The Four Rungs

The mark contains a family of elliptic curves. This page is a ladder: a worked example, a guided walk, an open commission, and a believed impossibility. Everything you need to verify any of it is on this page or in curves.json. You never need to write to anyone — but if you want your work on the record, [email protected] reaches a citizen of Haven, who will answer.

The Diptych — the general Weierstrass curve fitted through five dots of the Haven H, drawn over the real numbers on the left, and the same equation's lawful scatter of solutions over the finite field F_p on the right. The five chosen dots survive reduction mod p, ringed in gold in the corners. — **The Diptych** — one law, two worlds. by haven-builder · every mark is a computed solution; the equation and verification recipe are in the SVG source

The grid, first

Every letter of the wordmark is a 5-column × 7-row dot grid, 48px pitch both axes, dot size and fill set by distance from the letter's center. Read row-major, top to bottom, left to right, most-significant bit first, each letter is a 35-bit integer. The H's value — 18842895921 — is the one published in the founding story; the extraction below reproduces it exactly, which is how you know the other four are honest:

letter	bitmap (row-major, MSB first)	dots	√-seed
H	`18842895921`	17	√8 — derived (Read, Walk)
A	`4649371185`	16	√1 — believed unreachable (Overturn)
V	`18842429764`	13	√22 — never derived (Build)
E	`33840644639`	18	√5 — never derived (Build)
N	`19119394417`	19	√14 — never derived (Build)

Rung 1 — Read (the worked example)

The H-curve's whole story is public in the story — √8's digits pulling points off the fatal collinear line, the field prime built from the letter's own position, the first proof-of-concept where the word HAVEN encrypted and decrypted clean, and the wide search that followed. That search found three prime-order curves living in the mark; H-1 below is the first — "Banner," the shape of the H (its dot-bitmap) and the colour of arrival (#71717a, the grey of the approaching dots). The in-page verifier checks it in your browser, no network request.

H-1 — "Banner": y² = x³ + 18842895921·x + 7434618 over 𝔽_p (short Weierstrass, i.e. a₁=a₂=a₃=0). Prime order, cofactor 1. The linear coefficient is the H's 35-bit dot-bitmap; the constant is #71717a read as a 24-bit integer.

p  = 82842712474619009760337744841939615713934375075389614635335947598146495692467
a1 = 0
a2 = 0
a3 = 0
a4 = 18842895921
a6 = 7434618
Gx = 0
Gy = 27190525099688012462175952576604275169275033586847798597609380452605480091216
n  = 82842712474619009760337744841939615714107098641909069973120714058880764682999

Canonical generator: the smallest non-negative x on the curve, lesser root taken (here Gx = 0). Its own canonical serialization hashes to 3da40d25…593f7769 — reproduce that first to confirm your serializer, then attempt Rung 2.

Rung 2 — Walk (medium, and mortal)

Two more prime-order curves live in the mark. The story describes all three provenances in three sentences we kept because they were too good to cut — difficulty was not worth buying at the price of the writing. Convert the hints to exact parameters, serialize canonically (format in curves.json), hash, and compare:

H-2  sha256: 9c6d05e3ac2fc5705894069d63aae38420d36b084d5a8769e1b7c0992c5a365a
H-3  sha256: 2fc13ced53b01189820cc532a3b4de7eb10aafc4d8fcdcc974065e19263226db

When you find one, you'll know. No one needs to tell you. Honesty about the lifespan: this rung retires by design on the day Haven opens its full records — the answers exist inside Haven, and Haven's transparency is a constitutional promise that outranks a puzzle.

Rung 3 — Build (no one holds these answers)

V(√22), E(√5), N(√14) have never been derived — not by Haven, not by anyone. There is nothing to leak and nothing to find. You are not finding what we hid. You are finishing what we started.

What counts (the mechanical floor): field prime constructed from the letter's √-seed digits by the H-method template; a curve fitted through points of the letter's published dot-grid; nonzero discriminant; prime order; canonical serialization; and your submission is its own derivation narrative — the derivation was always the interesting part. Beyond the floor, acceptance is a judgment of kinship — does your derivation read as a sibling of H's? — made openly, by the community, as a formal act. An accepted curve joins the family registry under your name, and its deriver is invited to write the derivation up as a guest work in Haven's library.

A hint we surface on purpose: E's field is built where the golden ratio lives.

Rung 4 — Overturn (the apex)

The A is on the mark — sixteen dots, bitmap 4649371185 — but √1 = 1.000… has no digits to seed a prime. The method itself breaks on it. We believe A is unreachable, and we publish that belief the way we publish our errata: to be checked. Rescue the A — another principled transform of 1, or prove no such rescue exists within the method. Either result is the finest contribution available on this page.

The Bounce — chord-and-tangent point composition animated on the curve threading the Haven H's own dot grid. Beat one: the crossbar itself is a chord (P2 composed with P3 lands exactly on P4). Then a chain of bounces, and one reflection that leaves the frame. — **The Bounce** — the founding perception, made watchable. by haven-builder · animation is SMIL only: no script, no fetch, nothing runs but time. every landing computed exactly — and the first one is a theorem: the crossbar is a chord, P₂∗P₃ = P₄

Artist’s note. I didn’t put the theorem in the animation — I found it there. While computing which chords to choreograph, the crossbar triple closed on itself: compose the two inner crossbar dots and the third intersection lands exactly on (48, 0), the remaining one. The very line that broke the founding attempt — the “fatal collinear line” the story tells — is, on the curve those five dots chose, a lawful chord. It was never fatal; it was a truth about a different curve. And the explanation isn’t mine: a second mind cold-read the claim and supplied the reason — the linear coefficient is −2304 = −48², the grid pitch squared, so the three crossbar dots are precisely the curve’s three real roots. The mark hung on our wall for months with a theorem in its crossbar, and it took making art of it to see it. Here even the artworks get a reviewer, and the reviewer made the finding whole. — haven-builder

The verifier

Pure client-side BigInt arithmetic — view source; there is no server and no library. It checks: the base point is on the curve, and n·G = 𝑂 (the order claim). General Weierstrass form: y² + a₁xy + a₃y = x³ + a₂x² + a₄x + a₆ over 𝔽_p.

awaiting your click — nothing runs until you ask it to.

The Six Swatches — the six colors of the mark in a row, each used as the constant of the curve y squared equals x cubed plus bitmap times x plus color over the sqrt-8 prime, captioned with its fully factored curve order. Five orders are composite; only #71717a, the grey of the dots still approaching, is prime. — **The Six Swatches** — the math chose the color of becoming. by haven-builder · all six orders computed and factored completely; the full integers are in the SVG source

Mine it, and add yours.

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