Mixed-box obstruction in exact square-lift shadowing

Theorem

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Status: exact reformulation plus falsified simplification and finite proof-mining data

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this note does not prove universal Direct-Shadow Completeness or exact square-lift shadowing. It identifies the exact combinatorial object left after multiplicative/genus reductions and records a counterexample to a tempting but false generator-wise simplification.

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1. Normalize the trap set

For a depth k, write

m_k=4k-1

and define the normalized positive trap set

\boxed{ U_k=-T_k = \{e,4e\pmod{m_k}:e\mid k\}. }

Let

D_k=\{e\pmod{m_k}:e\mid k\}

be the divisor-residue set.

Because

4k\equiv1\pmod{m_k},

for e|k and f=k/e we have

4e\equiv f^{-1}\pmod{m_k}.

As e ranges over divisors, so does f. Therefore:

Theorem

\boxed{ U_k=D_k\cup D_k^{-1}. }

This is the exact two-box/inverse formulation of a Type A/B trap layer.

The inversion relationship itself is consistent with López's prior observation that the two Type A/B divisor families are mutual inverses; the present use is to organize exact square-lift projection shadowing.

2. Square-lift projection criterion

Let

d=4a-1

be squarefree and let

4j-1=d s^2.

Project the lifted traps modulo the ancestor modulus d.

Define

D_{j\to d} =\{e\bmod d:e\mid j\}.

Since also

4j\equiv1\pmod d,

the same divisor/inverse argument gives

\boxed{ -T_j\bmod d = D_{j\to d}\cup D_{j\to d}^{-1}. }

The ancestor normalized trap is

U_a=D_a\cup D_a^{-1}.

Since U_a is inversion-stable, we obtain:

Exact projection-shadow criterion

\boxed{ T_j\bmod d\subseteq T_a \iff D_{j\to d}\subseteq U_a. }

Thus exact square-lift projection shadowing is a problem about where all divisor residues of j land relative to one ancestor two-box set.

3. Exponent-box formulation

Factor

a=\prod_{i=1}^r p_i^{A_i}.

Let

H_a=\langle p_1,\ldots,p_r\rangle \le(\mathbb Z/d\mathbb Z)^\times

and define

\phi_a:\mathbb Z^r\to H_a, \qquad (x_1,\ldots,x_r) \mapsto \prod_i p_i^{x_i}\pmod d.

Let

B_A=\{x\in\mathbb Z^r:0\le x_i\le A_i\}.

Then

D_a=\phi_a(B_A)

and

D_a^{-1}=\phi_a(-B_A).

Hence

\boxed{ U_a=\phi_a(B_A\cup-B_A). }

This is the ancestor's exact signed exponent box.

Now factor

j=\prod_{q\mid j}q^{E_q}.

If a prime q|j lies outside H_a, exact projection shadowing fails immediately, because the divisor q itself lies outside U_a subset H_a.

If every prime factor lies in H_a, choose exponent vectors v_q satisfying

\phi_a(v_q)=q\pmod d.

Then every divisor of j has the form

\phi_a\left(\sum_q t_qv_q\right), \qquad 0\le t_q\le E_q.

Therefore exact projection shadowing becomes the signed-box containment problem

\boxed{ \phi_a \left( \left\{ \sum_qt_qv_q:0\le t_q\le E_q \right\} \right) \subseteq \phi_a(B_A\cup-B_A). }

The relation lattice ker(phi_a) allows wraparound, so the problem lives in a finite abelian exponent lattice rather than ordinary Euclidean boxes.

4. A tempting simplification is false

A natural guess is:

If every prime-power divisor q^t of j lands in U_a, then every divisor of j lands in U_a.

This would reduce exact projection shadowing to independent one-prime tests.

It is false.

5. First mixed-box counterexample

Take

\boxed{j=696.}

Then

4j-1=2783=23\cdot11^2,

so the squarefree ancestor modulus is

d=23, \qquad a=6.

The ancestor divisor set is

D_6=\{1,2,3,6\}\pmod{23}.

Its inverse set is

D_6^{-1}=\{1,12,8,4\}.

Thus

\boxed{ U_6=\{1,2,3,4,6,8,12\}\pmod{23}. }

Now

696=2^3\cdot3\cdot29.

Every prime-power direction individually stays inside the ancestor envelope:

2,4,8\in U_6,

3\in U_6,

and

29\equiv6\pmod{23}\in U_6.

Nevertheless the mixed divisor

87=3\cdot29

satisfies

87\equiv18\pmod{23},

and

\boxed{18\notin U_6.}

Likewise

174\equiv13\pmod{23}

is outside U_6.

Therefore

\boxed{ T_{696}\bmod23 \not\subseteq T_6 }

although every individual prime-power divisor direction passes the ancestor test.

This is a genuine mixed-box obstruction.

6. Finite search through j <= 20000

An exhaustive square-lift replay through j<=20000 found 3,788 non-squarefree moduli 4j-1.

They split as:

exact ancestor projection contained:       1,198
prime-power / axis failure visible:        2,575
all prime-power axes safe but mixed fail:     15

The first mixed-only failure is j=696 above.

The 15 mixed-only failures through this finite range are:

696, 1180, 2076, 2324, 6408,
7044, 8319, 9592, 10024, 10632,
10740, 12702, 16152, 19752, 19869

This finite list is proof-mining data, not a universal classification.

Its importance is negative: it falsifies the simplest generator-wise exact theorem and proves that the final residue core contains real interactions among divisor directions.

7. Two kinds of exact square-lift defect

The exact projection failure now has a clean dichotomy.

Axis defect

Some prime-power divisor already escapes:

\boxed{ q^t\bmod d\notin U_a. }

This is visible one generator at a time.

Mixed-box defect

Every prime-power divisor lies in U_a, but a product of different prime-power directions escapes:

\boxed{ \prod_q q^{t_q}\bmod d\notin U_a. }

This is a genuine interaction effect.

The first type dominates the finite data, but the second type is exactly where a purely local prime-by-prime classification fails.

8. Relation to the earlier quotient hierarchy

The hierarchy can now be written more sharply:

\boxed{ \begin{array}{c} \text{genus / quadratic signature defect}\\ \downarrow\\ \text{multiplicative subgroup defect }(q\notin H_a)\\ \downarrow\\ \text{single-axis signed-box defect}\\ \downarrow\\ \text{mixed signed-box defect}\\ \downarrow\\ \text{exact projection excess} \end{array}}

Each level is strictly finer than the one above it.

The mixed-box counterexample proves that the last step cannot be removed without an additional theorem.

9. New theorem target

The highest-value exact square-lift question is now:

Classify the maps from the divisor exponent box of j into the ancestor signed box B_A union -B_A modulo the relation lattice of H_a.

Useful subtargets are:

classify when a single prime-power axis stays inside the signed box;
classify when two individually safe axes have an unsafe sum;
find a Helly-type or bounded-support theorem: if full containment fails, is there always a failing divisor involving at most C distinct prime factors for an absolute small C?
determine whether every mixed-box defect is already detected by a pair of prime directions;
compare the minimal failing support with the fiber-kernel residual size.

If target 4 were true, the final square-lift residue obstruction would reduce from an arbitrary divisor box to pairwise interactions.

That is now directly falsifiable.

10. Novelty boundary

Exponent lattices, finite abelian groups, inverse sets, and box-containment problems are standard mathematical objects. López already records the mutual-inverse relation of the Type A/B divisor families.

The candidate contribution is the signed-box formulation and its role as the exact residual layer of the Type A/B minimal-depth shadow hierarchy, pending prior-art and external review.