Typed network of exact interfaces

Source: locally hosted dossier OMEGA_NAUGHT_Typed_Network_Exact_Proof_Dossier_v1.0.pdf · 1105802 bytes · SHA-256 2e4c1f1afd3a6fa1b9b1076722d7589ba89cbea07be3a8ab258f5a44a142e3e8Copy. Network audit v1.0, 5 September 2026. Cubature certificate cited as v1.1, 3 September 2026. Not a Zenodo preprint. No DOI. Does not enlarge Papers I–VII.

Locally hosted proof dossier. Not a Zenodo preprint. No DOI. Does not enlarge or rewrite Papers I–VII, CLASS_I_CUBATURE/1.0, CLASS_I_CUBATURE/1.1, or G6_RELEASE/1.1. The cubature coefficient payload cited as LaTeX filecontents is not extractable from the hosted PDF; those original bytes are now hosted separately under /cubature/. The de-branded executable audit is a separate hosted certificate of maps, not the dossier’s embedded verifier and not a joining theorem. Independent third-party reproduction is currently recorded as none.

The corresponding machine object is typed-map-audit in /archive/MACHINE.json. Name identification is not a joining theorem.

A supplementary verifier, TGA 2.1, is the active page at /tga. It keeps the TGA 2.0 witness graph and adds scaling identities, path-equality certificates, and a six-mark membership test. Frozen TGA 2.0 bytes remain at /tga/2.0/graph-report.json. The new report is /tga/2.1/graph-report.json. An edge is admitted only with a supported witness. A path certificate is admitted only for an implemented proof schema. Shared constants stay proposals. It is not a general theorem prover, not a source paper, and it does not replace this dossier or the production geometry and field-decoder modules. The interactive mathematical atlas is a separate page. It does not replace this dossier.

The dossier types exact maps between distinct theorem packages. It is stronger than a statement that the papers are merely related. It is not a claim that those packages form one retrospective theory, and it is not a joining theorem. Historical G₆ packaging remains historical packaging. Frozen SRSD v1.0 is used only as an external encoder. The cubature existence theorems remain logically independent of Ω₀, the floor map, and SRSD.

Two diagrams are proved, each arrow with a declared domain, codomain, and identity:

The middle double arrow is conjugacy of restricted step maps on $D_{15}$, not equality of the complete contents of Papers I and II. The independent cubature junction is field inclusion plus proof dependence inside the cubature branch only:

Set $\varphi=(1+\sqrt5)/2$, $\varepsilon=\pi/648000$, $\delta=\varepsilon/2=\pi/1296000$, $s=\sin\delta$, $c=\cos\delta$, and

The relevant fields are $K_5=\mathbb{Q}(\sqrt5)$, $K_\Omega=\mathbb{Q}(\pi,\sqrt5)$, $K_{\mathrm{coord}}=\mathbb{Q}(\sqrt5,s,c)$, and $K_{\mathrm{VII}}^{\mathrm{const}}=\mathbb{Q}(\pi,\sqrt5,s,c)=K_\Omega(s,c)$.

Correction 1. The coordinate-coefficient field means the coefficients of the marked-point formula relative to the formal realization data $(\alpha,a,u,v)$. It does not include arbitrary Cartesian coordinates of $\alpha,u,v$ or an arbitrary real scale $a$. Those choices can enlarge the field of a particular realization and must not be inserted into the intersection theorem.

Correction 2. The quantity $\tau_\Omega=1/\Omega_0$ is a distinguished Omega constant. In Paper I the notation $\tau>0$ denotes an arbitrary quantization scale. Throughout this interface theorem only, Paper I’s published scale is renamed $h>0$ with $Q_h(y)=h\lfloor y/h\rfloor$. This is a notational relabeling, not a modification of Paper I. The geometric dummy parameter on a Paper-VII baseline segment is likewise not the contraction scalar $\lambda$; when needed it is called $t$.

With the definitions in the dossier, all of the following statements hold.

1. $K_{\mathrm{coord}}\cap K_\Omega=K_5=\mathbb{Q}(\sqrt5)$. The intrinsic full constant field of Paper VII instead contains $K_\Omega$ and satisfies $K_{\mathrm{VII}}^{\mathrm{const}}=K_\Omega(s,c)$ and $K_{\mathrm{VII}}^{\mathrm{const}}\cap K_\Omega=K_\Omega$.
2. The distinguished scalar packages $(\varphi,\varepsilon)$ and $(\tau_\Omega,\lambda)$ reconstruct one another exactly.
3. Once independent realization data $(\alpha,a,u,v)$ are supplied, the Paper-VII marked coordinates are reconstructed exactly from $(\tau_\Omega,\lambda)$.
4. For every $h>0$, the Paper-I quantized step restricted to the lattice $hD_{15}$ is conjugate to the Paper-II rational integer step by $J_h(d)=hd$.
5. The Paper-II quotient and Euclidean remainder are exactly the frozen scalar SRSD encoding of $Qd$ at scale $S$.
6. The cubature coefficient field $K_5$ embeds naturally into $K_\Omega$, but the certified $f=23,24$ cubature theorems remain logically independent of Omega arithmetic, the floor dynamics, and SRSD.

The summary is therefore: VII ↔ V exact scalar reconstruction; V → I identical contraction scalar; I ↔ II restricted lattice conjugacy on $D_{15}$; II → III exact external quotient–residual encoding; VII/V and IV/VI shared golden coefficient field, without proof dependence.

$s=\sin\delta$ and $c=\cos\delta$ lie in the cyclotomic field $\mathbb{Q}(\zeta_{2592000})$, hence are algebraic over $\mathbb{Q}$. Thus $K_{\mathrm{coord}}/K_5$ is algebraic.

Lindemann’s theorem implies that $\pi$ is transcendental over $K_5$, so $K_\Omega=K_5(\pi)$ is a rational-function field in the transcendental element $\pi$. Every element of $K_5(\pi)$ that is algebraic over $K_5$ already lies in $K_5$. Therefore

The second line is definitional: $\pi=648000\varepsilon$ is already present in Paper VII, so the intrinsic constant field meets $K_\Omega$ in $K_\Omega$ itself. The first equality is the nontrivial theorem; the second prevents extending it from algebraic coordinate coefficients to all of Paper VII’s intrinsic constants.

Define

On the image of this map the inverse is

Both compositions are the identity on the declared scalar packages. Consequently $\Omega_0=1/\tau_\Omega$, $\mu=1-\lambda$, $\varphi=\lambda/\tau_\Omega$, $\pi=(1-\lambda)/\tau_\Omega$, and $\sqrt5=2\lambda/\tau_\Omega-1$, so $K_\Omega=\mathbb{Q}(\tau_\Omega,\lambda)=\mathbb{Q}(\pi,\sqrt5)$.

Once independent realization data are supplied, the six ordered marks are

with $x_0=s$, $x_1=1+s$, $x_2=\varphi^2+s$. This is an exact realization interface. It does not say that Paper V alone proves the Paper-VII geometric theorems.

Paper I defines $\lambda_\Omega=\varphi/(\varphi+\pi)$. The Omega arithmetic package defines the same real number as $\lambda=\varphi/\Omega_0$. Therefore $\lambda_\Omega=\lambda$. This is equality of real numbers, not an approximation and not an import of one paper’s theorem package into another.

Set $S=10^{15}$, $Q=339949771344778$, $q=Q/S$, $D_{15}=\{0,1,\ldots,32767\}$, $F_q(d)=\lfloor Qd/S\rfloor$, and $F_\lambda(d)=\lfloor\lambda d\rfloor$. Paper I’s quantizer at scale $h>0$ is $T_h(z)=Q_h(\lambda z)$.

Exact rational enclosure of $\lambda$, using Machin’s formula and an integer square-root bound for $\sqrt5$, gives $q<L<\lambda<U<q+10^{-15}$ with 60-decimal endpoints printed in the dossier, and therefore $Q=\lfloor S\lambda\rfloor$. No floating-point value is used in those comparisons.

For every $d\in D_{15}$, $F_\lambda(d)=F_q(d)$. An independent boundary-margin certificate shows that the displacement from $qd$ to $\lambda d$ is smaller than the available upward gap on the whole domain, with minimum gap $\Delta_{\min}=34535026732/10^{15}$ attained at $d=20706$. Consequently, for every $h>0$,

and the same identity holds for every iterate. This is conjugacy of restricted step maps, not equality of Papers I and II outside that finite domain.

Write $E_S$ and $D_S$ for the frozen scalar SRSD encoder and decoder at scale $S$, $M_Q(d)=Qd$, and $\rho_q(d)=Qd-SF_q(d)$. Then for every nonnegative integer $d$,

Multiplication by $Q$ and the final division by $Q$ are external maps. No SRSD definition, ontology, or theorem is changed. Ordered remainder ledgers reconstruct $d_0$ exactly by the usual finite geometric sum.

The inclusion $\iota:K_5\to K_\Omega$ is the natural embedding $\mathbb{Q}(\sqrt5)\hookrightarrow\mathbb{Q}(\pi,\sqrt5)$. Cubature matrices, moments, and certified weights lie in $K_5$. They may be represented inside $K_\Omega$ through $\iota$. That extension of scalars preserves exact equalities. It is not used to prove them: the cubature verifier works directly in $K_5$ and makes no use of $\pi$, $\Omega_0$, $\lambda$, the finite floor map, or SRSD.

