Binary-prime completed layers collapse to one-dimensional power intervals

Geometry

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Status: proved exact internal normal form

Date: 2026-08-15

Depends on: ES-SQUARE-TRAP-SIGNED-BOX-IDENTITY.md, ES-SQUAREFREE-SEMIPRIME-MIXED-RESIDUES.md

Claim boundary: gives an exact cyclic-interval description of the completed signed box for layer indices 2^e p with p odd prime. It does not classify every shadow edge out of these layers or prove universal completed coverage.

1. Setup

Let

\boxed{j=2^e p}

with

\boxed{e\ge1}

and p an odd prime.

Put

\boxed{m=4j-1=2^{e+2}p-1.}

The completed centered signed box is

\boxed{ R_j = \{2^x p^y\pmod m: -e\le x\le e, \ -1\le y\le1\}.}

The exact trap is

S_j=-R_j.

2. Eliminate the odd prime

From the defining modulus identity

2^{e+2}p\equiv1\pmod m,

we obtain

\boxed{p\equiv2^{-(e+2)}\pmod m.}

Therefore

2^x p^y \equiv 2^{x-(e+2)y} \pmod m.

Hence the complete two-dimensional signed box lies in the cyclic subgroup generated by 2.

3. The three exponent intervals

The three possible values of y produce:

y = -1

z=x+(e+2), \qquad -e\le x\le e,

\boxed{2\le z\le2e+2.}

y = 0

z=x,

\boxed{-e\le z\le e.}

y = +1

z=x-(e+2),

\boxed{-2e-2\le z\le-2.}

For e=1, the three integer intervals are

[-4,-2],\ [-1,1],\ [2,4],

which join consecutively.

For e>=2, the middle interval overlaps the outer intervals at ±2 or beyond.

Thus their union is exactly

\boxed{[-2e-2,2e+2]_{\mathbb Z}.}

4. Exact interval theorem

Theorem — binary-prime interval collapse

For every

j=2^e p, \qquad e\ge1, \qquad p\text{ odd prime},

one has

\boxed{ \mathcal R_{4j-1}(j) = \{2^z\pmod{4j-1}: -2e-2\le z\le2e+2\}.}

Therefore the complete Type-II trap is

\boxed{ S_j = \{-2^z\pmod{4j-1}: -2e-2\le z\le2e+2\}.}

The nominally two-dimensional square-completed layer is a single contiguous exponent interval in the cyclic subgroup generated by 2.

Residue collisions may identify different exponents if the order of 2 is short; the set identity remains exact.

5. Squarefree semiprime case e = 1

For

j=2p,

the theorem becomes

\boxed{ R_{2p} = \{2^z:-4\le z\le4\}.}

Thus the two mixed Type-II directions complete the López boundary into a nine-step symmetric interval.

This gives a particularly simple internal model for the first nontrivial mixed-support family.

6. Shadow criterion for later prime directions that are powers of 2

Let a later squarefree layer k lie on an ancestry edge into j=2^e p.

Suppose every prime factor r_i of k satisfies

\boxed{r_i\equiv2^{c_i}\pmod m.}

Then the later signed box reduces to

\left\{ 2^{\sum_i\epsilon_i c_i}: \epsilon_i\in\{-1,0,1\} \right\}.

Every such exponent satisfies

\left|\sum_i\epsilon_i c_i\right| \le \sum_i|c_i|.

Therefore a sufficient shadow condition is

\boxed{ \sum_i|c_i| \le 2e+2.}

Under this inequality,

\boxed{S_k\bmod m\subseteq S_j.}

So direct shadow becomes a one-dimensional signed-length budget.

7. Example j = 10

Take

j=10=2\cdot5, \qquad m=39.

The theorem gives

\boxed{R_{10}=\{2^z:-4\le z\le4\}.}

Consider the later squarefree layer

\boxed{k=8083=59\cdot137.}

Its modulus satisfies an ancestry relation with j=10.

Modulo 39,

59\equiv20\equiv2^{-1},

and

137\equiv20\equiv2^{-1}.

Thus the two later prime directions have

c_1=c_2=-1.

Their total signed-length budget is

|c_1|+|c_2|=2\le4.

Hence

S_{8083}\bmod39\subseteq S_{10}.

This explains a shadow that is not a simple factor lift and does not use a nontrivial stabilizer.

8. Relation to the canonical modulus identity

The interval collapse comes from the universal relation

\boxed{2^{e+2}p\equiv1\pmod{2^{e+2}p-1}.}

In exponent-lattice language, this is the kernel vector

\boxed{(e+2,1)}

for the generators (2,p).

The completed signed box fills both signs of the p coordinate, so quotienting by this relation folds the rectangle

[-e,e]\times[-1,1]

onto one interval.

This is an exact example of the more general idea that modulus relations can lower the effective dimension of a completed layer.

9. Research consequence

There are now three explicit internal geometries:

prime-power layers: one-dimensional from the start;
binary-prime layers 2^e p: two-dimensional exponent boxes that fold to one cyclic interval;
general multi-prime layers: genuinely higher-dimensional unless additional modulus relations collapse them.

This suggests classifying completed layers by their effective signed-box dimension modulo 4j-1, rather than only by omega(j).

A low effective dimension can make nonmultiplicative shadow containment a finite interval or polytope problem even when the raw factorization has several prime directions.