Full-stabilizer compression of FAB Kneser defects

Corridor

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Status: proved universal strengthening of FAB-KNESER-DIVISOR-DEFECT.md

Date: 2026-08-15

Depends on: FAB-KNESER-DIVISOR-DEFECT.md

Claim boundary: compresses every failed fixed-k FAB product box once its full stabilizer is taken. It does not prove that some auxiliary k must always succeed and therefore does not prove Erdős-Straus.

1. Setup

Let

C=\frac{p+k}{4}=\prod_i r_i^{e_i}

and let

R=\prod_i \{r_i^{-e_i},\ldots,r_i^{-1},1,r_i,\ldots,r_i^{e_i}\} \subseteq G_k=(\mathbb Z/k\mathbb Z)^\times

be the fixed-k signed divisor box.

Assume its exact FAB target is missed.

Let

\boxed{H=\operatorname{Stab}(R)}

be the full stabilizer, and put

\bar G=G_k/H, \qquad n=|\bar G|=[G_k:H].

Write

\bar R=R/H.

By construction,

\boxed{\operatorname{Stab}_{\bar G}(\bar R)=\{1\}.}

For each prime factor define

d_i=\operatorname{ord}_{\bar G}(r_iH).

2. No local factor may fill its projected subgroup

Suppose r_i notin H, so d_i>1.

Its local signed set in the quotient is

\bar A_i =\{(r_iH)^z:-e_i\le z\le e_i\}.

2e_i+1\ge d_i,

then the consecutive exponent interval contains representatives of every residue modulo d_i. Therefore

\boxed{\bar A_i=\langle r_iH\rangle.}

But then

\bar R =\bar A_i\prod_{j\ne i}\bar A_j

is invariant under multiplication by the nontrivial subgroup

\langle r_iH\rangle.

That contradicts the trivial stabilizer of \bar R.

Hence:

Theorem — projected-order gap

For every prime-power factor outside the full stabilizer,

\boxed{ \operatorname{ord}_{G_k/H}(r_iH) >2e_i+1. }

Equivalently,

\boxed{d_i\ge2e_i+2.}

This conclusion uses the exact stabilizer, not merely the Kneser size inequality.

3. Universal exceptional-valuation bound

FAB-KNESER-DIVISOR-DEFECT.md proves the Kneser budget

\sum_i \left( \min(2e_i+1,d_i)-1 \right) \le n-2.

For every r_i notin H, the projected-order theorem gives

2e_i+1<d_i,

so its contribution is exactly

2e_i.

For every r_i in H, the contribution is zero.

Therefore the entire Kneser budget collapses to

\boxed{ 2\sum_{r_i\notin H}e_i \le n-2. }

Hence:

Theorem — full-stabilizer defect mass

Every failed fixed-k FAB signed divisor box satisfies

\boxed{ \sum_{ r^e\parallel C, r\notin H }e \le \left\lfloor\frac{[G_k:H]-2}{2}\right\rfloor. }

Thus the total prime-factor valuation visible outside the stabilizer is universally bounded by half the quotient size.

No primality assumption on the quotient index is needed.

4. Interpretation

Write

C=C_H\,C_{\mathrm{exc}},

where C_H contains all prime powers with residue class in H, and C_exc contains the rest.

Then every failed exact placement has two simultaneous properties:

Large hidden background

All prime-power factors of C_H are invisible in the quotient defect.

Tiny visible defect

The total valuation

\Omega(C_{\mathrm{exc}}) =\sum_{r^e\parallel C_{\mathrm{exc}}}e

satisfies

\boxed{ \Omega(C_{\mathrm{exc}}) \le\left\lfloor\frac{n-2}{2}\right\rfloor. }

Moreover every exceptional prime has projected order strictly larger than twice its exponent plus one.

So a target miss is possible only when almost all shifted-factor mass collapses into one stabilizer subgroup and the visible quotient is supported by a short list of high-order atoms.

5. Recovery of earlier classifications

Index 3

The mass bound gives

\Omega(C_{\mathrm{exc}})\le0,

so every factor lies in H.

Index 5

\Omega(C_{\mathrm{exc}})\le1.

Thus there is at most one simple exceptional factor.

Index 6

\Omega(C_{\mathrm{exc}})\le2.

The projected-order gap excludes quotient orders 2 and 3 for every exceptional factor. Only order 6 remains. The external-nonresidue parity then reduces the possibilities further to the unique-simple-defect theorem in FAB-KNESER-INDEX6-CLASSIFICATION.md.

Index 10

\boxed{\Omega(C_{\mathrm{exc}})\le4.}

Every exceptional factor must have quotient order exceeding 2e+1. In the cyclic order-ten quotient this immediately excludes:

all order-two projections;
order-five projections of exponent at least two;
order-ten projections of exponent at least five.

This sharply reduces the next mixed classification before any case analysis begins.

6. Contrapositive expansion criterion

For any candidate subgroup H<=G_k, if the prime factorization of C=(p+k)/4 has too much valuation outside H, namely

\boxed{ 2\sum_{r^e\parallel C,\ r\notin H}e >[G_k:H]-2, }

then H cannot be the full stabilizer of a failed divisor box.

Likewise, if any factor outside H has

\operatorname{ord}_{G_k/H}(rH)\le2e+1,

then H cannot be the full defect stabilizer.

Thus a proof can eliminate candidate defect subgroups using only:

quotient orders of the actual shifted prime factors;
their valuations.

No exhaustive enumeration of all signed divisors is required.

7. Entropy-or-descent formulation

At an external-nonresidue factor-cycle vertex, the same shifted integer carries both:

a guaranteed external nonresidue factor for the descent edge;
a hypothetical Kneser defect subgroup if exact placement fails.

The full-stabilizer theorem says that the failure subgroup must hide almost all factor valuation, leaving only a bounded collection of high-order visible atoms.

Therefore the universal program can be stated more sharply as:

\boxed{ \text{either shifted-factor residue entropy escapes every proper stabilizer,} \quad\text{or the factor cycle transports a tiny high-order defect.} }

The remaining all-prime task is to prove that such a tiny high-order defect cannot persist around every external-nonresidue cycle.