Single-active first-shell collapse — retracted proof attempt

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Status: REVISE / RETRACTED PROOF

Date: 2026-08-14

Project: Free Computation Foundation / CENTL

Coordinator: Operator-01 / primary research lead

Claim boundary: the former proof in this file is invalid. The q in {3,5,9} hard-class collapse remains an independently verified finite theorem-certificate through k<=100000, not a universal theorem. The earlier universal theorem |N^act|=1 => q=p or p^2 remains unaffected.

Read with:

SINGLE-ACTIVE-EXCESS-PRIME-POWER.md — still proved;
SINGLE-ACTIVE-HARD-COLLAPSE-K100000.md — verified finite result;
CLASS-C-CENSUS-K1500.md;
operator-02/DIRECTIVE-FIRST-SHELL-REVIEW.md.

1. Retraction

An earlier version of this file claimed the universal implication

|\mathcal N^{\rm act}_{k,r}|=1 \Longrightarrow s\in\{3,5\} \Longrightarrow q\in\{3,5,9,25\},

followed by a hard-class elimination of q=25.

That proof used a false intermediate assertion.

The assertion was:

if m=d s^2 is a fixed-negative squareclass layer, then every earlier layer d u^2 in the same squarefree tower has the same negative Jacobi sign.

The correct statement is

\left(\frac r{d u^2}\right) = \left(\frac r d\right)

only when

\gcd(r,u)=1.

If a prime dividing u also divides r, then the Jacobi symbol is 0, not -1.

The invalid proof compared the active shell with unrelated odd values such as N, N-2, 3, 5, and 7 without first proving those values are coprime to r. Therefore uniqueness of the active fixed-negative core does not imply that all those comparison shells are inactive. Some may simply fail to be fixed-negative because their Jacobi symbol is zero.

The product/lcm interval contradictions built from those unrelated shells are therefore unsupported.

2. What remains proved

The theorem in SINGLE-ACTIVE-EXCESS-PRIME-POWER.md survives this correction.

Its argument only replaces the actual active square parameter s by a divisor such as s/p.

Because the original layer has Jacobi sign -1,

\gcd(r,s)=1.

Hence every divisor of s is also coprime to r, and removing an actual factor preserves the negative squareclass sign.

Thus the universal statement remains:

\boxed{

|\mathcal N^{\rm act}_{k,r}|=1 \Longrightarrow q=p\text{ or }p^2 }</div>

for one prime p.

The Class-B corollary also remains:

\boxed{ \text{Class B single-active} \Longrightarrow q=p^2,\quad p\nmid L. }

3. What remains independently verified finite evidence

The two-construction GitHub workflow through

k\le100000

examined

8,021,288

hard-compatible Type A/B target candidates and found

419,123

single-active candidates, with exact quotient distribution

q=3: 252,832
q=5:   4,173
q=9: 162,118
other:      0
Class B:    0

The independent verifier had zero mismatched fields.

Workflow provenance:

run:      31854964168
artifact: 9238743256
artifact sha256:
f390c20afe0c8fc97d9046c34117f4e0b2c8e56f255d6a31c732b337d16d2159

Therefore the sharp statement

\boxed{ \text{hard-compatible }|N^{act}|=1 \Longrightarrow q\in\{3,5,9\}\text{ and Class A} }

is a strong theorem candidate with exact finite verification through k<=100000, not a proved theorem.

4. Correct proof target

The missing ingredient must use information that distinguishes a hard-compatible Type A/B target residue from an arbitrary reduced residue.

For a target layer

M=4k-1,

the candidate residue is not arbitrary. It satisfies

r\equiv -e\quad\text{or}\quad -4e\pmod M

for some

e\mid k.

At the same time, an active fixed-negative layer

m=d s^2

satisfies

\left(\frac r d\right)=-1, \qquad \gcd(r,s)=1, \qquad q=m/\gcd(L,m)=p\text{ or }p^2.

The new theorem search must exploit the target trap residue together with quadratic reciprocity / divisor arithmetic, rather than assuming arbitrary shells in the same squarefree tower are negative.

5. Revised attack directions

A. Target-trap reciprocity

Substitute

r\equiv-e\text{ or }-4e\pmod M

into the fixed-negative condition modulo the squarefree kernel d.

Because

e\mid k, \qquad 4k\equiv1\pmod d

whenever d|M, divisor residues may force local quadratic constraints on the excess prime p.

The key question is whether the condition

(r/d)=-1

can coexist with a first active shell at a prime p>=7 or a Class-B square once the target residue is a Type A/B trap.

B. Actual-divisor shell reductions only

All universally valid square-tower reductions should use divisors of the actual active shell s, because those preserve coprimality with r automatically.

No comparison with an unrelated u may be called fixed-negative without proving gcd(r,u)=1.

C. Explain the observed absence of Class B

For Class B,

q=p^2, \qquad p\nmid L.

Since p is absent from both 840 and the target modulus M, it is a genuinely new square coordinate in the earlier fixed-negative layer.

The finite data suggest Type A/B target compatibility forbids such a coordinate in a unique active core. That is now a clean separate theorem target.

D. Preserve the finite falsifier

The k<=100000 double-construction test remains valuable and should be extended only as a regression/falsifier, not used as a substitute for proof.

6. Scientific correction protocol

This correction was made immediately upon finding the gap, before publication-grade promotion.

The invalid proof remains recoverable in Git history at commit

a7a0ccb17d59c8797e6c6f0b315ccc08db66ba87

but must not be cited as a theorem.

The repository is the canonical record of both the proposed proof and its retraction.

7. Current status table

|N^act|=1 => q=p or p^2:                    PROVED
Class-B single-active => q=p^2, p not in L: PROVED
hard single-active q in {3,5,9}:            FINITE-CERTIFIED through k<=100000
hard single-active Class-A only:             FINITE-CERTIFIED through k<=100000
former first-shell proof:                    RETRACTED / INVALID
universal hard-collapse theorem:             OPEN

This is the active proof frontier.