C1: single-active Class-C attack — Free Computation Foundation

Shadow

Fix a directly novel Type A/B candidate

Status: coordinated theorem target and falsification design

Date: 2026-08-14

Coordinator: primary research coordinator

Partner input: Operator-02 DIAMOND-CLASS-C-NODE.md, DIAMOND-FIXED-NEGATIVE-PULLBACK-SPLIT.md, DIAMOND-VALUATION-CRITERION.md

Claim boundary: this file defines an attack. It does not prove universal Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture.

1. Target

Fix a directly novel Type A/B candidate

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

For each earlier layer j<k, put

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

Let

\mathcal N_{k,r} =\{j<k:\sigma(m_j)\in F_k,\ (r/m_j)=-1\}

be the fixed-negative character core, and define Operator-02's active subcore

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

The first coordinated special case is

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

Call the unique active fixed-negative layer j0.

2. Why C1 is the correct first target

Character-shield completeness says the free quadratic signs can make every earlier layer outside the fixed-negative core Jacobi-positive. Fixed-negative layers with q_j=1 are parameter-inactive and direct novelty already guarantees that their locked residue is not an exact Type A/B trap.

Thus when the active core has one member, all remaining exact difficulty is concentrated at one moving fixed-negative layer, together with reducedness and any finer local compatibility required to realize the chosen character assignment.

The central question is therefore:

Does direct novelty at the unique active layer always leave a residue choice that is compatible with the global character shield and reducedness?

A proof would establish a genuine infinite special case of DSC-P rather than a finite-range statistic.

3. Valuation source split

For the unique active layer,

q_{j_0}>1 \iff \exists p:\ v_p(m_{j_0})>v_p(L).

The excess primes are classified as:

Class A: p|L with a higher power in m_j0 than in L;
Class B: p∤L occurring to even valuation in m_j0, hence invisible to the squareclass/Jacobi row but still active on the parameter line.

The C1 census must separate these two mechanisms.

4. Exact finite census to run first

On the already verified k<=1500 candidate bundle, the primary analyzer will compute for every directly novel candidate:

|N| and |N_act|;
the unique active layer when |N_act|=1;
q_j0 and its prime-power factorization;
Class A and Class B valuation witnesses;
the exact pulled-back forbidden set R_j0;
whether the candidate's fiber kernel is empty;
if nonempty, its residual prime signature;
whether all residual kernel primes are explained by Class A/B valuation witnesses from N_act;
bounded-selector radius from the independently generated selector construction where available;
the exact number of safe residues modulo q_j0 before and after reducedness.

The output is a census, not a theorem.

5. Falsifiers

The C1 theorem route is weakened or falsified in its proposed form if the census finds any of the following:

a single-active candidate whose post-character exact constraints require an additional active fixed-negative layer not captured by N_act;
a residual kernel coordinate with no valuation-source explanation from the active layer and no separately identified non-fixed-negative source;
a unique active layer whose exact safe set is nonempty by direct novelty but every safe choice conflicts with the necessary character/reducedness conditions;
evidence that the local compatibility problem cannot be represented on the active layer's valuation-excess coordinates.

None of these would falsify DSC-P itself. They would falsify this proof route.

6. Candidate proof route

For |N_act|=1, attempt to show:

all other earlier layers can be made Jacobi-positive by the proved character-shield extension;
inactive fixed-negative layers are exact-safe by direct novelty;
the unique active layer has a proper forbidden pullback set R_j0 because direct shadow is absent;
the remaining character choices restrict only signs or higher-power lifts on the prime-power coordinates of q_j0;
the two-box / multiplicative trap structure prevents those compatible choices from exhausting the complement of R_j0;
choose one reduced local residue and reverse the fiber/CRT construction;
Dirichlet gives infinitely many exact-depth primes.

The high-value lemma is Step 5.

7. Relationship to Operator-02

Operator-02 is asked to attack C1 independently from the formulation side:

classify possible Class A/B valuation patterns when |N_act|=1;
search for a direct local lemma forcing at least one safe residue;
identify any hidden source of residual constraints omitted by the primary formulation;
adversarially test any Coordinator proof attempt.

The Coordinator owns the canonical census, proof promotion, workflow integration, and theorem claim boundary.

8. Promotion rule

C1 is promoted from THEOREM-CANDIDATE to PROVED only after a universal proof is written and independently checked. A clean k<=1500 census, even with zero exceptions, remains finite evidence only.