About the work
Given P = R⁵ with sig(3,2), S₁ sig(3,1), S₂ sig(2,2), proves the decomposition is forced. Five steps: (1) dim(S₁ ∩ S₂) = 3, (2) W nondegenerate via Gram determinant, (3) sig(W) = (2,1) by intersection of embedding constraints, (4) orthogonal complements V₁ = S₁ ∩ W⊥ positive, V₂ = S₂ ∩ W⊥ negative, (5) adapted basis with off-diagonal entry α = ⟨V₁,V₂⟩ determined by fixed S₁, S₂. The downstream chain (Proof 21) depends only on W, not on α. Algebraic, no computation. Dependencies: Proofs 3, 4, 5, 16.
AI Availability Declaration
The work is allowed to be accessed by AI systems.
Creativity declaration
No AI has been used in the creative process of this work
Print work information
Work information
Title CCFU Proof 20 — Parent Decomposition: P = S₁ ⊕_W S₂
Given P = R⁵ with sig(3,2), S₁ sig(3,1), S₂ sig(2,2), proves the decomposition is forced. Five steps: (1) dim(S₁ ∩ S₂) = 3, (2) W nondegenerate via Gram determinant, (3) sig(W) = (2,1) by intersection of embedding constraints, (4) orthogonal complements V₁ = S₁ ∩ W⊥ positive, V₂ = S₂ ∩ W⊥ negative, (5) adapted basis with off-diagonal entry α = ⟨V₁,V₂⟩ determined by fixed S₁, S₂. The downstream chain (Proof 21) depends only on W, not on α. Algebraic, no computation. Dependencies: Proofs 3, 4, 5, 16.
Work type Research papers, Thesis, Lecture notes
Tags mathematics, invariant form, ccfu, proof, signature, explicit construction, spectrum, companion matrix
-------------------------
Registry info in Safe Creative
Identifier 2605245765172
Entry date May 24, 2026, 12:16 PM UTC
License All rights reserved
-------------------------
Copyright registered declarations
Author 100.00 %. Holder Captain Cookie Face Universe. Date May 24, 2026.
Information available at https://www.safecreative.org/work/2605245765172-ccfu-proof-20-parent-decomposition-p-s1-_w-s2
/
Registration: 2605245765172
