Class-C residual core: coordinated primary formulation

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Status: primary synthesis of proved reductions and Operator-02 diamond intake

Coordinator: primary research lead / Operator-01

Parallel input: Operator-02

Date: 2026-08-15

Claim boundary: universal DSC-P, universal López Type A/B coverage, and the Erdős-Straus conjecture remain open.

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1. Target candidate

Fix a directly novel Type A/B candidate at target depth k:

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

For each earlier layer j<k, write

m_j=4j-1, \qquad q_j=\frac{m_j}{\gcd(L,m_j)}.

Direct novelty means no single earlier layer already covers the entire target progression.

2. Active fixed-negative core

Following the Operator-02 split, define

\mathcal N_{k,r} = \{j<k:\text{Jacobi sign is fixed by the target progression and equals }-1\}

and

\boxed{ \mathcal N^{\rm act}_{k,r} = \{j\in\mathcal N_{k,r}:q_j>1\}. }

If q_j=1, the target progression is fixed modulo m_j; direct novelty already certifies that its fixed residue is not in T_j. Such a layer can defeat the scalar character shield without imposing any free s-constraint.

Therefore only N^act from the fixed-negative core can participate in the residual parameter-covering problem.

The exact arithmetic criterion is

\boxed{ q_j>1 \iff \exists p:\ v_p(m_j)>v_p(L). }

For a character-fixed row, a prime absent from L can witness this only through even valuation. This matches the primary square-lift theorem.

3. Residual fiber kernel

Apply the exact fiber-peeling theorem to the pulled-back forbidden sets.

If every coordinate peels, a reduced avoiding class is constructed and the candidate is solved.

Class C begins only when a nonempty residual coordinate kernel remains.

In every completed finite research range this kernel lies in a small explicit prime universe, but no universal finite prime bound independent of k is claimed.

4. Resolution hierarchy for an active row

For each active earlier layer, the unsafe region can be viewed at four resolutions:

\boxed{ \begin{array}{c} \text{Jacobi-negative half}\\ \supseteq\\ \text{local quadratic trap-signature coset}\\ \supseteq\\ \text{multiplicative trap coset }-D_j\\ \supseteq\\ \text{exact Type A/B trap set }T_j. \end{array}}

Here

D_j =\langle p\bmod m_j:p\mid j,\ p\text{ prime}\rangle.

The exact trap set inside -D_j is the image of two exponent boxes.

Thus a failure at any coarse shield is not a counterexample. It only means the next finer resolution must be used.

5. Squarefree-lift localization

For a character-fixed active row, write

m_j=d s_j^2, \qquad d=4a-1\text{ squarefree}.

Then a is the squarefree ancestor.

Every rational prime dividing j splits in

\mathbb Q(\sqrt{-d}),

and the projected multiplicative information lives in

K_a = \ker((\cdot/d)),

the Jacobi-positive subgroup modulo d.

Let

D_a =\langle p\bmod d:p\mid a\rangle.

The full multiplicative defect quotient is

\boxed{ \mathcal M_a=K_a/D_a. }

6. Full multiplicative defect conservation

j=\prod_q q^{e_q},

every divisor prime q has a defect class

\delta_a(q)=[q]\in\mathcal M_a.

The norm-form identity implies

\boxed{ \prod_{q\mid j} \delta_a(q)^{e_q}=1 \quad\text{in }\mathcal M_a. }

So an active Class-C row is not generated by arbitrary independent prime residues. Its higher multiplicative defects are globally constrained by a finite-abelian zero-product law.

7. Zero-product atom reduction

The defect sequence decomposes into minimal zero-product atoms.

Every atom has length at most the Davenport constant

\boxed{D(\mathcal M_a)}.

Consequently, for each fixed ancestor, an arbitrarily large lifted factorization decomposes into bounded-complexity multiplicative packets.

Each neutral atom corresponds to a divisor e|j satisfying

e\bmod d\in D_a.

The remaining exact question is whether the corresponding projected residue falls inside the ancestor's two-box trap set, or whether some still-earlier layer removes it.

This is the current bridge between group-theoretic conservation and exact shadowing.

8. Coordinated Class-C definition

A directly novel candidate belongs to the coordinated Class-C residual core when all of the following hold:

exact fiber peeling leaves a nonempty residual prime-power kernel;
the active fixed-negative core N^act is nonempty;
the coarse character/signature shields do not already produce a reduced avoiding class;
at least one active constraint remains after multiplicative-coset and squarefree-ancestor reductions.

The remaining problem is:

\boxed{ \text{find residual coordinates }s \text{ that avoid every exact }T_j, \quad j\in\mathcal N^{\rm act}_{k,r}, }

while satisfying reducedness.

The important point is that each remaining row now arrives with a finite algebraic certificate package:

ancestor a and squarefree modulus d;
residual prime-power valuations;
local signature quotient;
multiplicative defect group M_a;
zero-product atom decomposition;
exact ancestor two-box target.

9. What would prove universal DSC-P on this route

It suffices to prove a local escape theorem of the following shape:

For every directly novel coordinated Class-C system, the finite collection of active rows, after decomposition into multiplicative defect atoms, admits compatible residual prime-power choices avoiding every exact Type A/B two-box trap while preserving reducedness.

Potential stepping stones:

prove the result whenever |N^act|=1;
prove it for each of the smallest residual kernel signatures;
prove it whenever every active ancestor has trivial multiplicative defect quotient;
prove it for cyclic M_a using zero-sum atom classification;
classify all minimal atom types that fail ancestor two-box membership and show each is shadowed by another earlier layer.

The fifth item is currently the most direct bridge to exact Direct-Shadow Completeness.

10. Operator-02 contribution boundary

Operator-02's contribution here is the identification and naming of the active fixed-negative split, valuation-excess classification, finite residual-support envelope, and Class-C formulation.

The primary lane supplies the later multiplicative quotient, squarefree-lift, norm-form, defect conservation, zero-product atom, and exact two-box refinements.

This document deliberately records both provenance lines rather than collapsing them into one anonymous derivation.