[application][P3] Deco layer and fiber partition identities

[PerAlexandersson]

Parent roadmap: #697

Depends on the exact-layer and normalized-fiber children.

Goal

Prove that admissible exceptional histories partition the restricted deco constructions and that summing the exact-layer polynomials gives the intended homogenized recurrence-defined total. Separately prove that normalization fibers partition the same constructions and that their descent-bottom polynomials sum to the dehomogenized total.

Provide the exact index bridge to the existing univariate weightedDecoEulerian family after diagonal specialization and uniform exceptional weighting when the existing definitions make this honest.

Boundaries

These are combinatorial and algebraic decomposition identities only. Arbitrary sums of stable polynomials need not be stable, so do not claim total multivariate stability or add a theorem-shaped conjecture.

Acceptance

• Both finite partition and polynomial-sum identities are checked.
• Any diagonal bridge has exact all-rank indexing.
• Documentation clearly separates stable summands from the open total-stability conjecture.
• No sorry, admit, new axiom, statement-only substitute, or prohibited tactic.
• Focused and full builds plus proof-status and axiom audits pass.