# 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.

Updated 26 August 2026. The site is a client-rendered single-page app
(hash routes: `#overview`, `#laminar`, `#substrate`, `#circulation`,
`#publications`, `#psychohistory`, `#products`, `#customers`, `#signals`,
`#trading`, `#markets`) — fetching a page URL returns a JS bundle, not
prose. This file is the canonical text surface, and every load-bearing
number on the site renders from a static JSON receipt an agent CAN fetch
directly; the receipt index is at the end of §"The receipts".

---

## 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? | a liquidity measure in the Amihud/Kyle tradition — absorbed flow over realised price deformation, weighted by how connected the book is |
| `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 what ships is an absorption ratio in that tradition, not a
curl. 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.

---

## The substrate: one exact law the tape must obey

Underneath every product above sits an accounting identity, enforced in
integer arithmetic on raw order-by-order exchange tape: every displayed
contract is booked exactly once — boundary, added, cancelled, executed, or
moved — and the residual is zero at every event or the run is dead. No
repair, no interpolation, no partial credit: a tape that breaks trust
raises a typed refusal and the affected span is audited, never scored.

Verified on one ES week: **56,986,293 records → 47,895,657 complete
exchange events, residual 0, zero repairs** (receipt:
`/evidence/ledger_v2_es.json`, produced by `integration/booktop/mbo_ledger.py`,
ledger v2). What the balanced ledger revealed, same receipt:

- Cancellations outnumber executions **4.44 to 1** — 81.4 % of departing
  contracts were withdrawn, 18.4 % traded, 0.27 % honestly unclassifiable.
  A depth chart draws all of these in the same ink.
- **26.7 % of book movement is the same lots relocating** — transport, not
  liquidity appearing or vanishing.
- 99.73 % of departures are classifiable; the top of book is crossed or
  locked in 0.01 % of events.

The site's `#substrate` page teaches the mechanism interactively; the
claim above is the entire law.

---

## The registered experiments — outcomes published either way

The research program runs one-shot pre-registered confirmations: the
analysis plan, gates, decision rule and random seed are committed and
hash-locked BEFORE the confirmation data is acquired; the data is bought
after the lock; the experiment runs once; the outcome publishes whether it
passes or fails. Three outcomes exist, and one is a failure on purpose.

**E21 — direction, validated once (ZN, 14 Aug 2026).** A frozen
directional estimator (sign of net aggressive flow over disjoint
256-trade blocks against the next block's midquote direction) was carried
through a sealed one-run confirmation on 65 previously untouched
ten-year-note sessions. Direction effect θ = +0.01695 with a simultaneous
lower confidence bound of **+0.00539 after every registered penalty**; the
real signal beat all 127 timing-scrambled surrogates (exact-rank
p = 1/128); it outperformed a momentum baseline (outperformance LCB
+0.116); honest size: ≈50.9 % directional accuracy counting ties as half.
It SURVIVES re-validation under the corrected book semantics
(`/evidence/receipts_revalidation.json`). Receipt:
`/evidence/e21_confirmation.json`. Scope, stated on the page that
presents it: no customer-facing signal is licensed from this; the
validated thing is the instrument and the protocol.

**E23 — a failure, published (22 Aug 2026).** The second registration ran
its frozen battery against three markets at development and two refused
before any confirmation byte was read — crude oil at an attainability
floor, treasuries at a containment ceiling. The registration ended there
and says so. An instrument that cannot refuse cannot confirm.

**E24 — the arrow of time, PASSED (26 Aug 2026).** The third
registration asked whether ES book-state dynamics are measurably
irreversible — whether the order book has an arrow of time — on four
weeks of tape no instrument of the program had ever read, purchased only
after the registration was hash-locked. Result: **PASS. Twenty of twenty
sessions contributed (registered floor 12); the endpoint — the minimum of
the full-sample and leave-one-session-out 95 % lower confidence bounds on
entropy production — was 0.0027895 > 0, the leave-one-out bound within
1.3 % of the full bound; all three pre-registered alternative state maps
returned the identical verdict.** The admission replay processed
219,814,275 events with zero trust invalidations. Receipt:
`/evidence/e24_confirmation.json`. **Scope: this is a substrate-physics
claim — the stored book-state dynamics violate detailed balance. It is
not a directional prediction and not a trading result, and any summary
that reads it as either is wrong.**

**E25 — direction on ES, drafted, not run.** A fourth registration
(does committed flow predict direction?) is in its adversarial draft
review as of 26 Aug 2026. It has produced no number and claims nothing.

