The \(\zeta\) Coordinate Framework: Normalized-to-Analytic Conversion and \(\beta - \eta\) Parity Decomposition

Scott Gonzales, Kevin Patch, Andrew Menard, Ketan Patel

Avadari Institute of Algebraic Logic

In collaboration with Olly (Autonomous Research Agent, Gemini 3.7 Flash)

Category: Pearls / \(\zeta\) Framework • Working Reference Note

Abstract
We unify the primary \(\zeta\) coordinate framework with the \(\beta - \eta\) parity difference decomposition. Table 1 defines the conversion between normalized integer coordinates and continuous analytic L-function evaluations across the \(\zeta\) Parent Series, the Even Parity Sector, and the Odd Parity Sector via the scale \(4^n/\zeta\), yielding the Coordinate Ratio \(K(n)\). Table 2 isolates the parity difference sector to extract the Coordinate Remainder \(k(n) = K(n) - 2^n\).

Table 1. The \(\zeta\) Coordinate Framework: Normalized-to-Analytic Conversion via the Scale \(4^n/\zeta\)

Normalized Values for \(\zeta(4)\) with Associated Formulas

Even Parity \(\zeta\)		Parent Series \(\zeta\)		Odd Parity \(\zeta\)
8	7	128	112	120	116.9566528
15	1	240	16	236.9566528	3.043347237
16	14	256	224	240	233.9133055
	224		57344		56139.19333
	14.96662955		239.4660728		236.9371084
Coordinate Ratio \(K(4) = \dfrac{233.9133055}{14} = 16.70809325\)
\(2^{n-1}\)	\(2^{n-1} - 1\)	\(2^{n-1} \cdot 2^n\)	\(2^{n-1}(2^n - 2)\)	\(2^n \cdot 2^{n-1} - 2^{n-1}\)	\((2^{n-1} - 1)(2^n + k(n))\)
\(2^n - 1\)	\(1\)	\(2^n \cdot 2^n - 2^n\)	\(2^n\)	\(2^{n-1}(1 + k(n)(1 - 2^{-(n-1)}))\)	\(2^{n-1}(1 - k(n)(1 - 2^{-(n-1)}))\)
\(2^n\)	\(2^n - 2\)	\(2^n \cdot 2^n\)	\(2^n(2^n - 2)\)	\(2^n(2^n - 1)\)	\((2^n - 2)(2^n + k(n))\)
	\(2^n(2^n - 2)\)		\(2^n \cdot 2^n \cdot 2^n(2^n - 2)\)		\(2^n(2^n - 1)(2^n - 2)(2^n + k(n))\)
	\((2^n(2^n - 2))^{1/2}\)		\(2^n(2^n(2^n - 2))^{1/2}\)		\((2^n(2^n - 1)(2^n - 2)(2^n + k(n)))^{1/2}\)
Coordinate Ratio \(K(n) = \dfrac{(2^n - 2)(2^n + k(n))}{2^n - 2} = 2^n + k(n)\)

Analytic Values for \(\zeta(4)\) with Associated Formulas

Even Parity \(\zeta\)		Parent Series \(\zeta\)		Odd Parity \(\zeta\)
0.033822601	0.029594776	0.541161617	0.473516415	0.507339016	0.494472276
0.063417377	0.004227825	1.014678032	0.067645202	1.001811292	0.012866740
0.067645202	0.059189552	1.082323234	0.947032829	1.014678032	0.988944552
	0.004003889		1.024995634		1.003460311
	0.063276293		1.012420681		1.001728661
Coordinate Ratio \(K(4) = \dfrac{0.988944552}{0.059189552} = 16.70809325\)
\(\frac{1}{2}\cdot 4^{-n}(\zeta(n, 2/4) + \zeta(n, 4/4))\)	\(\frac{1}{2}\cdot 4^{-n}(\zeta(n, 2/4) - \zeta(n, 4/4))\)	\(\frac{1}{2}\zeta\)	\(\frac{1}{2}\eta\)	\(\frac{1}{4}(\zeta + \eta)\)	\(\frac{1}{4}(\beta + \beta)\)
\(4^{-n}\zeta(n, 2/4)\)	\(4^{-n}\zeta(n, 4/4)\)	\(\frac{1}{2}(\zeta + \eta)\)	\(\frac{1}{2}(\zeta - \eta)\)	\(\frac{1}{4}((\zeta + \eta) + (\beta + \beta))\)	\(\frac{1}{4}((\zeta + \eta) - (\beta + \beta))\)
\(4^{-n}(\zeta(n, 2/4) + \zeta(n, 4/4))\)	\(4^{-n}(\zeta(n, 2/4) - \zeta(n, 4/4))\)	\(\zeta\)	\(\eta\)	\(\frac{1}{2}(\zeta + \eta)\)	\(\frac{1}{2}(\beta + \beta)\)
	\(\eta \cdot 4^{-n}\zeta\)		\(\zeta \cdot \eta\)		\(\frac{1}{2}\beta(\zeta + \eta)\)
	\((\eta \cdot 4^{-n}\zeta)^{1/2}\)		\((\zeta \cdot \eta)^{1/2}\)		\((\frac{1}{2}\beta(\zeta + \eta))^{1/2}\)
Coordinate Ratio \(K(n) = \dfrac{\frac{1}{2}(\beta(n) + \beta(n))}{4^{-n}(\zeta(n, 2/4) - \zeta(n, 4/4))} = \dfrac{\beta(n)}{2^{-n}\eta(n)} = 2^n \dfrac{\beta(n)}{\eta(n)}\)

