Exact field interface

Let $F=\mathbb Q(\sqrt5)=\mathbb Q(\varphi)$ and $K=\mathbb Q(\sqrt2,\sqrt3,\sqrt5)$. Then $F\subset K$, $[K:\mathbb Q]=8$, and $[K:F]=4$. The eight products $1,\sqrt2,\sqrt3,\sqrt5,\sqrt6,\sqrt{10},\sqrt{15},\sqrt{30}$ span $K$. Their independence over $\mathbb Q$ is the standard basis theorem for the distinct square-free primes $2,3,5$. The checker records a nonzero determinant of the sign matrix of the eight embeddings.

This places the golden algebra of $N(x,y)=x^2+xy-y^2$ inside a larger exact coefficient field. It does not put OMEGA coordinates, or any constant that depends on $\pi$, into $K$. Transcendence of $\pi$ is a classical theorem. This note does not reprove it.

Every element of $K$ has one expression

The coefficient order is exactly the displayed basis order, as recorded in basis_order and the checker’s BITS tuple. An exact record keeps those eight rationals, each as a reduced numerator and a positive denominator, in integer strings. coefficient_record.schema.json is a JSON Schema for omega-exact-coefficient/1: eight coefficient objects, an integer-string numerator, and a positive-integer-string denominator. The checker also rejects an unreduced fraction and any zero other than 0/1. Trace and norm do not replace the coefficient vector.

For $x=u+v\varphi$ with rational $u,v$,

Each golden conjugate occurs four times among the eight embeddings of $K$. The larger norm drops the sign. $N_{F/\mathbb Q}(1)=1$ and $N_{F/\mathbb Q}(\varphi)=-1$, but both have larger norm $1$. Also $1$ and $\varphi^2$ are distinct and share golden norm $1$. Neither norm identifies the element by itself.

$K$ contains $\mathbb Q$, so it is dense in $\mathbb R$. A finite extension is not automatically a lattice. The length of $n=(2,3,0)$ is $\sqrt{13}$, and $\sqrt{13}\notin K$. The quadratic subfields are $\mathbb Q(\sqrt m)$ for $m\in\{2,3,5,6,10,15,30\}$. Those are square-free radicands, not field discriminants. That list is the Galois correspondence for this extension. The checker only confirms that $13$ is not one of those seven radicands.

Reflection through the plane perpendicular to $n$ is

The entries are rational, and $R^2=I$, $R^{\mathsf T}R=I$, $Rn=-n$. The reflection preserves $K^3$ without taking $\sqrt{13}$ as a coefficient. Coordinates, lengths, and operations are separate specifications.

For integer observations, when the set is nonempty,

Both the agreement on $0\le d<d_*$ and the inequality at $d_*$ are required. On the nonnegative integers that minimum is attained. The sequences $(0,1,2,3)$ and $(0,1,2,4)$ give $d_*=3$.

On a continuous domain the infimum need not be a disagreement point. If $P_A=0$ and $P_B(t)=0$ for $t\le 0$, $P_B(t)=t$ for $t>0$, the disagreement set is $(0,\infty)$. Its infimum is $0$, where the two functions still agree. Do not replace a missing value by an unspecified nearest element of $K$: the field is dense, and a nearest element need not exist.

Let $V$ be a finite-dimensional space of continuous functions on the sphere. If a positive-weight cubature rule integrates every product from $V$ against normalized spherical area, then

Vanishing samples force the integral of $f^2$ to vanish, hence $f=0$. Sampling is injective on $V$. For polynomials of degree at most $d$, exactness through degree $2d$ is sufficient. It is not a universal necessity, and it says nothing about nodes that are free to move.

The hosted twelve-node degree-five rule is exact on products from $\mathcal H_{\le 2}$, which has dimension 9. The previous paragraph therefore supports unique reconstruction on that space. This note does not add a cubature rule. The rule remains the one on the boundary-reconstruction note.

Under the specified alignment, $J(x,y,z)=z(x^2+xy-y^2)$ vanishes at the twelve nodes, while

Those are different facts:

• $J$ is an invisible harmonic cubic.
• $J^2$ is a degree-six integration obstruction.
• One extra sample with $J\ne 0$ identifies the coefficient on $\mathcal H_{\le 2}\oplus\mathbb R J$.
• That does not reconstruct an arbitrary cubic.

The minimal scalar-point form is the ten-sample decoder.

For $s_i>0$, $d_i=\lfloor y_i/s_i\rfloor$ and $\rho_i=y_i-s_id_i$, so $y_i=s_id_i+\rho_i$. The identity is exact for that division. A real residual need not have a finite exact encoding.

The chain is the declared field space, then injective samples, then exact sample records, then reconstruction. The 399 floor map is one integer encoding already in use. It is not the field $F$, and $Q/S$ is not $\lambda$.

From an empty directory: download the checker and the schema beside it, then run python3 verify_exact_interface.py and python3 -O verify_exact_interface.py. Both should print TOTAL 29 PASS. The optimized transcript is the same bytes as the normal one.

This section is a 15 October 2026 supplement. It does not change the 25 September note, the 399 ledger, the thesis file, or frozen G₆. The sine is named and is not evaluated.

Not an additional source paper, not a DOI, and not a change to the eight counted works. Not an enlargement of Paper VI, Paper VII, or paper 8. Not a new cubature rule and not a maximality claim. Not a physical lattice, a mass spectrum, or a validation of another framework. Not peer review and not independent reproduction. Frozen releases are unchanged.