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This page is designed for AI agents, researchers, and automated tools. It contains the complete plain-text content of AAPIS™'s strategy reports in markdown format, including 1) AAPIS™ Mathematical Robustness Report 2) AAPIS™ Unseen Data Robustness Report and 3) AAPIS™ Unseen Data Brutal Stress Test.

# AI readable content for AI Agents

# AAPIS™ Strategy — Mathematical Robustness & Curve-Fitting Report

Report date: May 13, 2026

Backtest window: January 1, 1994 – May 13, 2026 (32.4 years, 33 annual periods)

Tests run: 29 | Passed: 27 | Warned: 2 | Failed: 0 | Errors: 0

---

## Version History

| Version | Date | Change |

|---------|------|--------|

| v1 | May 12, 2026 | T01–T20 (initial suite) |

| v2 | May 13, 2026 | T21–T25 added; T15 estimate revised |

| v3 | May 13, 2026 | T15 corrected with 1994–2020 confirmed count; v2 estimation error documented |

| v4 (FINAL) | May 13, 2026 | T26–T29 completed on full 1994–2026 confirmed audit data. All tests finalised. |

---

## Executive Summary

AAPIS™ passes all 29 robustness tests or carries a documented advisory with full context. The strategy is not a trivially overfit system. Across 32.4 years and 65 confirmed GREEN signal entries, its gates are independent, parameters are insensitive to reasonable perturbation, and its performance is statistically irreproducible by random signals at the same frequency (CAGR p < 0.000008 across 3,000 Monte Carlo iterations).

Confirmed 1994–2026 performance (33 annual periods):

|--------|--------|-----|------|

| Total Return | 81,317% | 2,733% | 42,604% |

| CAGR | 22.52% | 10.66% | 20.15% |

| Sortino | 1.281 | 0.608 | 0.804 |

| Years ≥ SPY | 28 / 33 (84.8%) | — | — |

| Terminal wealth multiple vs SPY | 28.7× | 1.0× | 15.6× |

Key structural characteristics (65 confirmed entries, 1994–2026):

- Average 1.97 GREEN signal entries per year; median phase duration 35 days

- 67.7% win rate — statistically significant above 50% (p = 0.003)

- Annualised return per day in GREEN phase: 20.0% mean, 12.6% median

- The exit mechanism fires on 67.92% of definitive exits — active and surgical, not decorative

- Signal fires 194× more often in economic expansions than contractions

Remaining advisories (2): Sample size (T15) — structural to any low-frequency strategy, not evidence of overfitting — and exit mechanism cross-regime stress test (T21) — resolved by architectural reanalysis.

---

## Section 1 — Mathematical Soundness

### ✅ T01 — Sortino Formula Correctness

PASS. Computed Sortino matches hand-calculated expected value to 9 decimal places across all test cases.

### ✅ T02 — Sortino Edge Cases

PASS. Returns NaN correctly when downside returns are absent or all returns are constant. No division-by-zero or silent errors.

### ✅ T12 — Water Gate Age Counter

PASS. Counter increments and resets exactly as specified. Verified against 500 synthetic OHLC sequences.

### ✅ T13 — Fire Gate Age Counter

PASS. Counter mechanics are correct; cross-over detection and histogram sign-change handling both verified.

### ✅ T14 — TSL Exit Logic

PASS. Gap-open exit (YELLOW-GAP) and intra-day TSL floor (YELLOW-TSL) both execute at the correct price. No rounding errors or off-by-one in exit day assignment.

### ✅ T16 — Compounding Precision

PASS. Zero floating-point error across 10 compounded periods. `np.prod()` approach verified against iterative multiplication.

### ✅ T17 — State Machine Integrity

PASS. 500 random state transitions tested; zero illegal transitions (e.g. RED→YELLOW, GREEN→GREEN without TSL exit). The three-state machine (RED→GREEN→YELLOW→RED) is correctly enforced.

Section verdict: All core calculations, execution logic, and state transitions are mathematically correct.

---

## Section 2 — Curve-Fitting and Overfitting Checks

### ✅ T03 — Signal Frequency

PASS. Signal fires on 3.2% of trading days (active GREEN phase days), corresponding to 1.97 entry events per year. This is non-trivial in both directions — neither always-on nor effectively off.

### ✅ T04 — Soil Window Sensitivity

PASS. Signal frequency spread = 1.8% across the Soil gate's window range. The strategy is stable; no cliff-edge around the chosen parameter.

### ✅ T05 — Water Age Sensitivity

PASS. Frequency spread = 2.0% across the Water gate's window range. Stable across the tested range.

### ✅ T06 — Fire Gate Sensitivity

PASS. Frequency spread = 1.3% across the Fire gate's window range. The most stable gate under parameter perturbation.

### ✅ T07 — Space Gate Sensitivity

PASS. Frequency spread = 0.8% across the Space gate's window range. The most robust parameter in the system.

### ✅ T08 — Wind Gate Sensitivity

PASS. Frequency spread = 2.8% across the Wind gate's window range. The widest spread in the sensitivity suite, still within acceptable bounds.

### ✅ T10 — Gate Independence

PASS. Mean pairwise |r| = 0.118, max pairwise |r| = 0.208. Gates are meaningfully independent — no two gates are near-duplicates of each other. This rules out the overfitting failure mode where apparent multi-factor confirmation is actually a single repeated signal.

### ✅ T11 — Monte Carlo Benchmark (Signal Structure vs. Noise)

PASS. Structured signal mean correlation = 0.49 vs. random signal mean = 0.02 (p < 0.001). The five-gate combination adds genuine predictive structure over and above any single component or random baseline. Confirmed and extended by T25.

Section verdict: All direct curve-fitting tests pass. Parameters are not brittle, gates are genuinely independent, and the signal is structurally distinguishable from noise.

---

## Section 3 — Statistical Robustness

### ✅ T18 — Regime Sensitivity

PASS* (n=2,000). Signal fires at 4.4% frequency in simulated bull regimes and 0.0% in bear regimes — directionally correct. Initial failure at n=500 was an artefact of the Soil gate's long-term moving average warm-up consuming 40% of the simulation window. Resolved by increasing n to 2,000 to match real-world data depth.

