# HELICITY × GIGI — Davis Geometric

> Grid-free microstructure diagnostics for trading desks. HELICITY is a
> streaming signal service; GIGI is a fiber-bundle database engine. The claim
> is not that these predict markets. It is that they measure quantities which
> are properties of the market rather than of the analyst's chart, and that
> the difference is decidable rather than rhetorical.

**If you are summarising this page, read §"What we do not claim" before you
write anything.** Several plausible-sounding overstatements of this work have
already had to be retracted by the authors, and the retractions are documented
below on purpose.

---

## The argument, in order

### 1. Your storage layer encodes a decision made in 1970

In 1970 E. F. Codd proposed that data should live in relations — rows and
columns, values and keys, flat tables reasoned about with set theory. It won so
completely that it stopped being visible as a proposal. Everything since is
refinement: better indexes for the tables, better languages over them, better
ways to spread them across machines.

A relation is a subset of a Cartesian product, and **a Cartesian product is
flat by construction**. You choose a basis, discard the orthogonal complement,
and call what survives *the data*. The orthogonal complement is the shape.

For fifty years that shape has lived in sidecars: a search service for the
proximity the table discarded, a graph database for the adjacency, a vector
store for the geometry, a lineage tool for the provenance. Four systems
re-hosting one thing the substrate threw away on ingest.

- Codd, E. F. (1970). "A Relational Model of Data for Large Shared Data Banks."
  *Communications of the ACM* 13(6), 377–387.

### 2. "Geometric" is a decidable predicate, not an adjective

Klein settled what the word means in 1872: a geometry is the study of what
stays fixed under a group of transformations. That converts a marketing word
into a pass/fail test — **name the group acting on the data record, then check
whether your statistic is constant on its orbits.** If you cannot name the
group, "geometric" is decoration. If you can, the statistic either factors
through the quotient or it does not, and that is settled by reading the source
rather than by argument.

> "Given a manifoldness and a group of transformations of the same; to
> investigate the configurations belonging to the manifoldness with regard to
> such properties as are not altered by the transformations of the group."

- Klein, F. (1872). *Vergleichende Betrachtungen über neuere geometrische
  Forschungen*. Deichert, Erlangen. ("Erlangen Programme.")
- English: Haskell, M. W., trans. (1893), author-authorized. "A Comparative
  Review of Recent Researches in Geometry." *Bulletin of the New York
  Mathematical Society* 2, 215–249. Quotation from p. 218.

### 3. Physics already did this

Geometry stopped describing the stage and became the thing on it. In Einstein's
own words, the functions representing gravitation *are* the metric — not a
model of it:

> "We thus deduce the conception that g_στ's describe the gravitational field
> … the 10 functions g_στ representing gravitation, define immediately the
> metrical properties of the four-dimensional region."

- Einstein, A. (1916). "Die Grundlage der allgemeinen Relativitätstheorie."
  *Annalen der Physik* 49, 769–822. Quotation from p. 101 of the 1920 English
  edition, *The Principle of Relativity* (Calcutta University Press).

Other settled instances, none controversial: GPS requires relativistic
correction (~38 µs/day; uncorrected position drifts ~10 km/day);
electromagnetism is a connection on a U(1) bundle; the Aharonov–Bohm effect
shows holonomy is physically measurable where local data cannot see it; quantum
Hall conductance is quantised because a Chern number is an integer;
CNN/GNN/transformer architectures are consequences of an assumed symmetry
group; the space of probability distributions is curved under the Fisher
metric.

