Golden Shell Moment Discretization

Abstract

Golden geometric shells recover every positive finite defect moment within a fixed factor.

Theorem 1.1 (Golden shells give a multiplicative moment sandwich).

$$\begin{array}{c}\forall \iota ,\\ m,\delta :\iota \to \mathbb{R},\\ n:\iota \to \mathbb{N},s\in \mathbb{R},\\ \mathrm{Finite}(\iota )\wedge \left(0<s\wedge \left(\left(\forall i\in \iota ,0\le m\left(i\right)\right)\wedge \left(\forall i\in \iota ,\omega \left(n\left(i\right)+1\right)<\delta \left(i\right)\wedge \delta \left(i\right)\le \omega \left(n\left(i\right)\right)\right)\right)\right)\Rightarrow \\ {\phi }^{-2s}\cdot \left({\mathcal{G}}_{\perp }\right)\left(s\right)\le \left({\zeta }_{\perp }\right)\left(s\right)\wedge \left({\zeta }_{\perp }\right)\left(s\right)\le \left({\mathcal{G}}_{\perp }\right)\left(s\right).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Analytic/SeriesInequalities/GoldenShellMomentDiscretization.golden_shell_moment_sandwich (✓ std3). ∎

Source. Repository-derived.

Commentary.

Let a finite family of defects carry nonnegative real weights. Assign each defect delta(i) to the unique supplied shell n(i), between the successive radii omega(n(i)+1) and omega(n(i)). For every positive real exponent s, its exact weighted moment lies between the golden shell transcript and phi^(-2s) times that transcript.

The shell radius is omega(n)=(1/2) phi^(-2n). Consecutive radii differ by the positive ratio phi^(-2). Positive real powers preserve each pointwise shell inequality, nonnegative weights preserve order, and finite summation gives the displayed sandwich.

Finite indexing is the finite-support specialization of the source’s shell charges. It removes convergence assumptions without changing the regrouped weighted sum. The positive-exponent and nonnegative-weight hypotheses are explicit because reversing either sign can reverse the claimed inequalities.

The module also proves the exponent-two factor phi^(-4). A singleton at delta=1/2 computes both second moments as 1/4; moving it to delta=1 breaks the shell premise and computes the exact moment as 1 while the transcript remains 1/4, so the upper conclusion is false.

References

• Truth anchor: D5/S3/Analytic/SeriesInequalities/GoldenShellMomentDiscretization.golden_shell_moment_sandwich