q=3 Pointwise Absorption — Free Computation Foundation

Theorem

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Status: proved universal theorem

Date: 2026-08-15

Depends on: Q3-ABSORPTION.md, Q3-WEAK-REDUNDANCY.md

Claim boundary: sharpens q=3 direct-shadow detection from whole-layer ancestry to the actual trap point used by a candidate. Does not by itself prove that pointwise-primitive base traps cannot cover the corrected parameter domain, universal DSC-P, López-all-primes, or Erdős-Straus.

Setup

Fix a candidate progression

x(s)=r+Ls.

Let j be an earlier layer with

m_j=4j-1

and

q_j=\frac{m_j}{\gcd(L,m_j)}=3.

Put

\boxed{n_j=m_j/3.}

Then

\boxed{n_j\mid L.}

A class a mod 3 belongs to R_j precisely when there is a trap residue u in T_j such that

r+La\equiv u\pmod{m_j}.

In particular

\boxed{u\equiv r\pmod{n_j}.}

Theorem (pointwise q=3 absorption)

Assume a in R_j, witnessed by u in T_j.

If there is any earlier layer i<j satisfying

\boxed{m_i\mid n_j}

and

\boxed{u\bmod m_i\in T_i,}

then the entire candidate progression is directly shadowed by layer i.

Proof

Since

m_i\mid n_j\mid L,

the progression is frozen modulo m_i:

x(s)\equiv r\pmod{m_i} \qquad\text{for all }s.

The witness relation gives

u\equiv r\pmod{n_j},

hence also

u\equiv r\pmod{m_i}.

By hypothesis u mod m_i in T_i. Therefore

r\bmod m_i\in T_i.

Since every point of the progression has that same residue modulo m_i, every point lies in the earlier Type A/B layer i. Thus the candidate is directly shadowed. QED.

Definition — pointwise-primitive q=3 trap

For a q=3 layer j, call a trap residue u in T_j pointwise primitive if

\boxed{ \forall i<j\text{ with }m_i\mid m_j/3, \quad u\bmod m_i\notin T_i. }

This is a property of the actual trap point, not of the whole layer.

Corollary

On a directly novel candidate, every trap witness responsible for every nonempty q=3 pullback must be pointwise primitive.

Equivalently:

\boxed{ \text{non-primitive q=3 trap witness} \Longrightarrow \text{direct shadow}. }

Relation to strong absorption

Strong absorption assumes one ancestor i satisfies

T_j\bmod m_i\subseteq T_i.

That makes every trap of j non-primitive and therefore kills the entire layer whenever R_j is nonempty.

Pointwise absorption is strictly finer: even a base layer with no whole-set reducing ancestor can be dead for a particular candidate if the specific trap point selected by that candidate reduces into an earlier frozen trap.

Thus the true residual q=3 threat is smaller than the base-layer population:

\boxed{ \text{base layers} \supseteq \text{base layers with primitive traps} \supseteq \text{primitive traps actually alignable on one candidate}. }

Corrected q=3 cover target

Because 3|840|L, the exact Dirichlet parameter domain at q=3 is all of

\mathbb Z/3\mathbb Z.

After strong absorption, weak redundancy, and pointwise absorption, a directly novel q=3 obstruction would require pointwise-primitive trap witnesses whose pullbacks jointly cover

\boxed{\{0,1,2\}.}

This is the next exact search/theorem target.