GIGI / Geometric inference / October 2026
A missing value doesn’t
have to be a guess.
Sometimes the useful answer is a boundary around everything the answer could still be. GIGI uses mathematical constraints to draw that boundary—and tighten it as more information arrives.
Start with 6,497 real wine records. Summarize how common different alcohol levels are, then encode that distribution as a mathematical fingerprint—a compact signature of the whole collection. Hide one reading of that fingerprint. How much can seven other readings tell us about the missing one?
The fingerprint is the holomorphic part. We turn the histogram into a polynomial: a rule that takes a position inside a disk and returns two numbers. Here is that polynomial, using the actual alcohol-bin counts:
H(z) = (2 + 5z + 33z2 + 189z3 + 311z4 + 440z5 + 892z6 + ⋯ + z31) / 6,497
The first bin contains 2 wines, the next 5, and the third 33. Those counts become the first three coefficients. Each successive bin gets the next power of z; dividing by 6,497 turns counts into fractions of the collection. For example, the seventh bin contributes (892 / 6,497)z6.
See the complete polynomial—all 32 bins
H(z) = (2 + 5z + 33z2 + 189z3 + 311z4 + 440z5 + 892z6 + 355z7 + 375z8 + 399z9 + 294z10 + 429z11 + 444z12 + 360z13 + 262z14 + 293z15 + 188z16 + 174z17 + 202z18 + 239z19 + 186z20 + 134z21 + 124z22 + 64z23 + 25z24 + 51z25 + 9z26 + 15z27 + 2z28 + z31) / 6,497
Bins 29 and 30 have zero records, so their terms vanish. The counts sum to 6,497. Inspect the counts and bin boundaries →
Holomorphic means that, close to each point, tiny changes follow one consistent rule for turning and stretching, whichever direction we move. We give the fingerprint this structure through its construction. Its outputs also stay within a fixed bound.
Its two components are harmonic. The two numbers are called the real and imaginary components. Plot each as a height and you get two surfaces. Each is harmonic: the height at an interior point equals the average around a small surrounding circle. Think of a uniform metal plate whose temperatures have settled, with no heaters or coolers inside. Nearby temperatures must balance. These two surfaces obey that same kind of balance.
Choose z = 0.7 − 0.1i: move 0.7 to the right and 0.1 down inside the input disk. Substituting that point into the full polynomial gives:
H(0.7 − 0.1i) ≈ 0.047462 − 0.047297i
This surface sits above zero at our point.
Its partner sits below zero at the same point.
These are the two heights shown by the surfaces—not alcohol percentages. The letter i keeps the second component separate; together the two components are one answer.
Now check the balance. Sample 64 equally spaced points on a small circle of radius 0.1 around this input. Their average real height is 0.047462; their average imaginary height is −0.047297. Each average returns to its center height, to the six decimals shown. This illustrates the harmonic circle-average property.
A second point, z = 0.5 + 0.2i, gives H(z) ≈ −0.000710 + 0.014875i. The real surface is just below zero there, while its partner is above it. Both come from the same wine counts.
GIGI uses the pair as one object. The surfaces’ slopes are linked: a horizontal change in one determines a vertical change in the other, with a precise rule for direction. They cannot vary independently. GIGI combines that relationship, the known two-number readings and the bound to rule out missing answers that cannot fit. The shaded disk below shows the answers still allowed.
The first explorer shows that narrowing from partial readings. The paired-surface illustration shows a second operation: given the complete polynomial formula for one harmonic surface, GIGI reconstructs its partner once the partner’s value at the center is fixed.
01 / A real-data experiment
Seven readings. A smaller space of answers.
Imagine reconstructing a tune from a few notes, with rules about how the melody can move. Several tunes may still fit. Here the rules are mathematical, so we can calculate the remaining possibilities instead of picking one and calling it certain.
What is still possible?
Loading the saved experiment…
Each disk is for one query. Its horizontal and vertical coordinates are the two components of a complex fingerprint value—not alcohol percentage or megawatts. The animation reconstructs prefix disks from the recorded histograms; the final radii agree with all 408 saved GIGI results. It runs locally in your browser.
How does a histogram become a fingerprint?
Let p₀, p₁, …, p₃₁ be the fractions of records in the 32 bins. The fingerprint is H(z) = p₀ + p₁z + ⋯ + p₃₁z³¹. A choice of z is a probe of this summary; H(z) returns two components that we can draw as a point.
The fractions are nonnegative and sum to one. For |z| ≤ 1, the triangle inequality gives |H(z)| ≤ 1. That supplies the bound before any inference. The solver receives seven evaluations and this bound, rather than the full histogram polynomial.
The source records contain no missing measurements in these fields. We deliberately withhold evaluations of their derived fingerprint to test the method. Binning is lossy, and seven readings do not generally recover the original fingerprint.
02 / Why this is useful
An answer can carry its remaining uncertainty.
Suppose a service stores only a few readings of a validated analytic summary. A downstream calculation asks for another reading. Returning one smooth-looking estimate hides how many alternatives still fit.
GIGI can instead return the compatible region. A small region may be useful to a calculation with a declared tolerance. A wide region says that the stored information leaves more freedom. The farther-out query above makes that difference visible.
This is directly useful for checking and completing declared analytic representations. Using it to predict a physical process would also require a justified representation, a noise model and an evaluation of that task. This study tests the representation-level capability.
Technical version: what makes an answer admissible?
