Golden Local Euler Trichotomy

Abstract

The neutral and quadratic charge denominator specializes to split, inert, and ramified golden local Euler forms.

Theorem 1.1 (Split Charge Gives a Squared Linear Denominator).

$$\begin{gathered} \forall X:\mathbb{R}, \\ (\mathrm{goldenLocalDenominator}\left(1,X\right)={1-X}^2). \end{gathered}$$

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

Source. Repository-derived.

Commentary.

Substituting charge plus one makes both linear denominator factors equal, yielding the square of one minus X.

The equality is polynomial and remains independent of any convergence interpretation of X.

Theorem 1.2 (Inert Charge Gives a Quadratic Denominator).

$$\begin{gathered} \forall X:\mathbb{R}, \\ (\mathrm{goldenLocalDenominator}\left(-1,X\right)=1-X^2). \end{gathered}$$

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

Source. Repository-derived.

Commentary.

Substituting charge minus one multiplies one minus X by one plus X, giving one minus X squared.

This algebraic factor fusion does not assert that X is a prime monomial.

Theorem 1.3 (Ramified Charge Leaves One Linear Denominator).

$$\begin{gathered} \forall X:\mathbb{R}, \\ (\mathrm{goldenLocalDenominator}\left(0,X\right)=1-X). \end{gathered}$$

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

Source. Repository-derived.

Commentary.

At zero charge the quadratic-channel factor is one, leaving the neutral factor one minus X.

The statement records the ramified specialization only at the level of the totalized real denominator.

Theorem 1.4 (The Split Local Factor Is the Inverse Squared Denominator).

$$\begin{gathered} \forall X:\mathbb{R}, \\ (\mathrm{goldenLocalFactor}\left(1,X\right)={{1-X}^2}^{-1}). \end{gathered}$$

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

Source. Repository-derived.

Commentary.

The totalized split local factor is the reciprocal of the squared linear denominator.

Because inversion is totalized over the reals, no nonvanishing premise is claimed or required.

Theorem 1.5 (The Inert Local Factor Is the Inverse Quadratic Denominator).

$$\begin{gathered} \forall X:\mathbb{R}, \\ (\mathrm{goldenLocalFactor}\left(-1,X\right)={1-X^2}^{-1}). \end{gathered}$$

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

Source. Repository-derived.

Commentary.

The totalized inert local factor is the reciprocal of one minus X squared.

The equality specializes the definition and makes no analytic assertion about an Euler product.

Theorem 1.6 (The Ramified Local Factor Is the Inverse Linear Denominator).

$$\begin{gathered} \forall X:\mathbb{R}, \\ (\mathrm{goldenLocalFactor}\left(0,X\right)={1-X}^{-1}). \end{gathered}$$

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

Source. Repository-derived.

Commentary.

The totalized ramified local factor is the reciprocal of one minus X.

This completes the three charge specializations without adding a prime-classification claim.

References

• Truth anchor: D5/S3/PrimeForms/GoldenEuler/GoldenLocalEulerTrichotomy.inert_local_denominator
• Truth anchor: D5/S3/PrimeForms/GoldenEuler/GoldenLocalEulerTrichotomy.inert_local_factor
• Truth anchor: D5/S3/PrimeForms/GoldenEuler/GoldenLocalEulerTrichotomy.ramified_local_denominator
• Truth anchor: D5/S3/PrimeForms/GoldenEuler/GoldenLocalEulerTrichotomy.ramified_local_factor
• Truth anchor: D5/S3/PrimeForms/GoldenEuler/GoldenLocalEulerTrichotomy.split_local_denominator
• Truth anchor: D5/S3/PrimeForms/GoldenEuler/GoldenLocalEulerTrichotomy.split_local_factor