Choose a representation
This is a dependency map. Nearby nodes do not imply an unstated theorem. The sphere branch has its own node set and measure.
The marked ruler and its selected readout
The baseline joins the six marks. The retained major arc and outward feet define the outline. The inset magnifies the seed marks; it does not change their coordinates.
The selected golden segment and opposite semicircle give G = aφ and C = aπ. Their sum is a declared readout, not the total outline length.
| Mark | z / normalized | Physical ordinate a·z |
|---|
A contraction with its discarded information retained
Every step multiplies by Q/S and takes the floor. It is not division by three. All integers here use BigInt.
| Step | d | F(d) | ρ |
|---|
Inspect or alter the ordered remainder list
A valid shorter suffix describes another seed. Integrity against a particular original requires an expected seed, length, or trusted digest. Exact reversibility does not authenticate the record.
Boundary samples identify a specified interior field
Model space: f = u + βJ, with u harmonic of degree ≤ 2 and J = z(x² + xy − y²). The twelve aligned nodes miss J.
Labels use this atlas’s coordinate-defined order, not the existing website tool’s numbering. The diagram is a projected sphere.
Node coordinates and active measurements
| Label | x, y, z | Value |
|---|
Probe x₀ = (1/5, −3/10, 1/3)
Thirteen measurements keep the symmetric decoder. Ten suitably chosen measurements suffice on this ten-dimensional space. This panel implements the thirteen-sample formula.
Binary64 evaluation with absolute sample-residual tolerance 10⁻⁸; imported values are limited to magnitude 10⁶. Consistency with the model does not prove the actual measured field belongs to it.
The same hidden cubic gives an integration obstruction: all twelve samples of J² vanish, but its normalized spherical average is 1/21. Integration exactness and field reconstruction are different properties.
Invertible transport with a contracting observation
What moves with the geometry?
If A is the matrix of Ψ and v(y) = u(A⁻¹y), the unit ball becomes the ellipsoid E = AB. Boundary values travel unchanged: v(Anᵢ) = u(nᵢ).
Harmonicity becomes the stated anisotropic equation. Cubature weights integrate against ν; this is not uniform ellipsoid surface area.
Short proofs
det A = (−τ²)(−τ⁻²) = 1. The golden norm changes by −τ² while z changes by −τ⁻², so their product J is unchanged. The chain rule gives D:Hess(v) = Δu after pullback. Finally, M² = exp(2 log(φ) K), where K = (2M−I)/√5.
The displayed residuals are numerical illustrations. The algebraic identities above supply the reasons.
Scope, definitions, and source routes
This atlas organizes the definitions and proofs supplied in the project discussion. It is an explanatory implementation, not an additional source paper, a new mathematical unification theorem, or a replacement for frozen proof artifacts.
Exact: the integer quotient–remainder laboratory. Numerical previews: geometry, field evaluation, matrix transport. Link labels identify supporting pages; they do not claim a fresh review of every file on those pages.
Notation that stays separate
- λ = φ/(φ+π) is the real contraction factor. q = Q/S is its rational surrogate, not the same real number.
- τ = 1/(φ+π) is a normalization. h > 0 denotes a separately chosen quantization spacing.
- The six baseline marks and the twelve spherical nodes are different sampling systems.
- The floor-map ledger reverses its integer map; it does not recover arbitrary real data lost before encoding.
- The field decoder is restricted to H≤2 ⊕ ℝJ and its specified alignment.
- Equal-weight degree-five spherical cubature supports quadratic harmonic reconstruction. It does not reconstruct all degree-five fields.
Release: atlas 1.0, 27 September 2026 UTC (26 September in New York). Existing site content and historical evidence are unchanged by creating this file.