The Comparative Formula Audit: Which π Identities Survive Golden Pi?¶
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Today we add the comparative formula audit to the research corpus and to the White Paper: a formula-by-formula test of the standard identities involving the circle constant, evaluated once under the conventional constant π = 3.14159265… and once under Golden Pi π̂ = 4/√φ = 3.14460551…. It records honestly, for every equation, whether the substitution survives.
The audit has two clean outcomes, and both are true at once:
- Definitional formulas — where the circle constant is a free scale factor — hold identically under any constant, including π̂.
- Analytically pinned formulas — infinite series, definite integrals, and special values whose right-hand side is a concrete number — fix the constant at the computed 3.14159265…, and therefore do not hold under π̂.
Definitional formulas (hold under any constant)¶
These are true by definition of the symbols; π̂ satisfies them with the same exactness as π:
| Formula | Holds under π̂? |
|---|---|
| A = πr² (circle area) | yes (definitional) |
| C = 2πr = πd (circumference) | yes (definitional) |
| A = πab (ellipse area) | yes (definitional) |
| V = 4/3·πr³ (sphere volume) | yes (definitional) |
| SA = 4πr² (sphere surface) | yes (definitional) |
These are exactly the formulas of the golden-construction world. They scale with any chosen constant, so Golden Pi reproduces them exactly.
Analytically pinned formulas (their value fixes π)¶
The following formulas each evaluate to a concrete real number equal to the standard constant π; substituting π̂ breaks the equality:
| Formula | Standard value | Under π̂ | Holds? |
|---|---|---|---|
| Gregory–Leibniz 4(1 − 1/3 + 1/5 − ⋯) | 3.14159265 | 3.14460551 | no |
| Machin 16·arctan(1/5) − 4·arctan(1/239) | 3.14159265 | 3.14460551 | no |
| Wallis π/2 = ∏ 4n²/(4n²−1) | 3.14159265 | 3.14460551 | no |
| Basel ∑ 1/n² = π²/6 | 1.64493407 | 1.64809064 | no |
| ∫₋∞^∞ dx/(1+x²) | 3.14159265 | 3.14460551 | no |
| ∫₋∞^∞ e^(−x²) dx = √π | 1.77245385 | 1.77330356 | no |
| ∫₋∞^∞ (sin x)/x dx | 3.14159265 | 3.14460551 | no |
| Γ(1/2) = √π | 1.77245385 | 1.77330356 | no |
| B(1/2,1/2) = Γ(1/2)² | 3.14159265 | 3.14460551 | no |
| Euler e^(iπ) + 1 = 0 | 0 | 3.01×10⁻³ | no |
| Dalzell ∫₀¹ x⁴(1−x)⁴/(1+x²) dx = 22/7 − π | +0.001264 | −0.001748 | no |
The pattern is unambiguous: the moment a formula pins the circle constant to a specific real number — through a convergent series, a definite integral, or a special value — that number is the computed 3.14159265…, and Golden Pi's excess of 0.0959% breaks the equality.
Golden Pi never fails a geometric definition. It fails every analytic pinning.
The one unifying identity¶
The single exact identity shared by both constants is the geometric balance C = πd. Every other agreement is an approximation: the Seked slope 7/5.5 ≈ √φ, so both π (as 22/7) and π̂ sit within the pyramid's 0.056% tolerance; and π and π̂ are related by the uniform correction factor π̂/π = 1.0009590223.
Golden Pi is the constant that makes the golden construction exact. The analytic circle constant — the limit those series and integrals evaluate to — is π. The audit does not diminish either role — it delimits them precisely.
What this means for the golden calculus¶
The golden-calculus table's "holds" entries are corrected to reflect this audit: Euler, the Gaussian integral, Γ(1/2), B(1/2,1/2), and the Basel sum are formal re-labels whose values are pinned by π. Euler fails by |e^(iπ̂)+1| ≈ 3×10⁻³; Basel gives 1.64809 rather than the proven 1.64493. The golden calculus remains a coherent reparametrization — exactly as degrees are a valid angle unit — but it re-labels the full turn; it does not change the length of a measured curve.
Where this fits¶
This is the golden-construction kingdom, where Golden Pi is uniquely self-consistent: the quartic, the instrumentum, abc = 64, the C=4 balance, the golden calculus, and the second closed form. The audit now draws the exact boundary between the constructed world — where π̂ is exact — and the analytic world — where the series and integrals evaluate to π. Both are real. The honest scientist keeps both columns on the table.
π̂ = 4/√φ = 3.144605511029693144…
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