, it is the dyadic theorem.

Suppose now that k has an odd prime factor. Write

k=2^a n, \qquad n>1\text{ odd}.

Case 1: a>=1

Because 2|k, the subgroup D_k contains every power of 2, in particular

x=2^{a+3}.

Since n>=3,

2^{a+3}<2^{a+2}n-1=4k-1.

So x is already the canonical residue representative modulo 4k-1.

The normalized trap set

-T_k=\{e,4e:e\mid k\}

has representatives of two forms:

a divisor e|k, whose 2-adic valuation is at most a;
4e for a proper divisor e, whose 2-adic valuation is at most a+2;
the exceptional value 4k, which reduces to 1.

But

v_2(x)=a+3.

Therefore x belongs to D_k but not to -T_k. Hence the trap set cannot fill -D_k.

Case 2: a=0

Then k is odd and greater than 1. Since 4 in D_k, also

16\in D_k.

If k>=5, then

16<4k-1.

Every divisor e|k is odd, while every proper 4e is congruent to 4 mod 8. Thus the canonical residue 16, which is 0 mod 8, lies in neither family. So again 16 in D_k but 16 notin -T_k.

The only remaining odd case is k=3. Here

m=11, \qquad -T_3=\{1,3,4\}, \qquad 16\equiv5\pmod{11},

so saturation also fails.

Therefore the only saturated layers are powers of two. QED.

5. Dyadic modulus ancestry

For dyadic depths 2^a and 2^b,

m_a=2^{a+2}-1, \qquad m_b=2^{b+2}-1.

The classical Mersenne divisibility identity gives

\boxed{ m_a\mid m_b \iff a+2\mid b+2.}

This creates an exact divisibility lattice on the dyadic Type A/B layers.

6. Dyadic shadow theorem

Theorem

Let

1\le a<b.

a+2\mid b+2,

then every Type A/B trap at depth 2^b is automatically a Type A/B trap at depth 2^a after reduction modulo the earlier modulus:

\boxed{ T_{2^b}\bmod m_a =T_{2^a}. }

Proof

By the exact dyadic trap theorem,

T_{2^b}=-\langle2\rangle\pmod{m_b}.

Because m_a|m_b, reduction modulo m_a is a group homomorphism. The image of the subgroup generated by 2 is the subgroup generated by 2 modulo m_a, which has order a+2. Therefore

T_{2^b}\longmapsto-\langle2\rangle=T_{2^a}.

QED.

Thus dyadic modulus ancestry gives an infinite exact family of direct shadow relations.

Why a=0 is excluded

Although m_0=3 divides m_b whenever 2|(b+2), D_1 is trivial while the reduction of <2> modulo 3 is not. For example,

T_4\bmod3=\{1,2\} \not\subseteq T_1=\{2\}.

So k=1 is a saturation endpoint but not the first node of the cyclic dyadic shadow lattice.

7. Dyadic irredundant nodes

Within the dyadic subsystem with earlier cyclic nodes a>=1, depth 2^b for b>=1 has an earlier dyadic shadow ancestor exactly when b+2 has a proper divisor at least 3.

Therefore:

k=1 is an isolated exceptional endpoint;
k=2 (b+2=3) has no earlier dyadic ancestor;
k=4 (b+2=4) also has no earlier cyclic dyadic ancestor;
for b>=3, the dyadic node 2^b is irredundant with respect to earlier dyadic nodes exactly when b+2 is prime.

Equivalently, apart from the exceptional nodes 1 and 4, the dyadic irredundant family is

\boxed{2^{q-2}\text{ with }q\text{ prime}.}

This does not say that such a layer cannot be shadowed by a non-dyadic earlier layer.

8. Example shadow chains

Exact cyclic-dyadic chains include:

k=2   (m=7)   -> k=16   (m=63)   -> ...
k=4   (m=15)  -> k=64   (m=255)  -> ...
k=8   (m=31)  -> k=256  (m=1023) -> ...

For example,

31=2^5-1

divides

1023=2^{10}-1,

so the entire Type A/B trap set at k=256 reduces onto the exact trap set at k=8.

9. Why this matters

This gives the shadow program an especially transparent infinite lattice:

\boxed{ \text{nontrivial powers of two} \leftrightarrow \text{Mersenne-type moduli} \leftrightarrow \text{exact multiplicative trap cosets} \leftrightarrow \text{divisibility shadow lattice}. }

It also shows that multiplicative quotient saturation is exceptionally rigid: it happens only at dyadic depths, with k=1 as a trivial endpoint and k>=2 giving the cyclic family.

So for every non-dyadic layer, the exact Type A/B trap set is a proper subset even of its multiplicative coset envelope. The residual geometry inside that coset is genuinely necessary.

10. Regression and falsification

The first automated regression correctly caught the k=1 endpoint subtlety because the initial draft incorrectly used <2> as the divisor-generated subgroup at k=1. That failure was a tooling success, not a mathematical counterexample to the corrected theorem.

The corrected regression now treats k=1 separately and tests the cyclic dyadic shadow theorem only for a>=1.

11. Next theorem targets

classify how non-dyadic layers project into dyadic trap cosets;
determine which dyadic shadows remain relevant after Mordell-hard-class compatibility is imposed;
characterize direct shadows produced by odd-square square-lift ancestry and compare them with the dyadic Mersenne lattice;
search for further exact saturated subquotients even when the full multiplicative coset is not saturated;
use the dyadic lattice as a regression family for a general algebraic shadow theorem.

12. Novelty boundary

Mersenne divisibility and multiplicative orders are classical. López's Type A/B congruences are prior art. The candidate contribution is the exact dyadic Type-A/B trap-coset saturation classification and its induced infinite direct-shadow lattice inside the minimal-depth/shadow framework.

A targeted arXiv search on 2026-08-14 did not locate this exact formulation. That negative search does not establish publication priority.