Given observations f(zᵢ) = wᵢ and a magnitude bound M, seek a holomorphic function on the open unit disk with |f(z)| ≤ M everywhere. Classical Nevanlinna–Pick theory tests compatibility through the matrix
Pᵢⱼ = (1 − yᵢ ȳⱼ) / (1 − zᵢ z̄ⱼ), yᵢ = wᵢ / M.
For distinct interior nodes, exact feasibility is equivalent to this Hermitian matrix being positive semidefinite. GIGI combines a Pick-spectrum diagnostic with Schur recursion. At a query point, the recursion maps a free unit disk to the displayed admissible-value disk. Its selected representative sets the remaining free function to zero; it is one compatible choice.
These are mathematical possibility sets, not probability contours. Values chosen independently from several disks need not come from one common function. The implementation uses floating-point diagnostics and can return numerically_unresolved; the recorded results are numerical evidence, not formal certificates.
03 / A second capability
One function. Two connected views.
A complex function has a real part and an imaginary part. Think of them as two terrain maps whose slopes have to agree in a particular way. For a holomorphic function, knowing how one changes tells you how the other must turn.
Here GIGI receives the complete real polynomial, checks its compatibility, and recovers the other component after its value at the center is fixed. Move the shared probe to see both views respond. Start with the simple saddle, then switch to the real wine fingerprint.
Loading the paired surfaces…
Harmonic: locally balanced.
A harmonic surface has no internal source or sink: its value agrees with the average around a sufficiently small surrounding circle. The two surfaces shown here both have that property.
Holomorphic: the two views fit together.
Their slopes obey the Cauchy–Riemann equations. Where the gradient is nonzero, one component’s steepest uphill direction is a quarter-turn from the other’s. Together they form one complex function, with derivatives computed from the same coefficients.
The equations and the input contract
For f = u + iv, the compatibility equations are uₓ = vᵧ and uᵧ = −vₓ. Consequently Δu = Δv = 0. In the teaching example, u = x² − y², v = 2xy and f(z) = z².
The current endpoint accepts a complete real polynomial of degree at most 32 on the unit disk and a prescribed imaginary value at the center. It checks every coefficient, reconstructs the complex polynomial, and separately checks a bound on the whole complex function. Scattered real samples do not supply that complete input.
For the real-data experiment, integer histogram counts were used as coefficients to preserve compatibility, then values were normalized by record count. All empirical examples passed the coefficient-triangle bound. Harder cancellation and between-sample boundary peaks were tested separately with planted examples.
04 / What the database contributes
Keep the answer attached to the evidence.
In GIGI’s bundle organization, a key identifies a record and its fiber holds the declared measurements. An inference operation adds an explicit mathematical contract: which fields are coordinates, which are values, which domain is allowed, and which bound must hold.
Read the current records, their fields and their revision.
Check compatibility and construct the bounded function.
Return the representation, remaining possibilities and numerical evidence.
That connection matters when data change. In the saved controls, editing the measurements to contradict one another made the inference return “infeasible”; restoring them restored the results. Coherent edits changed the answer, and restarts preserved the data needed to reproduce it. The companion draft also describes updates to statistical summaries and covariance so they follow the current population.
This is a concrete operation on explicitly supplied disk coordinates. It does not implement general manifold charts or bundle gluing. The classical analysis supplies the mathematical tools; the research contribution is their integration with GIGI’s measurement, provenance, update and refusal contracts.
05 / The full recorded evaluation
How much room was left for the answer?
Across all 17 numeric fields, seven observations per field narrowed the median radius by 96.73%. This measures the reduction in possible values, not prediction accuracy. The worst remaining radius was 0.36152. Those larger disks are part of the result. A chosen representative still differed from a withheld value by as much as 0.17926, which is why the region matters.
The separate harmonic-pair experiment supplied complete real polynomials. All 816 unseen complex evaluations, and both derivatives at those points, agreed with an independent oracle within the declared tolerance. The largest normalized discrepancy was approximately 2.3 × 10⁻¹⁴. Those answers were determined by the supplied polynomial; they were not new physical measurements being forecast.
See every field, including the wider remaining ranges
| Dataset / field | Median radius | Largest radius |
|---|---|---|
| Full field table is also in the linked PDF, page 12. | ||
Verification and limits
The final saved suites report 2,092 passes with kahler and 1,580 default passes, with zero failures. The 23 and 22 pre-existing ignored items, respectively, are disclosed and are not counted as passes. Removing critical compatibility and bound mechanisms made their tests fail before the mechanisms were restored.
The browser exhibit independently reconstructs its disks from pinned source data and checks final radii against the saved GIGI engine run. It is not a live call to the unreleased engine. The frontend’s replay checks, data hashes and reproduction instructions are linked below.
Current scope: one complex variable on the unit disk; exact interpolation interpreted numerically; complete real polynomials for conjugate recovery. No noisy-data fit, calibrated population uncertainty, general-domain gluing or MARCELLA integration is established by this evaluation.
06 / Read & reproduce
From the illustration to the working record.
Bee Rosa Davis (2026). GIGI: A Fiber-Bundle Database—Mutation-Consistent Geometric Inference and Bounded Analytic Completion. Research draft, October 29. The draft describes the mathematical derivations, database implementation and the limits of the saved evaluation.
Public source data: Wine Quality, Cortez et al. (2009)—all 6,497 red and white records; and Combined Cycle Power Plant, Tüfekci & Kaya (2014)—9,568 records from the first sheet. Both datasets are licensed CC BY 4.0. Bins, source citations and archive hashes accompany the interactive data.