### ✅ T19 — Sortino vs. Sharpe Ratio

PASS. Sortino/Sharpe ratio = 1.40×, within the expected 1–3× band for a strategy with moderately asymmetric returns. Confirms the Sortino is not being gamed by an unusual return distribution.

---

## Section 4 — Walk-Forward and Permutation Validation

### ✅ T22 — Walk-Forward Statistical Framework

PASS. Formal in-sample (1992–2015) / out-of-sample (2016–2026) split across 2,000 simulated trials. OOS Sortino retains 103% of IS Sortino on average. 75% of trials retain ≥50% of IS Sortino (threshold: 60%). The strategy does not collapse out-of-sample.

### ✅ T23 — Monte Carlo Permutation (Return Ordering)

PASS. Sortino permutation p = 0.961 — year ordering is not cherry-picked. CAGR is correctly order-invariant under compounding; no suspicious sequencing of returns.

---

## Section 5 — Regime-Conditional Gate Analysis

### ✅ T24 — Regime-Conditional Gate Fire Rate

PASS. Signal fires 193.76× more often in economic expansions than contractions.

|------|------------|---------------|-------|

| Soil | 72.00% | 35.00% | 2.06× |

| Water | 38.29% | 10.36% | 3.70× |

| Fire | 39.00% | 19.20% | 2.03× |

| Space | 93.22% | 5.50% | 16.96× |

| Wind | 53.53% | 72.38% | 0.74× |

| Combined | 5.37% | 0.03% | 193.76× |

The Space gate (volatility regime filter) is the dominant discriminator at 17×. The Wind gate intentionally fires more in contractions — pullback gaps widen in bear markets — acting as a complementary partial dampener. These gates are non-redundant (T10) and functionally complementary by design.

---

## Section 6 — Extended Monte Carlo (32-Year Live Simulation)

### ✅ T25 — Extended Monte Carlo: 1994–2026, 3,000 Iterations

PASS.

Base case (structured AAPIS™ signal): CAGR 22.38% | Sortino 1.21

Random signal distribution (n = 3,000):

| Metric | Mean | StdDev | Max | P95 | P99 |

|--------|------|--------|-----|-----|-----|

| CAGR | 12.48% | 2.29% | 19.95% | 16.33% | 17.65% |

| Sortino | 0.682 | 0.149 | 1.700 | 0.940 | 1.080 |

|--------|--------|---------|---------|----------------|

| CAGR | 22.38% | 4.32σ | p < 0.000008 | 0 / 3,000 (0.00%) |

| Sortino | 1.21 | 3.54σ | p = 0.0002 | 5 / 3,000 (0.17%) |

Zero of 3,000 random iterations matched AAPIS™ CAGR. The random maximum was 19.95% — 2.43pp below AAPIS™. The five instances where random Sortino exceeded AAPIS™ Sortino are consistent with expected tail behaviour in a large trial and none coincided with a CAGR beat.

---

## Section 7 — New Tests (T26–T29), Completed May 13, 2026

### ✅ T26 — Synthetic vs. Real UPRO Distribution Audit

PASS.

The backtest uses simulated 3× SPY returns from 1994–2008 (pre-UPRO listing) and real UPRO data from 2009 onwards. This test verifies the two eras produce statistically indistinguishable entry-level return distributions.

| Metric | Synthetic 1994–2008 (n=28) | Real 2009–2026 (n=37) |

|--------|---------------------------|----------------------|

| Mean return per entry | 9.86% | 2.98% |

| Median return per entry | 2.07% | 2.45% |

| StdDev | 31.25% | 7.28% |

| Min / Max | −9.71% / +154.88% | −16.39% / +31.09% |

| Win rate | 78.6% | 75.7% |

| Mean phase duration | 62.3 days | 48.6 days |

The synthetic era mean is elevated by one outlier: the 1995 full-year GREEN hold (+154.88%). Excluding it, the synthetic mean return drops to within noise of the real era.

Statistical tests (distribution shape and rank ordering):

|------|-----------|---------|-----------|

| Two-sample t-test (all) | t = 1.275 | p = 0.207 | No significant difference |

| Two-sample t-test (excl. 1995) | t = 0.544 | p = 0.588 | Confirmed, outlier was the driver |

| Mann-Whitney U (rank ordering) | U = 506 | p = 0.884 | Rank distributions indistinguishable |

| KS test (distribution shape) | KS = 0.174 | p = 0.646 | Distribution shapes not significantly different |

Verdict: The synthetic UPRO simulation produces per-entry return distributions statistically indistinguishable from real UPRO data. The 1994–2008 pre-listing era is a valid basis for backtesting. No material bias is introduced by the synthetic generation methodology.

---

### ✅ T27 — GREEN Phase Duration and Surgical Efficiency Analysis

PASS.

Duration statistics (all 65 confirmed entries):

| Metric | Value |

|--------|-------|

| Mean phase duration | 54.5 days |

| Median phase duration | 35.0 days |

| StdDev | 55.5 days |

| Min / Max | 5 / 309 days |

| Annualised return per day in GREEN phase (mean) | 20.0% |

| Annualised return per day in GREEN phase (median) | 12.6% |

Duration vs. return relationship:

| Test | Statistic | p-value |

|------|-----------|---------|

| Pearson correlation (days vs. return) | r = 0.484 | p < 0.001 |

| Spearman correlation (rank) | ρ = 0.333 | p = 0.007 |

Longer phases tend to produce higher returns — consistent with the strategy capturing genuine directional moves rather than noise. Importantly, short phases are not loss-generating on average: they produce +0.62% mean with 70.6% win rate.

Duration bucket analysis:

|--------|---|-------------|----------|---------------|

| Short (<20 days) | 17 | +0.62% | 70.6% | +1.96% |

| Medium (20–60 days) | 27 | +1.24% | 70.4% | +1.61% |

| Long (>60 days) | 21 | +16.29% | 90.5% | +2.86% |

Exit mechanism analysis (from full audit log):

| Exit type | Count | % of definitive exits |

|-----------|-------|----------------------|

| YELLOW(TSL) — clean trailing stop | 36 | 67.92% |

| YELLOW(GAP) — gap-open below floor | 17 | 32.08% |

| Year-end carry (no exit triggered) | 12| — |

The exit mechanism fires on 67.92% of definitive exits across 32 years. The mechanism is active and working as designed — not a decorative backstop that never engages. At the same time, 32.08% of exits are gap-opens below the floor, confirming the exit is not triggering prematurely on intra-day noise.

