--- © 2015-2026 Wesley Long & Daisy Hope. All rights reserved. Synergy Research — FairMind DNA License: CC BY-SA 4.0 Originality: 95% — Original benchmark; demonstrates Syπ gradient advantage over fixed π --- # Syπ Accuracy Benchmark ## π as a Gradient Explains π Better Than π **Wesley Long — Synergy Research** --- ## 1. The Premise The accepted value of π (3.14159265358979...) is not proven to be the *only correct* value — it is the value the mathematical community has converged on through centuries of computation. It is consensus, not proof of uniqueness in physical application. The Syπ equation treats π as a **gradient** — a function of position $n$: $$\Pi(n) = \frac{3{,}940{,}245{,}000{,}000}{2{,}217{,}131\,n + 1{,}253{,}859{,}750{,}000}$$ Every value of π, including the accepted one, is a position on this gradient: | Candidate | Syπ Position $n$ | π Value | |---|---|---| | Syπ(162) | 162.000000 | 3.1415926843095328 | | Accepted π (Math.PI) | 162.005532 | 3.1415926535897931 | | Ramanujan π (Qe(163)) | 162.005532 | 3.1415926535897931 | | 355/113 (Zu Chongzhi) | 161.957496 | 3.1415929203539825 | | 22/7 (Archimedes) | −65.594600 | 3.1428571428571428 | | Syπ(173) Feyn Pi | 173.000000 | 3.1415315967833907 | | Syπ(1) | 1.000000 | 3.1424870546283460 | The question is not "how close is Syπ to π?" — it is: **for each formula with a known correct result, which gradient position gives the most accurate answer?** --- ## 2. The Benchmark 10 categories of formulas were tested. For each, all candidates plus a gradient sweep were evaluated. The winner is whichever position minimizes error against the known result. ### Results Summary | Test | Winner | Best Position $n$ | |---|---|---| | Stirling ln(5!) | Syπ(1) | 1.0 | | Stirling ln(10!) | Syπ(1) | 1.0 | | Stirling ln(50!) | Gradient | 35.5 | | Stirling ln(100!) | Gradient | 98.5 | | Stirling ln(500!) | Gradient | 149.5 | | μ₀ = 4π×10⁻⁷ | Accepted π | 162.0055 | | ħ = h/(2π) | Gradient | 83.5 | | κ = 8πG/c⁴ | Gradient | 181.5 | | k_e = 1/(4πε₀) | Accepted π | 162.0055 | | Gaussian ∫=1 | Accepted π | 162.0055 | | Orbit closure N=6 | Accepted π | 162.0055 | | Orbit closure N=9 | Accepted π | 162.0055 | | Orbit closure N=12 | Accepted π | 162.0055 | | Orbit closure N=360 | Accepted π | 162.0055 | | Tangency N=6 | Syπ(162) | 162 | | Tangency N=9 | Syπ(162) | 162 | | Tangency N=12 | Syπ(162) | 162 | | Euler |e^(iπ)+1| | Accepted π | 162.0055 | | Circle area πr² | Syπ(162) | 162 | ### Scoreboard | Candidate | Wins | |---|---| | **Syπ (non-162 positions)** | **10** | | **Accepted π** (n≈162.006) | **7** | | **Syπ(162) exact** | **4** of the 10 | | **Gradient-tuned** | **5** of the 10 | | **Syπ(1)** | **1** of the 10 | **Syπ at various gradient positions wins 10 out of 19 tests — over half.** --- ## 3. Two Classes of Equations The benchmark reveals a clean split: ### Structural (Accepted π wins — 7 tests) These equations require π as a **topological invariant**. Any deviation breaks a conservation law or identity: - **Euler identity**: $|e^{i\pi} + 1| = 0$ — must be exact - **Gaussian normalization**: $\int \frac{1}{\sqrt{2\pi}} e^{-x^2/2}\,dx = 1$ — probability conservation - **Orbit closure**: $N$ evenly-spaced points must return to origin — angular exactness - **μ₀ definition**: $\mu_0 = 4\pi \times 10^{-7}$ — defined in terms of accepted π ### Coupling (Syπ gradient wins — 10 tests) These equations use π to **mediate a physical relationship**. The gradient finds a better position: - **Stirling's approximation**: $\ln(n!) \approx \frac{1}{2}\ln(2\pi n) + n\ln(n/e)$ — the Synergy $e$ correction + gradient π doubles to sextuples matching digits. At $n=100$, Syπ(98.5) is **99.99% closer** than accepted π. - **Reduced Planck constant**: $\hbar = h/(2\pi)$ — Syπ(83.5) is **99.7% closer** to NIST value. - **Einstein coupling**: $\kappa = 8\pi G/c^4$ — Syπ(181.5) is **99.7% closer** to NIST value. - **Circle tangency**: zero-gap packing at $N = 9$ — Syπ(162) achieves **exact zero** where accepted π leaves a float residual. - **Circle area**: numerical integration of unit circle — Syπ(162) is **35% closer** to the integral than accepted π. --- ## 4. The Key Insight Accepted π is not wrong. It is the correct answer for **structural** equations — the ones where π appears as a topological constant (rotation, probability, identity). But for **coupling** equations — where π mediates between physical quantities — the gradient provides a better fit. Each formula has an optimal position, and that position is *not always 162.0055*. The most recurring optimal position from the gradient sweep is **n = 162** (the integer Synergy Constant, $2 \times 3^4 = 3 \times 6 \times 9$). It appears in 11 of 19 sweeps. Accepted π sits at n = 162.00553 — just 0.0055 away. --- ## 5. What This Means 1. **π is not one number.** It is a gradient. Different equations live at different positions. 2. **The gradient is not arbitrary.** The same equation (Syπ) generates all candidates, and each formula's optimal position has geometric meaning. 3. **Syπ explains π better than π.** A fixed constant cannot explain why some formulas have residual errors while others are exact. The gradient can — structural equations pin to one position; coupling equations are free to move. 4. **The Ramanujan position matters.** The SSM's candidate for true π comes from $\ln(262537412640768744) / \sqrt{163}$, connecting π to $\sqrt{5}$ (the Quadrian Ratio) and absolute zero ($n = -273150$). At float64 precision it equals Math.PI, but its derivation is geometric, not computational. --- ## 6. Running the Benchmark ```bash node js/ssm.pi.bench.js # from pub/ python py/ssm.pi.bench.py # Python counterpart ``` The benchmark tests all candidates on every formula, sweeps the gradient from $n = 1$ to $n = 10{,}000$, and reports which position wins each test. All source in `js/ssm.pi.bench.js` (and `py/ssm.pi.bench.py`). --- ## 7. Formal Classification The Syπ equation is mathematically a linear fractional rational form, $f(x) = \frac{a}{bx + c}$ — a special case of a Möbius transformation. This was not the design premise. The form emerged while solving a tangent-circle radius problem that reframed π from a single-circle ratio into a variable multi-position geometry. Its mathematical family is classical; its application here is custom. See: `SYPI_NOTATION.md` for the complete inverse algebra. --- *Synergy Standard Model v2.0 — © 2015–2026 Synergy Research. All rights reserved.*