Table 2. The \(k(n)\) Coordinate Remainder: \(\beta - \eta\) Parity Difference Decomposition

Normalized Values for \(\zeta(4)\) with Associated Formulas

Even Parity \(\zeta\)		Odd Parity \((\beta - \eta)\)
8	7	8	4.956652763
15	1	12.95665276	3.043347237
16	14	16	9.913305527
	224		158.6128884
	14.96662955		12.59416089
Coordinate Remainder \(k(4) = \dfrac{9.913305527}{14} = 0.70809325\)
\(2^{n-1}\)	\(2^{n-1} - 1\)	\(2^{n-1}\)	\((2^{n-1} - 1)k(n)\)
\(2^n - 1\)	\(1\)	\(2^{n-1}(1 + k(n)) - k(n)\)	\(2^{n-1}(1 - k(n)) + k(n)\)
\(2^n\)	\(2^n - 2\)	\(2^n\)	\((2^n - 2)k(n)\)
	\(2^n(2^n - 2)\)		\(2^n(2^n - 2)k(n)\)
	\((2^n(2^n - 2))^{1/2}\)		\((2^n(2^n - 2)k(n))^{1/2}\)
Coordinate Remainder \(k(n) = \dfrac{(2^n - 2)k(n)}{2^n - 2} = k(n)\)

Analytic Values for \(\zeta(4)\) with Associated Formulas

Even Parity \(\zeta\)		Odd Parity \((\beta - \eta)\)
0.033822601	0.029594776	0.033822601	0.020955861
0.063417377	0.004227825	0.054778462	0.012866740
0.067645202	0.059189552	0.067645202	0.041911722
	0.004003889		0.002835127
	0.063276293		0.053245910
Coordinate Remainder \(k(4) = \dfrac{0.041911722}{0.059189552} = 0.70809325\)
\(\frac{1}{2}\cdot 4^{-n}(\zeta(n, 2/4) + \zeta(n, 4/4))\)	\(\frac{1}{2}\cdot 4^{-n}(\zeta(n, 2/4) - \zeta(n, 4/4))\)	\(\frac{1}{4}(\zeta - \eta)\)	\(\frac{1}{4}\cdot 2(\beta - \eta)\)
\(4^{-n}\zeta(n, 2/4)\)	\(4^{-n}\zeta(n, 4/4)\)	\(\frac{1}{4}((\zeta - \eta) + 2(\beta - \eta))\)	\(\frac{1}{4}((\zeta - \eta) - 2(\beta - \eta))\)
\(4^{-n}(\zeta(n, 2/4) + \zeta(n, 4/4))\)	\(4^{-n}(\zeta(n, 2/4) - \zeta(n, 4/4))\)	\(\frac{1}{2}(\zeta - \eta)\)	\(\frac{1}{2}\cdot 2(\beta - \eta)\)
	\(\eta \cdot 4^{-n}\zeta\)		\(\frac{1}{2}(\zeta - \eta)(\beta - \eta)\)
	\((\eta \cdot 4^{-n}\zeta)^{1/2}\)		\((\frac{1}{2}(\zeta - \eta)(\beta - \eta))^{1/2}\)
Coordinate Remainder \(k(n) = \dfrac{\beta(n) - \eta(n)}{4^{-n}(\zeta(n, 2/4) - \zeta(n, 4/4))} = \dfrac{\beta(n) - \eta(n)}{2^{-n}\eta(n)} = 2^n \left(\dfrac{\beta(n)}{\eta(n)} - 1\right)\)