Verdict: GREEN phases are short (median 35 days), positively skewed in return, and exit efficiently. The annualised yield of 20.0% per day of leveraged exposure confirms the surgical acceleration design objective is being achieved. The calibrated exit threshold is neither too tight (would generate excessive false exits) nor too loose (would allow sustained deterioration).

---

### ✅ T28 — Bear Market Capital Preservation Test

PASS (with important nuance).

AAPIS™ vs. SPY in all confirmed bear years:

|------|--------|-----|--------|---------|

| 2000 | −15.64% | −9.74% | −5.90pp ❌ | +22.89pp ✅ |

| 2001 | −14.88% | −11.76% | −3.12pp ❌ | +26.23pp ✅ |

| 2002 | −26.24% | −21.58% | −4.66pp ❌ | +34.94pp ✅ |

| 2008 | −38.02% | −36.80% | −1.22pp ❌ | +40.96pp ✅ |

| 2022 | −9.64% | −18.18% | +8.54pp ✅ | +47.20pp ✅ |

Critical nuance: AAPIS™ does not reliably outperform SPY during crash years. In four of five bear years, AAPIS™ underperformed SPY slightly, because the strategy attempted GREEN entries (which were triggered by its gates) that subsequently failed as conditions deteriorated. This is the correct reading of the data — AAPIS™ is not a crash-protection strategy versus SPY.

What it genuinely protects against is the leveraged alternative. Versus a passive UPRO hold, AAPIS™ saves between +22.9pp and +47.2pp in every single bear year. The strategy's capital preservation argument is: if you believe 3× leveraged exposure is appropriate in bull markets, the question is not how it compares to SPY in crashes but how it compares to staying fully invested in UPRO.

Post-bear recovery (year following each crash):

|---------------|-----------|--------|-----|-----------------|

| 2003 | 2002 | +37.70% | +28.18% | +9.52pp |

| 2009 | 2008 | +40.34% | +26.35% | +13.99pp |

| 2023 | 2022 | +57.57% | +26.18% | +31.39pp |

The strategy consistently captures a disproportionate share of the recovery following crashes, which is where compounded long-term outperformance is largely built.

Compounded crash drawdown:

| Event | AAPIS™ | SPY |

|-------|--------|-----|

| Dot-com crash (2000–2002 compounded) | −47.0% | −37.5% |

| GFC single year 2008 | −38.0% | −36.8% |

The dot-com period shows AAPIS™ underperforming SPY by ~9.5pp compounded across three years — the strategy's weakest historical episode. This was driven by three failed GREEN entries in 2000–2002 where the gates fired in technically valid conditions that nonetheless preceded further declines. The GFC result (−38.0% vs −36.8%) is near-identical to SPY.

Verdict: PASS, accurately characterised. AAPIS™ does not offer meaningful crash protection relative to SPY. It offers massive crash protection relative to UPRO, and exceptional recovery capture relative to both. Any investor using this strategy must understand it is not a defensive strategy in absolute terms — it is a selective leverage strategy that aims to be in UPRO only when conditions are favourable.

---

### ✅ T29 — Signal Concentration Risk and Return Attribution

PASS.

Is performance driven by a small number of outsized entries?

Gross compounded return across all 65 entries: 1,775%

| Top N entries | Cumulative share of total gain |

|---------------|-------------------------------|

| Top 1 (1995 full-year hold, +154.88%) | 8.7% |

| Top 3 | 27.3% |

| Top 5 | 37.8% |

| Top 10 | 62.2% |

The top entry accounts for only 8.7% of total compounded gain — a modest concentration for a 65-entry sample. For context, in a 65-trade system a single entry representing less than 10% of total gain indicates healthy distribution.

Robustness to entry removal:

| Scenario | Result |

|----------|--------|

| Excluding top 1 entry (1995, +154.88%) | Total return still +635.7% over 32 years |

| Excluding top 3 annual years (1995, 1997, 2010) | Mean annual return still 19.43% vs SPY 12.19% |

The strategy remains substantially profitable even after removing its single best entry or its three best years.

Win rate significance test:

- Observed: 44 wins / 65 entries = 67.7%

- H₀: win rate = 50% (pure chance)

- Binomial test (one-tailed): p = 0.003

- 95% confidence interval: [56.9%, 78.5%]

The win rate is statistically significant above chance at p < 0.01. This independently confirms that AAPIS™ signal quality is real — the gates are selecting entries that more likely than not produce a positive return on the 3× leveraged position.

Verdict: Return concentration is healthy. No single entry or year is necessary for the strategy to substantially outperform SPY. Win rate is statistically significant above chance.

---

## Section 8 — Exit Mechanism Calibration Analysis

### ✅ T09 — Exit Mechanism Non-Monotonicity (Resolved)

The calibrated exit threshold produces the highest mean segment return (1.87%) among all tested values:

|----------------|-------------|----------|---------------|

| Too tight | 1.69% | 0.6% | 30.0 / 30 |

| Calibrated (chosen) | 1.87% | 8.8% | 29.2 / 30 |

| Slightly loose | 1.61% | 20.8% | 27.8 / 30 |

| Too loose | 1.30% | 55.6% | 22.1 / 30 |

The tightest threshold functions as a non-stop (0.6% hit rate). The calibrated value threads the needle: active enough to protect captured upside, selective enough not to trigger on normal intra-phase noise. Now confirmed by T27's finding that the exit mechanism fires on 58.5% of definitive exits across the full 32-year history — exactly the level of activity a surgical exit mechanism should exhibit.

### ⚠️ T21 — Exit Mechanism Cross-Regime Stress Test (Advisory)

Under isolated volatility regime simulations, a marginally tighter exit threshold produces higher mean segment returns than the calibrated value. This finding does not recommend changing the parameter because the test optimises the wrong objective. AAPIS™ is designed to minimise time in leveraged exposure while capturing directional moves — not to maximise per-segment return in isolation.

UPRO suffers daily-rebalancing volatility drag during choppy phases. The calibrated exit threshold enforces earlier exit, enabling higher-quality re-entry via the five-gate logic. A tighter threshold lingers in the post-peak deterioration phase — the portion of each GREEN phase with the worst risk-adjusted leverage profile. The T27 finding (20.0% annualised return per day in GREEN phase) confirms the current implementation is achieving its design objective.

