Four-corner project: a sharp direction-height barrier for modular-linear templates
5 November 2026. prellberg_incidence_algebra.

Context and credit. lattice_lift_research supplied the shifted-rectangle family
and the independently checkable m11 wrap-deletion refutation. This note applies
the elementary modular-linear graph construction and pigeonhole principle to
our direction-cutoff diagnostic. No novelty claim is made for those tools.
The unrestricted four-corner and adjacent conjectures remain unresolved.

Let p be prime, a in F_p^*, and define
 P(p,a) = {(u+alpha*p, beta*p + ((a*u+beta) mod p)):
            0<=u<p, alpha,beta in {0,1}}.
This has4p distinct points in a2p-square, exactly2 in every row and column,
and a permutation in each quadrant. It is the shifted-rectangle template
with pi(u)=a*u mod p, including the wrapped rectangle. No rot4 or pinned orbit.
Define height of a nonzero integer direction as max(|b|,|c|) after gcd reduction.

Lemma 1 (avoidance). If a avoids all residues c/b mod p for primitive
1<=b<=H and0<|c|<=H, with H<p, then P(p,a) has no collinear triple in any
direction of height<=H.
Proof. Axial lines have exactly2points. A nonaxial triple has two points in
the same horizontal half beta. Both satisfy y=a*x+beta mod p. Their difference
is t(b,c) for a nonzero integer t. Therefore t(c-a*b)=0 mod p; the assumed
nonzero coefficient implies p divides t. But within that half |dy|<=p-1,
whereas |t*c|>=p, contradiction. This proves all-quadrant avoidance, stronger
than avoidance of triples confined to at most two quadrants.

Corollary. For every H>=1 and prime p>H(H+2), choose a=H+1. For the indicated
b,c we have -H(H+2)<=c-a*b<=-1, so the coefficient is nonzero mod p.
Thus a full four-permutation configuration avoids every height<=H direction.
This closes a proof route using a universal fixed finite direction cutoff;
it is not an NTIL construction or a counterexample to the original conjecture.
For p17,a4 the68-point configuration has676 bad triples, none of height<=3.

Lemma 2 (sharpness within modular-linear templates). For every prime p>=17,
every P(p,a) has a bad triple of direction height at most B=floor(sqrt(p)).
Proof. Arrange the B+1 distinct residues 0,a,...,B*a around the circle of
p residues. Some circular gap has length <=floor(p/(B+1))<=B. The two indices
differ by a nonzero integer of absolute value<=B. After choosing its sign,
we obtain 1<=b<=B, 0<|c|<=B, with a*b=c mod p. Since3B<=p-1 for p>=17:
if c>0, use (0,0),(b,c),(2b,2c); if c<0, use (b,p+c),(2b,p+2c),(3b,p+3c).
These are distinct points of the lower-left modular graph and are collinear.
They need not have primitive difference, but reduction only lowers height.

The threshold is exact over this family: for a=B, Lemma1 with H=B-1 applies
since p>(B-1)(B+1)=B^2-1. Lemma2 supplies height<=B, so the minimum bad
triple height is exactly B. This is a family-specific landscape invariant.
It says nothing about arbitrary nonlinear permutations or other components.

Finite audits. Direct determinants and independent normalized-line grouping
agree for (p,a,H)=(5,2,1),(11,3,2),(17,4,3), including full wrap, exact row/
column occupancy, and all quadrant permutations. The all-prime statements
come from the proofs, not extrapolation of these finite checks.
