Exact reduced-parameter domain — Free Computation Foundation

Theorem

For an admissible target candidate write

Status: proved elementary theorem

Date: 2026-08-15

Claim boundary: this corrects the local parameter domain used in some C1/C2/CN proof-mining notes. It does not prove universal Type A/B coverage or Erdős-Straus.

Setup

For an admissible target candidate write

x(s)=r+Ls, \qquad L=\operatorname{lcm}(840,4k-1).

Let

Q=\operatorname{lcm}\{q_j:R_j\ne\varnothing\}.

The exact Dirichlet condition needed for a prime progression is

\boxed{\gcd(r+Ls,LQ)=1.}

The parameter itself need not be a unit modulo Q.

Every admissible CRT base satisfies

\boxed{\gcd(r,L)=1.}

because the hard residue is coprime to 840 and every Type A/B target trap is a unit modulo its target modulus.

Theorem

Assume gcd(r,L)=1. For every prime p|Q:

if p|L, then

r+Ls\equiv r\not\equiv0\pmod p

for every s;

if p\nmid L, then L is invertible modulo p, and

p\mid r+Ls \iff s\equiv-rL^{-1}\pmod p.

Therefore

\boxed{ \gcd(r+Ls,LQ)=1 \iff s\not\equiv-rL^{-1}\pmod p \text{ for every }p\mid Q,\ p\nmid L. }

Higher powers of p in Q do not change the gcd condition.

Define

\mathcal D_{r,L}(Q) =\{s\bmod Q:\gcd(r+Ls,LQ)=1\}.

Then

\boxed{

|\mathcal D_{r,L}(Q)| =Q\prod_{\substack{p\mid Q\\p\nmid L}} \left(1-\frac1p\right). }</div>

Important correction

The auxiliary set

(\mathbb Z/Q\mathbb Z)^*

is generally not the exact reduced parameter domain.

In particular, for the hard-class program

3\cdot5\cdot7\mid840\mid L.

Hence any local coordinate supported only on 3,5,7 carries the full residue-ring parameter domain as far as Dirichlet reducedness is concerned.

For q=3 specifically,

\boxed{\mathcal D_{r,L}(3)=\mathbb Z/3\mathbb Z.}

Thus two singleton rows forbidding {1} and {2} do not form a reduced obstruction: s=0 mod3 remains prime-compatible.

Combined with Q3-SINGLETON-PULLBACK.md, a genuine q=3 local cover requires three distinct singleton rows covering

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

The constructive core in DSC-COUNTEREXAMPLE.md shows that such a three-row cover can in fact occur.

Research rule

Future local escape and covering arguments must distinguish:

primes already dividing L, where parameter residues are unrestricted by reducedness;
primes outside L, where exactly one affine class modulo p is excluded.

Do not replace this exact affine condition by gcd(s,Q)=1 unless an equivalence has been proved for the stated special case.