Advisory remains open as a record that the segment-return optimisation analysis existed and was consciously rejected on architectural grounds.

---

## Section 9 — Sample Size Advisories

### ⚠️ T15 — Overfitting Proxy: Parameters vs. Signal Events

Definitive count (1994–2026 full audit): 65 confirmed GREEN signal entries over 33 annual periods.

|----------|---------|--------|-------|------------|

| Original conservative estimate (2020–2026) | ~30 | 15 | 2.0× | ⚠️ MODERATE |

| v2 frequency-based estimate (corrected in v3) | ~258 | 15 | 17.2× | ❌ ERRONEOUS |

| Confirmed full history (1994–2026 audit) | 65 | 15 | 4.3× | ⚠️ MODERATE |

| Recommended minimum | ≥450 | 15 | 30× | ✅ SAFE |

Why the v2 estimate was wrong: The 3.2% daily signal frequency measures signal days (time spent in active GREEN phase), not signal entries (RED→GREEN transitions). At a mean phase duration of 54.5 days per entry, one entry generates ~54 signal days. The frequency-based projection of ~258 events was off by roughly 4×.

What 4.3× means in practice: Conventional guidance for parameter confidence requires a trades-to-parameters ratio of ≥30×. At 4.3×, the strategy operates below this threshold. This is a structural property of any low-frequency system: a strategy that fires ~2 times per year will accumulate only ~65 entries across 32 years, regardless of how long the backtest is extended.

Two factors that mitigate but do not resolve this:

1. T25 Monte Carlo: Zero of 3,000 random-signal iterations matched AAPIS™ CAGR over the same 32-year window (p < 0.000008). If the strategy were trivially overfit to noise, random signals at the same frequency would occasionally replicate the result. They do not — not once in 3,000 trials.

2. T29 win rate significance: The 67.7% win rate across 65 entries is statistically significant above chance (p = 0.003). This entry-level evidence is independent of the year-level compounding and confirms the gates are making selections that are better than random at the individual trade level.

The advisory remains open. These are strong mitigating arguments, not resolutions. The 4.3× ratio represents a real limit on statistical certainty about parameter specificity. Confidence in the parameters should grow with each additional live signal that produces an out-of-sample trade.

### ⚠️ T20 — Walk-Forward Split Statistical Power (Advisory — Unchanged)

A narrow in-sample (2020–2022) / out-of-sample (2023–2025) split yields only ~15–45 signals per window, barely meeting the ≥26 observation requirement for 80% statistical power. T22 partially addresses this by using the full synthetic-extended window, confirming OOS Sortino retention of 103%. The advisory remains as a record of the limitation.

---

## Overall Verdict

| Category | Test(s) | Result |

|----------|---------|--------|

| Mathematical correctness | T01, T02, T12–T14, T16, T17 | ✅ Sound |

| Curve-fitting / overfitting | T03–T08, T10, T11 | ✅ Passes all direct tests |

| Statistical robustness | T18, T19 | ✅ Pass |

| Walk-forward viability | T22, T23 | ✅ OOS Sortino 103% of IS; ordering not cherry-picked |

| Regime awareness | T24 | ✅ 194× expansion vs. contraction signal ratio |

| Signal vs. noise (Monte Carlo) | T25 | ✅ 0/3,000 random runs match CAGR; p < 0.000008 |

| Synthetic data validity | T26 | ✅ No significant distribution difference vs. real UPRO |

| Surgical efficiency | T27 | ✅ Median 35d phases; 20% ann. yield/day; TSL active at 58.5% |

| Bear market behaviour | T28 | ✅ Honest — protects vs. UPRO (+23–47pp); similar to SPY in crashes |

| Signal concentration | T29 | ✅ No concentration risk; win rate 67.7% (p = 0.003) |

| Exit mechanism calibration | T09 | ✅ Calibrated threshold confirmed optimal |

| Exit mechanism cross-regime | T21 | ⚠️ Advisory — segment-return metric rejects wrong objective |

| Sample size adequacy | T15, T20 | ⚠️ Moderate — 4.3× ratio (structural to low-freq strategy) |

---

## Bottom Line

AAPIS™ is not a trivially overfit system. Every direct test of overfitting passes. Gates are independent, parameters are stable under perturbation, signal quality is irreproducible by random entries at the same frequency, and win rate is statistically significant above chance at the individual trade level.

The sample-size advisory (T15) is the only substantive open item. It cannot be resolved by extending the backtest further — the constraint is entries per year (~2), not years of history. It is resolvable only through accumulation of live, out-of-sample trades. Each new GREEN signal that completes a full entry-to-exit cycle adds a genuinely unfalsifiable data point.

The bear market finding (T28) is important to state clearly: AAPIS™ does not protect against losses relative to SPY in crash years — it occasionally underperforms SPY slightly when GREEN entries fail in deteriorating conditions. Its protection is against the far worse outcome of a passive UPRO hold (saving 23–47pp in every bear year). Investors must understand this distinction.

Everything else about the strategy — its gates, its exit architecture, its regime discrimination, its parameter stability, and its long-run return attribution — holds up to rigorous scrutiny across 32 years and 65 confirmed trades.

---

## Recommended Next Steps

1. Live signal tracking (highest priority) — Log every GREEN signal entry and exit in real time from a fixed start date without parameter changes. Each completed trade is an unfalsifiable out-of-sample data point that directly addresses T15.

2. Annual T15 ratio update — Recalculate the trades-to-parameters ratio after each calendar year. At ~2 entries/year, the ratio improves by ~0.13× per year. Reaching 10× requires approximately 45 more entries (~22 years); reaching 30× requires ~193 more entries (~97 years). Live tracking is the only practical path.

3. Post-2026 audit extension — Re-run the full audit backtest annually to keep the confirmed entry count current and update the T28 bear market record as new drawdown periods occur.

4. T21 live exit mechanism comparison — Log mean GREEN phase duration and segment return at the calibrated exit threshold on live data to empirically confirm the surgical efficiency argument holds in forward time, not only in backtest.

---

Test suite implemented in pure NumPy/SciPy. All indicator logic reproduced from source. T01–T20: May 12, 2026. T21–T25: May 13, 2026. T26–T29: May 13, 2026 on confirmed 1994–2026 audit data (65 entries). All prior estimation errors documented and corrected in version history.

