Published geometry to ellipses and operators

The input is the marked construction in Paper VII v1.6.0: $\varphi=(1+\sqrt5)/2$, $\varepsilon=\pi/648000$, $s=\sin(\varepsilon/2)$, $c=\cos(\varepsilon/2)$, and the six center-relative marks. The deposit is 10.5281/zenodo.22233573. Byte identity with that deposit was not checked for this package. The note does not use the outer scaffold.

Give the six center-relative columns equal weight and keep the circle center as the origin. Their second-moment matrix is diagonal. The filled set is the ellipse

$R^{2}=a^{2}\bigl[s^{2}+\tfrac{2}{3}(\varphi+2)s+\varphi^{2}\bigr],\quad r^{2}=a^{2}(1-s^{2}).$

At unit scale the supplied high-precision cross-check gives $R=1.6180357955446143$ and $r=0.9999999999970619$. $R$ is not $\varphi$. The limit as the seed tends to zero is $(a\varphi,a)$, which is not the published one-arcsecond seed. Subtracting the mean of the six marks instead produces a rank-one segment, not this ellipse.

The diagonal map $D=\mathrm{diag}(R/a,r/a)$ carries the generating circle onto the ellipse boundary and the retained arc onto the corresponding elliptical arc. The inverse metric $H=(DD^{\mathsf T})^{-1}$ recovers source angles and lengths. Ordinary Euclidean angles in the target are not generally preserved. The feet are mapped as feet. They are not replaced by ellipse-boundary points.

The filled ellipse, not the marked curve, is the numerical range of

$T_{0}=\begin{pmatrix}d&2r\\0&-d\end{pmatrix},\quad d=\sqrt{R^{2}-r^{2}}.$

The note records $T_{0}^{2}=d^{2}I$ and $\|T_{0}\|=R+r$. A translated ellipse keeps its numerical range when the omitted tail is the center times the identity, not zero. A disk of radius one centered at three is the stated counterexample to a zero tail.

Weights $2^{1-n}$ specify an infinite direct sum with the same numerical range, infinite rank, and tail norm $(R+r)2^{-N}$. The finite matrices in the code are truncations. They do not execute infinitely many blocks. An infinite assembly of independent copies of the curve is a further definition in the note. It is not forced by Paper VII.

Inside the declared golden-step family, the axis ratio recovers $s$ and the scale. That does not recover an arbitrary shape. The note also states an elementary matrix-polynomial bound with condition number $\kappa=\sqrt{(R+r)/(R-r)}\approx 2.058168728992495$ for the canonical numbers. A constant-two consequence quoted from an external catalogue is labeled there as not verified by this package.

The eight names in SHA256SUMS match the supplied checksum listing. sha256sum -c SHA256SUMS was run in this directory after the files were copied. The transcription PDF is the typeset Julia listing and checksum block. It is not one of those eight names.

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• omega_geometry_operator_bridge.jl · 19,070 bytes
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• verify_reference.py · 15,615 bytes
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• README.txt · 3,646 bytes
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• requirements-tested.txt · 185 bytes
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• execution/EXECUTION_RECORD.txt · 2,129 bytes
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• execution/python-optimized.txt · 2,311 bytes
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• SHA256SUMS · 723 bytes
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• omega_bridge_transcription.pdf · 157,577 bytes
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The supplied record says both Python processes exited 0, with 23 groups and 7,539 require calls in each mode. The logs differ only by the optimization-mode header. There are no Python assert statements. Selected website node coordinates are marked as transcriptions. The maximum absolute difference reported for that comparison is 0 in binary64. Floating-point thresholds are conformance tolerances, not certified enclosures. The 80-digit values are a cross-check, not a rigorous enclosure.

This archive did not rerun that checker and did not execute the Julia file. The Julia source targets 1.9 or later and contains 21 check groups, including exact checks in $\mathbb Q(\sqrt5)[s]$. No Julia PASS is claimed. No Lean build, infinite simulation, or physical experiment is claimed.

Commands recorded for a recipient, not run here:

python verify_reference.py
python -O verify_reference.py
julia --startup-file=no omega_geometry_operator_bridge.jl
julia --startup-file=no --check-bounds=yes omega_geometry_operator_bridge.jl