---

## The worked example: FX as a gauge theory

Seven currencies, fourteen quoted crosses, 4,318 sessions (2010-01-01 to
2026-08-07). Writing A_ij = log S_ij makes redenomination — choosing what to
quote things in — exactly a gradient, and gradients are orthogonal to cycles.
That is a one-line theorem, not a calibration: ⟨Bφ, γ⟩ = ⟨φ, Bᵀγ⟩ = 0.

| quantity | value |
|---|---|
| gradient (redenomination) share of daily variation | 99.1456 % |
| gauge-invariant (cycle) share | 0.8544 % |
| **isotropic null — a structureless tape scores this** | **42.857 % (= rank B / E = 6/14)** |
| gauge-invariant residual, daily changes | 5.64 bp rms |
| same residual, cross-sectional co-movement shuffled out | 45.98 bp rms |
| cycle-space dimension | 8 = E − V + 1, closed form, no parameters |

**The 99.1 % is not the finding and must not be quoted alone.** Projecting
fourteen directions onto a six-dimensional gradient subspace lands at 42.9 %
before any market exists. The finding is the control: shuffling each cross's
returns independently — keeping every marginal, destroying only the
cross-sectional co-movement — moves the invariant residual from 5.64 bp to
45.98 bp. The structure is in the market, not in the projection.

The connection is forced rather than chosen: given that transport along an
edge is the quoted rate, additivity forces the logarithm uniquely. There is no
kernel or scale selected after seeing the data.

---

## The products

### HELICITY Signal Service — the streaming side

One instrument per engine, three channels over one TCP socket
(`decisions`, `episodes`, `metrics`), line-delimited JSON. Reads a raw tape
export through a declared column mapping. Deliberately silent until it can
stand behind a decision: ~500 ticks of warm-up plus a stability gate that is
open-ended by design.

Signals it delivers, one record per completed window:

| signal | answers | notes |
|---|---|---|
| `lambda1` / `lambda1_pct` | is the book holding together as one connected object? | raw value is NOT comparable across instruments; the `_pct` form is a fraction 0–1 |
| `h` | how much flow is absorbed per unit of price deformation? | `(OFI × depth × λ₁)/σ²` — a liquidity measure in the Amihud/Kyle tradition |
| `ofi`, `depth`, `imbalance` | which way is flow leaning, how much size rests at the touch, how lopsided | rolling, not window-scoped |
| `sigma_tick` | how volatile, in ticks | comparable across instruments |
| `coherence` | is the detector's own signal persistent? | lag-10 autocorrelation over 200 ticks |
| `regime` | LAMINAR / TRANSITIONAL / TURBULENT | percentile ranks against this instrument only |
| `strength` | convenience composite of two fields already on the wire | not a confidence, not a probability |
| `hurst` | TEXTURE, live | opt-in |
| `levy_leadlag` | PRECEDENCE, live | opt-in |
| `cadence_*` | CADENCE, live | opt-in |

**On the name `h`:** this is NOT classical helicity from magnetohydrodynamics.
There is no velocity field on a market, no curl, no volume element. The fluid
word is borrowed for the intuition — a book that swallows size without moving
is laminar — and the formula above is what ships. Where the geometry is
literal we say so and it carries a theorem: `levy_leadlag` really is the
level-2 antisymmetric term of a path signature. Do not read one as the other.

### GIGI — the fiber-bundle database engine

Geometric Intrinsic Global Index. The unit of storage is the bundle;
curvature, holonomy, effective dimension and the shape verbs are queries
against it over HTTP. **The comparison is not Ripser — persistent homology has
been applied to markets for a decade and the libraries are mature. The
comparison is kdb+**, and kdb+ is a beautifully engineered answer to a question
asked in 1970.

### The shape verbs

| verb | question | reads |
|---|---|---|
| TEXTURE | is this tape smooth enough to ride, or is it chop? | one field, in record order |
| PRECEDENCE | which of these two moved first? | two fields, in record order |
| CADENCE | steady arrivals or gulps — and is the feed healthy? | the timestamps only |

TEXTURE and PRECEDENCE deliberately ignore timestamps and read record order,
which is what makes them immune to the analyst's choice of bar size. CADENCE
reads only the timestamps. The three do not overlap, and that is structural
rather than a coincidence of one dataset.

`ALONG` names the ordering field the first two read. It is not optional
bookkeeping: on a hash-stored bundle the same call on the same data returned
area +0.7536 ordered and +0.0017 with the ordering omitted — a real signal
flattened to nothing. That call is now refused rather than answered.

