C1 Partial Theorems Toward DSC-P — corrected

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Status: proved local C1 theorems + explicit remaining global obstruction

Date: 2026-08-15

Claim boundary: does not prove Erdős-Straus, López Type A/B coverage, universal DSC-P, or full C1. It does prove that the unique active fixed-negative row is never by itself a reduced covering obstruction.

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1. Correction to the previous draft

An earlier version of this file labeled the following as a theorem:

if q=p^a and R is a proper subset of Z/qZ, then a reduced class exists outside R.

That statement is false in general. A proper forbidden set can contain every reduced class while omitting only non-reduced classes.

The previous text itself noticed this possibility, so the theorem label was inconsistent with its own proof.

The false general statement is withdrawn.

A second correction: at one exact Type A/B layer, the compatible-trap pullback does not multiply one trap into several parameter classes. The affine pullback map is injective on the compatible trap fiber.

Both points are now handled exactly in SINGLE-ACTIVE-REDUCED-ESCAPE-THEOREM.md.

2. C1 setup

Let a directly novel target candidate have

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

Assume

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

Let j0 be the unique active fixed-negative layer and put

m=4j_0-1, \qquad g=\gcd(L,m), \qquad q=m/g.

Let

R\subseteq\mathbb Z/q\mathbb Z

be its exact Type A/B forbidden pullback.

Direct novelty gives

R\ne\mathbb Z/q\mathbb Z.

3. Prime-power shape theorem

The previously proved unique-active valuation theorem gives

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

for one odd prime p.

If p∤L (Operator-02 Class B), then fixed-squareclass parity forces

\boxed{q=p^2.}

Since 840|L, a Class-B prime satisfies

\boxed{p\ge11.}

4. Exact pullback injection

Let

U=\{u\in T_{j_0}:u\equiv r\pmod g\}.

The map

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

is injective on U.

Therefore

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

This is an exact fact, not a heuristic fiber-size assumption.

5. Class-A local theorem

p\mid L,

then reducedness at p does not depend on the parameter:

r+Ls\equiv r\pmod p.

Because gcd(r,L)=1, every parameter class is reduced at p.

Since R is proper, choose any

s_0\notin R.

Then the unique active row is avoided exactly and reducedness holds automatically.

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

6. Class-B local theorem

Assume

p\nmid L.

Then

q=p^2, \qquad m=g p^2=4j_0-1.

Because p is odd,

g\equiv3\pmod4

and

j_0=\frac{g p^2+1}{4}.

For either trap family

-e \quad\text{or}\quad -4e, \qquad e\mid j_0,

compatibility modulo g forces e into one fixed residue class modulo g.

Any residue class modulo g contains at most

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

integers in the interval 1<=e<=j_0.

Therefore the two trap families together give

\boxed{|R|\le\frac{p^2+3}{2}.}

Since p∤L, non-reduced parameters form exactly one class modulo p, so the number of reduced parameter classes modulo p^2 is

\boxed{p^2-p.}

For p>=5,

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

Class B has p>=11, so the forbidden set is strictly too small to contain every reduced class.

Hence every Class-B unique active row also has a reduced local escape.

7. Universal local C1 theorem

Combining the two cases:

\boxed{

|\mathcal N^{\rm act}_{k,r}|=1 \Longrightarrow \text{the unique active fixed-negative row admits a reduced exact local escape.} }</div>

This is now proved universally.

The former local “two-box pullback gap” is therefore closed in the single-active regime.

8. Why full C1 is still open

The k<=1500 independently verified census showed:

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

So after the local active row is understood, the actual residual system is dominated by nonfixed exact rows.

The remaining theorem is not

can the active row leave one reduced class?

That is solved.

The remaining theorem is:

\boxed{ \text{Can one choose a parameter that simultaneously} \text{ realizes the guaranteed active-row escape and avoids} \text{ every surviving nonfixed exact row?} }

This is an interaction/coordination problem among the residual rows.

9. Correct C1 target

A sufficient theorem would be:

C1 coordination theorem candidate

For every directly novel candidate with |N^act|=1, after exact fiber peeling of the nonfixed rows, at least one reduced assignment survives that is compatible with the guaranteed local active-row escape.

If proved, this would establish C1 and give infinitely many exact-depth primes for every single-active directly novel candidate by Dirichlet.

It still would not prove universal DSC-P, since |N^act|>1 cases would remain.

10. Finite evidence

Through k<=1500, every one of the 2,770 single-active candidates is globally resolved.

Among the 1,480 nonempty residual kernels, the fixed selector menu

\{0,\pm1,\ldots,\pm64\}

found a reduced global escape in

\boxed{1,480/1,480}

cases, with maximum radius 48.

That is strong finite evidence for the coordination theorem, not a proof.

11. Next proof actions

classify the nonfixed residual rows that survive in C1;
prove why the unique active row is peeled in 1,462/1,480 nonempty finite kernels;
solve the two smallest residual signature cases {11,13} exactly;
solve the recurring {3,11,13} signature structurally;
search for a local product/character invariant forcing overlap among the nonfixed rows;
only then promote from C1 to bounded |N^act|>1.

12. Wall statement

Erdős-Straus remains open. López universal Type A/B coverage remains open. Universal DSC-P remains open. Full C1 remains open.

The local active-row obstruction is no longer open.