Tau-Weighted Boundary Constants

pi_tau := pi * iota_tau and e_tau := e * iota_tau: tau-weighted boundary constants for the two lemniscate lobes. pi_tau + e_tau = 2*iota_tau. Ratio pi_tau/e_tau = pi/e encodes lobe asymmetry.

Tau-Weighted Boundary Constants

Summary

pi_tau := pi * iota_tau and e_tau := e * iota_tau: tau-weighted boundary constants for the two lemniscate lobes. pi_tau + e_tau = 2*iota_tau. Ratio pi_tau/e_tau = pi/e encodes lobe asymmetry.

Statement

%
\label{def:pi-tau-boundary}
% Depends: I.D34
Define the \textbf{$\tau$-weighted boundary constants}:
\[
    \boxed{%
    \pi_\tau := \pi \cdot \iota_\tau
    = \frac{2\pi}{\pi + e}
    \approx 1.07262,}
    \qquad
    e_\tau := e \cdot \iota_\tau
    = \frac{2e}{\pi + e}
    \approx 0.92738.
\]
These satisfy the \textbf{normalization identity}:
\[
    \pi_\tau + e_\tau = 2\iota_\tau.
\]

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-01.jsonl line 250
• Manuscript source: 2nd-edition/book-i-categorical-foundations/02_mainmatter/part10/ch41-master-constant.tex lines 340-358

Lean / Formalization Notes

• Formalization: planned
• Module: None
• Name: None

Dependencies

• Canonical: I.D34, I.D20

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.