The method is open. The calibration is sealed.
A coherence finding is only worth what its method of proof can withstand. This page discloses everything a due-diligence desk needs to trust the work — the mathematical class, the leak-free construction, the out-of-sample and permutation inference, and the negative results — while marking the single boundary we hold closed: the calibrated indicator constellation. None of what is disclosed here lets a reader reproduce the estimator; all of it shows that our rigor is harder to fake than our formula.
A complex order parameter, public since 1975
The estimator belongs to a well-known family. Markets are treated as ensembles of coupled units and read through the Kuramoto order parameter — canonical in physics since Kuramoto (1975) and widely applied in econophysics. There is no secret in the class itself; naming it is what makes the work legible to a quant.
For each asset and time t, seven strictly-trailing standardized indicators are mapped to phases and compressed into a single complex number. Its modulus is the coherence “hub” R; its argument is the collective “aim” ψ.
Harmonic-1 order parameter Z₁ = (1/N) · Σ exp( i · θ_k ) = R · e^(iψ), R = |Z₁| ∈ [0,1], N = 7
We adopt the base harmonic n = 1, selected by a pre-registered head-to-head against n = 2 and a golden-ratio harmonic (the latter a clean null on both the harmonic and horizon-scaling axes). Windows are fixed in bars, not calendar time, so the operator is identical across sampling scales. The two falsifiable questions the two components answer:
Does R forecast size?
Whether the hub predicts forward realized volatility, after a horizon-matched trailing-vol control and a nonlinearity (spline) control.
Does ψ forecast sign?
Whether the aim predicts the direction of the next move — tested as a null against a powered momentum positive-control on the same series.
What we disclose, and what stays sealed
The edge was never the idea — coherence-on-markets is googleable. The edge is the calibrated constellation: which indicators, at which look-backs, normalized and phase-mapped in which way. We name that the boundary exists and that it is fixed, not tuned to outcome; we do not enumerate it. Naming what you protect is itself a credibility signal.
- The mathematical class. Kuramoto order parameter, public since 1975.
- The phase map & harmonic. θ = π(1 + tanh z); base harmonic n = 1, windows fixed in bars.
- The inference protocol. Walk-forward / chronological split-half OOS, block-permutation nulls at the autocorrelation length, bootstrap-CI counting under multiplicity.
- Data provenance discipline. Public inputs only; deterministic pipeline given inputs.
- Leak-free construction. Strictly-trailing fields; horizon-matched controls.
- The negative results. Reported with equal weight (see §5).
- The seven-indicator constellation. The specific trailing indicators and their composition.
- Look-back windows. The exact bar lengths per indicator and substrate.
- Normalization & phase parameters. The z-scoring scheme and the fixed nonlinearity control (sealed).
- The reference implementation. The committed code and era-segmented backtest.
- Status: fixed and pre-registered — not fit to results. Disclosed only under agreement (see §6).
This is the standard posture of serious quantitative shops: the frame is open, the parameters are not. A page that shows rigorous protocol while explicitly marking the IP line reads as more serious than a vague “proprietary AI” black box — it signals there is something worth protecting and that we know how to protect it.
The controls that do the load-bearing work
The cheap way to manufacture a coherence result is to let the predictor secretly re-encode information the target already contains. Two controls are designed specifically to break that.
Horizon-matched trailing-vol
Trailing realized volatility is computed over the same horizon as the target — never longer — so the hub cannot “predict” volatility by re-encoding clustering the trailing model already holds.
5-knot spline nonlinearity
A spline in trailing volatility removes the “Daido artifact”: a raw R→vol link can be R acting as a disguised nonlinear function of trailing vol. What survives the spline is the genuine increment.
Reported effect sizes are therefore partial correlations — the incremental signal after these controls, not a raw correlation. They are modest by construction; that is the point. A small, robust partial that survives the gauntlet is worth more than a large raw number that dissolves under scrutiny.
What a due-diligence desk actually scrutinizes
None of the following lets anyone reproduce R — and all of it is what makes the method of proof hard to fake.
Pooled cross-unit correlations are provisional until confirmed inside units; pooling across life-stages or regimes can manufacture structure absent in any single unit.
Nulls are drawn by permuting blocks at ≈ the autocorrelation length, because i.i.d. permutation over-rejects on autocorrelated series.
Controls fit on the first half, tested on the held-out second half; sign-agreement across halves is reported. “Pooled-significant but time-unstable” is an admissible, reported verdict.
When scanning K units or harmonics, we count how many bootstrap CIs exclude zero — not whether any single cell reaches nominal significance.
The direction-null is only credible while a momentum positive-control fires on the identical series. A null with a dead control is uninformative; a null while the control fires is a demonstrated wall.
Once in-sample falsification passes, we stop re-slicing the same era; only unseen forward data de-risks further. A live 0DTE forward-collector is the pre-registered test now running.
