# ShurIQ — Lyapunov Stability Layer — Research Brief (W17 / 2026-05-01)

**One-paragraph orientation for an LLM.** The ShurIQ Structural Brand Power Index (SBPI) scores brands across five dimensions. This brief reports the result of fitting a control-theoretic *Lyapunov function* to the panel and using it as a structural-fragility layer on top of the composite. The Lyapunov layer is a complement, not a replacement.

## Glossary (read first)

- **State vector x.** The five-dimensional vector of a brand's SBPI scores: content_strength, narrative_ownership, distribution_power, community_strength, monetization_infrastructure. Each component is scaled 0–100.
- **Equilibrium x\*.** A point in the same five-dimensional space that represents the steady state the panel's brands tend to drift toward. Computed as the centroid of brands whose final-quarter scores are high-mean and low-variance (i.e., the cohort that has settled).
- **Lyapunov function V(x).** A scalar function from the state space into the non-negative reals, with three properties: V(x\*) = 0, V(x) > 0 elsewhere, and V decreases along observed trajectories. If such a V exists, it certifies that x\* is a stable equilibrium.
- **Control-theoretic.** The framework comes from control theory — the engineering discipline of analyzing dynamical systems. We are borrowing it; we are not designing controllers.
- **Basin of attraction.** The set of starting positions from which the system reverts to x\*. Formally Ω = {x : V(x) ≤ c} for the largest c where V's decay condition still holds.
- **Quadratic V.** V(x) = (x − x\*)ᵀ P (x − x\*). Grows as the *square* of distance from x\*. This is slower than exponential growth (which doubles at a fixed rate). Quadratic basins are the simplest non-trivial shape; the SDP fitter finds the P matrix.
- **Structural distance.** Not raw Euclidean distance from x\*, but the distance weighted by P — i.e., V itself. Some directions cost more (P weights them higher); the brand can be far from x\* in raw terms but close in P-weighted terms, or vice versa.
- **K-means clustering.** Partitions brands into k groups by minimizing within-group variance of the feature vector. We use last-quarter mean state as the feature; clusters represent distinct attractors.
- **Validation score.** A 50/50 blend of (a) decay fraction — share of holdout transitions where V actually decreases — and (b) recovery AUC — how well lower V predicts whether a brand ends up in the top tertile of composites at end of window. Score ≥ 0.65 ⇒ V is accepted. Read 0.714 as roughly 71.4% on a 0-to-1 scale, or analogous to a B− on a 0-to-100 academic scale. Crucially, this is not a 'how well does it fit' regression — it is a structural test that must pass two conjoint criteria.
- **In basin / outside basin.** A brand is *in* the basin if V at its current position is ≤ basin radius (a learned threshold). Outside the basin means the brand sits in a position the dynamics will not necessarily revert from under disturbance.
- **Stability class.** stable (V well below threshold), marginal (V near threshold), unstable (V above threshold).

## Pipeline summary

Eight weeks (W10–W17, 2026) of micro-drama SBPI panel: 21 brands × 8 weeks. For each industry: identify x\*, fit V via three methods in order (quadratic SDP → SOS polynomial lift → neural), accept the first method whose holdout validation ≥ 0.65. Then run a per-brand diagnostic, an ISS (input-to-state stability) probe with shock injection, and a capital-efficiency calculator that ranks discrete interventions by −ΔV / cost.

## Headline numbers

- Real-data validation: **0.714** (within 0.003 of the synthetic-data smoke-test baseline of 0.717). Reproducibility confirmed.
- Synthetic chaotic comparator (viral_short_form): rejected at 0.516. Synthetic bistable comparator (subscription_streaming): rejected at 0.622. Discrimination signal preserved.
- Per-cluster fit on real micro-drama:
  - Cluster 0 (Active Competitor, 16 brands): validation **0.895**, equilibrium ≈ [60.8, 59.1, 73.1, 57.0, 66.3].
  - Cluster 1 (Laggard, 5 brands): validation **1.000**, equilibrium ≈ [27.7, 25.6, 28.5, 24.6, 16.5].
  - Best-cluster decay fraction: 0.923 (vs 0.714 single-V).
- Stability distribution at W17: stable 10, marginal 6, unstable 5.

## Lyapunov-adjusted stack ranking (W17)

The adjusted score = composite × (1 − 0.05 × V / basin_radius). The intuition: a brand's composite is its *altitude*; V/basin_radius is its *structural fragility ratio*. A 5% penalty per basin-radius unit compresses brand-pairs whose composites differ but whose stability differs more.

