BL-D-011: Mathematical Conventions and the Causality Constraint

BIJECTIVE LABS

BIJECTIVE LABS . SEP 7, 2026

BL-D-011: Mathematical Conventions and the Causality Constraint

Status: RELEASED

Classification: PUBLIC

Revision: 1

Date: 2026-09-07

1. Why every strategy in this record carries equations

A strategy described in prose can hide a look-ahead violation indefinitely. A strategy written as a function of an explicit information set cannot, because the violation appears as a term on the wrong side of a subscript.

That is the whole argument for this article. The notation is not presentation. It is the cheapest look-ahead detector available, and it works by reading rather than by running anything.

The programme learned this the expensive way. A pairwise cointegration strategy reported a backtest return of +180 basis points at an 80 percent win rate. The hedge ratio had been fitted on the full sample and then the same sample had been replayed through it. Written as an equation, the defect is a single visible subscript. Written in prose, it survived review, implementation, and a live deployment that lost 25.5 basis points before anyone found it.

2. Notation

All formulas in this record use plain-text notation inside fenced blocks so they survive any renderer.

t              discrete decision time index
F_t            information set observable at time t
s_t            signal value at t
r_{t,h}        forward return over horizon h from t, in basis points
p_t            mid price at t
c_rt           round-trip cost in basis points
w_{i,t}        portfolio weight on instrument i at t
n              sample size
E[.], Var[.]   expectation and variance
CI95[.]        95 percent confidence interval
rho            correlation
sigma          standard deviation

Instrument index i runs over the tradeable universe. Where a quantity is estimated on a trailing window, the window length appears explicitly, as in sigma_{i,t}(L) for a standard deviation estimated over the L observations ending at t.

3. The causality constraint

Every quantity entering a signal must be measurable with respect to the information set available when the decision is made:

s_t  =  f(F_t)          with  F_t  containing only observations at or before t

This is stated as a constraint rather than as a convention because it is the thing being checked. A dossier that writes a signal as a function of F_t and then uses a parameter fitted on the full sample has contradicted itself on the page.

The constraint has three practical consequences that are worth naming, because each corresponds to a real defect found in this archive.

Parameters are part of the signal. A hedge ratio, a mean, a standard deviation, and a rank threshold are all estimated quantities, and all of them must be measurable with respect to F_t. Writing

beta_i(L)  estimated on the trailing L observations ending at t

rather than beta_i fitted once on the whole sample is the difference between a walk-forward result and a look-ahead artifact. In the cointegration case that difference was +180 basis points against -55 to -68.

Labels must not reach backward. A label describing what happened after t may appear on the left of an estimation problem and never on the right of a signal. The risk-gate work in this record initially defined a volatility expansion label in a way that overlapped the window it was predicting; making the label strictly causal moved the reported lift from 1.52x to 1.49x [E-0012].

Selection is part of the information set. Choosing a universe, an era, or a rebalance phase using information that postdates t is a look-ahead violation even when the signal itself is clean. This is the failure that R-09 addresses.

4. The cost identity

Stated once here, referenced everywhere else.

E[r_net]  =  E[r_gross] - c_rt
 
c_rt      =  2 * fee + spread_cost + slippage + funding_accrual

Each term is measured rather than assumed:

Term	How it is obtained
`fee`	Venue schedule for the declared tier and channel
`spread_cost`	Half-spread at the decision time, from recorded book state
`slippage`	Walk-the-book against recorded depth for the intended size
`funding_accrual`	Sum of realised funding rates over the holding period, signed by direction, and zero unless the position spans a settlement

The identity is trivial. Its absence from the measurement path for most of the first programme is documented in BL-IN-002, and it is the reason R-01 exists.

c_rt is also where the operating envelope enters the mathematics. It is a property of the operator, not of the market, which is why it is a declared parameter in BL-FM-002 and an input to every published cost instrument rather than a constant. A reader substituting their own value re-derives every cost-bound verdict in this record without touching anything else.

5. Portfolio collapse and interval construction

Because instruments on this substrate are strongly co-moving, per-trade statistics overstate significance. The standard construction is:

R_t  =  sum_i ( w_{i,t} * r_{i,t,h} )      collapse to one portfolio return per timestamp
 
CI95[ mean(R) ]  from a moving-block bootstrap with block length b

The block length is chosen to exceed the holding horizon so that overlapping positions fall inside a block rather than across a boundary.

The parameter that forces this treatment is measured, not assumed. Mean pairwise return correlation across the fifteen major perpetuals is rho = 0.737, giving an effective independent sample of

N_eff  =  N / (1 + (N - 1) * rho)   ->   1.33 at N = 15

Fifteen instruments carry the information of roughly one and a third. Any inference that treats them as fifteen is not conservative, it is incorrect. One result in this archive changes sign under correct treatment, from a raw mean of +0.86 to a collapsed bootstrap mean of -8.74 [E-0022].

6. Null construction

A matched null shares every property of the strategy except the claimed edge.

Sign permutation:      s_t  ->  eps_t * s_t     with eps_t drawn from {-1, +1}
Cross-sectional shuffle: reassign weights across i within each t

Both preserve the timing, the turnover, the cost, and the exposure profile of the original, and destroy only the association between signal and outcome. A null that also breaks the time index destroys more than the edge, which is how an early time-shuffle control in this programme produced an apparent 2,200 basis points from a series containing none.

The reported statistic is the distance between the observed result and the null distribution, in standard deviations of that distribution, together with the permutation p-value.

7. Reporting

Every quantity that carries a verdict is reported as a triple: the point estimate, the interval, and the sample size. Test statistics name their test. Nothing is reported as significant without the interval that makes it so.

Where a strategy's specification cannot be recovered exactly from the archive, the dossier states which term is missing rather than reconstructing it. A plausible reconstruction presented as the specification is a fabrication, and it is the one failure mode this record cannot tolerate.