Prime-power shadow kernel

Status: theorem note plus finite exact bound

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this note does not prove universal Direct-Shadow Completeness, Lopez Type A/B coverage, or the Erdos-Straus conjecture.

This note develops the prime-power kernel hinted at by ODD-COVERING-BRIDGE.md and the candidatewise results in DIRECT-SHADOW-K1000.md.

The main point is exact: a large class of prime-power coordinates can be removed from the union-shadow satisfiability problem by a local load inequality. Any genuine counterexample to Direct-Shadow Completeness must therefore survive inside a much smaller prime-coordinate kernel.

1. Pullback system

Fix an admissible hard-class Type A/B candidate (k,h,t) and write

For every earlier layer j<k, let

and let

be the forbidden parameter residues satisfying

Let

The parameter space decomposes as the product of the prime-power coordinates p^{A_p}.

2. Coordinate load

Fix a prime p|Q. For every active constraint with p|q_j, put

Define the exact local load

This quantity is not the global cover mass. It measures only how much of the p^{A_p} coordinate can be excluded after every other prime-power coordinate has been fixed.

3. Prime-power peeling lemma

Lemma

If

then the p coordinate is peelable: every assignment of all other prime-power coordinates that satisfies the constraints not involving p can be extended to a value modulo p^{A_p} satisfying every constraint involving p.

Proof

Fix arbitrary values of every coordinate except p^{A_p}.

Consider one constraint (q_j,R_j) involving p, with

Once the other coordinates are fixed, each residue in R_j can forbid at most one residue modulo p^{a_{j,p}}. Each such residue has exactly

lifts modulo p^{A_p}.

Therefore constraint j excludes at most

values of the full p coordinate.

By the union bound, all constraints involving p exclude at most

If lambda_p<1, this number is strictly smaller than p^{A_p}. Hence at least one value of the p coordinate survives. QED.

4. Satisfiability-preserving elimination

The lemma gives an exact elimination rule.

When lambda_p<1:

1. remove the p coordinate;
2. remove every constraint involving p;
3. solve the residual constraint system;
4. extend the solution back to p using the peeling lemma.

Loads can only decrease after constraints are removed. Therefore this process may be iterated.

The coordinates that remain when no further lambda_p<1 move is possible form the prime-power shadow kernel of the candidate under this peeling rule.

If the kernel is empty, the candidate is proved not union-shadowed without searching over the complete period Q.

5. Reduced prime-realization load

For DSC-P we need more than an avoiding integer. We need a reduced progression.

For every prime p|Q with p not dividing L, the condition

forbids exactly one residue class modulo p, namely

This removes a fraction 1/p of the p^{A_p} coordinate.

Define the augmented load

If

the same proof peels p while preserving the possibility of a reduced final progression.

For an admissible candidate, r is already coprime to L, because its residues modulo 840 and 4k-1 are prime-compatible units. Thus if every coordinate can be removed by augmented peeling, the resulting avoiding class is reduced modulo LQ and Dirichlet supplies infinitely many exact-depth primes.

6. Candidate-independent upper load

The local load can be bounded without knowing the candidate.

Since R_j is an affine pullback of the trap set,

Also, if p|q_j, then necessarily

Because a_{j,p}>=1, we obtain

For reducedness, a uniform sufficient bound is therefore

Whenever

the prime coordinate p is guaranteed peelable for every admissible candidate at target depth at most k, regardless of h, t, or the detailed gcd reductions.

This is deliberately conservative: candidate-specific q_j omit many of the terms counted by B_p(k), higher p-adic exponents reduce load further, and multiple trap residues can project to the same local residue.

7. A crude analytic kernel bound

Let

Since

and 4j=1 mod p selects at most one residue class of j mod p, the number of indices j<k contributing to B_p(k) is at most approximately ceil((k-1)/p).

Thus the simple sufficient estimate

shows that all sufficiently large prime coordinates are automatically peelable.

Using ceil(x)<=x+1, a convenient sufficient inequality is

So any hypothetical union-shadow counterexample is forced into primes below a scale on the order of

before any candidate-specific information is used.

This is not yet the sharp kernel. It is a universal elementary bound.

8. Exact universal bound through k = 1000

For k<=1000, exact trap enumeration gives

More importantly, evaluating the sharper finite quantity B_p(1000) for every odd prime shows that

for every prime

Therefore, for every admissible Type A/B candidate through depth 1000, every parameter prime coordinate p>=113 is universally peelable even after the reducedness condition is included.

Only the following 28 primes can survive this candidate-independent first kernel bound:

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,
53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109

The actual candidate-specific kernels are usually smaller because the universal bound intentionally overcounts.

Exploratory exact peeling of difficult certified candidates has already produced residual kernels supported only on substantially smaller primes, for example at most 67 or 89 in selected high-depth cases. These observations are proof-mining data, not a universal p<=89 theorem.

9. Why this matters

This changes the shape of the proof search.

A union-shadow counterexample no longer needs to be imagined as a gigantic arbitrary covering system over all prime factors of Q.

Large prime coordinates are provably disposable.

The obstruction, if it exists, must condense into a small-prime kernel.

This is strongly consonant with classical covering-system theory, where small prime divisors play an essential role in possible covers, but the present lemma is specialized directly to the Type A/B pullback system and handles multi-residue constraints.

The universal DSC-P problem can now be attacked as:

10. Next theorem targets

1. sharpen B_p(k) using the exact trap-cardinality formula and the arithmetic progression 4j=1 mod p;
2. include exact candidate-specific gcd reductions and p-adic exponents in automated kernel certificates;
3. classify the small-prime kernels appearing through k<=1200 and beyond;
4. determine whether the kernel belongs to finitely many isomorphism types after quotienting by residue relabeling;
5. prove a local satisfiability theorem for those kernel types;
6. combine the kernel theorem with Direct-Shadow Completeness to obtain a constructive proof of exact-depth realization.

11. Current interpretation

The shadow-cover problem appears to have two scales:

• a large-prime exterior that can be removed by an elementary local-load argument;
• a small-prime interior where the genuine overlap geometry lives.

That interior is now the highest-value object in the proof search.