Square-lift core after quadratic character saturation

Geometry

Read with:

Status: proved structural reduction and active proof direction

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Claim boundary: this note does not prove Direct-Shadow Completeness, universal López Type A/B coverage, or the Erdős-Straus conjecture. It isolates the exact kind of arithmetic freedom that remains after the quadratic character obstruction has been compressed.

Read with:

1. Setup

Fix a target Type A/B candidate at depth k and put

M=4k-1, \qquad L=\operatorname{lcm}(840,M).

For an earlier layer j<k, write

m_j=4j-1.

Let sigma(n) be the squareclass vector of an odd integer:

\sigma(n)=(v_p(n)\bmod2)_p.

Let F_k be the squareclass coordinate subspace generated by the odd primes dividing L.

An earlier layer is character-fixed when

\sigma(m_j)\in F_k.

These are precisely the rows whose Jacobi sign is completely determined by the target progression at the squareclass level.

2. Square-lift theorem

Theorem

\sigma(m_j)\in F_k,

then every prime p not dividing L occurs in m_j with even exponent:

\boxed{ p\nmid L \Longrightarrow v_p(m_j)\equiv0\pmod2. }

Consequently there exist positive integers a_j,b_j such that

\boxed{ m_j=a_j b_j^2,}

with

\operatorname{rad}(a_j)\mid\operatorname{rad}(L)

and

\gcd(b_j,L)=1.

Proof

The statement sigma(m_j) in F_k means exactly that every prime coordinate outside the support of L has coefficient zero in the squareclass vector. Therefore, for every p not dividing L,

v_p(m_j)\equiv0\pmod2.

Collect all prime powers outside L into

b_j=\prod_{p\nmid L}p^{v_p(m_j)/2}.

Then b_j^2 contains the complete part of m_j supported outside L. The quotient

a_j=m_j/b_j^2

has prime support entirely inside L. QED.

3. Pullback form

The parameter pullback for the candidate uses

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

The same parity statement survives this reduction for every prime outside L, because gcd(L,m_j) contains no factor of such a prime.

Thus

\boxed{ p\nmid L \Longrightarrow v_p(q_j)\text{ is even}.}

Equivalently, q_j admits a canonical decomposition

\boxed{ q_j=c_j s_j^2, }

where every prime factor of c_j divides L and

\gcd(s_j,L)=1.

The factor s_j^2 is the complete part of the pullback modulus supported on new primes outside the target modulus.

4. Interpretation

The character-shield completeness theorem says that collective quadratic inconsistency introduces no obstruction beyond character-fixed negative rows.

The square-lift theorem now says something stronger about those rows:

any prime coordinate that is still genuinely new after the target modulus is fixed can enter a character-fixed obstruction only through an even prime-power exponent.

So once the quadratic information is exhausted, the remaining freedom is not another independent squareclass bit. It is higher-order p-adic lifting inside an already fixed squareclass.

The proof search therefore has a natural filtration:

\boxed{ \text{raw residue system} \to \text{quadratic squareclasses} \to \text{fixed-negative rows} \to \text{square-power lifts} \to \text{exact trap residues}. }

5. Why this matters

This removes a large class of hypothetical obstructions.

A prime p outside L cannot appear to an odd exponent in the unresolved character core. If it did, the Jacobi sign at p would be a free quadratic variable, and the layer would not be character-fixed in the first place.

Therefore the unexplained arithmetic beyond the character shield is necessarily higher-order:

extra p^2,p^4,... information at genuinely new primes;
or additional powers of primes already dividing L.

That suggests replacing a generic covering-system viewpoint with a p-adic lifting viewpoint.

6. Finite k <= 1200 diagnostic

A replay of the frozen k<=1200 candidate bundle gives the following proof-mining signal among the 11,056 candidates not immediately solved by the quadratic character shield.

The number of character-fixed negative rows whose modulus is not already completely fixed by L has:

median: 1
mean:   about 2.274
maximum: 24

For those variable fixed-negative rows, the observed pullback moduli q_j were confined to

3, 5, 7, 9, 11, 13, 15, 25, 27, 49, 81, 121, 169, 289, 361

in the finite range.

More importantly, the genuinely new primes p not dividing L occurred only through the square factors

11^2, 13^2, 17^2, 19^2

in that replay, exactly as the theorem requires.

These particular finite lists are not universal claims. Their purpose is to show how sharply the exact residual problem has already compressed in the certified range.

7. Next theorem target: square-lift avoidance

The immediate question is now:

Given a directly novel candidate, can the remaining fixed-negative square-lift rows always be avoided by choosing the higher p-adic digits without destroying the character shield on the other rows?

A positive theorem would replace a large global congruence cover by a small collection of local lifting problems.

The desired architecture is

\boxed{ \text{direct novelty} \to \text{character saturation} \to \text{square-lift core} \to \text{local p-adic escape} \to \text{reduced avoiding progression}. }

That is now one of the highest-value proof routes toward DSC-P.