Class-C active-core census through k = 1500 — Free Computation Foundation

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Status: exact finite theorem-certificate result

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Coordinator: Operator-01 / primary research lead

Partner framework: Operator-02 active fixed-negative core and valuation-source split

Claim boundary: this is an exact finite-range result. It does not prove universal Direct-Shadow Completeness, López Type A/B coverage, or the Erdős-Straus conjecture.

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1. Provenance

The census replays the already frozen candidate bundle from the completed direct-shadow workflow:

source workflow run:  31849103304
source artifact id:   9238241616
source candidate set: 53,240 directly novel candidates through k<=1500

The census itself ran in a separate replay-only workflow:

census workflow run: 31854324273
workflow head:       d330e20082297d21f3005f5a173eaebfdc40ea9b
census artifact id:  9238613961
artifact sha256:
9ba6c7425356dac272821a71b677811ed697c3dd062040cf78302b5f272031ba

The artifact contains:

class-c-census.json
class-c-census-report.md
class-c-census-independent-verifier.json
provenance.txt
SHA256SUMS

Its internal SHA-256 manifest was checked successfully.

2. Independent verification

The independent verifier reconstructs the fixed-negative core, active subcore, valuation sources, exact pullbacks, residual fiber kernels, and bounded selectors with a different peeling control flow: local fiber loads are recomputed from scratch after every peel rather than maintained incrementally.

It returned:

{
  "direct_novel_candidates_checked": 53240,
  "independent_control_flow": "fiber loads recomputed from scratch after each peel",
  "k_limit": 1500,
  "mismatched_sections": [],
  "single_active_candidates_checked": 2770,
  "verdict": "VERIFIED"
}

Therefore the counts below are independently reproduced finite statements.

3. Active-core census

Among all 53,240 directly novel candidates:

character shield solvable, N empty:             38,658
N^act empty:                                     43,968
N nonempty but N^act empty:                       5,310
exactly one active fixed-negative layer:          2,770

The 5,310 inactive-only cases are especially important conceptually: scalar character shielding can fail even though every fixed-negative row is already parameter-inactive. Direct novelty then certifies exact safety at those locked rows.

4. Single-active population

For all

\boxed{2,770}

candidates with

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

the unique active quotient was:

q = 3: 1,322
q = 5:    34
q = 9: 1,414

No other q occurred.

Every one of the 2,770 cases was Operator-02 Class A only:

Class A only: 2,770
Class B only:     0
mixed A/B:        0

Thus every observed active valuation excess came from a higher power of a prime already dividing the target progression modulus L. No even-powered free-prime source occurred in the single-active population through k<=1500.

This 3,5,9 / Class-A-only collapse is a finite result, not yet a universal theorem.

5. Exact pullback size

For the unique active row j0, the exact forbidden pullback R_j0 was tiny:

|R_j0| = 0: 2,644 candidates
|R_j0| = 1:   126 candidates

The number of locally reduced and exact-safe residues modulo q_j0 was never zero. The minimum was

\boxed{2}.

Distribution:

2 reduced-safe residues:   12 candidates
3 reduced-safe residues: 1310
5 reduced-safe residues:   34
8 reduced-safe residues:  114
9 reduced-safe residues: 1300

This finite behavior is consistent with the universal local lemma in SINGLE-ACTIVE-LOCAL-ESCAPE.md.

6. Fiber-kernel interaction

The 2,770 single-active candidates split as:

fiber kernel empty:      1,290
fiber kernel nonempty:   1,480

Among the 1,480 nonempty residual systems, the unique active fixed-negative row itself survived fiber peeling only

\boxed{18}

times.

It was removed before the final residual kernel in

\boxed{1,462}

cases.

Residual edge-source census:

nonfixed earlier rows:                 69,672
unique active fixed-negative row:          18
other fixed-negative rows:                   0
fixed-positive rows:                         0

This is a decisive organizational result:

\boxed{ \mathcal N^{\mathrm{act}} \text{ and the final fiber residual are different resolutions.} }

The active fixed-negative core identifies where scalar character shielding fails while the parameter still moves. It does not identify the complete exact residual row set after fiber peeling.

7. Residual kernel signatures in C1

Among the 1,480 nonempty single-active systems, the observed residual signatures were:

{3,5,11,13,17,19,23}:           680
{3,11,13}:                       336
{3,11,13,19,23}:                 210
{3,5,11,17,19,23}:               160
{3,11,13,17,19,23}:               54
{3,5,11,13,17,19,23,29,31}:      16
{3,5,13,17,19,23}:                12
{3,5,11,13,19,23}:                 6
{3,11,13,19}:                       4
{11,13}:                             2

The two {11,13} cases are the smallest nontrivial residual systems in this C1 range and are priority exact laboratories.

8. Independent bounded-selector result

Every nonempty C1 residual kernel was solved by the fixed bounded selector menu

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

Result:

\boxed{1,480/1,480}

with maximum required absolute selector

\boxed{48}.

This is finite evidence only. It does not prove a universal bounded-selector theorem.

9. What the census changed

The original Class-C intuition risked treating N^act as the exact residual obstruction.

The census rejects that simplification.

The correct coordinated picture is:

\boxed{ \begin{array}{c} \text{fixed-negative character core }\mathcal N\\ \downarrow\\ \text{parameter-active subcore }\mathcal N^{act}\\ \downarrow\\ \text{exact fiber peeling}\\ \downarrow\\ \text{mostly nonfixed residual rows} \end{array}}

So the next theorem must coordinate two structures:

the valuation/character mechanism controlling N^act;
the nonfixed exact rows that survive into the small-prime fiber kernel.

10. New theorem targets

The verified census gives four immediate targets:

single-active excess theorem: prove as much as possible about why a unique active fixed-negative row has a prime-power excess quotient of very low complexity;
hard-class 3,5,9 collapse: prove or falsify the observed restriction q in {3,5,9} and absence of Class B in the hard-class single-active regime;
two-prime residual theorem: solve the two {11,13} C1 systems structurally rather than by selector lookup;
active-to-nonfixed coordination: identify why the unique active row usually peels away while the remaining nonfixed rows still admit a common local escape.

The first of these already admits a universal prime-power reduction, recorded separately in the next theorem note.