[application][P3] Deco descent-bottom stable layers and normalized fibers

[PerAlexandersson]

Status and scope

The manuscript proves two decomposed multivariate stability results for the deco descent-bottom polynomial. The stability of their total sum remains an open conjecture. This issue is only for the proved decompositions and must not present the total-stability conjecture as solved.

Target A: exact exceptional layers

For height h and an admissible nonconsecutive set R of exceptional construction positions, define the homogenized exact-layer polynomial

Btilde_(h,R)(s,u)
  = sum_tau s^(h-1-des(tau)) product_(descent bottoms r of tau) u_r.

Formalize the construction from the seed s using:

• a normal step f -> s(1+u_1 D)f, where D=partial_s+sum_j partial_(u_j) after the required relabeling;
• an exceptional step given by relabeling and multiplication by s u_2.

Prove every exact layer is MvRealStable, then derive stability of its specialization at s=1.

Target B: normalized-code fibers

For a normalized code c, define the fiber over subsets of its disjoint eligible (0,2) pairs and prove the descent-bottom enumerator factors as a monomial, a power of two, and factors

u_(h-j) + u_(h-j+1).

Conclude that every fiber is real stable. If inexpensive, retain independent nonnegative pair weights through the corresponding partial symmetrizations.

Existing infrastructure

The library already has MvRealStable, multiplication, renaming, partial derivative, boundary specialization, and finite-product stability support. The main work is finite-variable indexing, relabeling, and the exact combinatorial decomposition; finite checks are not substitutes.

The existing Applications/EulerianVariations.PeakValues stability theorem concerns a different family and should only be used as an API/style precedent.

Explicit exclusions

• Do not claim stability of the sum of the layers or fibers. Stability is not closed under arbitrary addition.
• Do not add the total polynomial as a proved theorem or theorem-shaped scaffold.
• The all-rank total-stability conjecture and the affine-slice interval-preserver problem remain open mathematics.

Acceptance criteria

• Admissible exceptional sets, the two construction operators, and their indexing/relabeling laws are defined exactly.
• Every exact exceptional layer has a checked MvRealStable theorem and a checked s=1 specialization.
• Every normalized fiber has a checked factorization and MvRealStable theorem.
• The decomposition statements prove that the layers/fibers sum to the intended recurrence-defined total, without using that sum to infer stability.
• Proved layers and the open total-stability conjecture are clearly distinguished in documentation.
• No sorry, new axiom, or theorem-shaped statement scaffold is counted as completion.
• Focused and full builds pass without warnings, and public endpoints receive the standard axiom audit.

[PerAlexandersson]

Decomposed this application roadmap into four bounded lanes after checking the authoritative manuscript proof and the current multivariate-stability APIs:

• [foundation][P1] Stability of nonnegative directional-derivative pencils #775 — application-neutral nonnegative directional-derivative stability pencil; this is the first foundation task.
• [application][P2] Deco exceptional histories and stable exact layers #776 — admissible exceptional histories, exact layer recursion, and layerwise stability; depends on [foundation][P1] Stability of nonnegative directional-derivative pencils #775.
• [application][P2] Deco normalized-code fibers and stable product factorization #777 — normalized-code fibers, commuting adjacent swaps, product factorization, and fiber stability.
• [application][P3] Deco layer and fiber partition identities #778 — the two partition/sum identities and exact bridge back to the recurrence-defined total; depends on [application][P2] Deco exceptional histories and stable exact layers #776 and [application][P2] Deco normalized-code fibers and stable product factorization #777.

The existing Wagner/matrix interlacing infrastructure remains useful for univariate preserver clients, but it does not replace #775: the layer proof is an upper-half-plane stability argument for a multivariate directional derivative. The total multivariate stability conjecture remains explicitly excluded.

[PerAlexandersson]

#776 is complete through closed children #780 and #781. The exact exceptional layers are now defined on canonical finite coordinates and proved homogeneous, multivariate real stable, and stable after s=1 dehomogenization. The implementation reuses #775/#779 and does not invoke the univariate Wagner/matrix route. Remaining independent application work is #777 (normalized-code fibers), followed by #778 (partition/sum and recurrence identification). Total multivariate stability remains open and excluded.