Research proof index

Lemma 2.4 (paper-facing form). Assuming global convexity of μ, this provides the two inequalities labelled (2.2) and (2.3) in the file, packaged as a conjunction. API note: the proof reduces to the interval-local lemmas in HarderNarasimhan.Convexity.Impl by using the equivalence ConvexI TotIntvl μ ↔ Convex μ.

Lengths of Jordan–Hölder filtrations agree under modularity. Assuming ℒ is modular and μ satisfies the standard hypotheses (including affinity), any two Jordan–Hölder filtrations for μ have the same finite length.

Proposition 2.6 (paper-facing form). For x<y<z, the statement consists of: - the unconditional monotonicity μA (x,z) ≤ μA (y,z), and - under convexity, parts (a), (b), (c) giving refined comparisons/equalities involving μA. API note: the proposition is packaged as a nested conjunction/implication structure mirroring the paper.

Semistability of the initial segment and the “no improvement to the right” property. If x ∈ St μ, then: 1. the restriction of μ to the interval (⊥, x) is semistable, and 2. for any y > x, the μA-slope on (⊥, x) is not dominated by the slope on (x, y). This packages the two internal statements impl.prop3d7₁ and impl.prop3d7₂. API note:...

Totality/maximality consequences and the slope decomposition formula. Assuming convexity and an additional admissibility hypothesis (either totality of ≤ on S, or an attainment condition on μ), the internal results show: 1. St μ is totally ordered and, under DCC, admits a greatest element; and 2. for any stable breakpoint x and any y > x, the...

Proposition 4.16: a TFAE package relating the three equalities for μmin/μmax on TotIntvl, and (under chain conditions) equivalence with Nash equilibrium.