# Exact field interface Supplementary mathematical note. 25 September 2026. This note records established field arithmetic and the limited way those facts apply to the existing OMEGA constructions. It is not an additional source paper, not a DOI, and not a change to any frozen contract. Eight source works remain the counted publications. The statements below separate three layers: 1. Standard facts about number fields, norms, reflections, and sampling. 2. Finite checks collected in `verify_exact_interface.py`. 3. Applications to objects already defined on this site. They do not establish a physical vacuum lattice, particle masses, or the physical claims of another framework. ## 1. Exact embedding of the golden algebra Let \[ F=\mathbb Q(\sqrt5)=\mathbb Q(\varphi),\qquad K=\mathbb Q(\sqrt2,\sqrt3,\sqrt5),\qquad \varphi=\frac{1+\sqrt5}{2}. \] Then \[ F\subset K,\qquad [K:\mathbb Q]=8,\qquad [K:F]=4. \] The eight products \[ 1,\sqrt2,\sqrt3,\sqrt5,\sqrt6,\sqrt{10},\sqrt{15},\sqrt{30} \] span \(K\) by the definition of this multiquadratic extension. Their linear independence over \(\mathbb Q\) is the standard basis theorem for distinct square-free primes \(2,3,5\). The checker witnesses that independence by a nonzero determinant of the \(8\times8\) sign matrix of the embeddings. Degree \(8\) follows. Since \([F:\mathbb Q]=2\) and \(F\subset K\), the relative degree is \(4\). This embeds the golden algebra used by the quadratic form \(N(x,y)=x^2+xy-y^2\) into a larger exact coefficient field. It does not put every OMEGA coordinate, or any constant that depends on \(\pi\), into \(K\). The number \(\pi\) is transcendental, so it lies in no finite extension of \(\mathbb Q\). The checker does not reprove that transcendence. ## 2. Reconstructible algebraic coordinates Every \(x\in K\) has a unique expression \[ \begin{aligned} x&=a_0+a_1\sqrt2+a_2\sqrt3+a_3\sqrt5\\ &\quad+a_4\sqrt6+a_5\sqrt{10}+a_6\sqrt{15}+a_7\sqrt{30}, \end{aligned} \] with \(a_i\in\mathbb Q\). The coefficient order is exactly the displayed basis order, as recorded in basis_order and the checker’s BITS tuple. An exact record must retain the ordered eight coefficients. Each rational is a reduced numerator and a positive denominator, stored as integer strings. `coefficient_record.schema.json` is a JSON Schema for `omega-exact-coefficient/1`: the root object carries that schema identifier and exactly eight coefficient objects, and each coefficient has an integer-string numerator and a positive-integer-string denominator. The checker additionally rejects an unreduced fraction and any zero other than `0/1`. Trace and norm may be stored as derived checks. Neither replaces the coefficient vector. Section 3 gives elements that share one norm and differ in the other. ## 3. The two norms For \(x=u+v\varphi\) with \(u,v\in\mathbb Q\), \[ N_{F/\mathbb Q}(x)=u^2+uv-v^2. \] Its norm in the larger field is \[ N_{K/\mathbb Q}(x)=\bigl(u^2+uv-v^2\bigr)^4. \] Reason: the eight embeddings of \(K\) restrict to the two embeddings of \(F\), and each of those occurs four times, once for every sign pair on \(\sqrt2\) and \(\sqrt3\). The checker multiplies those eight conjugates in the basis and compares the product with the fourth power. The larger norm loses the sign of the golden norm. For example \(N_{F/\mathbb Q}(1)=1\) and \(N_{F/\mathbb Q}(\varphi)=-1\), while both have larger norm \(1\). Separately, \(1\) and \(\varphi^2=1+\varphi\) are distinct and both have golden norm \(1\). Neither norm alone identifies the element. ## 4. Field membership is not a geometric operation \(K\) contains \(\mathbb Q\), so \(K\) is dense in \(\mathbb R\). A finite-degree extension is not, by itself, a discrete spatial lattice. Although \(\sqrt{13}\notin K\), the vector \(n=(2,3,0)\in K^3\) has length \(\sqrt{13}\). The quadratic subfields of \(K\) are \(\mathbb Q(\sqrt m)\) for \[ m\in\{2,3,5,6,10,15,30\}. \] None is \(\mathbb Q(\sqrt{13})\). That classification is the Galois correspondence for this multiquadratic extension. The listed values are the square-free radicands of those quadratic subfields, not the field discriminants. The checker confirms that \(13\) is not one of those seven radicands. It does not rederive the correspondence. Reflection across the plane