Unisona Sharpe Certificate — v1 (DRAFT · Status: Proposed)

0. Definitions (so the theorems are unambiguous)

For a strategy or portfolio with excess-return stream r_t (return above the risk-free rate):

• Sharpe ratio S = E[r] / σ(r), annualized by S_ann = S · √P where P is

periods/year (P =for daily). Sharpe is the object we maximize.

• Ex-ante Sharpe = expectation under our model. Ex-post Sharpe = realized

in data. The certificate optimizes ex-ante; §4 requires ex-post verification.

• Correlation ρ_ij = corr(r_i, r_j) between two strategies' return streams.
• All returns are net of the modeled cost COST_BPS per turnover event.

1. The theorems (the certified math)

Theorem— Diversification raises Sharpe (the core result). Proved here.

Statement. Let N strategies each have excess-return mean μ, standard deviation σ, Sharpe s = μ/σ, and equal pairwise correlation ρ. An equal-weight combination has Sharpe

$$ S(N,\rho) = s \cdot \frac{\sqrt{N}}{\sqrt{1 + (N-1)\rho}}. $$

Proof. Equal weights w_i = 1/N. Portfolio mean = μ. Portfolio variance

$$ \mathrm{Var} = \frac{1}{N^2}\Big(N\sigma^2 + N(N-1)\rho\sigma^2\Big) = \frac{\sigma^2}{N}\big(1+(N-1)\rho\big). $$

So portfolio std = σ·√((1+(N-1)ρ)/N), and S = μ / std = s·√(N/(1+(N-1)ρ)). ∎

Consequences (the design's whole reason to exist).

• ρ → 0 (uncorrelated): S → s·√N. **Ten independent Sharpe-0.5 strategies

combine to Sharpe ≈ 1.58.** This is the free lunch.

• ρ → 1 (identical): S → s. Correlated bets add nothing.
• ⇒ **The design must pay for low correlation, not just more strategies.**

A trend follower + a mean-reverter + a carry harvester beats three momentum clones of the same thing.

Theorem— The maximum-Sharpe portfolio is the tangency portfolio. Cited (Markowitz 1952); proof sketch.

Statement. For return vector μ and covariance Σ, the weights maximizing (wᵀμ)/√(wᵀΣw) are w* ∝ Σ⁻¹μ.

Proof sketch. Maximizing the Sharpe is scale-invariant in w; setting the gradient of (wᵀμ)/√(wᵀΣw) to zero yields μ ∝ Σw, i.e. w ∝ Σ⁻¹μ. ∎ Design consequence: allocation must be covariance-aware — size by inverse covariance, not equally, once correlations are estimated with enough data.

Theorem— Volatility targeting improves realized Sharpe under vol clustering. Cited: Moreira & Muir (2017), "Volatility-Managed Portfolios," J. Finance.

Scaling exposure inversely to forecast volatility (leverage_t = σ*/σ̂_t) leaves Sharpe unchanged under i.i.d. returns, but raises realized Sharpe when volatility is forecastable and clustered (empirically true in markets), because exposure is cut before high-variance, low-return regimes. Design consequence: position size targets a constant portfolio vol σ*, not a constant dollar/share.

Theorem— Higher Sharpe implies shallower drawdown. Cited: Magdon-Ismail & Atiya (2004).

For a return process with drift μ and vol σ, expected maximum drawdown over a horizon scales (in the diffusion approximation) inversely with the Sharpe ratio. Design consequence: optimizing Sharpe is not at odds with client drawdown-aversion — it is the same objective. The Calmar (return/maxDD) improves with Sharpe.

Theorem— Fractional Kelly is the Sharpe-favorable growth point. Cited: Kelly (1956); MacLean/Thorp/Ziemba.

Full-Kelly sizing maximizes long-run log-growth but with punishing variance. Half-Kelly captures ≈ 75% of the growth at ≈ 25% of the variance — a strictly better risk-adjusted operating point. Design consequence: the sizing layer caps at a fraction of Kelly (f ≤ ½), never full Kelly.

Theorem— Regime gating raises unconditional Sharpe iff the classifier beats its base rate. Proved here (conditional).

Statement. Partition time into "risk-on" and "risk-off" regimes. If a classifier identifies risk-off periods (lower/negative drift, higher vol) with better-than-base-rate accuracy, then conditionally moving to cash during predicted risk-off raises the unconditional Sharpe.

Proof. Unconditional excess return is the regime-probability-weighted mean; excising a segment whose conditional mean is below the cash rate raises the numerator and (since risk-off vol is higher) lowers the denominator — both move Sharpe up. The inequality reverses if the classifier is at or below base rate (whipsaw cost with no informational gain). ∎

This is the honest hinge of the whole system. Theoremdoes not assert our 200-day / GEM regime gate has edge — it states the precise condition under which it helps, and thereby converts a hope into a measurable hypothesis: does our gate classify risk-off better than base rate, net of switching cost? That number is produced by the harness, not asserted here. (Current 10-yr harness run: the 200d gate cut max-drawdown from −33.7% to −19.5% and lifted Sharpe 0.89 → 1.01 — consistent with Theoremholding over that window. One window is not proof; §4.)

2. Design — how chat + trader optimize client Sharpe

The theorems compose into a single pipeline. Each stage cites the theorem it implements.

