Corridor
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Status: proved universal theorem on the Mordell-hard prime lane
Date: 2026-08-15
Depends on: FAB-UNBOUNDED-DIVISOR-RATIO-CERTIFICATE.md, FAB-HARD-NONRESIDUE-BRIDGE.md
Claim boundary: this rules out a broad class of tempting fixed-k constructions. It does not by itself prove that a rescue always exists and therefore does not prove Erdős-Straus.
1. Setup
Let p be a Mordell-hard prime. In particular
Let C>0 satisfy
and put
Assume k>1. Then
and
The fixed-k divisor-square theorem says that this k supplies a sufficient fab certificate exactly when there is a divisor
such that
We now show that a C assembled entirely from quadratic residues modulo p can never do this.
2. Reciprocity transfer lemma
Lemma
For every odd prime r|C,
where the left symbol is Jacobi when k is composite.
Proof
Because r|C,
Since k\equiv3 mod4, quadratic reciprocity gives
But
The two (-1/r) factors cancel, so
Finally p\equiv1 mod4, hence reciprocity between p and r contributes no sign:
QED.
The factor 2
If 2|C, then C is even and
Therefore
On the hard-prime lane p\equiv1 mod8, so also
Thus the same residue sign is preserved for the dyadic factor as well.
3. Mirror obstruction theorem
Theorem
Assume every prime factor r of C is a quadratic residue modulo p:
Then there is no divisor u|C^2 satisfying
Consequently this k cannot supply a fixed-k sufficient fab certificate.
Proof
By the reciprocity-transfer lemma, every prime divisor of C is also Jacobi-positive modulo k. Hence every divisor
satisfies
If instead
then, since 4 is a square modulo odd k,
But k\equiv3 mod4, so
This contradicts (u/k)=+1. Therefore no such divisor exists. QED.
4. Structural interpretation
The theorem says that the fixed-k rescue mechanism cannot be obtained by simply taking a nearby integer C whose complete prime support has already been forced onto the quadratic-residue side of p, and then reflecting it through
That construction preserves the residue sign of every prime factor of C, while the fixed-k target
is Jacobi-negative.
Thus any successful interior divisor-square rescue must import genuine nonresidue support:
This is the fixed-k mirror of the external-nonresidue theorem in FAB-HARD-NONRESIDUE-BRIDGE.md.
5. Exact corollaries for the current counterexample sieve
The theorem kills several seductive but structurally impossible one-line constructions.
Corollary A — mirror of the p+1 spine
Let
If the simplest p+1 filter has failed, every odd prime factor of C is 1 mod4; hence every such factor is a quadratic residue modulo hard p. The factor 2, when present, is also a residue because p\equiv1 mod8.
Therefore the reflected choice
cannot rescue a hard-prime survivor through the fixed-k divisor-square criterion.
Corollary B — mirror of the Eisenstein neighbour
Let
If the exact k=3 filter has failed, every prime factor of A is 1 mod3. For hard p, reciprocity gives those factors quadratic-residue sign relative to the corresponding mirror construction. Hence taking a fixed-k construction obtained merely by reflecting this already-residue-safe support cannot supply the missing nonresidue target.
The same principle applies to the other shifted-factor filters whenever their failure theorem has already forced every prime divisor of the chosen C to be a quadratic residue modulo p.
6. Research consequence
This theorem prunes an entire family of false proof strategies.
The next successful construction must not be a mirror of an already-safe shifted factor. It must deliberately incorporate a prime or factor carrying
For Mordell-hard primes the small shield gives
so the first possible genuinely new prime support begins at the external boundary
depending on p.
This explains why the one-shot search should now construct k from external nonresidue data, rather than reflect any of the already-proved residue-safe neighbouring forms.