# Gleim–Markowitz optimality and the GM! Frontier **Catalog date:** 2026-08-25 **Slug:** `2026-08-25-gleim-markowitz-frontier` **Journal trailhead:** `dfai/publications/journal/2026-08-25-gleim-markowitz-frontier/` **Kind:** working paper · definition supplement Proposed for a later edition of *GM Futures Conviction Theory*. Does **not** amend the frozen First Edition PDF (July 2026). Author lineage: William Lloyd Gleim. “GM” records the Gleim–Markowitz research lineage (treatise endnote). This note does not claim Harry Markowitz’s co-authorship, review, or endorsement. Status: theory. **Not a product** until `GET /v1/gm/frontier` is real. Do not advertise a 402 that 404s. GM! Index remains the published conviction print. Lattice is an internal cross-section; **omega market line** is the public CAPM successor and is not the Index. `GET /v1/gm/omega_book` remains hurdle-net budgeting, not this envelope. --- ## The gap The First Edition already does the replacement work: | Classical | First Edition object | Still missing in the PDF | |---|---|---| | Return distribution → (μ, σ) | Whole distribution → hurdle net (G, L) and Ω (Def 4.1–4.3) | — | | Variance as risk | Tail-aware / downside axis; loss mass L as the risk report (Ch 6) | A numbered risk coordinate for efficiency | | Mean–variance dominance | Fig 5.1 schematic: A* omega-efficient vs A mean–variance | A numbered dominance relation | | Efficient frontier (σ, μ) | §5.2 and Table 6.1: “mean–omega frontier on a tail-aware risk axis” | A numbered frontier | | Investor / tangency choice | Theorem 6.1 omega security line; Table 6.1 “omega-efficient mix” | A numbered optimality definition | This supplement numbers those four missing objects and proves the algebraic bridge that makes omega a *ranking primitive on the same plane* whose risk axis is the loss mass. --- ## Setup Fix a declared hurdle τ (Principle 6.2: no omega figure without its hurdle) and a feasible set W of books. For w ∈ W write R(w) for the book return and G(w; τ) = E[max(R(w) − τ, 0)], L(w; τ) = E[max(τ − R(w), 0)], Ω(w; τ) = G/L when L > 0, exactly Definition 4.2. Let μ(w) = E[R(w)]. Proposition 4.2 (net-mean identity) is load-bearing: G − L = μ − τ. The two primitives of Principle 4.1 stay independent: (G, L) measures; Ω ranks. --- ## The bridge **Proposition S.1 (ranking identity).** From the net-mean identity, μ = τ + G − L = τ + L(Ω − 1) (L > 0). Hence: 1. At **fixed loss mass** L, ranking books by μ is ranking them by G, and is ranking them by Ω. 2. At **fixed Ω**, μ is affine in L. Rays of constant omega are straight lines in the (L, G) plane; their slope is Ω itself. 3. Mean–variance’s (σ, μ) plane is therefore replaced, like-for-like, by the tail-aware plane (L(τ), μ). The omega ratio is the ranking primitive on that plane. That is why Fig 5.1 can call the omega-efficient envelope the true envelope once the risk axis prices the left tail. This is the closed gap: whole-distribution omega is not a rival league-table statistic sitting beside Markowitz. It *is* the ranking of the full-distribution successor plane. --- ## Definitions **Definition S.1 (Tail-aware risk).** Tail-aware risk at a declared hurdle τ is the loss mass ρ_τ(w) := L(w; τ) = E[max(τ − R(w), 0)]. Any strictly increasing transformation of L that still prices the left tail is an equivalent risk axis. No omega figure is published without its hurdle. **Definition S.2 (Mean–omega dominance).** Book w mean–omega dominates w′ at τ when ρ_τ(w) ≤ ρ_τ(w′) and μ(w) ≥ μ(w′), with at least one inequality strict. Variance is not an admissible risk coordinate for this relation. **Definition S.3 (Omega-efficiency / Gleim–Markowitz efficiency).** A feasible book w* is omega-efficient (Gleim–Markowitz efficient) at τ on W when no w in W mean–omega dominates it. This is the full-distribution replacement of Markowitz mean–variance efficiency. **Definition S.4 (GM! Frontier).** The GM! Frontier (mean–omega frontier, omega-efficient frontier) at (W, τ) is the set of omega-efficient books, or its image in (ρ_τ, μ). It is the Table 6.1 successor to the (σ, μ) allocation menu. It is not an index, feed, or product. **Definition S.5 (Gleim–Markowitz optimality).** A feasible book w* is Gleim–Markowitz optimal at (W, τ) when it is omega-efficient and maximises Ω(w; τ) = G/L on {w in W : L(w; τ) > 0}. An interior optimum satisfies the omega security line g_i − Ω_p ℓ_i = κ (Theorem 6.1). S.3 is the *menu* (every undominated book). S.5 is the *selection* on that menu — the full-distribution analogue of a max-Sharpe / tangency choice, now characterized by the omega security line rather than by a covariance beta. Table 6.1’s trustee rule (“hold the omega-efficient mix at the declared τ ladder”) is S.3 applied rung by rung; it does not by itself pick the global S.5 book. --- ## Dual picture: the (G, L) Omega frontier Kapsos, Zymler, Christofides, and Rustem (and related LP treatments) already call the Pareto set in gain–loss coordinates the Omega frontier: no attainable point lies above it; a ray from the origin is a constant-Ω locus; the steepest feasible ray is max Ω. Proposition S.1 maps that set onto the GM! Frontier. Because μ = τ + G − L, Pareto improvement in (G, L) at fixed τ is mean–omega dominance in (L, μ) once the risk axis is L. The house contribution is not the discovery that Omega can be optimized. It is: 1. the two-primitive split (net vs ratio); 2. the canonical triad (M, D, ΔΩ) and the mirror (Theorem 4.1); 3. the declared-hurdle convention; 4. the security-line characterization (Theorem 6.1); 5. the time-inherent polarity (Principle 1.1). Cite the LP literature for computation. Cite this supplement for the named efficiency, frontier, and optimality objects in the GM program. --- ## What this is not - **Not GM! Index.** The index publishes conviction cells. The frontier is an allocation envelope over books. Lattice is an internal cross-section; the sold CAPM name is the omega market line (`GET /v1/gm/market_line`). That line is not the Index. - **Not the harvest ladder.** That prospectus “frontier” is an activity–quality trade-off for events, not a portfolio envelope. - **Not `/v1/gm/omega_book`.** That route allocates by lattice-Ω leave-one-out marginals (Table 6.1 hurdle-net budgeting). It does not compute the GM! Frontier and does not certify Gleim–Markowitz optimality. - **Not a claim that Markowitz endorsed this construction.** - **Not an information coefficient.** IC may score a published print later. It is not the optimality criterion. --- ## Edition status The July 2026 PDF stays byte-identical. These definitions are the working-paper lock for a second-edition §5.2 / §6.5 insertion (Definitions 5.2–5.5 in treatise numbering, if that edition accepts them). Machine lock: `dfy_iq.fci.gm_frontier`. House note: `ops/notes/2026-08-25-gleim-markowitz-frontier.md`.