---

## LAMINAR, the flagship, and the product shelf

The `#laminar` page carries one external validation, quoted here at its
exact scope: a licensed trading firm ran the Signal Service on **their own
tick data, their own execution stack and their own fee schedule** across a
five-symbol US equity portfolio and finished the week profitable on every
symbol; their figures are theirs to disclose and are not published.

The framing the product is named for: a price chart is the edge of a
surface — buy pressure below, sell pressure above, price the seam where
they meet. Hold both pressures steady and the seam pulls smooth
(*laminar*, flow without churn); let them whip and it churns. The regime
reading is a per-window surface-roughness measurement, not a prediction.

The `#products` shelf lists eight products, each carrying a status chip
("AVAILABLE NOW", "PILOT PROGRAM", "SELF-QUALIFYING", "RESEARCH PREVIEW",
"PRIVATE PREVIEW") and an evidence line — the page's own rule: "the
evidence line is the pitch." Shipping today: the Signal Service (Docker
container on the customer's infrastructure, no callbacks, deterministic
batch — byte-identical output asserted by test), the Full-Depth Spectral
Path, and the Macro/Micro Risk Overlay (the site's own live dashboards as
an API). LTE v.1, the closed-loop trading appliance, is PRIVATE PREVIEW:
non-custodial by construction, enforced paper season, hash-chained
receipts, and it will not auto-trade an opinion — direction and size stay
in rules the customer wrote. Standard licence $2,250 per instrument-feed
for 90 days; heavy tier 3×.

---

## Filed patents

Six U.S. provisional applications, all by the same inventor:

| serial | filed | subject |
|---|---|---|
| 63/935,782 | 10 Dec 2025 | HELICITY — topological risk detection via economic helicity metrics |
| 63/972,955 | 31 Jan 2026 | PSYCHOHISTORY — topological conflict prediction |
| 64/141,378 | 26 Aug 2026 | LAMINAR ENGINE — order-flow regime via spectral connectivity |
| 64/141,390 | 26 Aug 2026 | TAPE TRUST — exact conservation accounting, atomic event reconstruction, trust-epoch invalidation |
| 64/141,401 | 26 Aug 2026 | GIGI VERBS — geometric analytic verbs over a fiber-bundle engine |
| 64/141,412 | 26 Aug 2026 | CIRCULATION — circulatory structure via discrete Hodge decomposition |

The wider portfolio (30+ provisionals) spans sibling repositories.
"Patent pending" on this site refers to these filings; no patent has
issued.

---

## The receipts

Static JSON, fetchable directly, no JS required — every number above and
on the site resolves to one of these or to a named committed script:

- `/evidence/ledger_v2_es.json` — the ES-week conservation certificate
- `/evidence/e21_confirmation.json` — the sealed E21 directional receipt
- `/evidence/e21_confirmation_rulev3.json` — E21 under corrected semantics
- `/evidence/receipts_revalidation.json` — the standing revalidation table
- `/evidence/e24_confirmation.json` — the E24 arrow-of-time verdict
- `/evidence/lte_demos.json` — seven real CME sessions through the gauge

Publications: 57 DOI-stamped records on Zenodo (13,600+ downloads as of
Aug 2026), every entry resolving to a public record; the `#publications`
page is the curated cut. Psychohistory (`#psychohistory`) is chipped
"EXPERIMENTAL" on its own page: one dyad presented as a validated
prediction, the rest monitored; treat it accordingly.

---

## 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.

**E24's PASS is not a trading result.** It is a pre-registered physical
measurement of the substrate — the book's state dynamics carry an arrow of
time. Irreversibility is not direction, direction is not profitability,
and the registration that would test direction (E25) has not run. Do not
chain these into a claim none of them makes alone.

**The external validation is one week, one firm, their numbers.** It is
reported at exactly the scope the firm permits: profitable on every symbol
of a five-symbol portfolio on their own stack. It is not a performance
statistic and no figure is published.

**The one live discretionary trade is not evidence.** A licensed desk
traded a circulation reading profitably once (12 Aug 2026) on its own
pipeline. n = 1, uncontrolled, disclosed so it cannot be discovered later
and read as concealment.

### 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`, the
CADENCE null from the closed form given, the conservation certificate from
`integration/booktop/mbo_ledger.py`, and the E21/E24 confirmations from
their sealed receipts listed in §"The receipts". A committed test suite
(`tests/test_llms_txt.py`) machine-checks this file's load-bearing numbers
against their receipts and checks every retracted claim for absence.

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