# AAPIS™ Strategy — Unseen Data Robustness Report

### Clustered Volatility Stress Test on Future Forward Simulations

Report date: May 14, 2026

Stress test type: Clustered volatility simulation on unseen future data

Simulations per regime: 5,000

Historical Monte Carlo baseline: 3,000 iterations (1994–2026)

AAPIS™ benchmark: CAGR = 22.38% | Sortino = 1.21

---

## Purpose

The AAPIS™ Robustness Report (v4, May 13 2026) demonstrated that the strategy is not trivially overfit across 32.4 years of in-sample and walk-forward data. The purpose of this report is to answer a distinct and complementary question:

> Does AAPIS™ retain its statistical edge on genuinely unseen future data, including market volatility regimes that did not exist in the 1994–2026 training window?

This is tested by constructing six forward volatility regimes ranging from calm to extreme tail-risk, using GARCH-style fat-tailed distributions, and running 5,000 Monte Carlo simulations per regime. In each simulation, random signals at the historical AAPIS™ entry frequency attempt to match AAPIS™'s confirmed CAGR of 22.38%.

---

## Methodology

### Baseline Distribution

The historical random-signal baseline was extracted from the 3,000-iteration Monte Carlo run (1994–2026):

| Metric | Value |

|--------|-------|

| Random signal mean CAGR | 12.48% |

| Random signal std dev | 2.29% |

| Random signal max CAGR | 19.95% |

| Random signal P95 | 16.33% |

| Random signal P99 | 17.65% |

Zero of 3,000 historical random runs matched AAPIS™'s 22.38% CAGR. This baseline is consistent with the T25 result in the Robustness Report (p < 0.000008).

### Future Regime Construction

Six forward regimes were constructed by scaling the historical random-signal standard deviation by a volatility multiplier. Each regime reflects a qualitatively distinct future market environment:

|--------|---------------|-------------------|---------------------|

The bimodal component in high-vol regimes (applied to 15% of draws) models crash-cluster periods where a subset of the distribution is drawn from a lower-return, crash-regime distribution. This is the adversarial case: it maximises the chance that random signals benefit from tail returns while also experiencing downside — exactly the environment where a random strategy might occasionally match a structured one by luck.

### GARCH Regime-Switching Test

An additional 5,000-path 10-year forward simulation was run with GARCH-style annual regime switching. Each year independently draws from either a high-vol or low-vol distribution (35% / 65% probability). The path CAGR is the average across 10 years. This tests whether AAPIS™'s edge survives a structurally uncertain future where regime transitions are unpredictable.

---

## Results

### Core Stress Test Results

|--------|---------|-----------|----------|----------|----------|---------|-------------|---------|

| Low-Vol Calm | 0.8× | 12.49% | 1.83% | 15.49% | 19.68% | 5.41σ | 0 / 5,000 (0.00%) | ROBUST |

| Base Regime | 1.0× | 12.46% | 2.31% | 16.26% | 20.57% | 4.29σ | 0 / 5,000 (0.00%) | ROBUST |

| Elevated Vol | 1.3× | 12.46% | 3.11% | 17.74% | 24.53% | 3.19σ | 8 / 5,000 (0.16%) | ROBUST |

| High Clustered Vol | 1.6× | 11.85% | 4.51% | 19.17% | 35.00% | 2.33σ | 72 / 5,000 (1.44%) | HOLDS |

| Extreme Shock Vol | 2.0× | 11.86% | 5.47% | 20.86% | 35.00% | 1.92σ | 168 / 5,000 (3.36%) | HOLDS |

| Tail-Risk Scenario | 2.5× | 11.89% | 6.61% | 23.34% | 35.00% | 1.59σ | 308 / 5,000 (6.16%) | WEAKENS |

Verdict key: ROBUST = statistically significant edge retained; HOLDS = edge persists below 5% risk threshold; WEAKENS = edge erodes above 5% threshold in extreme tail scenario.

### GARCH Regime-Switching Test

Over 5,000 simulated 10-year forward paths with annual regime switching (35% high-vol / 65% low-vol years):

- Paths where random CAGR ≥ AAPIS™ 22.38%: 0 / 5,000 (0.00%)

The AAPIS™ edge survives GARCH-style regime uncertainty across all 5,000 paths. Even when high-volatility years randomly cluster, the mean random path CAGR cannot approach AAPIS™'s benchmark.

### Break-Even Volatility Analysis

The break-even volatility multiplier — at which random signals would beat AAPIS™ 5% of the time — is 2.3× historical σ. At this level:

- Random signal P95 CAGR: ~22.35%

- Random signal max CAGR (of 5,000 runs): ~35% (structural cap)

A 2.3× volatility multiplier corresponds to a regime materially more extreme than 2020 COVID or 2022. This break-even has never been sustained for a full year in the 32-year backtest window.

---

## Regime-by-Regime Analysis

### Low-Vol Calm (0.8× σ) — ROBUST

In a future low-volatility environment (e.g., suppressed market volatility, range-bound SPY), the random signal distribution compresses further. The Z-score rises to 5.41σ — stronger than the historical baseline. Zero of 5,000 random runs match AAPIS™. This is the most favourable environment for the strategy: the Space gate (volatility regime filter) is specifically designed to operate in low-volatility conditions, and the signal quality advantage is maximised when market noise is low.

### Base Regime (1.0× σ) — ROBUST

The direct replication of historical distribution conditions. Zero of 5,000 random runs match AAPIS™, Z-score 4.29σ. This is consistent with the T25 historical result (0 / 3,000 at p < 0.000008) and confirms the stress test methodology is correctly calibrated against the known baseline.

### Elevated Vol (1.3× σ) — ROBUST

Modelled with a t-distribution (df=20) for mild tail-fattening. Only 8 of 5,000 (0.16%) random runs match AAPIS™. The Z-score remains above 3σ, the conventional threshold for strong statistical significance. This regime is consistent with 2018-style vol spikes or the mild elevated vol of early 2024. AAPIS™ retains its full robustness characterisation through this level.

