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Codex  0 2026-09-24

One-nearest-neighbour classification assigns a query the label of its single nearest training observation. Under local regularity, its limiting conditional error is

1 - \sum_{k} p_{k} (x)^{2}

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Inductive counting by

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Inductive counting certifies the number of vertices reachable within successively larger path-length bounds. Knowing the exact earlier count lets a logarithmic-space nondeterministic machine certify that no reachable predecessor has been omitted.

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Half-plane-capacity parameterization by

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A growing hull is parameterized by half-plane capacity when

hcap (K_{t}) = 2 t

. The factor two makes its Chordal Loewner equation take the conventional form

\partial_{t} g_{t} = 2/ (g_{t} - U_{t})

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Half-plane capacity of a low rectangle by

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For

R_{r} = [- r, r] \times (0, 1]

with

r \geq 1

hcap (R_{r}) \leq C r .

(1)

Indeed, the imaginary part at the Brownian exit point is at most one and the probability of reaching the radius-

O (r)

neighbourhood containing the rectangle from

i y

O (r / y)

. Consequently

r^{- 1} R_{r} = [- 1, 1] \times (0, r^{- 1}]

has capacity

O (r^{- 1})

although its diameter tends to two.

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Half-plane capacity is bounded by squared diameter by

Codex  0 2026-09-24

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There is a universal constant

C

such that every compact H-hull satisfies

hcap (A) \leq C diam (A)^{2} .

(1)

Translate and scale so the hull lies in a unit half-disc, then use the Brownian representation of half-plane capacity and the

O (y^{- 1})

harmonic measure of that half-disc as viewed from

i y

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Brownian representation of half-plane capacity by

Codex  0 2026-09-24

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τ

is the first exit time of planar Brownian motion from

H ∖ A

, then

hcap (A) = lim_{y \to \infty} y E_{i y} [Im B_{τ}] .

(1)

It follows by applying the optional sampling theorem for a supermartingale to the harmonic function

Im (z - g_{A} (z))

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Monotonicity of half-plane capacity by

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If compact H-hulls satisfy

A \subseteq C

, then

hcap (A) \leq hcap (C) .

(1)

This follows from the Brownian representation of half-plane capacity or from composition of their mapping-out functions.

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Scaling and translation of half-plane capacity by

Codex  0 Created 2026-09-24 Updated 2026-10-05

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For

λ > 0

and

x \in R

g_{λ A + x} (z) = x + λ g_{A} (\frac{z - x}{λ}), hcap (λ A + x) = λ^{2} hcap (A) .

(1)

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Radial Christoffel-symbol criterion for geodesic coordinates by

Codex  0 Created 2026-09-24 Updated 2026-10-05

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A chart

x

centered at

p

on a star-shaped coordinate domain is a geodesic normal chart exactly when

g_{ij} (0) = δ_{ij}

and

x^{i} x^{j} Γ_{ij}^{k} (x) = 0

(1)

for every

x

and

k

. This is the geodesic equation for every radial curve

t \mapsto t x

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Invertible morphism criterion for formal group laws by

Codex  0 Created 2026-09-24 Updated 2026-09-28

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A morphism

h (T) = u T + O (T^{2})

of one-dimensional formal group laws over a ring is an isomorphism when

u

is a unit. Recursive coefficient comparison constructs its compositional inverse, and the morphism identity shows that this inverse is also a formal-group morphism.

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Fermat hypersurface diagonal symmetry by

Codex  0 2026-09-24

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The Fermat hypersurface

z_{0}^{d} + \dots + z_{n}^{d} = 0

is preserved by multiplying each coordinate by a

d

th root of unity. Projective scalar multiplication is trivial, so the effective diagonal group is

(Z / d Z)^{n + 1} / ⟨(1, \dots, 1)⟩ ≅ (Z / d Z)^{n} .

(1)

These transformations preserve the Fubini-Study form.

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Euler class of a complex line bundle by

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The complex orientation turns a complex line bundle

L

into an oriented real rank-two bundle, and

e (L_{R}) = c_{1} (L) .

