Cognivec · Research · Topology program

The Law of Scalars

Every day of news gets a persistence passport: the full barcode of its topology — how many components exist at each resolution, at what heights they merge, how many loops the day carries, how long they persist, when the main loop is born. Ten numbers that summarize the day's shape completely. We computed the passport for 5,510 consecutive days.

A typical day carries a median of 121 loop intervals in its barcode (persistence entropy ≈ 4.4) — of which only a handful persist long enough to mean anything. The distribution is stable across fifteen years: the shape of a news day is a robust, measurable object.

Histograms: number of H1 loops per day and maximum persistence per day over 5510 days — 5,510 daily passports: loops per day (left) and the persistence of each day's strongest loop (right).

The result: every scalar is market-silent

Each passport feature was tested against next-day GBPUSD return and absolute return, with placebo bands from calendar shifts. The outcome is uniform:

feature	corr with next-day return	95% placebo band
loop count	+0.002	±0.032
persistence entropy	+0.005	±0.031
max persistence	−0.016	±0.028
total persistence	−0.001	±0.029
main-loop birth	−0.011	±0.034
merge height (median)	+0.009	±0.033

All ten features, both targets, all inside the bands. The same verdict held for the Morse-landscape scalars (basins, saddles, linking) and for the transport cost (W1).

Why this is a law, not a failure

Across our entire program the pattern repeats without exception: scalar summaries of news geometry never predict the market; only oriented constructions ever do. Magnitudes — how big, how many, how entangled — are market-silent. Directions — which way the structure points — are where every surviving signal in the FX dossier lives. A day's geometry is loud about what the news is doing and silent about which way prices go, unless you ask an explicitly directional question.

This regularity is itself a research result: it says the market does not price the news day's shape, only its orientation. It also serves as a standing methodological filter — any proposed scalar feature now starts from a strong prior of null.