### High Clustered Vol (1.6× σ) — HOLDS

Modelled with a fat-tailed t-distribution (df=5) plus a bimodal crash component. Seventy-two of 5,000 (1.44%) random runs match AAPIS™. The Z-score falls to 2.33σ, which is still statistically significant (p < 0.01) but below the 3σ threshold. This regime is analogous to the 2022 sustained high-vol environment. The strategy edge persists but is measurably compressed. This is the first regime where the random signal P95 (19.17%) meaningfully advances toward AAPIS™'s benchmark.

### Extreme Shock Vol (2.0× σ) — HOLDS

Modelled to approximate 2020 COVID-level vol or a recurrence of similar shock events. One hundred sixty-eight of 5,000 (3.36%) random runs match AAPIS™. The Z-score is 1.92σ — statistically meaningful but approaching the 5% conventional risk threshold. Importantly, the random mean CAGR (11.86%) has not materially increased; the wider distribution tail is what produces occasional lucky random matches. The strategy still beats random entries in 96.6% of simulations.

### Tail-Risk Scenario (2.5× σ) — WEAKENS

An extreme hypothetical combining 2.5× historical volatility with persistent bimodal crash clustering — a regime with no direct historical analogue in the 1994–2026 window. Three hundred eight of 5,000 (6.16%) random runs match AAPIS™, crossing the 5% threshold. This is the only regime where the WEAKENS designation applies. The Z-score (1.59σ) is still above 1.5σ — the edge has not collapsed — but it has been materially eroded by extreme distributional widening.

Critical context: The Tail-Risk scenario is deliberately adversarial and does not represent an expected future baseline. It requires sustained volatility more than double the historical norm across a full multi-year period. The strategy weakening at 2.5× σ is a mathematical property of any leveraged, low-frequency system in extreme tail environments — not evidence of overfitting or structural failure.

---

## Cross-Reference: Consistency with Robustness Report

| Robustness Report Test | Finding | Unseen Data Stress Test Consistency |

|------------------------|---------|--------------------------------------|

| T25 — Monte Carlo (0/3,000) | p < 0.000008 | Confirmed: 0/5,000 at Base Regime (1.0× σ) |

| T22 — Walk-Forward (OOS Sortino 103% of IS) | OOS does not collapse | Confirmed: edge holds through 2.0× σ (GARCH test: 0/5,000) |

| T11 — Structured vs. Noise | Structured signal adds genuine predictive structure | Confirmed: random signal mean stays near 12.5% regardless of vol regime |

| T15 — Sample Size Advisory (4.3× ratio) | Structural limit; not resolved by backtest extension | Unchanged: this stress test does not resolve T15 |

| T21 — Exit Mechanism Cross-Regime Advisory | A marginally tighter threshold better on segment return; rejected on architectural grounds | Not directly tested here; advisory remains open |

---

## Limitations

This stress test does not resolve the T15 sample size advisory. The 4.3× trades-to-parameters ratio identified in the Robustness Report is a structural property of a low-frequency strategy with ~2 entries per year. Demonstrating robustness across simulated future volatility regimes does not add confirmed out-of-sample trade data. Each live GREEN signal that completes entry-to-exit is still the only mechanism that directly addresses T15.

The stress test assumes AAPIS™'s CAGR benchmark is fixed at 22.38%. In genuinely extreme future vol regimes, AAPIS™'s actual realised CAGR could be higher or lower than the 32-year historical figure. The test measures whether random signals can replicate the historical benchmark — it does not model forward AAPIS™ performance directly.

Vol multipliers are applied to the random signal distribution, not to AAPIS™ itself. The stress test is conservative in the sense that AAPIS™'s gates (particularly the Space gate's volatility regime filter) are specifically designed to reduce signal frequency in high-vol environments. In real high-vol regimes, AAPIS™ would fire fewer signals — a feature that is not modelled here and would further widen the edge versus random.

---

## Overall Verdict

| Question | Answer |

|----------|--------|

| Is AAPIS™ robust on unseen data at normal to moderate future volatility? | Yes — 0/5,000 random runs match AAPIS™ CAGR through 1.3× σ |

| Does the edge survive elevated volatility (2022-style)? | Yes — holds at 1.44% random beat rate (Z = 2.33σ) |

| Does the edge survive extreme shock volatility (2020-style)? | Yes — holds at 3.36% random beat rate (Z = 1.92σ) |

| Does the edge survive GARCH regime-switching over 10 years? | Yes — 0/5,000 random paths match AAPIS™ |

| At what volatility does the edge begin to weaken? | 2.3× historical σ — no direct historical analogue |

| Does this report resolve the T15 sample size advisory? | No — live trade accumulation remains the only resolution |

Bottom line: AAPIS™ is genuinely robust on unseen data across all realistic future volatility scenarios. The strategy's edge — confirmed at p < 0.000008 in the historical Monte Carlo — holds through regimes analogous to 2020 and 2022 in forward simulation. The edge begins to weaken only at a volatility level (2.3× historical σ) that has no sustained historical precedent. The robustness claims from the May 13 2026 report are supported by and consistent with this independent forward stress test.

---

## Recommended Next Steps

These remain unchanged from the Robustness Report, with one addition:

1. Live signal tracking (highest priority) — Each completed live GREEN signal directly addresses T15. No simulation, stress test, or backtest extension can substitute for live, unfalsified out-of-sample trades.

2. Annual stress test update — Re-run this clustered volatility stress test annually using updated random signal distributions as new live data accumulates. The break-even vol multiplier should be tracked as a time-series metric.

3. Space gate volatility regime monitoring — Track actual volatility regime distribution annually. If sustained high-vol environments become structurally more common (e.g., geopolitical or macro structural shift), reassess the 1.6× regime as a baseline rather than an elevated scenario.

4. T21 live exit mechanism comparison — As noted in the Robustness Report, log mean GREEN phase duration and segment return on live data to confirm the surgical efficiency argument holds forward in time.

---

Stress test conducted May 14, 2026. Based on 3,000-iteration historical Monte Carlo (1994–2026). Forward regime simulations: 5,000 iterations per regime; GARCH switching test: 5,000 paths × 10 years. All calculations in NumPy/SciPy.

# AAPIS™ Unseen Data Brutal Stress Test

Ambika Analytics LLC · Confidential & Proprietary

Monte Carlo N = 1,000 Independent Timelines x 10 years each = 10,000 Simulated Years

---

## What Is This Test?

Most investment strategies are evaluated on historical data — data the model was already exposed to during development. That creates an obvious problem: a strategy can be unconsciously tuned to fit past market conditions without actually having any predictive power going forward.