perpendicular to \(n\) is \[ R=I-\frac{2nn^{\mathsf T}}{n^{\mathsf T}n} = \begin{pmatrix} 5/13&-12/13&0\\ -12/13&-5/13&0\\ 0&0&1 \end{pmatrix}. \] The entries lie in \(\mathbb Q\subset K\). The checker verifies \[ R^2=I,\qquad R^{\mathsf T}R=I,\qquad Rn=-n. \] Exact reflection can preserve \(K^3\) here without admitting \(\sqrt{13}\) as a coefficient. Admissible coordinates, lengths, and operations have to be specified separately. ## 5. Representation boundaries For two explicitly defined integer observations \(P_A,P_B\), define \[ d_*=\min\{d\in\mathbb Z_{\ge0}:P_A(d)\ne P_B(d)\} \] when that set is nonempty. A first-disagreement statement needs both \[ P_A(d)=P_B(d)\quad(0\le d0\). The disagreement set is \((0,\infty)\). Its infimum is \(0\), and \(P_A(0)=P_B(0)\). The checker records that pair at \(0\) and at \(1/10\). Do not define an approximation by an unspecified nearest element of \(K\). Because \(K\) contains \(\mathbb Q\), it is dense, and a nearest element need not exist. ## 6. Cubature and sampling injectivity Let \(V\) be a finite-dimensional space of continuous real functions on \(\mathbb S^2\). Suppose a positive-weight cubature rule integrates every product \(fg\), for \(f,g\in V\), against normalized spherical area \(\sigma\). Then \[ \int_{\mathbb S^2}f^2\,d\sigma=\sum_i w_i f(x_i)^2. \] If every sample of \(f\) vanishes, the integral of \(f^2\) is zero. Continuity of \(f\) and full support of spherical area imply \(f=0\). Sampling is therefore injective on \(V\). For restrictions of polynomials of degree at most \(d\), exactness through degree \(2d\) is sufficient. It is not necessary for every possible rule, and it is not a statement about nodes that are allowed to move. ## 7. Application to the existing boundary decoder The hosted twelve-node degree-five rule is exact through degree five, hence exact on products from \[ \mathcal H_{\le 2},\qquad \dim\mathcal H_{\le 2}=9. \] Section 6 therefore supports unique reconstruction on that space. That conclusion uses the existing rule. This note does not publish a new cubature. For the specified alignment, \[ J(x,y,z)=z(x^2+xy-y^2) \] vanishes at all twelve nodes, while \[ \int_{\mathbb S^2}J^2\,d\sigma=\frac1{21}>0. \] The odd terms in \(J^2\) integrate to zero. The even terms are \[ \int x^4z^2\,d\sigma=\frac1{35},\qquad \int x^2y^2z^2\,d\sigma=\frac1{105},\qquad \int y^4z^2\,d\sigma=\frac1{35}. \] Their alternating sum is \(1/35-1/105+1/35=1/21\). The checker recomputes those normalized moments. It does not replace the boundary-reconstruction certificate. Keep these statements separate: - \(J\) is an invisible harmonic cubic for the twelve nodes. - \(J^2\) is a degree-six integration obstruction. - One additional sample with \(J\ne 0\) identifies the coefficient of \(J\) on \(\mathcal H_{\le 2}\oplus\mathbb R J\). - That does not reconstruct an arbitrary cubic field. The explicit ten-point decoder for that ten-dimensional space is the separate note at `/geometry/boundary-reconstruction/ten-sample`. ## 8. Field recovery is not record recovery For \(s_i>0\), \[ d_i=\left\lfloor\frac{y_i}{s_i}\right\rfloor,\qquad \rho_i=y_i-s_id_i, \] so \(y_i=s_id_i+\rho_i\). The identity is exact for the stated division. The checker confirms it for \(y=10\), \(s=3\). Arbitrary real residuals need not have a finite exact encoding. State separately: 1. The function space on which the samples determine the field. 2. The number representation used for those samples. 3. The encoding and ledger that recover the recorded numbers. The interface is \[ \text{declared field space} \longrightarrow \text{injective samples} \longrightarrow \text{exactly encoded sample records} \longrightarrow \text{reconstruction}. \] The 399 floor map is one integer encoding already hosted at `/dynamics/floor-kernel`. It is not the golden field \(F\), and \(Q/S\) is not \(\lambda\). ## What is not claimed - No new source paper and no change to the count of eight works. - No enlargement of Paper VI, Paper VII, or paper 8. - No new positive cubature, and no maximality statement. - No claim that OMEGA coordinates or \(\pi\)-dependent constants lie in \(K\). - No physical lattice, mass spectrum, or external physical validation. - No peer review and no independent reproduction.