Learn the construction before the tools

1. Start with the geometry page: the unit circle, opening, six marks, pose, and selected readouts.
2. Read geometric arithmetic compatibility: geometric scaling, the rational quantizer, the forced discrepancy, and exact history recovery.
3. Read reference recovery: what observations determine, which relationships survive, and when extra measurements are needed.
4. Read the reference language and use the decoder. A scalar rank is not a 120-coordinate residue witness.
5. Read the 600-cell signal network. It is not the typed interface graph on /network.
6. Follow the proof-source and execution-record links and compare them with the stated checks.

Open the atlas

• Construction

  Input: radius a = 1, one-arcsecond seed ε = π/648000, identity pose.

  Result: six labeled marks. Selected measurements G = aφ and C = aπ. The seed chord is not the outline.

  Open the construct tool

• Integer ledger

  Input: seed 399. Q = 339949771344778, S = 10^15, scale unit 1/100.

  Result: quotient 135, remainder 639958766566422, then 45, 15, 5, 1, 0. Radius 3.99.

  Open the recover tool

• Spherical average

  Input: the monomial x²y²z² on the twelve icosahedral nodes, weight 1/12.

  Result: continuous mean 1/105; twelve-node sample 0. Degree 6 is not guaranteed.

  Open the sphere tool

• Boundary field

  Input: u = 2 + 3x + x² − y² with β = 4, probe (1/5, −3/10, 1/3).

  Result: quadratic part 2.55, J = −11/300, value 2.40333333. Twelve samples miss β; the extra sample recovers it.

  Open the boundary-field tool

The mathematical atlas is a separate interactive map of these constructions. It is not a new source paper, and it does not replace the four tools.

A worked step

Take radius count 399 and scale unit 1/100. The radius is 3.99. One step of the rational map Q = 339949771344778, S = 10^15 produces quotient 135 and remainder 639958766566422. Repeating gives 45, 15, 5, 1, 0. The remainders, the scale unit, and the pose are what the record keeps. λ = φ/(φ+π) is a different number and is not this map.

On the sphere, x²y²z² is the degree-six obstruction: the continuous average is the exact rational 1/105, and the twelve-node sample is 0. On the boundary field, the same probe moves under transport while the reconstructed value of the default field with β = 4 stays 2.40333333. Those are separate checks. None of them is a new source paper.

Seed

The one-arcsecond opening ε = π/648000. A 12° drawing is not this seed.

Remainder

ρ = Qd − S⌊Qd/S⌋. Stored remainders recover d.

λ and Q/S

λ = φ/(φ+π) is the irrational contraction parameter. Q/S is the rational floor map. They are not the same number.

J

The cubic z(x² + xy − y²) preserved by the transport family and invisible to the twelve aligned nodes.

Pose

The center and quaternion stored in a ledger record. Camera yaw is not pose.

Certificate

Data supplied to support a precisely stated claim and checked against explicit conditions. A verifier checks the certificate; PASS records the outcome of a particular execution.