Golden Charge Tomography

Abstract

Neutral and quadratic charge channels invert exactly to split and inert channels by the C2 Fourier transform.

Theorem 1.1 (The Split Channel Is Reconstructed Exactly).

$$\begin{gathered} \forall split:\mathbb{R},inert:\mathbb{R}, \\ (splitFromChannels(\mathrm{neutralChannel}\left(split,inert\right),\mathrm{chargeChannel}\left(split,inert\right))=split). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/PrimeForms/GoldenEuler/GoldenChargeTomography.split_channel_reconstruction (✓ std3). ∎

Source. Repository-derived.

Commentary.

Adding the neutral and signed charge channels isolates twice the split component.

Division by two gives exact finite Fourier inversion and requires no analytic assumptions.

Theorem 1.2 (The Inert Channel Is Reconstructed Exactly).

$$\begin{gathered} \forall split:\mathbb{R},inert:\mathbb{R}, \\ (inertFromChannels(\mathrm{neutralChannel}\left(split,inert\right),\mathrm{chargeChannel}\left(split,inert\right))=inert). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/PrimeForms/GoldenEuler/GoldenChargeTomography.inert_channel_reconstruction (✓ std3). ∎

Source. Repository-derived.

Commentary.

Subtracting the signed charge channel from the neutral channel isolates twice the inert component.

The identity is purely algebraic and does not make a statement about Dirichlet series or zeros.

Theorem 1.3 (The Indicators Partition Unit Mass).

$$\begin{gathered} \forall charge:\mathbb{R}, \\ (\mathrm{splitIndicator}\left(charge\right)+\mathrm{inertIndicator}\left(charge\right)=1). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/PrimeForms/GoldenEuler/GoldenChargeTomography.split_add_inert_indicator (✓ std3). ∎

Source. Repository-derived.

Commentary.

The positive and negative charge indicators add to one for every real charge value.

This partition identity is algebraic; it does not require the charge to equal plus or minus one.

Theorem 1.4 (The Signed Indicator Difference Recovers Charge).

$$\begin{gathered} \forall charge:\mathbb{R}, \\ (\mathrm{splitIndicator}\left(charge\right)-\mathrm{inertIndicator}\left(charge\right)=charge). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/PrimeForms/GoldenEuler/GoldenChargeTomography.split_sub_inert_indicator (✓ std3). ∎

Source. Repository-derived.

Commentary.

The split indicator minus the inert indicator reproduces the original real charge.

Together with the sum identity, this records the two-coordinate inverse transform only.

Theorem 1.5 (Positive Charge Selects the Split Indicator).

$$(\mathrm{splitIndicator}\left(1\right)=1).$$

Proof. Machine-checked in Lean as D5/S3/PrimeForms/GoldenEuler/GoldenChargeTomography.split_indicator_pos_charge (✓ std3). ∎

Source. Repository-derived.

Commentary.

At charge plus one, the split indicator has value one.

The endpoint evaluation identifies the split channel without asserting a classification of inputs.

Theorem 1.6 (Positive Charge Vanishes in the Inert Indicator).

$$(\mathrm{inertIndicator}\left(1\right)=0).$$

Proof. Machine-checked in Lean as D5/S3/PrimeForms/GoldenEuler/GoldenChargeTomography.inert_indicator_pos_charge (✓ std3). ∎

Source. Repository-derived.

Commentary.

At charge plus one, the inert indicator has value zero.

This is the complementary endpoint evaluation to the positive split indicator.

Theorem 1.7 (Negative Charge Vanishes in the Split Indicator).

$$(\mathrm{splitIndicator}\left(-1\right)=0).$$

Proof. Machine-checked in Lean as D5/S3/PrimeForms/GoldenEuler/GoldenChargeTomography.split_indicator_neg_charge (✓ std3). ∎

Source. Repository-derived.

Commentary.

At charge minus one, the split indicator has value zero.

The theorem evaluates the finite indicator and introduces no local Euler hypothesis.

Theorem 1.8 (Negative Charge Selects the Inert Indicator).

$$(\mathrm{inertIndicator}\left(-1\right)=1).$$

Proof. Machine-checked in Lean as D5/S3/PrimeForms/GoldenEuler/GoldenChargeTomography.inert_indicator_neg_charge (✓ std3). ∎

Source. Repository-derived.

Commentary.

At charge minus one, the inert indicator has value one.

This completes the two endpoint evaluations of the charge-channel transform.

References

• Truth anchor: D5/S3/PrimeForms/GoldenEuler/GoldenChargeTomography.inert_channel_reconstruction
• Truth anchor: D5/S3/PrimeForms/GoldenEuler/GoldenChargeTomography.inert_indicator_neg_charge
• Truth anchor: D5/S3/PrimeForms/GoldenEuler/GoldenChargeTomography.inert_indicator_pos_charge
• Truth anchor: D5/S3/PrimeForms/GoldenEuler/GoldenChargeTomography.split_add_inert_indicator
• Truth anchor: D5/S3/PrimeForms/GoldenEuler/GoldenChargeTomography.split_channel_reconstruction
• Truth anchor: D5/S3/PrimeForms/GoldenEuler/GoldenChargeTomography.split_indicator_neg_charge
• Truth anchor: D5/S3/PrimeForms/GoldenEuler/GoldenChargeTomography.split_indicator_pos_charge
• Truth anchor: D5/S3/PrimeForms/GoldenEuler/GoldenChargeTomography.split_sub_inert_indicator