Small-selector hypothesis for residual Type A/B fiber kernels

Theorem

This note continues the reduction developed in SHADOW-KERNEL.md and FIBER-SHADOW-KERNEL.md.

Status: active proof-mining experiment

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

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

This note continues the reduction developed in SHADOW-KERNEL.md and FIBER-SHADOW-KERNEL.md.

1. Motivation

The candidatewise search through k<=1200 found reduced avoiding progressions for all 41,470 directly novel candidates. A separate coordinate analysis showed that every certified solution lies within at most nine guided prime-power coordinate changes of a simple unary-safe basepoint.

Fiber peeling goes further: it removes prime-power coordinates by theorem, without consulting the stored avoiding witness. The remaining obstruction is a small-prime residual kernel.

The next question is deliberately simple:

Are those residual kernels already satisfiable by a tiny fixed menu of ordinary integer parameter values?

If so, the global-looking covering problem may admit a surprisingly small selector mechanism after the fiber exterior is removed.

2. Residual kernel

Fix a directly novel candidate

x=r+Ls.

After exact augmented fiber peeling, suppose the residual prime set is

P_{\rm ker}=\{p_1,\ldots,p_m\}

with residual forbidden constraints

s\bmod q_j\in R_j.

An integer selector s0 solves the residual kernel if

s_0\bmod q_j\notin R_j

for every residual edge and, for each residual prime p not dividing L,

r+Ls_0\not\equiv0\pmod p.

Once such a residual assignment is fixed, the exact fiber-peeling theorem extends it backward across the peeled coordinates. Thus a selector that solves the residual kernel gives an independent constructive proof of a reduced avoiding class for that candidate.

3. Finite small-selector experiment

The automated analyzer tests the fixed menu

\mathcal S_B=\{0,\pm1,\pm2,\ldots,\pm B\}

in increasing absolute value, with a deterministic tie order.

For each directly novel candidate it records:

whether fiber peeling already empties the kernel;
whether some s0 in S_B solves the residual kernel;
the first selector found;
the smallest absolute selector radius needed;
the distribution of selectors across all nonempty residual kernels;
any residual kernels not solved by the finite menu.

The default experimental bound is B=64.

A failure of this finite selector menu is not a counterexample to DSC-P. It would only mean that this particularly simple post-peeling mechanism is insufficient.

4. Why a positive result would matter

Suppose a tiny fixed selector set solved every residual kernel over a large exact range. That would suggest a much stronger theorem architecture than raw witness search:

\boxed{ \text{direct novelty} \to \text{fiber peel} \to \text{tiny residual kernel} \to \text{small selector} \to \text{reduced avoiding class}. }

The computational objective is not to replace proof with a lookup table. It is to identify which local residue values repeatedly survive, then derive the arithmetic reason.

A particularly strong pattern would be the existence of a very small universal set such as

\{0,1,-1,2,-2\}

or another bounded menu independent of k. Such an observation would become a theorem target, not a theorem by itself.

5. Falsification value

This test can fail cleanly in several useful ways:

a residual kernel requires a selector outside the tested range;
selector radii grow rapidly with k;
different kernel signatures require unrelated selector values;
the small-selector phenomenon disappears past the current finite frontier.

Any of those outcomes would teach us where the apparent low local complexity breaks.

6. Automation

The implementation is shadow_small_selector_analyzer.py. It reconstructs the pullback system and performs fiber peeling independently of the stored reduced witness. It then tests only the residual kernel against the fixed selector menu.

The direct-shadow GitHub Actions workflow runs this stage before CENTL certification and freezes its JSON/report into the hashed research artifact.

7. Current theorem target

The larger target remains DSC-P:

\text{not directly shadowed} \Longrightarrow \text{reduced avoiding class}.

The small-selector experiment asks whether the missing proof can be compressed still further:

\boxed{ \text{every residual fiber kernel belongs to a locally solvable family with a bounded selector mechanism.} }

If the data supports that statement, the next task is to classify the selector by kernel signature and prove why the surviving local residue exists.