Display of exact series

Multidimensional charts

Accurate here means: integer and rational series are computed exactly, then drawn. Pixel positions are binary64. These pictures are not theorems, not Lean, and not the paper 8 offline explorer.

What a chart can be

A screen is two-dimensional. Higher-dimensional objects appear here as exact tables, 2D charts of exact sequences, orthogonal projections, or a 3D rotation of a 3D point set. Ambient dimensions 4, 5, and 8 are a parameter of the deformation eigenvalues, not a volumetric rendering of $\mathbb R^m$ .

IEEE-754 is not used to decide floor images. It is used to place pixels and to evaluate $e^{\pm\eta}$ . The one-arcsecond seed on /geometry still needs a magnified inset; that is a separate drawing of $\Gamma$ .

Depth layers

First-hitting layers on {0,…,32767}

Exact integer counts from paper 8, Theorem 5.3. Sum = 32768. Bars are those integers; axis ticks are binary64 display.

Loading depth chart…

Exact first-hitting counts on {0,…,32767}
Depth	Right endpoint of nested preimage	Count at this depth
0	0	1
1	2	2
2	8	6
3	26	18
4	79	53
5	235	156
6	694	459
7	2044	1350
8	6015	3971
9	17696	11681
10	52057	15071

Full statement and checker: /fibers and paper 8.

Floor images

Exact floor images T(n) for n = 0…80

Computed in this browser with BigInt: T(n) = ⌊Q n / S⌋. Intermediate products are not IEEE-754. The plotted points are exact integers that fit in a JS safe integer.

Loading floor chart…

First disagreement with the real map is not in this window. It is seed 42,189,065. A single-seed lab remains on the home page.

Twelve-node sphere

Twelve-node even moments versus the sphere

Normalized surface measure. Degrees 0, 2, 4 agree exactly. Degree six is the exact obstruction 2/15 − 1/7 = −1/105. Bars are those rationals converted for display.

Loading moment chart…

Exact even moments, normalized measure
Monomial	Sphere integral	Twelve-node Q	Agree
1	1	1	yes
z²	1/3	1/3	yes
z⁴	1/5	1/5	yes
z⁶	1/7	2/15	no, −1/105

Twelve normalized icosahedral vertices

Exact template: cyclic permutations of $(0,\pm1,\pm\varphi)$ scaled onto the unit sphere. These scatter plots are binary64 projections of that template, not a new cubature theorem. True ambient dimension three; higher-dimensional localization is a slice of a different field, not these twelve points.

Loading xy projection…

Loading xz projection…

Drag to rotate. Display only; chord threshold is a drawing heuristic.

Dimension m

Deformation eigenvalues in dimension m

Closed form with scale c = 1: $e^{\eta}$ , $e^{-\eta}$ , and $e^{-2\log c/(m-2)}=1$ . This is the paper 8 / NS family, not an arbitrary compact-set motion. Pixel values are binary64 of that formula. The numerical guard in the companion rejects much larger η; this plot stays in a moderate range.

at η = 2.5: 12.1825 · 0.0820850 · 1.00000

Loading eigenvalue chart…

Localization formulas live in the NS extension and paper 8 §7. This chart does not draw a 5D or 8D velocity field.

What this is not

Not a source paper. Not peer review. Not a proof-assistant kernel check.

Not the paper 8 HTML explorer inside the companion ZIP. That file was not recertified as a visual proof.

Not a Navier–Stokes blowup picture, not frequencies 10–21, and not a joining theorem of Papers I–VII and paper 8.