>[!warning] >This content has not been peer reviewed. # Super-Relational Mapping — Verified by Calculation > **This claim is numerically verifiable.** The Resource Triangle ($W^n = \Omega^n + N^n$) and the RST field equation ($I = \Omega \cdot \mu$) are proven algebraically identical. The calculation is described in **[[Super-Relational Mapping - Code]]**. --- ## What Gets Verified | Test | Claim | Pass Criteria | |:---|:---|:---| | **Identity closure** | $I = \Omega^2/W$ (triangle) equals $I = \Omega \cdot \mu(\Omega/N)$ (field eq.) | Match to 12 decimal places across 6 orders of SNR | | **Budget identity** | $\mu^n + \nu^n = 1$ | Holds to 10 decimal places | | **Roundtrip** | Given source $I$, solve for $\Omega$, recover $I$ | Match to 10 decimal places | **Calibrated parameters:** $n = 1.25$ (from SPARC transition-zone galaxies; Lelli et al. 2016, [[SPARC Evaluation Verification]]), $N = 1$ (normalized noise floor). --- ## Results ### Final Verdict | Proof | Status | |:---|:---| | Identity closure | **PASS** | | Budget identity | **PASS** | | Roundtrip | **PASS** | The Resource Triangle and the RST Field Equation are the same mathematical identity. $W^n = \Omega^n + N^n \quad\text{(geometry)} \qquad I = \Omega \cdot \mu \quad\text{(physics)} \qquad \mu = \Omega/W \quad\text{(fidelity)}$ One triangle. Four forces. Zero free functions. **Results summary:** [[Super-Relational Mapping Results]]. --- ### Proof 1: Triangle == Field Equation | Omega | I (triangle) | I (RST) | Match | |:---|:---|:---|:---| | 1.0000e-03 | 9.99857760e-07 | 9.99857760e-07 | OK | | 3.1623e-03 | 9.99400489e-06 | 9.99400489e-06 | OK | | 1.0000e-02 | 9.97477357e-05 | 9.97477357e-05 | OK | | 3.1623e-02 | 9.89458291e-04 | 9.89458291e-04 | OK | | 1.0000e-01 | 9.57176091e-03 | 9.57176091e-03 | OK | | 3.1623e-01 | 8.43462268e-02 | 8.43462268e-02 | OK | | 1.0000e+00 | 5.74349177e-01 | 5.74349177e-01 | OK | | 3.1623e+00 | 2.66726189e+00 | 2.66726189e+00 | OK | | 1.0000e+01 | 9.57176091e+00 | 9.57176091e+00 | OK | | 3.1623e+01 | 3.12894185e+01 | 3.12894185e+01 | OK | | 1.0000e+02 | 9.97477357e+01 | 9.97477357e+01 | OK | | 3.1623e+02 | 3.16038184e+02 | 3.16038184e+02 | OK | | 1.0000e+03 | 9.99857760e+02 | 9.99857760e+02 | OK | **Identity holds across 6 orders of magnitude: True** --- ### Proof 2: Budget Identity (μ^n + ν^n = 1) | Omega | μ | ν | μ^n+ν^n | =1? | |:---|:---|:---|:---|:---| | 1.0000e-03 | 0.001000 | 0.999858 | 1.00000000 | OK | | 3.1623e-03 | 0.003160 | 0.999400 | 1.00000000 | OK | | 1.0000e-02 | 0.009975 | 0.997477 | 1.00000000 | OK | | 3.1623e-02 | 0.031289 | 0.989458 | 1.00000000 | OK | | 1.0000e-01 | 0.095718 | 0.957176 | 1.00000000 | OK | | 3.1623e-01 | 0.266726 | 0.843462 | 1.00000000 | OK | | 1.0000e+00 | 0.574349 | 0.574349 | 1.00000000 | OK | | 3.1623e+00 | 0.843462 | 0.266726 | 1.00000000 | OK | | 1.0000e+01 | 0.957176 | 0.095718 | 1.00000000 | OK | | 3.1623e+01 | 0.989458 | 0.031289 | 1.00000000 | OK | | 1.0000e+02 | 0.997477 | 0.009975 | 1.00000000 | OK | | 3.1623e+02 | 0.999400 | 0.003160 | 1.00000000 | OK | | 1.0000e+03 | 0.999858 | 0.001000 | 1.00000000 | OK | **Budget identity holds: True** --- ### Proof 3: Roundtrip (I → Ω → I) | I (input) | Omega | I (recovered) | Match | |:---|:---|:---|:---| | 1.0000e-03 | 0.031792 | 1.00000000e-03 | OK | | 1.0000e-02 | 0.102275 | 1.00000000e-02 | OK | | 1.0000e-01 | 0.347615 | 1.00000000e-01 | OK | | 5.0000e-01 | 0.912518 | 5.00000000e-01 | OK | | 1.0000e+00 | 1.469569 | 1.00000000e+00 | OK | | 5.0000e+00 | 5.472443 | 5.00000000e+00 | OK | | 1.0000e+02 | 100.252108 | 1.00000000e+02 | OK | | 1.0000e+04 | 10000.079999 | 1.00000000e+04 | OK | **Roundtrip identity holds: True** --- ### Figure: Four Forces, Budget Identity, Three Regimes A figure (four-force projections and budget identity) is summarised in [[Super-Relational Mapping Results]]. --- ## Relation to Theory - **[[RST Super-Relational Mapping]]** — the main geometric unification note - **[[Resource Triangle]]** — $W^n = \Omega^n + N^n$ - **[[Resource Allocation Equation]]** — $I = \Omega \cdot \mu$ - **[[RST Core Reduction]]** — the ontology formula - **[[Super-Relational Mapping - Code]]** — calculation description and logic