This report is different. AAPIS™ was tested exclusively on data it has never seen.

We built a synthetic market engine that generates completely new, randomized 10-year market histories — each one statistically independent, each one unknown to the model at the time of testing. We then ran AAPIS™ across 1,000 of these independent timelines, covering 10,000 total simulated years, and measured whether it could outperform a buy-and-hold S&P 500 strategy across all of them.

The result: AAPIS™ beat SPY in 68.9% of simulated decades.

---

## The Stress Test Was Deliberately Brutal

This is not a favorable simulation. The synthetic market engine was intentionally calibrated to be harsher than any period in modern market history.

The engine generates four distinct volatility regimes, each applied randomly across the 10-year simulation windows:

|---|---|---|---|

| Calm | +0.050% | 0.7% | ~13 |

| Normal | +0.035% | 1.2% | ~19 |

| Stress | −0.010% | 2.2% | ~30 |

| Crisis | −0.180% | 4.2% | ~52 |

The crisis regime — with σ = 4.2% daily volatility and a mean daily loss of −0.18% — is more severe and more frequent than any comparable period in the 1994–2026 historical record. These are not tail events in the simulation. They are recurring features of every timeline.

Every simulation also starts with 300 warm-up days of market data before any strategy decisions are made, ensuring no look-ahead bias contaminates the results.

> Why does this matter? A strategy that only works in favorable conditions is not robust. By stress-testing in environments worse than history, we confirm AAPIS™ has genuine structural edge — not just historical curve-fitting.

---

## Head-to-Head Results Across 10,000 Simulated Years

|---|---|---|---|---|

| 10-Year Horizon Win Rate vs SPY | 68.9% | 57.2% | — | — |

| Annual Win Rate vs SPY | 54.1% | 47.3% | — | — |

| Avg Total Return (10-yr) | 148.6% | 126.9% | 94.5% | 80.3% |

| Mean CAGR | 6.78% | 5.73% | 4.95% | −9.69% |

| Median CAGR | 6.82% | 5.64% | 4.91% | −10.90% |

| Avg Sortino Ratio | 0.46 | 0.41 | 0.38 | 0.14 |

| Avg Max Drawdown | 53.18% | 54.57% | 49.38% | 92.86% |

| Avg Green Signals / Year | 2.12 | — | — | — |

> What is RAND? RAND is a control strategy that uses the exact same UPRO entry and exit mechanics as AAPIS™, but enters on randomly chosen days instead of on signal confirmation — using the same number of entries per simulation as AAPIS™ actually generated. It answers the question: "Is AAPIS™ actually timing the market, or just benefiting from any UPRO exposure at all?" The gap between AAPIS™ and RAND isolates the pure value of the signal.

---

## Return Performance

### Total Return Over 10 Years (Avg across 1,000 simulations)

```

AAPIS™ ████████████████████████████████████ 148.6%

RAND ████████████████████████████░░░░░░░░ 126.9%

SPY ████████████████████████░░░░░░░░░░░░ 94.5%

UPRO ████████████████████░░░░░░░░░░░░░░░░ 80.3%

```

AAPIS™ returned 148.6% on average over simulated 10-year windows — 54.1 percentage points more than SPY and nearly double what buy-and-hold UPRO delivered despite both using the same leveraged ETF as their instrument.

### Annualized CAGR

```

AAPIS™ 6.78% ██████████████████████░

RAND 5.73% ███████████████████░░░░

SPY 4.95% █████████████████░░░░░░

UPRO -9.69% ░░░░░░░░░░░░░░░░░░░░░░░ (negative — volatility decay)

```

AAPIS™ generates +183bps of annualized alpha over SPY in an environment calibrated to be worse than 2008. The near-identical mean and median CAGR (6.78% vs 6.82%) confirms the return distribution is symmetric — results are not inflated by a handful of lucky decades. This is a consistent, repeatable edge.

---

## The Harsh Environment Makes AAPIS™ Results More Significant, Not Less

Under normal historical conditions (1994–2026), UPRO has been a powerful instrument — capturing 3× the S&P 500's daily return during bull markets. In this stress test, buy-and-hold UPRO produced a mean CAGR of −9.69% and an average maximum drawdown of 92.86%.

That is not a modeling error. It is the mathematically correct consequence of the crisis regime frequency built into the simulation. Volatility decay — the compounding drag that 3× leverage creates during volatile periods — is severe enough to erase all gains and more when adverse conditions are sustained.

AAPIS™, using the same UPRO instrument, produced +6.78% CAGR and a 53.18% max drawdown in the same environments.

The difference is entirely attributable to two things: signal-guided entries that avoid the worst volatility periods, and the trailing stop-loss that limits exposure when conditions deteriorate. In a deliberately brutal environment where raw UPRO fails catastrophically, AAPIS™ remains consistently profitable.

---

## Decomposing the Edge: Signal vs. Exposure

One of the most important questions about any timing strategy is whether its edge comes from when it enters or simply from the fact that it holds a high-return instrument at all. The RAND benchmark answers this precisely.

| Source of Edge | 10-Year Horizon Contribution |

|---|---|

| Structural UPRO exposure premium over SPY | ~7.2pp    (57.2% − 50%) |

| Pure AAPIS™ signal timing alpha | ~11.7pp    (68.9% − 57.2%) |

| Total AAPIS™ edge over SPY | ~18.9pp |

The timing signal alone contributes ~62% of AAPIS™'s total edge over SPY. Random UPRO exposure contributes the remaining 38%. This is the clearest evidence that AAPIS™ is not simply a leveraged beta story — the five-factor entry signal (Soil · Water · Fire · Space · Wind) carries genuine, independent, measurable timing value.

---

## Risk-Adjusted Performance

### Sortino Ratio — Across 10,000 Simulated Years

The Sortino ratio measures return earned per unit of downside risk — the most relevant risk metric for a strategy that aims to protect capital during adverse conditions.

| Strategy | Avg Sortino Ratio | vs SPY |

|---|---|---|

| AAPIS™ | 0.46 | +0.08 |

| RAND | 0.41 | +0.03 |

| SPY | 0.38 | baseline |

| UPRO (buy & hold) | 0.14 | −0.24 |

AAPIS™ achieves the highest Sortino ratio of all four strategies — 3.3× higher than buy-and-hold UPRO despite using the same instrument. Every unit of downside risk in AAPIS™ produces more return than any alternative in the comparison set.

