LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Convexity and relative closedness of forward sectors

Statement

Under Forward-sector hypotheses for a simple state domain, for every simple state $x$ , the forward sector $A_{x} = {y \in Γ_{i} : x ≺ y}$ is convex and closed relative to $Γ_{i}$ . For collinear $y = t x + (1 - t) z$ , $0 < t < 1$ , $y ≺ z$ implies $x ≺ y$ , and $x ≺ z$ implies $x ≺ y$ .

Facts & Assumptions

Given: A1–A7 and S1–S3 on a nonempty open convex domain $Γ_{i}$ .

[A1]

Forward-sector hypotheses for a simple state domain: A1–A7 include consistency, scaling, splitting, convex-combination access, and stability for positive spectators.

[F1]

Relative interiors of finite-dimensional convex sets: a nonempty convex set has nonempty relative interior, and segments from a relative-interior point to a closure point, excluding the closure endpoint, remain in the relative interior.

[F2]

Spectator cancellation without comparison completeness: any finite common spectator cancels when the two bases have equal mass, under A1–A6 without CH.

Proof

1.1A1algebra

If $x ≺ y$ and $x ≺ z$ , splitting gives $x \sim (t x, (1 - t) x)$ . Consistency, positive scaling, and transitivity then give $(t x, (1 - t) x) ≺ (t y, (1 - t) z)$ . The convex-combination access hypothesis in [A1] gives $(t y, (1 - t) z) ≺ t y + (1 - t) z$ , so the convex combination belongs to $A_{x}$ .

1.2A1F2algebra

Let $y = t x + (1 - t) z$ with $0 < t < 1$ . The convex-combination access hypothesis in [A1] gives $(t x, (1 - t) z) ≺ y$ . Splitting gives $y \sim (t y, (1 - t) y)$ . If $y ≺ z$ , scaling by $1 - t$ and adding $t y$ gives $(t y, (1 - t) y) ≺ (t y, (1 - t) z)$ . Compose and cancel the common $(1 - t) z$ by [F2], then scale by $1 / t$ , to get $x ≺ y$ . If instead $x ≺ z$ , scaling by $1 - t$ and adding $t x$ gives $(t x, (1 - t) x) ≺ (t x, (1 - t) z) ≺ y$ ; recombination gives $x ≺ y$ .

2.1A1F1step 1.1algebra

Take $y \in {\overline{A_{x}}}^{Γ_{i}}$ . If $y \in A_{x}$ , the claim is immediate. Otherwise choose $w \in ri A_{x}$ by [F1]. Because $Γ_{i}$ is open, choose $z = w + (1 + η) (y - w) \in Γ_{i}$ for some $η > 0$ . For every integer $n > 1 + η$ , $y_{n} = (1 + 1 / n) y - z / n = w + (1 - η / n) (y - w)$ lies in $ri A_{x}$ by [F1], so $x ≺ y_{n}$ . Also $y = \frac{n}{n + 1} y_{n} + \frac{1}{n + 1} z$ . The convex-combination access hypothesis in [A1] gives $(\frac{n}{n + 1} y_{n}, \frac{1}{n + 1} z) ≺ y$ ; scaling by $(n + 1) / n$ and using the reversible split of $y$ gives $(y_{n}, z / n) ≺ (y, y / n)$ .

3.1A1step 2.1algebra∎

Consistency adds the same spectator $z / n$ to $x ≺ y_{n}$ , and transitivity with step 2.1 gives $(x, z / n) ≺ (y, y / n)$ for every $n > 1 + η$ . Apply the stability hypothesis in [A1] to bases $x, y$ , fixed positive equal-mass represented spectators $z, y$ , and the tail sequence $ϵ_{m} = 1 / (m + n_{0})$ with $n_{0} > 1 + η$ . It follows that $x ≺ y$ . Thus every relative closure point belongs to $A_{x}$ .

Convexity and relative closedness of forward sectors

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources