Universal reduced escape for a unique active fixed-negative row

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Status: proved universal theorem

Date: 2026-08-15

Project: Free Computation Foundation / CENTL

Claim boundary: this theorem resolves the unique active fixed-negative row locally. It does not by itself solve the simultaneous nonfixed residual system, universal DSC-P, López Type A/B coverage, or Erdős-Straus.

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1. Setup

Fix a directly novel target candidate

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

Suppose the active fixed-negative core has exactly one layer j.

Put

m=4j-1, \qquad g=\gcd(L,m), \qquad q=m/g.

Let

R\subseteq\mathbb Z/q\mathbb Z

be the exact forbidden pullback of T_j to the parameter s.

Direct novelty gives

\boxed{R\ne\mathbb Z/q\mathbb Z.}

The previously proved unique-active valuation theorem gives

\boxed{q=p\text{ or }p^2}

for one odd prime p.

2. Exact pullback injection

The compatible trap residues are

U=\{u\in T_j:u\equiv r\pmod g\}.

The pullback map is

u\longmapsto \frac{u-r}{g} \left(\frac Lg\right)^{-1} \pmod q.

Lemma 1

This map is injective on U.

Proof

If two compatible trap residues u_1,u_2 have the same pullback class, then

\frac{u_1-u_2}{g}\equiv0\pmod q.

Hence

u_1-u_2\equiv0\pmod{gq}=0\pmod m.

But u_1,u_2 are residues modulo m, so they are equal. QED.

Therefore

\boxed{|R|=|U|.}

No trap residue “multiplies” into several parameter classes at one layer. The pullback is an affine relabeling of the compatible trap fiber.

3. Class A case

Assume

p\mid L.

This is Operator-02 Class A valuation excess.

Since R is proper, choose any

s_0\notin R.

Then the exact Type A/B condition at the unique active row is avoided.

Reducedness at p is automatic because

r+Ls_0\equiv r\pmod p

and

p\mid L, \qquad \gcd(r,L)=1.

Thus

p\nmid r+Ls_0.

So every Class-A unique active row has a reduced exact local escape.

This recovers and extends the earlier Class-A note.

4. Class B shape

Assume now

p\nmid L.

The unique-active prime-power theorem forces

\boxed{q=p^2.}

Because

840\mid L,

we also have

p\notin\{2,3,5,7\}.

All Type A/B moduli are odd, so p is odd and therefore

\boxed{p\ge11.}

We now bound the exact forbidden fiber.

5. Universal Class-B fiber bound

Since

m=g p^2=4j-1,

we have

\boxed{j=\frac{gp^2+1}{4}}.

Because p^2=1 mod 4 and m=3 mod 4,

g\equiv3\pmod4.

Hence

\frac jg = \frac{p^2}{4}+\frac{1}{4g}

lies strictly between

\frac{p^2}{4} \quad\text{and}\quad \frac{p^2}{4}+\frac1{12}.

Since p is odd,

p^2\equiv1\pmod4,

so every fixed residue class modulo g contains at most

\boxed{\frac{p^2+3}{4}}

integers in the interval 1<=e<=j.

Now split the Type A/B trap set into its two divisor families:

T_j=\{-e:e\mid j\}\cup\{-4e:e\mid j\}.

For a trap -e to lie in the compatible fiber u=r mod g, the divisor e must occupy one fixed residue class modulo g.

For a trap -4e, multiplication by 4 is invertible modulo odd g, so e again occupies one fixed residue class modulo g.

Therefore each family contributes at most

\frac{p^2+3}{4}

compatible traps.

By Lemma 1,

Lemma 2

\boxed{

|R| \le \frac{p^2+3}{2}. }</div>

This bound does not use the divisor condition beyond the fact that divisors lie in [1,j]; actual trap fibers are often much smaller.

6. Reduced parameter classes in Class B

Since p∤L, the non-reduced condition is

r+Ls\equiv0\pmod p.

Because L is invertible modulo p, this excludes exactly one residue class modulo p.

Thus among the p^2 parameter classes modulo p^2, the number that remain reduced is exactly

\boxed{p^2-p=p(p-1).}

For every

p\ge5,

\frac{p^2+3}{2}<p^2-p,

because the difference is

\frac{(p-3)(p+1)}2>0.

Class B has p>=11, so the inequality is strict.

Consequently the forbidden set R is too small to contain all reduced parameter classes.

There exists at least one

s_0\pmod{p^2}

such that simultaneously

s_0\notin R

and

p\nmid r+Ls_0.

7. Main theorem

Theorem

For every directly novel candidate with

\boxed{|\mathcal N^{\rm act}_{k,r}|=1,}

the unique active fixed-negative layer admits an exact reduced local escape.

This holds in both valuation regimes:

\boxed{ \begin{array}{ll} \text{Class A:}&\text{direct novelty + fixed reducedness},\\[1mm] \text{Class B:}&q=p^2,\ p\ge11,\ |R|<(p^2-p). \end{array}}

Therefore the unique active fixed-negative row is never by itself a reduced covering obstruction.

QED.

8. What this does and does not prove

This closes the local gap that had previously been phrased as:

could a proper Type A/B pullback still contain every reduced parameter class?

For a unique active fixed-negative row, the answer is now:

\boxed{\text{No.}}

But C1 contains a second layer of difficulty.

The independently verified k<=1500 census showed that after fiber peeling, the residual system is dominated by nonfixed earlier rows:

single-active candidates:                       2,770
nonempty final fiber kernels:                   1,480
unique active row survives final kernel:            18
nonfixed residual edge incidences:             69,672

So the remaining C1 problem is not local escape from the active row.

It is:

\boxed{ \text{coordinate the guaranteed active-row escape} \text{ with all surviving nonfixed exact rows.} }

That is the correct next theorem target.

9. Falsifier

A counterexample to this theorem would require either:

a unique-active quotient not of shape p or p^2, contradicting the earlier theorem;
a Class-A row where reducedness changes with s, contradicting p|L;
a Class-B exact trap fiber with more than (p^2+3)/2 compatible trap residues;
or an arithmetic error in the parameter-fiber injection.

Each condition is explicit and independently testable.

10. Significance

The Class-C program has now separated two logically different phenomena:

\boxed{ \text{fixed-negative activity} \ne \text{final exact residual obstruction}. }

The first is completely locally escapable in the single-active regime.

The unresolved structure therefore lives in the interaction graph of the remaining nonfixed exact rows, not in a mysterious local failure of the unique character-negative row.

This removes one entire candidate obstruction mechanism from C1.