OMEGA marked reference geometry

Source work: Paper VII, version 1.6. Zenodo preprint. DOI 10.5281/zenodo.22233573. Not peer reviewed. The companion generator evaluates the closed formulas; it is not a theorem checker.

This branch is separately defined Euclidean geometry. It does not join Papers I–VI into one theorem, and it does not modify frozen SRSD.

Machine objects omega-marked-kernel and construct-transfer-gate are listed in /archive/MACHINE.json. Reconstruct from the node log and declared constants; do not digitize the figure.

The input data are a center $\alpha$, a positive scale $a$, an oriented Euclidean plane through $\alpha$, and an ordered pair $d,b$ on the radius-$a$ circle with central separation $\varepsilon=\pi/648000=1''$. The chord direction is then extended outward by $a$ and by $a\varphi$ on both sides. Those extensions are declared rules. They are not derived from the arcsecond. Only the golden-ratio identity $\varphi^2=1+\varphi$ is used in the bilateral extension.

Version 1.6 proves that this rule set, together with the selected major arc, determines exactly one marked kernel: six baseline marks, every distance among those marks and from them to $\alpha$, all fifteen pairwise center angles, rank-two Gram data, and an oriented frame in $\mathbb{R}^3$. The six baseline marks are collinear (affine dimension one). The center-augmented configuration has affine dimension two.

The metric outer scaffold in Section 11 is an adopted transcription of the repeated $a,a\varphi$ notebook annotations. It defines Space, Echo, and Loco once that rule is adopted. The larger circle and additional curved links in the expanded notebook drawing are not required for Theorem 3.4 and Proposition 9.1. Genuinely nonplanar completion remains an open constraint stage.

Each inner straight segment has length $a$. Each outer golden segment has length $a\varphi$. Each complete foot has length $a\varphi^2$. The opposite semicircle has length $a\pi$. The selected golden readout is the average of the two outer segments. The selected circular readout is that opposite semicircle. Those are the measurements defined in the hosted proof source.

A receiving system rebuilds the labeled kernel from $\mathcal K=(\alpha,a,u,v,n,\varepsilon,\varphi)$ and the master formula $p_{j,\sigma}=\alpha+a(cu+\sigma x_j v)$. Scalar records do not place the figure. Scaffold data travel only if separately adopted.

Theorem 3.4. Fix $a>0$, $\varepsilon=1''$, the ordered seed $(\alpha,d,b)$, and the sequential bilateral extension $a$ then $a\varphi$. The Construction-to-OMEGA theorem proves that these data, together with the selected major arc, determine exactly one marked kernel: six baseline marks, every distance among those marks and from them to $\alpha$, all fifteen pairwise center angles, rank-two Gram data, and an oriented frame in $\mathbb{R}^3$. Equivalently, the following are uniquely determined up to orientation-preserving Euclidean isometry:

1. the six marks $d_2,d_1,d,b,b_1,b_2$;
2. all pairwise distances among them and from them to $\alpha$;
3. all fifteen pairwise angles subtended at $\alpha$;
4. every triangle with vertex $\alpha$ and two baseline marks;
5. the normalized configuration Gram matrix;
6. an oriented orthonormal frame $(u,v,n)$ in any oriented realization in $\mathbb{R}^3$.

The six baseline marks are collinear and have affine dimension one. The center-augmented configuration has affine dimension two.

The OMEGA shape on this page is a marked planar kernel, not a volume and not a unique outer shell. In any oriented realization it is a rigid figure in a plane through the distinguished center α: the Paper VII arc-and-feet curve Γ (the selected major arc joined to two outward golden feet), six collinear baseline marks on the line y = −a cos(ε/2), and the center off that line. Only the seed pair lies on the radius-a circle; those two points are joined by the retained major arc, not by a straight chord. This is the existing construction, not a different object. The third direction is only the oriented normal of that plane.

Structure, in the sense of Theorem 3.4: the six marks $d_2,d_1,d,b,b_1,b_2$ lie on one affine line. Adjoining $\alpha$ produces an affine plane. The master reconstruction is $p_{j,\sigma}=\alpha+a(cu+\sigma x_j v)$ with $x_0=\sin(\varepsilon/2)$, $x_1=1+\sin(\varepsilon/2)$, $x_2=\varphi^2+\sin(\varepsilon/2)$ (normalized coordinates $x/a$, in units of $a$). The complete curve is the existing arc-and-feet object $\Gamma=\gamma([\varepsilon/2,2\pi-\varepsilon/2])\cup E_-\cup E_+$, not a circle with marks on a diameter and not a new discovery. Distances, the fifteen pairwise center angles, the rank-two Gram factorization, and the oriented frame $(u,v,n)$ are then unique up to orientation-preserving Euclidean isometry and positive homothety about $\alpha$. That is the whole rigid object. A receiving constructor that is given those nodes has the kernel, and does not have a unique shell.

Construct mode and the node log are a transfer gate for that kernel. A receiving system can rebuild the unique labeled similarity class in R³ from the exported nodes, exact-expression strings, and binary64 coordinates. The gate is typed: kernel data may pass; the scaffold passes only if separately adopted; exclusions never become drawable nodes. Display exaggeration does not travel. The scalar map between (φ, ε) and (τ_Ω, λ) is a side channel of numbers, not a reconstruction of the geometric realization, and not a joining theorem.

Uses of the gate, and not more: interchange so another system can recreate the planar kernel in $\mathbb{R}^3$ from the node log; isolation of the 12° picture from the true one-arcsecond seed; a typed latch that will not silently promote the adopted scaffold or an excluded outer drawing to kernel status; and a scalar chart-change $(\varphi,\varepsilon)\leftrightarrow(\tau_\Omega,\lambda)$ that does not reconstruct a geometric realization without center, scale, frame, and construction data. The gate does not prove cubature, does not modify frozen SRSD, and does not join Papers I–VII.

Euclidean homothety of the six marks is the scaling checked in the de-branded typed-map audit. That audit identifies maps; it does not join Papers I–VII and does not transfer a unique shell. See Typed network. A Lean 4 formalization of the selected geometric-provenance measurement is inventoried at /formal; sources are hosted and hash-matched, lake build PASS on this archive 2026-09-13, and it is not a uniqueness theorem among all measurements.

Construct mode is the presentation and interchange layer of the Paper VII kernel. It reveals the marked planar kernel in fifteen sourced stages, exports OMEGA_KERNEL_NODES/1.0 at the true one-arcsecond seed, and keeps a 12° picture strictly non-normative. Automated replay stops at kernel completion. The adopted scaffold is a separate jump. This layer does not replace the Euclidean proof, does not introduce a second geometric evaluator, and does not determine an outer curve or signed depth.

Technical release paper, 6 September 2026, not a source work and not peer reviewed: OMEGA_Kernel_Construct_v1.1.0.pdf · 368491 bytes · SHA-256 6b2593e2ad232793296b8e8b3acadcc8a71d240b5e8c799342d90cd1815bd912Copy. Node contract OMEGA_KERNEL_NODES/1.0. Canonical unit log: omega-kernel-nodes.unit.true.json. Scalar certificate: verify_scalar_interface_exact.py. The live evaluator remains paper-vii-kernel.ts.

Write $\delta=\varepsilon/2$, $s=\sin\delta$, $c=\cos\delta$. Let $u$ be the unit internal bisector and $v$ the positively directed unit chord. Then

The first and second outward marks are $d_1=d-av$, $b_1=b+av$, $d_2=d-a\varphi^2 v$, $b_2=b+a\varphi^2 v$. The six marks have the uniform form

The quantities $x_j$ are normalized coordinates $x/a$ (units of $a$). Physical baseline abscissae are $\pm as,\;\pm a(1+s),\;\pm a(\varphi^2+s)$.

The total baseline width is $\|d_2-b_2\|=2a(\varphi^2+s)=2a\varphi^2+L_{\mathrm{chord}}$. The one-arcsecond chord survives exactly as the excess over the bilateral golden total $2a\varphi^2$.

Theorem 6.1. The three positive bearings from the bisector are

Numerically $\theta_2=69^\circ05'41.496848751\ldots''$. The outer golden increment is $\gamma=\theta_2-\theta_1$ with

Every center angle is a difference of signed bearings $(-\theta_2,-\theta_1,-\theta_0,\theta_0,\theta_1,\theta_2)$, or equivalently

Radial lengths: $\rho_0=a$, $\rho_1=a\sqrt{2(1+s)}$, $\rho_2=a\varphi\sqrt{3+2s}$. The last identity uses $1+\varphi^4=3\varphi^2$.

Collect the normalized center-relative landmark vectors as columns of $X_\Omega$ in the basis $(v,u,n)$. The physical configuration is $P_\Omega(a)=\alpha\mathbf{1}^T+a X_\Omega$. Write

The normalized Gram matrix is $\overline G_\Omega=c^2\mathbf{1}\mathbf{1}^T+zz^T=UU^T$ with entries $(\overline G_\Omega)_{ij}=c^2+z_i z_j$. For general scale, $G_\Omega=a^2\overline G_\Omega$.

Proposition 8.1. Because symmetry gives $\sum_i z_i=0$,

Both diagonal entries are positive, so $\operatorname{rank}U=\operatorname{rank}\overline G_\Omega=2$. That is the proof. A floating-point rank of the binary64 image is a numerical check of the same identity, not a substitute for it. The local kernel is a rigid marked plane, not a volume object. Rank two does not prevent exact use in $\mathbb{R}^3$: the third basis vector is the oriented normal.

Proposition 9.1. From an ordered seed in oriented $\mathbb{R}^3$ put $r_d=d-\alpha$, $r_b=b-\alpha$, and

The entire local kernel is reconstructed by the master point equation $p_{j,\sigma}=\alpha+a(cu+\sigma x_j v)$. Its circular body is $C(t)=\alpha+a(\cos t\, u+\sin t\, v)$ for $\delta\le t\le 2\pi-\delta$, and the two feet are $L_\pm$ of length $a\varphi^2$. Every point of the local kernel satisfies $(p-\alpha)\cdot n=0$.

The Oriented R³ panel materializes the same object: 500 samples of the major arc, 100 samples on each golden foot, the six landmarks, the six center rays, the Definition 11.1 scaffold, and the center. Drag to orbit. The companion Julia script evaluates the same binary64 image, including the Gram factorization $\overline G_\Omega=c^2\mathbf{1}\mathbf{1}^T+zz^T$ and all fifteen pairwise center angles. LinearAlgebra rank and floating-point comparisons check that image; they do not prove the theorem.

Positive homothety about $\alpha$ acts freely and transitively on $\{K_\Omega(a):a>0\}$. Lengths scale by the factor, Gram entries by its square, and the complete angular fan is independent of $a$. An element of $\mathrm{SE}(3)$ places the canonical object anywhere in oriented three-space without changing distances, center angles, incidence, or handedness.

Exact identities and numerical checks are different evidence. $\varphi^2-\varphi-1=0$ follows from $\varphi=(1+\sqrt5)/2$. A floating-point residual of zero checks an implementation. The seven kernel nodes are $\alpha$ and the six marks. Agreement with the stored binary64 coordinates in the node log, including a maximum coordinate difference of zero, establishes agreement with those stored values. It is not an exact-arithmetic certificate.

A view is a camera, not a second curve. Planarity is $(p-\alpha)\cdot n=0$. The orthogonal view along $n$ shows the full outline. A view along either in-plane axis collapses it. With $c=\cos(\varepsilon/2)$ and $A=\varphi^2+s$,

is the image of a view along $v$, and

is the image of a view along $u$. Naming either collapsed image “plan” or “side” records the chosen orientation. In the hosted identity pose, $\alpha=0$, $a=1$, $u=(0,-1,0)$, $v=(1,0,0)$, and $n=(0,0,1)$.

The marked curve and its selected readout are determined. Thickening, sweeping, extrusion, and revolution are not. If $\Gamma$ stays the centerline, the six marks and the spine quantities $G$, $C$, and the outline length stay the measurements of that spine. An offset boundary generally has a different length. A sweep also needs a cross-section orientation. At the nonsmooth joins $d$ and $b$ the curve does not supply a unique transverse normal plane. $W_{\mathrm{readout}}=G+C$ remains that selected sum. It is not a scaffold width and not an offset length.

Revolution is a valid additional construction once an axis is specified. Whether labeled features merge depends on that axis, and a point cross-section does not prevent a merger. Revolving the canonical kernel about the $y$-axis of the identity pose sends both feet onto one annulus

The circular component generates the spherical surface

Those are consequences of that chosen revolution, not source geometry. This account adds no thickness and no depth.

