C05.1 · Derivation

The golden identities form one algebraic chain

From \(\phi^2=\phi+1\), division by \(\phi\) gives \(\phi^{-1}=\phi-1\). Squaring these relations and adding yields \[ \phi^2+\phi^{-2}=3. \] The determinant and adjoint follow from the diagonal cubic: \[ N(J)=1,\qquad J^\#=J^{-1}. \] Consequently \[ B(J,J)=B(J^\#,J^\#)=4,\qquad B(J,J^\#)=3, \] and therefore \[ G=\begin{pmatrix}4&3\\3&4\end{pmatrix},\quad \det G=7,\quad \sin^2\theta_{\rm mismatch}=\frac7{16}. \] Every displayed value is printed by the `golden.*` output keys. These are linked consequences of the same specified golden pair, not independent empirical successes.

Imposing \(\det G=7\) as a selector hypothesis is a different logical operation from evaluating \(\det G\) after choosing the golden spectrum. This article performs the evaluation; it does not infer the necessity of the selector premise from its own successful evaluation.

Unit determinant alone is too weak

For the reciprocal family \(D(r)=\operatorname{diag}(r,1,r^{-1})\), the determinant stays unity, but \[ \det G(r)=\bigl(r^2+1+r^{-2}\bigr)^2-9. \] The script derives this from the actual adjoint and trace form. The exact control \(r=3/2\) gives \[ \det G=6025/1296,\qquad \sin^2\theta_{\rm mismatch}=6025/17689. \] These are not the golden values (`negative_control.r3over2.*`). A normalization premise is not already a spectrum-selection theorem.

A second control locates an omitted hypothesis in a source shorthand

Paper 3 p.59 says, as printed:

Paper 3 §5.4, sealed Rev32.7 PDF p.59:

> ∥Jvac ∥2 = 4 is a theorem from DET-7 + N (J) = 1,

Read literally as an implication for an arbitrary positive diagonal spectrum, this is missing a condition. The full symmetry-form selector in Appendix E p.164 also imposes inversion symmetry. The omission matters for the separate equal-norm statement, though it does **not** invalidate the internal mismatch ratio.

Here is an exact countermodel to the shorter equal-norm implication. Put \[ u^3=\frac{11+\sqrt{105}}4,\qquad D=(\sqrt u,\sqrt u,u^{-1}),\qquad u>0. \] The determinant is unity. Its squared trace norms are \[ U=2u+u^{-2},\qquad V=2u^{-1}+u^2. \] The defining polynomial \[ 2u^6-11u^3+2=0 \] implies \(UV=16\), hence \(\det G=7\), because \(B(D,D^\#)=3\). But the numerator of \(U-4\) has polynomial gcd \(1\) with the defining polynomial. Thus \(U\ne4\). The code certifies both the zero remainder for \(UV-16\) and the gcd statement under `DET7_equal_norm_control.*`.

The correct distinction is: \[ N(D)=1,\ \det G=7\ \Longrightarrow\ UV=16 \quad\Longrightarrow\quad \sin^2\theta_{\rm mismatch}=7/16, \] whereas \(U=V=4\) needs the additional reciprocal/selected-spectrum condition. **The statement of LIB2-356 survives this control.** The source's intermediate norm inference needs its suppressed selector hypothesis restored. This is reported for house review, not silently edited in the Registrar.

Dimension ratio is not a mass exponent

The declared octonion basis has dimension \(8\). The diagonal Jordan frame has rank \(3\), with the upper bound supplied by the cubic identity and the lower bound by the displayed orthogonal idempotents. Hence \[ \dim(\mathbb O_s)/\operatorname{rank}(J_3)=8/3. \] The dimensions and exact ratio appear under `dimensions.*`.

Appendix A §A.9, Rev29 correction, sealed Rev32.7 PDF p.148:

> Neither exact fact establishes, by itself, that these are the mass exponent and the normalization:

The dimension calculation therefore remains internal. It is not a derivation of a physical scaling exponent.

The elementary first-row weight and its stated inputs

For the real unit in one off-diagonal Hermitian block, the squared trace norm is \(2\). For the stipulated three-copy vector it is \(3\). Normalizing both gives \[ \frac1{\sqrt2}\frac1{\sqrt3}=\frac1{\sqrt6}. \] The actual block product, copy-vector norm and resulting product are computed under `normalization.*`. They reproduce the scalar factorization carried by LIB2-050.

This does not derive the use of the three-copy vector, the choice of a first-row route, or a universal block-weight functor. No kaon data or physical CKM angle is used in the calculation.

The two registered generator results are not re-encoded as trivial scalar tests

LIB2-051 reports the coefficient \(c=1/2\) in the registered decomposition \(T_{\rm half}=T_{\rm scalar}+cT_{\rm oct}\). LIB2-052 reports uniqueness within the normalized, registered Peirce sector. The extract cites A558 and A603/E110, respectively, but does not supply the defining matrices, the complete normalization constraints or the full sector parametrization.

We therefore do **not** define a new matrix with an inserted half coefficient and announce that the registered generator has been verified. Nor does a dimension count of an unrelated centralizer verify its uniqueness. Both rederivations remain **[NOT VERIFIED]**, with the specific missing inputs listed in `GENERATION_HOLDS`.

Finite trace normalization is another object

The ratio in LIB2-053 is a readout on a reflected module under a Frobenius compatibility constraint. It is not established by dividing the imaginary-octonion dimension by the Jordan rank, even though the scalar agrees. Companion C08.2 computes the actual constraint defects, including an unequal-weight negative control. This article invokes that separate calculation rather than relabelling a scalar coincidence as its proof.