Candidate-independent trap-fiber bounds

Status: theorem note with exact finite evaluations

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

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

This note sharpens FIBER-SHADOW-KERNEL.md by removing the candidate dependence from the fiber-width estimate. The key observation is that the affine pullback from a Type A/B trap set to the parameter line preserves CRT fiber multiplicities up to relabeling.

The result gives a much smaller universal finite prime kernel than the coarse |R_j| load bound.

1. Setup

For an earlier layer j, put

For a target candidate x=r+Ls, define

The compatible traps are

The pullback forbidden set is

Fix a prime p|q_j and write

2. Affine fiber invariance

Multiplication by the unit

is an automorphism of both CRT coordinates

Therefore it changes only the labels of the fibers, not their cardinalities.

Before that unit scaling, fixing the non-p coordinate means fixing

Equivalently, it fixes u modulo

Hence the candidate-specific fiber width f_{j,p} from the fiber peeling theorem satisfies

where

The quantity kappa depends only on the trap set T_j and the prime power dividing m_j. It does not depend on the target candidate (k,h,t).

3. Candidate-independent local contribution

The exponent a=v_p(q_j) depends on the target candidate because the gcd with L may remove some p-power from m_j.

To dominate every possible candidate, define

when p|m_j, and beta_{j,p}=0 otherwise.

For every target candidate and every active occurrence of p at earlier layer j,

4. Universal reduced fiber load

For a depth bound K, define

The initial 1/p is a worst-case allowance for the local reducedness condition. If p|L, that cost is actually zero, so this remains an upper bound.

Theorem

If

then the prime coordinate p is reduced-fiber-peelable for every admissible Type A/B target candidate at every depth

Proof

For a fixed candidate, sum the exact reduced fiber load

Here epsilon_p is either 0 or 1/p.

Each active summand is at most beta_{j,p}, and the target range k<=K only removes terms from the sum defining F_p(K). Therefore

The reduced fiber-peeling theorem applies. QED.

5. Exact finite evaluations

The quantities below are obtained by exact enumeration of the finite trap sets and exact rational arithmetic. They are finite theorem bounds, not heuristic estimates.

Through K = 1000

The only primes for which the candidate-independent bound does not already prove peelability are

The last non-automatically-peelable prime is 37. Therefore

This improves the earlier coarse universal threshold p>=113 dramatically.

Through K = 1200

The only primes not eliminated by the universal trap-fiber bound are

Thus

Through K = 1500

The universal finite kernel is contained in

Therefore every prime coordinate

is automatically peelable for every admissible candidate through k=1500.

Through K = 3000

Even at the full depth used in the original hard-prime shadow map, the candidate-independent trap-fiber bound leaves only

as possible first-stage kernel primes.

Hence

is universally reduced-fiber-peelable through k=3000.

This is before using target-specific gcd cancellation, iterative removal of incident constraints, or the exact candidate-specific fibers. All three make the actual kernel smaller.

6. Why this is important

The union-shadow problem originally involved parameter periods containing many prime factors and hundreds of active congruence constraints.

The coarse local-load theorem proved that sufficiently large coordinates can be peeled.

The fiber theorem improved the candidate-specific load.

The present bound adds the missing bridge:

Through k=3000, before looking at a particular target residue, every potential obstruction has already been forced onto only seventeen small primes.

This is not a universal-in-k bounded-prime theorem. The finite universal kernel can grow with K. But it is a major compression of the exact finite problem and a concrete route toward a structural proof.

7. New object: trap-fiber collision profile

For each earlier layer, the values

measure how strongly the divisor-generated Type A/B trap set collides when projected away from a prime-power coordinate.

This trap-fiber collision profile is a new natural object in the current framework.

The next analytic question is to bound or classify kappa_{j,p^a} directly from the divisor structure of j and the two trap maps

A sufficiently sharp uniform bound on these collision profiles could turn the finite small-kernel phenomenon into an asymptotic theorem.

8. Immediate theorem targets

1. derive closed bounds for kappa_{j,p^a} using the location of divisors e|j inside residue classes modulo m_j/p^a;
2. classify cross-collisions between the -e and -4e trap families inside one fiber;
3. determine whether iterative fiber peeling has an absolute residual-prime bound even though the first candidate-independent bound grows slowly with K;
4. classify the residual kernels on the small prime sets above;
5. prove those kernels always possess a reduced satisfying assignment when direct shadowing is absent.

The obstruction has now been compressed twice: first from congruence layers to prime-power coordinates, and then from arbitrary coordinate loads to exact trap-fiber collisions.