Definition 11.1 adopts the repeated $a,a\varphi$ outer annotations as total half-height $a\varphi^2$ above and below the baseline, using the already determined endpoints $d_2,b_2$ as side coordinates. Writing $w=-u$ and $m=(d+b)/2$,

Proposition 11.3. Scaffold width and height are $W=2a(\varphi^2+s)$ and $H=2a\varphi^2$, hence $W-H=2as=L_{\mathrm{chord}}$. The exact departure from a square is the one-arcsecond chord length. At $\varepsilon=0$, $s=0$ and $W=H$.

The same identities give $\|S-\alpha\|=a(\varphi^2-c)=a\varphi+h_{\mathrm{sag}}$. The companion Julia script and the browser check compute $W=\|T_+-T_-\|$, $H=\|T_--E\|$, and $L_{\mathrm{chord}}=\|b-d\|$ from the constructed coordinates, then compare $W\approx H+L_{\mathrm{chord}}$ at 64 machine epsilons. That is a binary64 identity check of the constructed points, not a re-typing of the closed forms.

This is an adopted transcription of a drawing rule, not a uniqueness theorem for the whole notebook figure. The larger circle and additional curved links remain optional Euclidean objects to be solved only after their incidences or radii are fixed. A genuinely nonplanar point is represented as $p=\bar p+hn$; signed-depth completion is the last stage of the hierarchy

The first three stages are explicit in this preprint. The last is not claimed here.

A reference generator accompanies the preprint. It exports the canonical landmark record as JSON and a sampled OBJ polyline. OBJ sampling affects only the displayed circular curve; landmark coordinates are direct evaluations of the closed formulas. Reconstructing from $\mathcal K=(\alpha,a,u,v,n,\varepsilon,\varphi)$ requires no individually stored coordinates.

• omega_reference_geometry.py · 10096 bytes · SHA-256 5151dcac227eae510dbcf26f0815c39e8ab66ef54482d8973c93347008117489Copy
• OMEGA_local_kernel_v1_6_0.json · 4281 bytes · SHA-256 b280522bcd2cf63927e8dfd6438f78fa3a2e7af0cbfd484ee9cd55994b9c3753Copy
• OMEGA_local_kernel_v1_6_0.obj · 203905 bytes · SHA-256 2654f85dab5881e9a0659985729450a89d7cc8df3b07035c22a4c4372b10a079Copy
• omega_kernel_oriented_r3.jl · 6235 bytes · SHA-256 760162f6470fde51c95cf30cca459e2e160825d73e790a46dc4580aca588d47cCopy
• verify_geometry.py · 7860 bytes · SHA-256 140a957e3c5bb26f3c88eefdfc21b4bceac132900b07cc2ef14aa03b702ffcaeCopy
• OMEGA_Paper_VII_v1.6.0_publication_ready_package.zip · 2923282 bytes · SHA-256 ac7f3ef23cd1fdc51e17ffab12762f1042a70085a3281eade8d5495dfee4118dCopy

Run: place the published JSON and OBJ in generated/ beside the scripts, then python3 verify_geometry.py (NumPy required). Independent generation: python3 omega_reference_geometry.py --json regenerated/OMEGA_local_kernel_v1_6_0.json --obj regenerated/OMEGA_local_kernel_v1_6_0.obj. Generator defaults write into the current working directory; those CWD files are not the published pair. Optional Julia visualization: julia --project=. omega_kernel_oriented_r3.jl after installing Plots in that separate environment. Plots is not a proof-companion dependency. The Julia script is a unit-scale binary64 materialization of the kernel, the 15-angle fan, the Gram factorization, and the outer scaffold. It is a numerical visualization and reconstruction tool, not a theorem checker for Theorem 3.4, and it is not the Julia mathematical proof companion.

The Julia proof companion (v0.1 / software 0.1.0) is indexed at /julia-proofs: written geometric proofs, exact descriptors, placement interfaces, and a reconstruction example, corresponding to the Lean source. Sixty-eight members hosted; SHA256SUMS match. Author-supplied O0/O3 records are not a site run. Companion software, not a source paper and not a unique shell.

Further locally hosted packages share Ω₀ constants, a four-dimensional rotation, or a forced fluid realization of a marked planar copy. None of these packages is a theorem of Paper VII, and none modifies frozen SRSD.

