Golden Reflection Transfer

Abstract

Reflection-paired golden gains are globally balanced, while pointwise neutrality occurs exactly at zero normal displacement.

Theorem 1.1 (Every Golden Transfer Gain Is Positive).

$$\begin{gathered} \forall \delta :\mathbb{R}, \\ (0<\mathrm{goldenTransferGain}\left(\delta \right)). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/Weil/GoldenCriticalSpectrum/GoldenReflectionTransfer.golden_transfer_gain_pos (✓ std3). ∎

Source. Repository-derived.

Commentary.

The transfer gain is a real exponential and is strictly positive for every real normal displacement.

This sign statement does not require the displacement to arise from a spectral point.

Theorem 1.2 (Reflected Displacement Gives Reciprocal Gain).

$$\begin{gathered} \forall \delta :\mathbb{R}, \\ (\mathrm{goldenTransferGain}\left(-\delta \right)={\mathrm{goldenTransferGain}\left(\delta \right)}^{-1}). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/Weil/GoldenCriticalSpectrum/GoldenReflectionTransfer.golden_transfer_gain_neg (✓ std3). ∎

Source. Repository-derived.

Commentary.

Negating a normal displacement turns its exponential golden gain into the reciprocal gain.

The identity expresses reflection symmetry pointwise and makes no neutrality claim.

Theorem 1.3 (A Reflected Gain Pair Has Product One).

$$\begin{gathered} \forall \delta :\mathbb{R}, \\ (\mathrm{goldenTransferGain}\left(\delta \right)\times \mathrm{goldenTransferGain}\left(-\delta \right)=1). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/Weil/GoldenCriticalSpectrum/GoldenReflectionTransfer.reflected_transfer_product_one (✓ std3). ∎

Source. Repository-derived.

Commentary.

For every displacement, its gain and the gain at the negative displacement multiply to one.

This determinant-like paired balance holds even when neither member has unit gain.

Theorem 1.4 (Unit Gain Characterizes Zero Displacement).

$$\begin{gathered} \forall \delta :\mathbb{R}, \\ ((\mathrm{goldenTransferGain}\left(\delta \right)=1)\iff (\delta =0)). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/Weil/GoldenCriticalSpectrum/GoldenReflectionTransfer.golden_transfer_gain_eq_one_iff (✓ std3). ∎

Source. Repository-derived.

Commentary.

The golden transfer gain equals one exactly when the real normal displacement is zero.

Strict positivity of the golden period makes the exponential coordinate injective at the unit value.

Theorem 1.5 (Both Reflected Gains Are Unit Exactly on the Fixed Axis).

$$\begin{gathered} \forall \delta :\mathbb{R}, \\ (((\mathrm{goldenTransferGain}\left(\delta \right)=1)\wedge (\mathrm{goldenTransferGain}\left(-\delta \right)=1))\iff (\delta =0)). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/Weil/GoldenCriticalSpectrum/GoldenReflectionTransfer.reflected_pair_pointwise_neutral_iff (✓ std3). ∎

Source. Repository-derived.

Commentary.

A displacement and its reflection both have unit gain exactly at zero displacement.

The conjunction enforces pointwise neutrality of both members, which is stronger than their automatic product balance.

Theorem 1.6 (Paired Balance Is Strictly Weaker Than Pointwise Neutrality).

$$((\mathrm{goldenTransferGain}\left(1\right)\times \mathrm{goldenTransferGain}\left(-1\right)=1)\wedge (\mathrm{goldenTransferGain}\left(1\right)\ne 1)).$$

Proof. Machine-checked in Lean as D5/S3/Weil/GoldenCriticalSpectrum/GoldenReflectionTransfer.paired_balance_strictly_weaker (✓ std3). ∎

Source. Repository-derived.

Commentary.

At displacement one, the reflected gain pair still has product one while the positive-displacement gain is not one.

This explicit witness separates global paired balance from the pointwise unit-gain condition.

References

• Truth anchor: D5/S3/Weil/GoldenCriticalSpectrum/GoldenReflectionTransfer.golden_transfer_gain_eq_one_iff
• Truth anchor: D5/S3/Weil/GoldenCriticalSpectrum/GoldenReflectionTransfer.golden_transfer_gain_neg
• Truth anchor: D5/S3/Weil/GoldenCriticalSpectrum/GoldenReflectionTransfer.golden_transfer_gain_pos
• Truth anchor: D5/S3/Weil/GoldenCriticalSpectrum/GoldenReflectionTransfer.paired_balance_strictly_weaker
• Truth anchor: D5/S3/Weil/GoldenCriticalSpectrum/GoldenReflectionTransfer.reflected_pair_pointwise_neutral_iff
• Truth anchor: D5/S3/Weil/GoldenCriticalSpectrum/GoldenReflectionTransfer.reflected_transfer_product_one
• Dependency: D5/S3/Weil/GoldenCriticalSpectrum/GoldenCriticalRadius