Finite shift corridor for the classical strong/Type-II conjecture

Corridor

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Status: proved elementary reduction

Date: 2026-08-15

Depends on: ES-TWO-TARGET-SIGNED-BOX-EQUIVALENCE.md, ES-TYPEII-ROOT-GEOMETRY.md, STRONG-ES-MIZONY-THEPAULT-PROVENANCE.md

Claim boundary: this is an elementary consequence of the standard Type-II factor form. It does not prove the strong conjecture or Erdős--Straus, and no novelty claim is made for the size bound itself.

1. Fixed-shift Type-II data

Let

p\equiv1\pmod4

be prime and let

q\equiv3\pmod4, \qquad \gcd(q,p)=1.

Put

\boxed{C=\frac{p+q}{4}.}

A standard Type-II hit at shift q is equivalent to a factorization

\boxed{C=BDT}

with positive integers B,D,T such that

\boxed{q\mid B+D.}

This is the normalized -1 target in the signed divisor box.

2. Universal upper bound on a successful Type-II shift

Because

q\mid B+D

and B+D>0,

q\le B+D.

For positive B,D,

B+D\le BD+1

because

(B-1)(D-1)\ge0.

Also

BD\le BDT=C.

Therefore

q\le C+1.

Substituting

C=\frac{p+q}{4}

gives

4q\le p+q+4,

hence

\boxed{3q\le p+4.}

Thus:

Theorem — finite Type-II shift bound

Every standard Type-II solution for prime p≡1 mod4 has an associated shift satisfying

\boxed{ q\le\frac{p+4}{3}.}

So the strong/Type-II existence problem for a fixed prime has a finite admissible shift corridor.

3. Consecutive-integer parameterization

Define

\boxed{A=\frac{p+3}{4}.}

Every positive shift congruent to 3 mod 4 has the form

\boxed{q_h=4h+3, \qquad h\ge0.}

The corresponding shifted integer is

\begin{aligned} C_h &=\frac{p+q_h}{4}\\ &=\frac{p+3}{4}+h\\ &=A+h. \end{aligned}

Thus the fixed-shift Type-II problem runs through consecutive integers:

\boxed{C_h=A+h.}

The shift bound becomes

4h+3\le\frac{p+4}{3}.

Equivalently,

12h+9\le p+4,

\boxed{h\le\frac{p-5}{12}.}

Hence only the initial corridor

\boxed{ A,\ A+1,\ \ldots,\ A+\left\lfloor\frac{p-5}{12}\right\rfloor }

can support a standard Type-II shift.

4. Exact defect corridor for a hypothetical strong counterexample

Let

R_{q_h}(C_h)

be the signed divisor box of C_h modulo q_h.

A Type-II hit is exactly

-1\in R_{q_h}(C_h).

Therefore a hypothetical prime counterexample to the strong/Type-II conjecture must satisfy simultaneously

\boxed{ -1\notin \mathcal R_{4h+3}(A+h) }

for every integer

\boxed{ 0\le h\le\left\lfloor\frac{p-5}{12}\right\rfloor }

except the irrelevant case q_h=p, where the shift is not coprime to p.

This is a long finite sequence of coupled factorization defects on consecutive integers.

5. Root form across the corridor

The root geometry says a Type-II hit at h is equivalent to a factorization

\boxed{A+h=sbt}

with s squarefree and

\boxed{b+t\equiv0\pmod{4h+3}.}

Thus a strong counterexample must make every consecutive integer in the corridor avoid this complementary-root congruence.

The modulus changes linearly with the position:

\boxed{q_h=4h+3.}

So the obstruction is not a collection of independent arbitrary moduli. It is a synchronized pair

\boxed{(A+h,\,4h+3)}.

6. First two corridor positions

h = 0

q_0=3, \qquad C_0=A=\frac{p+3}{4}.

For Mordell-hard p, the exact q=3 theorem already gives:

\boxed{ q=3\text{ misses} \iff \text{every prime factor of }A\text{ is }1\pmod3.}

h = 1

q_1=7, \qquad C_1=A+1=\frac{p+7}{4}.

The next exact factorization defect can be classified completely in the cyclic group of order six; see the companion q=7 filter note.

Thus the corridor begins with two consecutive integers carrying sharply constrained but different local splitting laws.

7. Strategic consequence

The strong/Type-II route can be attacked as a consecutive defect corridor rather than as an unbounded auxiliary search.

A hypothetical counterexample requires approximately

\frac p{12}

simultaneous signed-box misses on consecutive integers, each with a linearly changing modulus.

Potential proof tools now include:

exact small-shift factor restrictions (q=3,7,11,...);
sieve arguments across the consecutive integers A+h;
incompatibility of splitting conditions imposed by several small moduli;
average signed-divisor expansion as h ranges through the corridor;
identifying a short initial segment whose combined local conditions already force a hit.

The shift bound does not solve the strong conjecture, but it turns its fixed-prime search space into a highly structured finite interval.