OMEGA_REFERENCE_GEOMETRY/0.1 is a software state schema: pair coordinates $\Theta(x,y)$, frozen SRSD encode/decode, and canonical JSON. It is an architectural draft for iteration, not a new theorem and not a package-conformance run of a later geometry RFC.

OMEGA drift experiment v0.1 is a falsification-first numerical benchmark of an experimental 4D rotation $R(2\pi\lambda)\oplus R(2\pi\mu)$. That operator is a hypothesis layer. The published run finds no universal privilege of an OMEGA fixed anchor across uniformly random dynamics. An interactive view of the same pair is at /geometry/pair-dynamics.

OMEGA object exploration 02 derives a four-point cross-ratio inverse, affine/projective landmark decoding bounds, and a nonnegative degree-four obstruction for uniform baseline integration on the six marks. The 36-degree Gauss–Lobatto member is a related family instance. Canonical seed remains one arcsecond. Not a theorem of Paper VII.

Self-similarity, numerical difference, and exact recovery is a 13 October 2026 note. It states five conclusions and written arguments for them. This archive reran the accompanying checker and has not reviewed those arguments as a proof. Not a theorem of Paper VII.

Integration proof bundle is a 14 October 2026 ledger of 34 theorem records and 12 excluded claims. This archive reran its Python checker and has not reviewed the written arguments. Julia and Rust were not in the upload. Not a theorem of Paper VII.

Relative representation and recoverable information is a 15 October 2026 thesis. It states a compositional reconstruction claim. This archive reran a transcription of the printed checker and has not reviewed the written arguments. The auxiliary seed $1/2$ is not the canonical sine. Not a theorem of Paper VII.

Published geometry to ellipses and operators derives an equal-weight, circle-centered ellipse from the six center-relative landmarks, an affine chart that recovers source angles with a target metric, and a two-by-two operator whose numerical range is that filled ellipse. A weighted infinite direct sum is specified, not executed. The arc-and-feet curve is not identified with the filled ellipse, and $R$ is not $\varphi$. Not a theorem of Paper VII and not an amendment of the Zenodo deposit. Julia was not run.

Six-mark sampling and a conditional polar packing uses the same normalized marks as an invertible sampler on polynomials of degree at most five. An added sample bound gives a polar pair whose volume product is $256/45$. The two-ball packing is conditional on two external hypotheses that the note does not establish. Not a theorem of Paper VII.

Affine dynamics, sampling, and quadrature classifies nonnegative exactness on that family: degree 3 below $s_*=(\sqrt5-1)/4$ and degree 5 at and above. This-archive Python and Java match expected.txt. Author rust stdout is hosted; Julia and Rust sources were not received. Not a source paper.

Boundary reconstruction, volume-preserving transport, and an invariant cubic reconstructs harmonic quadratics from an aligned twelve-node icosahedron, transports the decoder by a volume-preserving matrix that preserves $J$, and recovers the cubic coefficient from one extra sample. The explicit ten-sample decoder is the minimal scalar-point form of that reconstruction. The exact field interface records the golden field inside $\mathbb Q(\sqrt2,\sqrt3,\sqrt5)$, the two norms, and the split between a sampled field and an encoded record. Not a theorem of Paper VII.

OMEGA–Navier–Stokes geometric transport, Version 3.0 is a locally hosted fluid realization of a marked planar copy. It is not Paper VII, not a blowup theorem, and not a claim that Navier–Stokes selected $\varphi+\pi$.

Canonical record: 10.5281/zenodo.22233573. Concept DOI 10.5281/zenodo.22233572 identifies all versions. Local mirror: /papers/OMEGA_Paper_VII_v1.6.0.pdf. SHA-256 b6369875753ba9fe29473850bcd5e92c6bb0a9c86d3226abf9439f7f31373b9eCopy. 1570493 bytes.

Version 1.4 remains on this host as a superseded local draft at /papers/OMEGA_Paper_VII_v1.4.pdf and is served noindex, noarchive. Version 1.2 at /papers/OMEGA_Paper_VII_Refined_Draft_v1.2.pdf and version 1.1 are likewise superseded. Version 1.6 is current.

The archive makes no blanket claim of novelty, priority, peer review, or independent validation. Some displayed identities are standard or definitional and are included to make the computational specification self-contained. Any claim of originality must concern a specifically stated formulation or derived result and must be evaluated separately against the relevant prior literature.