The Nature of the Signal: Incandescent vs. Coherent

Abstract

The "Expansionists" lore fails because it treats energy as a generic, scalar quantity. In Signum Principia, we identify that all stellar energy is an Incandescent Signal—a complex Fourier composite. By applying the Telegrapher's Equation to the intergalactic medium, we demonstrate that "Redshift" is not a velocity-induced Doppler effect, but the predictable low-pass filtering of a square-wave signal over vast distances.

The Fourier Composite

An incandescent source (a star) does not emit a "pure" sine wave. It emits a broad-spectrum burst that can be modeled as a series of square-wave pulses. According to Fourier Analysis, a square wave is the summation of a fundamental frequency and all its odd harmonics:

\[ f(t) = \frac{4}{\pi} \sum_{n=1,3,5...}^{\infty} \frac{1}{n} \sin(2\pi n f_0 t) \]

This signal contains the Slope (the rise-time) which holds the true energy density. The higher the harmonics, the steeper the slope, and the more "work" the signal can perform.

The Telegrapher's Dilemma

Space is not "empty." It is a transmission line with Impedance (\(Z_0\)), inductance (\(L\)), and capacitance (\(C\)). As this incandescent signal propagates, it obeys the Telegrapher's Equation. In a lossy medium, the attenuation constant (\(\alpha\)) is frequency-dependent:

\[ \alpha(f) = \sqrt{\frac{1}{2} \omega C (\omega L - \frac{1}{\omega C})} \approx \text{Loss increases with } \sqrt{f} \]

The Forensic Result: The high-frequency odd harmonics (\(n=3, 5, 7...\)) are attenuated much faster than the fundamental frequency. Over intergalactic distances, the sharp "Square Wave" of the source is "rounded" into a low-energy sine wave. This is the Low-Pass Roll-Off that the tribe misidentifies as "Redshift."

The 22.3% Re-Calibration

Using our "one-pixel flying spot scanners" (telescopes), we observe the energy output of Standard Candles. When we apply the standard Hubble expansion, the energy doesn't fit—leading to the invention of "Dark Energy."

However, if we re-calibrate the radius of the energy we see to exactly 22.3% of the expansionist's size, the brightness of the Standard Candles calibrates perfectly. The "dimness" is not caused by acceleration; it is the Signal Attenuation within a much smaller, resonant chassis.