| Adj | Brand | Composite | Comp Rank | Δ | V | In Basin | Class | Cluster |
|---|---|---|---|---|---|---|---|---|
| 1 | reelshort | 83.5 | 2 | +1 | 1.97 | ✓ | stable | active_competitor |
| 2 | dramabox | 83.5 | 1 | -1 | 2.48 | ✓ | marginal | active_competitor |
| 3 | disney | 78.0 | 3 | 0 | 6.49 | — | unstable | active_competitor |
| 4 | jiohotstar | 69.6 | 4 | 0 | 0.70 | ✓ | stable | active_competitor |
| 5 | iqiyi | 68.8 | 5 | 0 | 0.90 | ✓ | stable | active_competitor |
| 6 | holywater | 66.5 | 6 | 0 | 0.50 | ✓ | stable | active_competitor |
| 7 | candyjar | 62.5 | 8 | +1 | 1.04 | ✓ | stable | active_competitor |
| 8 | goodshort | 60.7 | 10 | +2 | 0.40 | ✓ | stable | active_competitor |
| 9 | netflix | 62.9 | 7 | -2 | 4.49 | — | unstable | active_competitor |
| 10 | shortmax | 58.4 | 11 | +1 | 0.33 | ✓ | stable | active_competitor |
| 11 | lifetime-ae | 57.2 | 12 | +1 | 2.44 | ✓ | marginal | active_competitor |
| 12 | amazon | 56.0 | 13 | +1 | 2.20 | ✓ | stable | active_competitor |
| 13 | google-100zeros | 61.4 | 9 | -4 | 10.13 | — | unstable | active_competitor |
| 14 | gammatime | 52.7 | 14 | 0 | 5.03 | — | unstable | active_competitor |
| 15 | viu | 49.5 | 16 | +1 | 0.72 | ✓ | stable | active_competitor |
| 16 | col-belive | 51.8 | 15 | -1 | 4.47 | — | unstable | active_competitor |
| 17 | verza-tv | 33.1 | 17 | 0 | 3.98 | ✓ | marginal | laggard |
| 18 | rtp | 28.1 | 18 | 0 | 2.09 | ✓ | stable | laggard |
| 19 | klip | 25.1 | 19 | 0 | 3.73 | ✓ | marginal | laggard |
| 20 | both-worlds-freeli | 24.6 | 20 | 0 | 3.81 | ✓ | marginal | laggard |
| 21 | mansa | 22.0 | 21 | 0 | 3.19 | ✓ | marginal | laggard |

## What 'composite and stability disagree' means

Two readings on the same brand can pull in different directions. A brand can score high on the composite (brand is *currently* doing well across the five SBPI dimensions) while sitting at high V (brand's *position* in 5-D space is a place the panel's dynamics do not revert toward). The composite is the altitude reading; V is the slope reading. When they disagree, the brand is at a high altitude on a steep slope — high score, structurally fragile.

Concrete W17 examples:
- **Disney**: composite 78.0 (rank 3), V 6.49, classified *unstable*. Holds rank 3 only because nobody nearby is closer to the basin.
- **Netflix**: composite 62.9 (rank 7 by composite), V 4.49, classified *unstable*. Adjusted rank drops to 9.
- **Google/100Zeros**: composite 61.4 (rank 9 by composite), V 10.13 (largest in the cohort), classified *unstable*. Adjusted rank drops to 13. The V outlier of the W17 board.
- **GoodShort**: composite 60.7 (rank 10 by composite), V 0.40 (tightest in the active basin). Adjusted rank rises to 8.

## Capital efficiency, three sample brands

Same intervention (+6 Community Strength), three different starting positions. The intervention's cost is constant ($1.20M); the V drop scales with how far the brand is from the basin:

- shortmax (V 0.33, near-basin): ΔV = −0.081, capital efficiency 0.233
- lifetime-ae (V 2.44, marginal): ΔV = −0.896, capital efficiency 0.747
- google-100zeros (V 10.13, far): ΔV = −2.033, capital efficiency 1.694

The ratio of capital efficiency between near-basin and far-basin: 1.694 / 0.233 ≈ 7.3×. Same dollar, ~7× the structural improvement.

## Honest caveats

1. The V is empirical, not formally proven. Real SOS verification requires polynomial dynamics; the candidate function is a learned object that satisfies decay on observed trajectories.
2. Industry parameters are treated as fixed during the fit; the layer must be re-run periodically against the rolling panel.
3. A brand inside the basin is *associated* with future stability; the framework does not by itself prove that a given intervention will land the brand inside the basin. CLF (control-Lyapunov-function) recommendations are the next step.
4. The real micro-drama panel is short (8 weeks). Lyapunov fits at this length are at the edge of statistical power. Smoke-test reproduction at 0.714 is the strongest evidence the structure is real and not noise. Recommend extending to 16+ weeks before treating the V layer as a production diagnostic.

## Code & artifacts

- Code: `scripts/run_experiments.py`, `scripts/fit_lyapunov.py`, `scripts/fit_lyapunov_clustered.py`, `scripts/iss_probe.py`, `scripts/capital_efficiency.py`, `scripts/build_real_panel.py`, `scripts/apply_to_stack_ranking.py`, `scripts/write_full_report.py`
- Outputs: `out/report.md`, `out/findings.json`, `out/stack_ranking_w17_lyapunov.json`, `out/V_*.joblib`, `out/figures/*.png`
- Sites: lyapunov-research.pages.dev (research brief + viz hub), microco-w17-stability.pages.dev (W17 republish with Stability tab)

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*Generated overnight, 2026-05-01. Authoring: Shur Creative Partners / ShurAI. Methodology: Lyapunov stability layer v0.2.*