Golden Modular Metric Criterion

Abstract

The golden modular flow reaches its positive-metric unitary boundary exactly at zero horizontal drift.

Theorem 1.1 (Positive metric realization is equivalent to zero modular drift).

$$\begin{gathered} \begin{array}{c}\forall \delta ,\gamma :\mathbb{R},\\ \mathrm{let}P:\mathbb{R}=2\cdot \log (\phi );\mathrm{let}a:\mathbb{C}=(\delta +i\gamma )\cdot P;\\ \mathrm{let}u:\mathbb{C}=\exp (a);\mathrm{let}v:\mathbb{C}=\exp (-(a));\\ \mathrm{let}M:\mathrm{Matrix}(\mathrm{Fin}(2),\mathrm{Fin}(2),\mathbb{C})=\mathrm{diagonal}([u,v]);\mathrm{let}\rho :\mathbb{C}=\frac{1}{2}+\delta +i\gamma ;\\ \mathrm{let}U:\mathbb{R}=\frac{1}{2}\cdot \mathrm{Re}(\mathrm{trace}(M^{\ast }\cdot M))-1;\\ \mathrm{ListTFAE}([\delta =0, \\ \forall \lambda :\mathbb{C},\lambda \in \mathrm{spectrum}(\mathbb{C},M),\Vert \lambda \Vert =1, \\ \exists H:\mathrm{Matrix}(\mathrm{Fin}(2),\mathrm{Fin}(2),\mathbb{C}),\mathrm{PosDef}(H)\wedge M^{\ast }\cdot H\cdot M=H, \\ U=0])\wedge \\ \delta \ne 0\Rightarrow \mathrm{xiReading}(\rho )=0\Rightarrow \mathrm{xiReading}(\mathrm{criticalLineMirror}(\rho ))=0\wedge \neg \exists H:\mathrm{Matrix}(\mathrm{Fin}(2),\mathrm{Fin}(2),\mathbb{C}),\mathrm{PosDef}(H)\wedge M^{\ast }\cdot H\cdot M=H.\end{array} \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/Algebra/GoldenModularMetricCriterion.golden_modular_metric_criterion (✓ std3). ∎

Source. Repository-derived.

Commentary.

The source matrix is constructed in its two-dimensional eigenbasis from arbitrary real drift and phase and the fixed golden logarithmic period.

Zero drift, unit norm of every spectral value, preservation of a positive definite Hermitian metric, and vanishing normalized trace defect are equivalent.

At nonzero drift, a zero of the completed reading retains its canonical same-height reflected zero, while no positive definite invariant metric exists.

References

• Truth anchor: D5/S3/ConceptDynamics/Algebra/GoldenModularMetricCriterion.golden_modular_metric_criterion
• Dependency: D5/S3/ConceptDynamics/Algebra/PositiveInvariantMetricSelection
• Dependency: D5/S3/Weil/ZetaLinear/ReflectedZeroModePhaseFlattening
• Dependency: D5/S3/Zeros/Symmetry/ZetaConjugationCovariance