Keyboard shortcuts

Press ← or → to navigate between chapters

Press ? to show this help

Press Esc to hide this help

PairedIdealIntersectionCongruence

Abstract

Actual ideal-intersection graphs and uniform higher-prime Laplacian obstructions.

The public mathematical input is unit-exponent.tex at immutable revision b93becea662a746badeadba42fdc3d3eb3db9382. General endpoint arguments follow the completed mathematical note supplied 2026-11-02. The formal proofs and this exposition are independently authored; no priority claim follows.

Theorem 1.1 (Nilpotent characteristic transport).

Lean statement: D5/S3/Combinatorics/SpectralGraph/PairedIdealIntersectionCongruence.shifted_tensor_charpoly_mod_three

Proof. Machine-checked in Lean as D5/S3/Combinatorics/SpectralGraph/PairedIdealIntersectionCongruence.shifted_tensor_charpoly_mod_three (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. the-omega-institute/ideal-intersection-laplacian contributors (2026). Six-support complement quotient, global endpoint spectrum, fixed-gap expansion and integer finiteness. URL: https://github.com/the-omega-institute/ideal-intersection-laplacian/blob/b93becea662a746badeadba42fdc3d3eb3db9382/paper/core/body.tex.

Commentary.

Over any commutative ring R of characteristic three, suppose b+1 and c+1 are nilpotent, with arbitrary nilpotency orders. Put X=(F_b+I) tensor I and Y=I tensor(F_c+I). Their squares are (b+1)I and (c+1)I, so X and Y are nilpotent. They commute and K2+I=X+Y-XY; the existing commuting nilpotent sum, product and subtraction laws give nilpotence of K2+I. Its characteristic polynomial differs from the fourth power of the polynomial variable by a nilpotent polynomial. The infinite mod-nine consumer specializes this general law to the reduced field F3, where that difference is zero. Nonreduced rings and unbounded nilpotent thickenings are included in the provider statement.

Theorem 1.2 (The whole infinite family excludes divisibility by 27).

Lean statement: D5/S3/Combinatorics/SpectralGraph/PairedIdealIntersectionCongruence.shifted_tensor_det_not_dvd

Proof. Machine-checked in Lean as D5/S3/Combinatorics/SpectralGraph/PairedIdealIntersectionCongruence.shifted_tensor_det_not_dvd (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. the-omega-institute/ideal-intersection-laplacian contributors (2026). Six-support complement quotient, global endpoint spectrum, fixed-gap expansion and integer finiteness. URL: https://github.com/the-omega-institute/ideal-intersection-laplacian/blob/b93becea662a746badeadba42fdc3d3eb3db9382/paper/core/body.tex.

Commentary.

For b=9s+5,c=9t+8, the symbolic determinant is 27W+18 for an integer polynomial W. Equivalently, with b=3u-1,c=3v-1, r=u+v,z=uv, it is 36(90z squared+57zr-5r squared+44z+r+2); the bracket is two modulo three. Hence 27 never divides the determinant in this entire family.

Theorem 1.3 (Infinite-family noninteger real spectrum).

Lean statement: D5/S3/Combinatorics/SpectralGraph/PairedIdealIntersectionCongruence.two_mod_nine_noninteger

Proof. Machine-checked in Lean as D5/S3/Combinatorics/SpectralGraph/PairedIdealIntersectionCongruence.two_mod_nine_noninteger (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. the-omega-institute/ideal-intersection-laplacian contributors (2026). Six-support complement quotient, global endpoint spectrum, fixed-gap expansion and integer finiteness. URL: https://github.com/the-omega-institute/ideal-intersection-laplacian/blob/b93becea662a746badeadba42fdc3d3eb3db9382/paper/core/body.tex.

Commentary.

For positive b=9s+5 and c=9t+8, suppose all real symmetric eigenvalues were integers. The nilpotent characteristic reduction makes every shifted integer root divisible by three. Their product would then be divisible by 3 to the fourth power, contradicting the preceding determinant formula. Positive diagonal transport supplies a noninteger real eigenvalue of the original weighted tensor. The pair (5,8) is the specialization s=t=0; exchanging the tail coordinates covers (8,5). No fixed mod-17 success certificate is used.

References