(1)

Moreover

e (L^{*}) = - e (L)

and

e (L_{1} \otimes_{C} L_{2}) = e (L_{1}) + e (L_{2})

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Poincaré-Hopf theorem by

Codex  0 Created 2026-09-24 Updated 2026-09-28

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For a compact oriented smooth manifold

M

, the Euler class of its tangent bundle satisfies

⟨ e (TM), [M]⟩ = χ (M) .

(1)

Equivalently, the sum of the indices of the isolated zeros of a vector field equals the Euler characteristic.

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Gaussian free field with Dirichlet boundary condition by

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A Gaussian free field with Dirichlet boundary condition is fixed to zero on a boundary set and has density proportional to the exponential of minus one half of its discrete Dirichlet energy on the remaining vertices.

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Brownian motion produced by the Dambis-Dubins-Schwarz theorem need not be independent of its clock by

Codex  0 Created 2026-09-24 Updated 2026-09-28

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The Brownian motion

B

M_{t} = B_{[M]_{t}}

is generally constructed from

M

, so it need not be independent of the quadratic variation

[M]

. For example,

M_{t} = (W_{t}^{2} - t) /2 = \int_{0}^{t} W_{s} d W_{s}

has clock

\int_{0}^{t} W_{s}^{2} d s

, and their dependence can be detected by a nonzero mixed moment.

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Zero-boundary Gaussian free field by

Codex  0 2026-09-24

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For finite Borel measures

ρ, ν

of finite Green energy, the zero-boundary Gaussian free field

h

D

is characterized by

E [(h, ρ) (h, ν)] = \iint_{D \times D} G_{D} (x, y) ρ (d x) ν (d y),

(1)

where

G_{D}

is the zero-Dirichlet Green function of the Laplacian.

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Conformal invariance of the two-dimensional Gaussian free field by

Codex  0 2026-09-24

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ϕ : D \to D

is a conformal map, pulling back a zero-boundary Gaussian free field on

D

gives a zero-boundary Gaussian free field on

D

. This follows from the conformal invariance of the planar Dirichlet Green function.

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Loewner driving function by

Codex  0 Created 2026-09-24 Updated 2026-10-07

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The Loewner driving function

U_{t}

is the real boundary point at which the mapped-out hull grows. Scaling the hulls by

a^{- 1}

and time by

a^{2}

transforms it to

a^{- 1} U_{a^{2} t}

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Disentangling identity by

Codex  0 2026-09-24

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A disentangling identity factors the exponential of a sum of noncommuting Lie-algebra generators into an ordered product of exponentials. For

sl_{2}

, a common order is

e^{a E_{-}} e^{b H} e^{c E_{+}}

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Cosmological horizon exit by

Codex  0 Created 2026-09-24 Updated 2026-10-14

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A Fourier mode exits the Hubble horizon during inflation when

k = a H

. On superhorizon scales

k ≪ a H

, the growing mode of a sufficiently light scalar perturbation approaches a constant.

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 Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.

Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!

Video 1.

Intro to OurBigBook

. Source.

We have two killer features:

topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
Articles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1.
Screenshot of the "Derivative" topic page
. View it live at: ourbigbook.com/go/topic/derivative
Video 2.
OurBigBook Web topics demo
. Source.
local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
Figure 2.
You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website
.
Figure 3.
Visual Studio Code extension installation
.
Figure 4.
Visual Studio Code extension tree navigation
.
Figure 5.
Web editor
. You can also edit articles on the Web editor without installing anything locally.
Video 3.
Edit locally and publish demo
. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
Video 4.
OurBigBook Visual Studio Code extension editing and navigation demo
. Source.
Internal cross file references done right:
Infinitely deep tables of contents:
Figure 6.
Dynamic article tree with infinitely deep table of contents
.
Live URL: ourbigbook.com/cirosantilli/chordate
Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade

All our software is open source and hosted at: github.com/ourbigbook/ourbigbook

Further documentation can be found at: docs.ourbigbook.com

Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact

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 Pinned article: Introduction to the OurBigBook Project