        ┌─────────────────────────── UNISONA CHAT (operator + verification) ──────────────────────────┐

        │  • states ex-ante Sharpe estimate + confidence for every proposed allocation                 │

        │  • surfaces the evidence [claim, evidence, confidence, source] per decision                   │

        │  • operator veto / override / rollback — ALWAYS (Collapse-Cert §5)                            │

        └───────────────────────────────────────────────────────────────────────────────────────────┘

                                              │  approved intent

                                              ▼

   Σ₀ TRADER — Sharpe-optimization stack (ex-ante → gated → sized → verified):

     1. STRATEGY ENSEMBLE        Σ₀ TA-core + trend + mean-reversion + carry   → diversify (Thm 1)

     2. CORRELATION-AWARE ALLOC  size by Σ⁻¹μ, penalize correlated bets        → tangency  (Thm 2)

     3. REGIME GATE              200d/GEM risk-on/off overlay, edge-gated        → drawdown  (Thm 6)

     4. VOL TARGETING            leverage_t = σ*/σ̂_t to a fixed portfolio vol   → realized S (Thm 3)

     5. FRACTIONAL-KELLY CAP     per-position f ≤ ½ Kelly, hard heat limit       → growth pt (Thm 5)

     6. VERIFY GATE              execute only if ex-ante S ≥ threshold AND

                                 evidence passes; else abstain                    → Σ₀ rigor

     7. AUDIT + ROLLBACK         every fill logged; reversible; operator veto     → Collapse-Cert

Why chat and trader are one loop, not two products. Chat is the Reason + operator-verification surface; the trader is the Act + Verify surface. The Sharpe estimate a client sees in chat is the same ex-ante number the sizing layer acts on — no divergence between what's shown and what's done. This is the CLAUDE.md single-loop constraint (Observe→Remember→Reason→Act→Verify→Converge) applied to capital.

The optimization objective, precisely. Maximize ex-ante annualized Sharpe subject to: (a) portfolio vol ≤ σ*; (b) per-position size ≤ ½-Kelly; (c) total portfolio heat ≤ operator limit; (d) no position without passing the verify gate; (e) full reversibility. Sharpe is the objective; the constraints are the non-negotiable client-protection envelope.

3. What is PROVEN vs. what is CLAIMED-pending-evidence

The certificate certifies C1–C6 and the design's faithful implementation of them. It explicitly does NOT certify E1–E4. Those are the deliverables of §4.

4. Verification protocol (what turns E-claims into evidence)

To promote each empirical claim from CLAIMED to VERIFIED, run and log:

1. Per-strategy net Sharpe (E1) — daily-backtest-harness on each strategy,

multi-window (include a bear window: 2000–02, 2008), costs on. Record Sharpe + 95% CI (Sharpe SE ≈ √((1+S²/2)/T)).

1. Correlation matrix (E2) — pairwise ρ of the strategies' daily return

streams. Thmonly pays if the off-diagonals are small.

1. Gate edge (E3) — classify each day risk-on/off, compare gate accuracy to

the base rate; compute switching-cost-adjusted Sharpe lift across ≥3 windows.

1. Aggregate ex-ante vs ex-post (E4) — does the combined stack's realized

Sharpe match the Thm-1/2 prediction? Gap = model error to investigate.

Each run writes a git-stamped record under data/trading/leaderboard/. The certificate's empirical table (§3) is updated only from those records — never from memory or a single paper (Noise-Sorting rule).

5. Collapse-Certificate alignment (client protection, non-negotiable)

Any strategy or allocation admitted under this certificate MUST satisfy:

• Explicit heat limits — max per-position and max portfolio exposure.
• ½-Kelly ceiling — no full-Kelly or beyond; no leverage past the risk budget.
• Human-in-the-loop — operator validates before live deployment; veto anytime.
• Full audit trail + rollback — every decision reversible in one commit/flag.
• Self-correction trigger — auto-flatten / halt if rolling realized Sharpe or

max-drawdown breaches a preset threshold (the ex-post safety canary).

• Abstention over gambling — if the verify gate fails, the system holds cash.

Not trading is a valid, Sharpe-preserving action.

• Two-condition sleeve-admission gate — a candidate sleeve is admitted to the

ensemble only after it demonstrates BOTH: (a) measured pairwise ρ < 0.4 to the current blend, AND (b) a positive standalone Sharpe whose 95% CI excludes 0 (a real, significant edge). Low correlation is necessary but not sufficient — a zero-edge uncorrelated sleeve dilutes return. The harness correlation matrix + Sharpe CIs are the sole arbiter; narrative difference is never enough. Binding precedents (2026-07-10): mean-reversion (ρ=0.61) and short-vol (ρ=0.73) rejected on (a); BOTH L/S momentum variants rejected on (b) despite being genuinely market-neutral — L/S sector (ρ≈0.09 to blend, Sharpe 0.18 [−0.44, 0.80]) and L/S single-stock (ρ≈0.13 to blend, −0.06 to SPY, Sharpe 0.24 [−0.38, 0.86], −59.7% momentum-crash drawdown). Each lowered COMBO Sharpe (1.12→1.06/1.07) though each cut maxDD to ~−9%. Finding: decorrelation is easy; a significant standalone edge is the scarce ingredient. COMBO3 remains the verified ensemble; the direct arbiter is whether a sleeve raises the blended Sharpe, which none of the four candidates did this window.

6. Signatures

This certificate is Status: Proposed. It carries no authority over live capital until the operator signs §6. The return door remains fully open.

References

• Markowitz, H. (1952). Portfolio Selection. J. Finance.
• Kelly, J. (1956). A New Interpretation of Information Rate.
• Magdon-Ismail, M. & Atiya, A. (2004). Maximum Drawdown. Risk.
• Moreira, A. & Muir, T. (2017). Volatility-Managed Portfolios. J. Finance.
• MacLean, Thorp, Ziemba (2011). The Kelly Capital Growth Investment Criterion.
• Companion evidence: scripts/daily-backtest-harness.js,

records under data/trading/leaderboard/.