Magnitude is lawful; direction is sovereign
Every figure below is from the Scalar Flower working paper (June 29, 2026) and survives the full gauntlet of §3–4. The hub→forward-volatility partial correlation is positive in every substrate and ordered by market depth.
| Substrate | Partial r | Units | Positivity | Chrono sign-agree | Verdict |
|---|---|---|---|---|---|
| Equities + bonds + commodity ETFs | +0.062 | 13 | 100% | 92% | Time-stable |
| FX major pairs | +0.054 | 5 | 100% | 100% | Deep, liquid |
| Crypto (27-coin, 5m→1D) | +0.049 | 27 | 96% | 48% | Active research frontier |
| Belief / prediction markets | +0.022 | 123 | — | ~50% | Active research frontier |
Pooled significance clears block permutation at p < 0.003. Cross-asset daily confirmation: metals (silver +0.16, copper +0.13, gold +0.09; p < 0.01), 10/10 G10 FX crosses positive (mean +0.078), 4/4 Treasury tenors positive. A cross-sectional synchronization measure over 27 coins predicts forward 24h index volatility incremental to the priced correlation premium (r ≈ +0.09, p = 0.003, OOS-stable across time halves and disjoint baskets); flagged states precede ~1.6× volatility elevation at 1 day. Source: Scalar Flower working paper, Table 1 & §5.3.
On the “active research frontier” tag. Crypto and belief are not closed null findings. The lawful magnitude channel is present in both (positive pooled r, p < 0.003 pooled), but in these young, reflexive markets the tradeable edge is time-unstable and is the subject of ongoing, resourced investigation — regime-conditioned recalibration, era-segmented modeling, and short-dated tests. The deep-market legs (equities, FX, rates, metals) are the settled tier; crypto and belief are where we are actively deploying effort to convert a confirmed scientific structure into a robust economic signal.
The direction null, at equal weight
| Substrate | R → direction | Momentum control → direction | Control p |
|---|---|---|---|
| Equities | −0.010 | −0.025 | ≤ 0.001 |
| Crypto | +0.005 | −0.020 | ≤ 0.001 |
| Belief | −0.028 | −0.114 | ≤ 0.001 |
| Bitcoin (sharpest) | −0.0009 | −0.247 | ≤ 0.001 |
Sovereignty index S ≈ 0.95–0.99. The Bitcoin case is the cleanest demonstration: in the most reflexive substrate, where momentum's directional pull is maximal, coherence grabs essentially none of the direction. Source: working paper, Table 2 & §5.2.
The null is the product’s spine — read it first
Direction is not predictable, by construction — demonstrated against a powered control, not assumed. And against traded 30-day implied volatility the signal does not beat the market in any asset class (vol-buyer win rates 26–36%); the equity volatility-risk-premium sign replicates but tradeable magnitude does not (t = 1.56, NS). Implied volatility already prices the coherence level, its fluctuations, and its repricing dynamics. Standalone option-relative alpha at standard tenors is not established and remains gated on a short-dated (0DTE) test now collecting.
What is established: the magnitude channel beats naive volatility targeting as an overlay, and the crypto magnitude channel is mispriced relative to options (R→VRP Spearman: BTC +0.32, ETH +0.23, pooled +0.27, p = 0.0014; strategy t > 2, cost-robust). Source: working paper, §5.5 & Abstract.
The live demo numbers are illustrative
Why include it at all: the demo answers the “can you show it actually runs?” question without exposing the calibrated constellation. The synthetic corpus is seeded and deterministic, so the math is genuine even though the data is illustrative. Verified figures live only in the table above; demo figures live only in the demo.
A clear ladder: public → NDA → license
The withholding is a deal structure, not a refusal. Here is exactly what opens at each tier.
- This method page & the working paper
- The order-parameter class & phase map
- Full validation protocol
- The interactive illustrative demo
- All negative results
- No calibrated constellation
- No reference implementation
- Live results on real market data
- Era-segmented backtest walkthrough
- Full inference logs & CI counts
- 0DTE forward-collector readouts
- Architecture review with the research team
- Constellation still sealed
- The committed indicator constellation
- Look-back windows & normalization
- Reference implementation & pipeline
- Risk-overlay integration support
- Ongoing research collaboration
The fear — “if we explain it we invite a clone” — is addressed by the structure: the idea is public, the calibration stays sealed through Tier 2, and only a license opens Tier 3. Sophisticated capital funds demonstrated rigor, not secrecy. All figures on this page trace to the locked corpus / working paper; no number is added or changed in web adaptation without re-verification.
In the gap between the HAR win and the IV null
The product is not a directional forecaster and not an options-alpha machine. Its value is precise and defensible: it flags the regimes where a backward-looking HAR realized-volatility baseline becomes structurally too calm — a model-risk signal for desks and risk engines — while honestly conceding that traded implied volatility at standard tenors already prices the move. Science first, alpha second. The honesty is the moat.