Prime-power Type A/B trap dichotomy — Free Computation Foundation

Shadow

Read with MERSENNE-SHADOW-LATTICE.md and MULTIPLICATIVE-TRAP-COSET.md.

Status: proved structural theorem

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this theorem concerns the internal structure of Type A/B trap layers. It does not prove universal Type A/B coverage or the Erdős-Straus conjecture. Literature priority remains under review.

Read with MERSENNE-SHADOW-LATTICE.md and MULTIPLICATIVE-TRAP-COSET.md.

1. General prime-power layer

Let

k=p^a

with p prime and a>=1, and put

m=4p^a-1.

The divisor-generated subgroup is

H=\langle p\rangle\le(\mathbb Z/m\mathbb Z)^\times.

Since

4p^a\equiv1\pmod m,

we have

4\equiv p^{-a}\pmod m.

The divisors of p^a are p^i, 0<=i<=a. Therefore the two trap families become

-p^i

and

-4p^i\equiv-p^{i-a}.

Hence

\boxed{ T_{p^a} =\{-p^j:-a\le j\le a\} \subseteq-\langle p\rangle. }

This is an exact exponent-window description.

2. Odd prime case

Assume p is odd.

Because 4 does not divide p^a, the exact trap-cardinality theorem gives

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

Let

r=\operatorname{ord}_{m}(p).

The exponent-window description shows

2a+1\le r.

Theorem

For every odd prime p and a>=1,

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

Consequently

\boxed{ T_{p^a}\subsetneq-\langle p\rangle. }

Proof

Suppose for contradiction that

r=2a+1.

Then

p^{2a+1}\equiv1\pmod m.

Write n=p^a. Since

4n\equiv1\pmod m,

we have

16n^2\equiv1\pmod m.

But

p^{2a+1}=pn^2\equiv1\pmod m.

Multiplying the latter congruence by 16 gives

p\equiv16\pmod m.

For all odd prime-power cases except (p,a)=(3,1),

m=4p^a-1>|p-16|,

so this congruence would force p=16, impossible for a prime. The exceptional numerical case has m=11 and 3 is not congruent to 16 mod 11 either.

Thus equality cannot occur. Since r>=2a+1, we obtain

r>2a+1.

Therefore the 2a+1 trap residues cannot exhaust the cyclic subgroup. QED.

3. Binary case

For p=2, the situation changes because

4=2^2

is itself a power of the subgroup generator.

The Mersenne theorem gives

\boxed{ \operatorname{ord}_{2^{a+2}-1}(2)=a+2 }

and

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

Thus binary prime powers saturate the divisor-generated multiplicative coset exactly, whereas odd prime powers never do.

4. Dichotomy

We therefore have the clean prime-power split

\boxed{ \begin{array}{ll} p=2 &: T_{2^a}=-H_{2^a};\\[1mm] p\text{ odd} &: T_{p^a}\subsetneq-H_{p^a}. \end{array} }

The infinite Mersenne shadow lattice is therefore a genuinely binary phenomenon inside the prime-power family, not a generic consequence of prime-power divisor structure.

5. Consequences for the theorem program

exact multiplicative-coset saturation has a distinguished infinite binary family;
odd prime-power layers retain an intrinsic divisor-sparsity gap even after the full multiplicative quotient has been used;
attempts to generalize the Mersenne shadow lattice to odd prime powers must exploit something other than full coset equality;
the exact density factor |T_k|/|H_k| from TRAP-QUOTIENT-FACTORIZATION.md is therefore essential, not merely a technical remainder.