Cyclic phase
Computes delta_p and sin^2(theta_p) for a few prime channels.
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Mathematical atlas
PT mathematics / phase
theorem
CRT, holonomy, and cyclic phase
How CRT and cyclic phase force channel products.
Plain
The idea
When several independent cycles coexist, each can be read separately and then recomposed. This is the intuition behind the Chinese Remainder Theorem.
PT adds that each cycle carries a phase. A persistent channel is not only an open slot: it is a stable cyclic orientation.
CRT → phase
Standard
Standard reading
CRT decomposes a composite module into coprime prime channels. Pontryagin duality turns this decomposition into a product of characters.
This mechanism makes PT product forms natural, especially in bridges where the active channels $3,5,7$ contribute multiplicatively.
Takeaways
- CRT gives channel independence.
- Pontryagin turns independence into a product.
- Cyclic phase carries persistent amplitude.
Technical
Technical formulation
Cyclic phase is encoded by $\theta_p$ with $\sin^2\theta_p=\delta_p(2-\delta_p)$. It turns channel depth into persistent amplitude.
The monograph treats the BA5 product form as locked by CRT + Pontryagin under the relevant bridge axioms.
Monograph: ch05_geometry, ch06_holonomy, ch09_bridge, BA5.
Formulas
$\mathbb{Z}/(p_1p_2p_3)\mathbb{Z}\cong\mathbb{Z}/p_1\mathbb{Z}\oplus\mathbb{Z}/p_2\mathbb{Z}\oplus\mathbb{Z}/p_3\mathbb{Z}$
$\sin^2\theta_p=\delta_p(2-\delta_p)$
$\alpha_{\rm sieve}=\prod_{p\in\{3,5,7\}}\sin^2\theta_p$
public code
Code and scripts
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