Squarefree-lift localization of the Type A/B character residual

Geometry

Read with:

Status: proved theorem family plus finite replay target

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. It refines the residual left by the character-shield theorems.

Read with:

1. Squarefree ancestor

For an earlier layer j, write

m_j=4j-1.

Let

d_j=\operatorname{sf}(m_j) =\prod_{v_p(m_j)\text{ odd}}p

be the squarefree kernel of m_j. Then there is a unique odd integer s_j>=1 with

\boxed{m_j=d_j s_j^2.}

Because every odd square is 1 mod 4 and m_j=3 mod 4,

d_j\equiv3\pmod4.

Hence

\boxed{a_j=\frac{d_j+1}{4}}

is a positive integer and

\boxed{d_j=4a_j-1=m_{a_j}.}

We call a_j the squarefree ancestor depth of j.

Moreover a_j<=j, with equality exactly when m_j is squarefree.

2. Squarefree character reduction

Theorem

For every integer x coprime to m_j,

\boxed{ \left(\frac{x}{m_j}\right) = \left(\frac{x}{d_j}\right). }

Proof

Write

m_j=\prod_p p^{e_p}.

The Jacobi symbol is

\left(\frac{x}{m_j}\right) = \prod_p\left(\frac{x}{p}\right)^{e_p}.

Every even exponent contributes 1; the odd exponents are exactly the primes occurring in d_j. QED.

Thus the scalar quadratic character of a layer depends only on its squarefree ancestor modulus.

3. Fixed-negative layers are square-lifts

Fix a target candidate (k,h,t) and write

x\equiv r\pmod L, \qquad L=\operatorname{lcm}(840,m_k).

An earlier layer j<k is character-fixed when every prime occurring to odd exponent in m_j divides L. Equivalently,

\boxed{d_j\mid L.}

For such a layer, the candidate progression fixes x mod d_j, and the previous theorem gives

\boxed{ \left(\frac{x}{m_j}\right) = \left(\frac{r}{d_j}\right) }

for every reduced member of the progression.

Therefore every immutable Jacobi-negative layer in the character residual is literally a square-lift over a smaller fixed modulus d_j=m_{a_j}.

4. Direct novelty at the ancestor

Suppose the target candidate is directly novel.

If d_j|L, then the earlier ancestor modulus m_{a_j}=d_j divides the target progression modulus. Therefore the residue modulo d_j is fixed to r mod d_j.

Direct novelty implies

\boxed{ r\bmod d_j\notin T_{a_j}.}

Otherwise the ancestor layer a_j itself would directly shadow the target candidate.

Hence every fixed-negative layer of a directly novel candidate sits over a residue satisfying

\boxed{ \left(\frac r{d_j}\right)=-1, \qquad r\bmod d_j\notin T_{a_j}. }

In words: the character residual is built from nontrap quadratic nonresidues at squarefree ancestor layers.

5. Projection excess

The exact trap set at the lifted layer can be projected to the ancestor modulus:

\pi_j(T_j)=T_j\bmod d_j.

Define the squarefree projection excess

\boxed{ E_j = \pi_j(T_j)\setminus T_{a_j}. }

Projection-excess theorem

Let a directly novel target candidate have d_j|L. If

r\bmod d_j\notin E_j,

then the candidate progression is automatically safe from the entire exact layer j.

Proof

Because the candidate is directly novel,

r\bmod d_j\notin T_{a_j}.

If also r mod d_j is not in E_j, then

r\bmod d_j\notin T_j\bmod d_j.

Every member of the candidate progression has the same residue r mod d_j. Therefore no member can lie in T_j mod m_j, since any such hit would project into T_j mod d_j. QED.

Corollary

\boxed{E_j=\varnothing,}

then every directly novel candidate for which d_j|L automatically avoids layer j exactly.

Equivalently,

T_j\bmod d_j\subseteq T_{a_j}

makes the lifted layer redundant over every directly novel progression that fixes its ancestor modulus.

This is an exact shadow theorem along the squarefree-lift ancestry

\boxed{m_j=s_j^2m_{a_j}.}

6. Why this matters

CHARACTER-SHIELD-COMPLETENESS.md proves that scalar-character inconsistency comes only from immutable negative earlier layers.

The present theorem shows that most of those immutable negative layers may still be removed exactly, without solving any global covering problem, by checking only whether their fixed ancestor residue lies in the projection excess E_j.

The hierarchy is now

\boxed{ \text{character shield} \to \text{fixed-negative square-lifts} \to \text{projection excess }E_j \to \text{genuinely active exact lift core}. }

The object E_j is therefore a sharper target than the entire Jacobi-negative half of the unit group.

7. Finite k <= 1200 replay signal

A replay against the frozen 41,470 directly novel candidates through k<=1200 gives the following proof-mining statistics:

layers through 1200:                         1,200
non-squarefree moduli m_j:                    224
layers with nonempty projection excess E_j:   115
layers with empty projection excess:         1,085

Among the 41,470 directly novel candidates:

character-shield inconsistent candidates:    11,056
of those, no fixed-negative layer remains
active after the exact projection-excess test: 7,608
candidates with >=1 active excess layer:       3,448
maximum active excess layers in one candidate:    10

Across all candidates, 38,022/41,470 have zero active squarefree projection-excess layers.

These numbers do not mean all 38,022 candidates are independently solved by this theorem alone. Collective character equations among non-fixed layers can still require additional choices. The result is a localization statement: the immutable exact-residue part of the character residual collapses to a much smaller family of projection-excess lifts.

The largest observed excess size through k<=1200 is 22, at

j = 700
m_j = 2799
squarefree ancestor modulus d_j = 311
ancestor depth a_j = 78

8. Immediate theorem target

The next target is to classify exactly when

\boxed{E_j=\varnothing.}

Because

m_j=s_j^2m_{a_j},

this asks for a divisor-theoretic criterion under which

T_j\bmod m_{a_j}\subseteq T_{a_j}.

A successful classification would produce an infinite family of exact square-lift shadow relations and isolate the exceptional lifts where new residue information is genuinely created.

A second target is to combine the projection-excess core with the full local quadratic-signature quotient. The likely proof architecture is

\boxed{ \text{direct novelty} \to \text{signature shield} \to \text{squarefree projection excess} \to \text{fiber kernel} \to \text{tiny exact residual}. }

9. Novelty boundary

Squarefree kernels and the identity of Jacobi symbols under deletion of even prime exponents are classical. The candidate contribution is their use as an exact squarefree-lift localization and projection-excess reduction inside the Type A/B minimal-depth/shadow framework.

Publication priority remains subject to broader literature review and independent mathematical scrutiny.