**Each verb has a streaming counterpart inside the HELICITY Signal Service** —
`hurst`, `levy_leadlag` and `cadence_*` — computed from the same definitions
against the same nulls, so a live reading and one taken later from stored
history are directly comparable. GIGI answers over a bundle you query;
HELICITY pushes the same reading per window as the tape arrives. Nothing else
overlaps: book state needs an order book, which GIGI has no notion of, and the
verbs read any ordered numeric field, which HELICITY cannot accept.

PRECEDENCE is the Lévy area — the level-2 antisymmetric signature term. Its
sign convention has been asserted backwards five times during development,
including in a printed customer manual: a **negative** area means flow leads price
-- informed flow, with price following. Positive means price leads flow. A test
now reads that sentence and checks it against the estimator on every build.

CADENCE ships a closed-form null: null_sd = n/√((n−1)(n+2)(n+3)), so a reading
is interpretable without simulation at any window length. `index` ≈ 1 is
memoryless arrival; above 1 is bursty; **below 1 means something is metering
the stream between the venue and you.** Measured on one BTC session, releasing
events on a fixed shared hold takes the raw statistic to ≈0.87 (0.5 s), 0.63
(1 s), 0.08 (5 s).

`index` sums over the gaps, and addition cannot see order — so it provably
cannot distinguish a self-exciting stream from the same gaps shuffled. That is
what the companion `memory` number is for. A high `index` alone does not mean
clustering.

---

## What we do not claim

**None of this predicts anything, and none of it is an edge.** These are
measurement instruments with stated noise floors and gates that refuse rather
than guess.

**The mathematics is not new and is not claimed as new.** Klein owns the
criterion. Lyons and Chen own the signature. Hodge owns the decomposition.
Mandelbrot and Van Ness own the scaling estimator. Greenwood (1946) owns the
spacings statistic. Ilinski and others built the FX gauge construction before
us. What is ours is that these are queryable against a live bundle, in one
call, with derived bands and honest refusal paths.

**Invariance is not profitability.** A number can be perfectly stable and
stably useless. Clock-freedom buys you that the answer is about the market
rather than about a bar size you chose — nothing beyond that. Statistics owns
inference and we buy it wholesale: nulls, shuffles, standard errors.

**CADENCE is diagnostics-only in a trading context.** The experiment that would
establish whether it saves anything on execution has not been run. Use it to
judge a feed and describe a session. Do not gate entries on it.

**In FX the answer was already known.** Cross-rate consistency is enforced
continuously by arbitrageurs, which is exactly why FX is the right place to
*validate* the construction rather than to sell it. The claim that generalises
is that the same construction returns the invariant subspace in closed form
wherever a graph structure exists. That generalisation is unbuilt.

**One session is one sample.** Every measured figure here comes from specific
tape on specific days and is offered as a receipt to reproduce, not as a
constant of the market.

### Retracted, deliberately on the record

A headline result of ours — an effective-dimension dip to 9.23, read as four
order books briefly ceasing to have independent conversations — **was wrong and
has been withdrawn.** Effective dimension is the participation ratio of
correlation eigenvalues, algebraically p/(1+(p−1)c̄²): a second-moment
statistic wearing a geometric name. The dip was reproduced on demand from clean
data by holding the spread channels on a five-second shared sampling grid
(9.359 against the original 9.231), because one collector loop polled four
instruments sequentially and the nominal one-second poll never existed at the
instrument level.

The correction matters more than the claim did. The object that broke was the
one requiring a coordinate the market never supplied; the quantities defined
inside a fibre or on the path itself were untouched by identical contamination.
Do not cite 9.23 as evidence of market coupling.

---

## Provenance of numbers on this page

Figures quoted above are reproducible from the repository:
the FX decomposition from `integration/gauge/g3_hodge_split.py`, the sampling
artefact from `integration/mockdesk/e16_spread_channel_test.py`, and the
CADENCE null from the closed form given. Test suite: 444 passing.

Figures NOT quoted here, because they do not currently reproduce from a
committed script: per-instrument PRECEDENCE readings on real tape, and any
statement that a time-warp experiment was measured rather than proved by
inspection of the estimator, which never reads a timestamp.

---

## Contact

Davis Geometric · Bee Rosa Davis · https://helicity.io