Printed finite sizes: $|X_{23}|=5292$, 56 orbits, 52 invariant rows, rank 52, affine dimension 4, target degree 71, minimum orbit $(21,1,1)$; $|X_{24}|=5762$, 61 orbits, 57 rows, rank 57, affine dimension 4, target degree 75, minimum orbit $(23,1,0)$. Modular rank witnesses use $p=1000000000000000031$.

A same-environment replay quoted in the dossier for 5 September 2026 reports PASS on both frequencies and 109 exact invariant moment identities. That is not an external clean-room reproduction and not peer review. The 117 exact $a+b\sqrt5$ masses, the runnable verifier, and the 109-row ledger are now hosted as original files under /cubature/; they are not extractable from this PDF. The printed extract remains a distinct artifact.

Cubature existence page · Printed finite-audit extract · SHA-256 354976dc65ddcb7194e8f46fe9651234b4bdaea51b25cc9f82fa0d0815bf4688Copy

The hosted Python script is a de-branded executable audit of maps already typed on this page: affine contraction, displacement conjugacy, finite-domain floor equality, remainder-ledger reconstruction, an external quotient–residual encoder, and Euclidean homothety of a finite marked configuration with a linear observable onto the scalar contraction. Name identification is not a joining theorem. The script’s own end-to-end statement requires no geometric, physical, or application-specific interpretation.

Source: typed_connection_proof.py · 31256 bytes · SHA-256 a0c32a3c37fbf217d8111aaa8dd5cc0050d0e302df3ae339b05d99556bb32281Copy. Named runtime: typed_connection_proof.runtime.txt · 2397 bytes · SHA-256 1335a532ae210dccae24532e9fccf0ae3c94f5f8ec83d3e81770309ebe6633a6Copy. Python 3.10.21 with SymPy 1.14.0. Exit 0. Not a Zenodo preprint. Not an additional source paper. Does not enlarge Papers I–VII.

Evidence levels are kept distinct in the script: symbolic identity, analytic argument, exact finite certificate, and regression check. Symbolic identities, checked exactly with SymPy, include affine inverse and difference scaling, displacement conjugacy, the one-parameter group law, the flow ODE, the integer-pair automorphism, homothety/scalar semiconjugacy, all fifteen marked-point squared-distance identities, affine dimension one, center-relative and Gram rank two, and radial-template derivative identities. Exact integer and rational claims, checked with Python integers and fractions.Fraction, include the encoder and decoder, rational grid quantization, and reconstruction from an ordered remainder ledger. The finite-domain claim involving $\pi$ is: for every integer $d\in D=\{0,\ldots,32767\}$,

certified by explicit rational enclosures of $\sqrt5$ and $\pi$ (integer square-root bounds and Machin’s formula), together with $q<\lambda$. No binary floating-point approximation is used in that certificate. The named runtime reports PASS on all 32768 states, exhaustive remainder reconstruction of every seed through 23 steps, and rejection of seven malformed-input classes.

Mapping de-branded names onto archive objects identifies maps. It does not collapse the papers into one theorem, does not import one paper’s proof package into another, and does not reconstruct a unique outer shell.

The script’s end-to-end statement is a statement about those maps: a linear observable of a homothety is the scalar contraction; on any declared integer domain where $F_\lambda=F_q$, quantization lands on the rational floor; ordered remainders reconstruct every finite trajectory. The script then states that no geometric, physical, or application-specific interpretation is required. This archive keeps that disclaimer. Python 3 with SymPy ≥ 1.12. Symbolic identities; exact rational enclosures of √5, π, and λ; exhaustive floor equality on D = {0, …, 32767} with q < λ certified; exhaustive 23-step remainder reconstruction on all 32768 seeds; seven malformed-input classes rejected. No binary64 comparison is used in the floor certificate. Homothety of the six kernel marks is Euclidean scaling of the Paper VII configuration, not a unique shell. Radial template identities in the script are algebraic and are not a physical theorem. Passing the named runtime is implementation conformance of that audit, not independent third-party reproduction and not a joining theorem.

requirements.txt · README.txt · Reproduce this certificate

The dossier pins SHA-256 digests of the TeX sources used for the audit. Those TeX files are not re-hosted here; the printed hashes are recorded in the extract. Paper VII is cited at version 1.6.0. Independent third-party reproduction is currently recorded as none.

A Lean 4 formalization of the selected geometric-provenance measurement is inventoried at /formal. Its documentation is hosted; executable .lean sources are hosted and hash-matched; lake build PASS on this archive 2026-09-13. That package is not a joining theorem and is not independent reproduction of Papers I–VII.

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