Shadow-cover geometry behind Direct-Shadow Completeness

Shadow

This note follows DIRECT-SHADOW-COMPLETENESS.md. The candidatewise run established that all 19,016 directly novel hard-compatible candidates through k=600 have reduced avoiding progressions. The question here is why that happens.

Source in the repository

Status: exact finite diagnostic from the verified k <= 600 candidate bundle

Date: 2026-08-14

Claim boundary: this note identifies a proof obstruction and a stronger theorem target. It does not prove universal Direct-Shadow Completeness.

This note follows DIRECT-SHADOW-COMPLETENESS.md. The candidatewise run established that all 19,016 directly novel hard-compatible candidates through k=600 have reduced avoiding progressions. The question here is why that happens.

For a candidate progression

x=r+Ls,

each earlier Type A/B layer j<k forbids a finite set

R_j\subseteq\mathbb Z/q_j\mathbb Z, \qquad q_j=\frac{4j-1}{\gcd(L,4j-1)}.

The candidate is union-shadowed exactly when these pulled-back forbidden residue systems cover every integer parameter s.

1. A cheap density proof does not explain the result

A first possible explanation would be a union bound. Define the raw cover mass

W(k,h,t)=\sum_{j<k}\frac{|R_j|}{q_j},

summing only nonempty forbidden systems.

If W<1, the union bound immediately proves that the forbidden systems cannot cover all integers. If most directly novel candidates had W<1, Direct-Shadow Completeness would have a simple density explanation.

That is emphatically not what the verified k<=600 bundle shows.

Across all 19,016 directly novel candidates:

minimum W:        0
median W:         8.500010768048488
mean W:           7.906791184249354
maximum W:        10.789829231455588
W < 1:            132 candidates
W > 1:         18,884 candidates

Thus

\boxed{18,884/19,016\approx99.306\%}

of directly novel candidates have raw cover mass greater than one, often much greater than one, yet every one of them still has an explicit avoiding parameter and a reduced Dirichlet progression.

Therefore the observed noncoverage is not explained by low total forbidden density.

The overlap and dependency geometry among the forbidden congruences is doing essential arithmetic work.

2. The constraint systems are genuinely dense

The same exact recomputation gives, among the directly novel candidates through k=600:

median number of active earlier constraints: 251
mean number of active earlier constraints:   245.6381994110223
maximum active earlier constraints:          477

So a typical candidate is not escaping because only a handful of earlier layers interact with it. Hundreds of earlier Type A/B layers can impose nonempty forbidden parameter classes.

Nevertheless the complete collection still fails to cover the parameter line in every tested directly novel candidate.

3. Nor is there a simple new-coordinate explanation

Another tempting explanation is that each successive constraint might introduce a fresh CRT prime-power coordinate, leaving enough freedom to escape.

That is also false in its simplest form.

Process the active constraints in increasing layer order and maintain the accumulated parameter modulus

Q_i=\operatorname{lcm}(q_1,\ldots,q_i).

Call a constraint new-coordinate if adjoining its q_j enlarges the accumulated lcm, and old-coordinate otherwise.

Across the verified candidate bundle, both kinds occur heavily. The median counts are approximately

new-coordinate constraints: 145
old-coordinate constraints:  108

and some candidates have more than 200 active old-coordinate constraints.

For the hardest first-run reduced witness

(k,h,t)=(500,529,1979),

there are 392 active earlier constraints:

207 new-coordinate constraints
185 old-coordinate constraints

So the finite theorem cannot be explained by saying that every forbidden layer acts on a fresh independent coordinate.

4. The stronger structural clue

We now know three things simultaneously in the tested range:

no single earlier layer covers a directly novel candidate;
the total nominal forbidden density is usually far above 1;
hundreds of dependent constraints nevertheless leave a reduced avoiding progression.

This points toward a special overlap theorem for Type A/B pullback constraints.

The important object is no longer merely the list of forbidden densities

|R_j|/q_j.

It is their structured intersection geometry.

A future proof of Direct-Shadow Completeness must explain why Type A/B forbidden residue systems overlap so strongly that collective coverage cannot arise without one layer already becoming a direct shadow.

That property would be special. General covering systems absolutely can cover the integers without any single constituent covering them, so no generic union-bound or generic CRT argument can establish the desired theorem.

5. Refined theorem target

For a directly novel candidate (k,h,t), define the finite pullback family

\mathscr R_{k,h,t} = \{(q_j,R_j):1\le j<k,\ R_j\ne\varnothing\}.

The experimental phenomenon can be stated as:

\boxed{ R_j\ne\mathbb Z/q_j\mathbb Z\ \forall j \quad\Longrightarrow\quad \bigcup_j\{s:s\bmod q_j\in R_j\}\ne\mathbb Z. }

for every tested Type A/B pullback family.

The prime-realization version adds that the complement contains a residue class s0 for which

\gcd(r+Ls_0,LQ)=1.

The candidate proof mechanism should therefore search for an invariant of the family of pullbacks, not merely of individual q_j or cardinalities |R_j|.

6. High-value proof directions

The next proof search should test the following mechanisms.

A. Shadow-of-shadow redundancy

Many old-coordinate constraints may themselves be redundant after pullback. Determine whether every apparently new forbidden class is contained in the union of a small ancestral basis while the basis itself necessarily leaves a residue uncovered.

B. Prime-power coordinate signatures

Factor every q_j into prime powers and study which local coordinates each R_j can occupy. Search for a prime-power coordinate on which all active Type A/B forbidden sets omit a common compatible value.

C. Multiplicative character obstruction

The residues in T_j are generated from divisors of j by the two maps e -> -e and e -> -4e. Their affine pullbacks may preserve a quadratic, multiplicative, or divisor-theoretic signature that prevents arbitrary covering-system behavior.

D. Minimal-cover contradiction

Assume a smallest union-shadowed but not directly shadowed candidate exists. Extract a minimal subcover of parameter residue systems. Classical covering-system constraints on repeated maximal moduli or prime-power divisibility may then collide with the special form q_j=(4j-1)/gcd(L,4j-1) and the trap-cardinality bounds.

E. Reduced-survivor strengthening

The finite data show more than an uncovered integer: every candidate has an uncovered parameter whose resulting arithmetic progression is reduced modulo LQ. A proof may be easier if constructed prime-power coordinate by prime-power coordinate while preserving reducedness from the outset.

7. Why this is a better clue than simply extending the search

The k<=600 experiment already rules out the obvious failure mode for 19,016 individual candidates. The cover-mass diagnostic now rules out the obvious easy proof.

That is progress in both directions:

the conjecture survived a much stronger falsification attempt;
the explanation cannot be a trivial density estimate.

The remaining mechanism must account for substantial, highly overlapping modular mass.

In project terms, this is the diamond inside the diamond: the shadow graph appears to carry a nontrivial overlap geometry that is invisible if one records only which layers are connected.

The next mathematical goal is to identify and prove that overlap invariant.