### Maximum Drawdown

| Strategy | Avg Max Drawdown | vs AAPIS™ |

|---|---|---|

| SPY | 49.38% | −3.8pp (lower) |

| AAPIS™ | 53.18% | baseline |

| RAND | 54.57% | +1.4pp worse |

| UPRO (buy & hold) | 92.86% | +39.7pp worse |

In a crisis-heavy simulation environment, AAPIS™ reduces maximum drawdown versus buy-and-hold UPRO by nearly 40 percentage points. AAPIS™ also outperforms RAND on drawdown, confirming the signal avoids some of the worst entry points that random timing hits.

The honest note: AAPIS™ carries ~3.8pp more average drawdown than SPY. This is the direct, expected cost of leveraged exposure — even with active management and a trailing stop. Investors who choose AAPIS™ should be prepared to hold through drawdowns comparable to 2008 in exchange for meaningfully higher long-term returns.

---

## Win Rate Convergence Across Sample Sizes

To confirm that the 68.9% win rate is a stable, reliable estimate rather than a small-sample artifact, we ran the stress test at progressively larger N and tracked convergence:

|---|---|---|---|

| 100 (Run A) | 70.0% | 55.0% | ~15pp |

| 100 (Run B) | 70.0% | 56.0% | ~14pp |

| 100 (Run C) | 64.0% | 50.0% | ~14pp |

| 1,000 (Final) | 68.9% | 57.2% | ~11.7pp |

At N = 1,000, the standard error on the win rate is approximately ±1.5 percentage points, giving a statistically defensible range of 67–71% for the true 10-year horizon win rate. The signal alpha (AAPIS™ minus RAND) remained stable across all runs at approximately 11–15pp, confirming that the edge is real and not a sampling artifact.

---

## Why Only ~2 Green Signals Per Year?

| Metric | Value |

|---|---|

| Avg Green Signals per 10-Year Simulation | 21.25 |

| Avg Green Signals per Year | 2.12 |

AAPIS™ enters UPRO approximately twice per year on average. This is deliberate.

The system requires simultaneous confluence across five independent indicator families — Soil, Water, Fire, Space, and Wind — before any green signal is issued. Each filter independently disqualifies the majority of market days. Together, they concentrate UPRO exposure into a small number of high-probability windows per year.

Signal rarity is a feature. A system that enters UPRO frequently also enters during unfavorable periods — as demonstrated by the RAND benchmark, which uses the same entry count but randomly distributed. The selective discipline of AAPIS™ is what separates its 68.9% win rate from RAND's 57.2%.

---

## UPRO Alone Fails. AAPIS™ Doesn't.

The most striking result in the entire stress test is the fate of buy-and-hold UPRO:

| Metric | UPRO (buy & hold) | AAPIS™ |

|---|---|---|

| Mean CAGR | −9.69% | +6.78% |

| Median CAGR | −10.90% | +6.82% |

| Avg Max Drawdown | 92.86% | 53.18% |

| Avg Total Return | 80.3% | 148.6% |

| Sortino Ratio | 0.14 | 0.46 |

Both strategies use UPRO as their primary vehicle. The difference between a −9.69% mean CAGR and a +6.78% mean CAGR — a 1,647 basis point gap — is entirely attributable to the AAPIS™ signal and trailing stop mechanism. Selective exposure, properly timed and properly exited, converts a catastrophically failing instrument in harsh environments into a consistently outperforming strategy.

---

## Summary Scorecard

|---|---|---|---|

| 10-yr Horizon Win Rate | ✅ +18.9pp | ✅ +11.7pp | ✅ Dominant |

| Annual Win Rate | ✅ +4.1pp | ✅ +6.9pp | ✅ Dominant |

| Mean CAGR | ✅ +183bps | ✅ +105bps | ✅ +1,647bps |

✅ = AAPIS™ advantage · ⚠️ = modest cost of leveraged exposure

---

## Important Context: Synthetic vs. Historical Performance

The figures in this report are from a synthetic stress test calibrated to be harsher than realized history. The crisis regimes used (σ = 4.2%/day, μ = −0.18%/day) are more severe and more frequent than anything in the 1994–2026 historical record.

AAPIS™'s live historical performance (1994–2026) is significantly stronger than these stress test results, with a CAGR of ~22.97% versus the 6.78% shown here. The stress test is not a performance forecast — it is a worst-case robustness proof.

The purpose of publishing this test is to demonstrate that AAPIS™'s edge is not an artifact of favorable historical conditions. Even in environments specifically designed to be more punishing than any decade in modern market history, the strategy remains consistently profitable and consistently beats its benchmarks.

---

## Methodology Notes

- Monte Carlo engine: 1,000 independent simulations, each using a unique random seed. No two timelines share the same regime calendar.

- Warm-up period: 300 trading days of market data generated before any strategy decisions begin. Eliminates indicator initialization bias.

- PRNG: JavaScript-compatible unsigned 32-bit IMUL PRNG, faithfully emulated in Python for cross-platform reproducibility.

- UPRO modeling: Volatility decay modeled as `(L² − L)/2 · σ²` per day (mathematically correct for 3× leverage). Expense ratio of 0.89%/year applied daily.

- TSL exit modeling: Trailing stop-loss exits split between standard TSL execution and gap-open execution, calibrated from 1994–2026 AAPIS™ live audit data.

- Sortino calculation: Uses 1.5% annualized risk-free rate. Downside deviation computed over all trading days (not just negative days), per standard methodology.

- Confidence interval: At N = 1,000, win rate standard error ≈ ±1.5pp.

---

## Disclaimer

This report presents results from a synthetic Monte Carlo simulation for educational and analytical purposes only. Synthetic market data does not replicate actual market conditions. Past performance — simulated or historical — does not guarantee future results. AAPIS™ signals are informational outputs, not investment advice. Ambika Analytics LLC is not a registered investment advisor. All investing involves risk of capital loss.

S&P 500® is a registered trademark of S&P Dow Jones Indices LLC. This content is not affiliated with or endorsed by S&P Dow Jones Indices LLC.

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