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PairedIdealIntersectionLift

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.

Definition 1.1 (The full paired exponent family).

Lean statement: D5/S3/Combinatorics/SpectralGraph/PairedIdealIntersectionLift.pairExponents

Formalization. D5/S3/Combinatorics/SpectralGraph/PairedIdealIntersectionLift.pairExponents (✓ std3).

Citation. 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.

The exponent vector is (a,a,b,c). Neither tail exponent is bounded above. The final actual graph theorems permit every prime permutation.

Theorem 1.2 (Injective paired support construction).

Lean statement: D5/S3/Combinatorics/SpectralGraph/PairedIdealIntersectionLift.paired_lift_nonzero

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

Citation. 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 each tail support T, assign h(T) to {0} union T and -h(T) to {1} union T; all other supports get zero. The positive member of each pair is a genuine nonempty proper support cell with positive weight, so any nonzero coordinate vector lifts to a nonzero graph vector.

Theorem 1.3 (Exact zero sum of actual paired vectors).

Lean statement: D5/S3/Combinatorics/SpectralGraph/PairedIdealIntersectionLift.paired_lift_sum

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

Citation. 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.

Each paired cell has the same positive multiplicity a times the product of the tail exponents in T. The two values cancel exactly, including the extension by zero across all universal vertices.

Theorem 1.4 (Exact actual paired action).

Lean statement: D5/S3/Combinatorics/SpectralGraph/PairedIdealIntersectionLift.paired_support_action

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

Citation. 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.

The support-complement Laplacian on this subspace is K_a-I. The proof evaluates the full four-coordinate support algebra with symbolic exponents, including all zero cells and both removed endpoints. It enumerates the fixed support carrier, not a finite set of exponents.

Theorem 1.5 (Actual paired spectral lift).

Lean statement: D5/S3/Combinatorics/SpectralGraph/PairedIdealIntersectionLift.actual_paired_eigenvalue

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

Citation. 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.

A nonzero eigenvector of K_a with eigenvalue kappa lifts to the actual ideal graph eigenvalue V+1-kappa, V=(a+1) squared (b+1)(c+1)-2. The paired vector has explicit zero sum, so the reflection is valid even if kappa-1 is zero.

References