The Golden Heat Spectrum and Its L2 Midline

Abstract

The excited golden Euler spectrum has heat abscissa one over phi squared and strict L2 threshold one over twice phi squared.

Theorem 1.1 (The golden spectrum has abscissa one over phi squared).

$$\mathrm{IsHeatAbscissa}(goldenSpectrum,\frac{1}{{\phi }^2}).$$

Proof. Machine-checked in Lean as D5/S3/Midline/GoldenHeatSpectrum.golden_heat_abscissa (✓ std3). ∎

Source. Repository-derived.

Commentary.

The golden Euler spectrum has convergence strictly to the right of its heat abscissa and divergence strictly to the left; no boundary behavior is asserted.

Theorem 1.2 (The golden heat coefficient is L2 right of the midline).

$$\forall s:\mathbb{C},\frac{1}{2\times {\phi }^2}<\Re (s)\Rightarrow \mathrm{MemLp}(\mathrm{heatCoefficient}(goldenSpectrum,s),2).$$

Proof. Machine-checked in Lean as D5/S3/Midline/GoldenHeatSpectrum.golden_heat_l2_mem (✓ std3). ∎

Source. Repository-derived.

Commentary.

Above the strict half-abscissa threshold, the universal heat-trace result gives L2 membership for the golden heat coefficient.

Theorem 1.3 (The golden heat coefficient is not L2 left of the midline).

$$\forall s:\mathbb{C},\Re (s)<\frac{1}{2\times {\phi }^2}\Rightarrow \neg \mathrm{MemLp}(\mathrm{heatCoefficient}(goldenSpectrum,s),2).$$

Proof. Machine-checked in Lean as D5/S3/Midline/GoldenHeatSpectrum.golden_heat_l2_not_mem (✓ std3). ∎

Source. Repository-derived.

Commentary.

Below the strict half-abscissa threshold, the universal heat-trace result excludes L2 membership for the golden heat coefficient.

References

• Truth anchor: D5/S3/Midline/GoldenHeatSpectrum.golden_heat_abscissa
• Truth anchor: D5/S3/Midline/GoldenHeatSpectrum.golden_heat_l2_mem
• Truth anchor: D5/S3/Midline/GoldenHeatSpectrum.golden_heat_l2_not_mem
• Dependency: D5/S3/Midline/UniversalHeatTrace