>[!warning] >This content has not been peer reviewed. # Structural Foundations The Resource Triangle, fidelity function, and graph dynamics rest on several structural claims. This note clarifies which are derived from axioms, which are conjectured, and which remain open — and where scripts provide computational support. --- ## Resistance and the $L^n$ norm The Resource Triangle $W^n = \Omega^n + N^n$ is derived from axioms ([[Fidelity Derivation]]). Axiom A5 defines relational distance as a resistance metric; the Micro-Graph and Reality Engine use edge weight $= 1/\mu$, graph distance $d_R = \sum 1/\mu$ along paths ([[Micro-Graph Generator Results#RST-mirroring (resistance metric)]]). The question was whether path resistance *itself* aggregates via $L^n$. It does not. Doyle & Snell (*Random Walks and Electric Networks*) establish that effective resistance obeys linear additivity in series: $R = R_1 + R_2$. Barlow–Bass and Kigami give resistance estimates on fractal graphs; no explicit $L^n$-additivity for path segments appears. The script `rst_verification_q1_resistance_lnorm.py` confirms that $R(A,C) = R(A,B) + R(B,C)$ on geodesic paths, while $L^n$ aggregation gives larger relative error (~8%). Scaling: $R_{\mathrm{eff}} \sim L^{1.0}$ on the backbone (vs $n = d_B \approx 1.22$). So $L^n$ applies to the signal/noise split ($\Omega$ vs $N$) in the Resource Triangle, not to the resistance metric along paths. --- ## Fidelity and Tsallis statistics The fidelity $\mu(\eta,n) = \eta/(1+\eta^n)^{1/n}$ is derived from axioms (Resource Triangle, homogeneity, symmetry). See [[RST Core Reduction]], [[Relational Substrate Theory (RST)]]. The algebraic relation $q = 1 - 1/n$ links the Tsallis index to RST $n$: $e_q(x) = (1+(1-q)x)^{1/(1-q)}$. RST $\mu(\eta,n)$ and Tsallis $e_q(-\eta)$ are different functions but share limiting behavior ($\eta \to 0$: linear; $\eta \to \infty$: saturate); $n$ controls transition sharpness in both. Literature (PLOS One 2021) supports Tsallis statistics on fractal networks. Whether the Kigami Laplacian Green's function on a fractal with $d_s = 4/3$ natively yields $\mu(\eta,n)$ remains open. [[rst_kigami_spectrum]] audits feasibility; the 1/$\mu$ decimation map produces eigenvalue ratios ~1.7, not ~207 — a direct Laplacian-to-mass-ratio derivation fails. The script `rst_verification_q2_tsallis.py` documents the algebraic relation and comparison; figure: `python calculations results/verification_q2_tsallis_comparison.png`. --- ## Local walk dimension The relation $d_s = 2d_H/d_w$ is standard (Alexander-Orbach). Implemented in [[graph_rule_utils]]: `walk_dimension_msd` returns a global $d_w$ from $\langle r^2 \rangle(t) \sim t^{2/d_w}$. The framework proposes that $d_w$ is not a global constant but a local variable dictated by relational density, with local spectral dimension shifting between the Euclidean limit ($d_s = 3$) and the Polya recurrence limit ($d_s = 2$). Standard literature treats $d_w$ as a global property (Barlow, Jonsson, Kigami). Inhomogeneous random graphs and variable-speed walks exist, but no standard definition of local $d_w$ as a per-vertex scalar field has been found. The framework introduces local $d_w$ as an innovation; no precedent in the cited literature. --- ## Isotropic projection (3D→1D) The noise floor $a_0 = c(\theta/3)/(2\pi) = c\theta/(6\pi)$ uses "Isotropic Projection": $\theta = 3H$ in FLRW cosmology is projected onto one signal dimension with factor $1/3$. See [[RST Baseline 1.0]]. The geometric claim: for isotropic expansion in $D$ dimensions with volumetric scalar $\theta$, the 1D flux along any signal path scales as $\theta/D$. FLRW $\theta = \nabla_\mu u^\mu = 3\dot{a}/a$ is standard. Isotropic diffusion and light transport show 1/D-type scaling in some contexts; a strict theorem (flux $= \theta/D$) has not been located. Verlinde, DGP derive $a_0 \sim cH$ with $2\pi$ or $6\pi$ factors; the factor $1/3$ from projecting $\theta = 3H$ onto one dimension is geometrically natural but not cited as a named theorem. Status: conjectured. --- ## Graph evolution and critical percolation The Pure Axiom Substrate (rst_axiom_pure.py) uses local Shannon cost (A2), $W = \log_2(k+2)$, Landauer prune. No $I_{\mathrm{conn}}$ or $I_{\mathrm{bal}}$. See [[further applications/Micro-Graph Generator/Micro-Graph Generator (RST)]]. It achieves $d_B \approx 1.21$, $d_s \approx 1.33$ on seed 42. See [[Micro-Graph Generator Results]], [[Genesis - Bare vs dressed]]. **Protocol:** N ≥ 3 seeds in band ($|d_B - 1.22| < 0.05$), var_d_B < 0.15; or FSS: $d_B(\infty) \in [1.17, 1.27]$. Rigorous verification uses thermalization (autocorrelation burn-in) and chemical-distance $d_B$ (burning method). Last run: 2025-03-11, result: FAIL (N_in_band=0, var_d_B=0.058; mean d_B ≈ 1.78). See [[Graph rule verification results]], [[Graph rule spec]]. Whether such a degree-biased, assortative-mixed rule asymptotes to critical percolation has not been formally proven. Preferential attachment (Barabási–Albert) and assortative mixing (Newman) affect network structure; biased percolation (Bazant) uses degree-dependent retention with non-universal exponents. Critical percolation (Stauffer & Aharony) focuses on $p_c$ as invariant; dynamic rules that reach $p_c$ are not standard. The script `rst_graph_rule_convergence.py` provides computational evidence (relaxation time, multi-seed variance). Status: empirically supported; open as a general theorem. --- ## Scripts These scripts support or clarify the claims above. | Claim | Script | Location | |:---|:---|:---| | Resistance vs $L^n$ | `rst_verification_q1_resistance_lnorm.py` | `expanded theory applied/further applications/Micro-Graph Generator/` | | Tsallis relation | `rst_verification_q2_tsallis.py` | `expanded theory/python calculations/` | | Ledger convergence | `rst_graph_rule_convergence.py` | `expanded theory applied/further applications/Micro-Graph Generator/` | | Finite-size scaling | `rst_graph_rule_fss.py` | `expanded theory applied/further applications/Micro-Graph Generator/` | Run from respective folders: `python rst_verification_q1_resistance_lnorm.py`; `python rst_verification_q2_tsallis.py`. --- ## Quick reference | Topic | Status | |:---|:---| | Resistance and $L^n$ | Clarified: $L^n$ applies to $\Omega$/N, not path resistance | | Fidelity and Tsallis | $\mu$ from axioms; Tsallis link algebraic; Laplacian-native link open | | Local $d_w$ | Framework innovation; no literature precedent | | Isotropic projection | Conjectured | | Graph attractor | Empirically supported; formal theorem open | --- ## References - J. Kigami, *Analysis on Fractals*, Cambridge University Press (2001). - M. Barlow, "Random walks on supercritical percolation clusters," *Ann. Probab.* **32**, 3024 (2004). - S. Alexander and R. Orbach, "Density of states on fractals: 'fractons'," *J. Phys. Lett.* **43**, L625 (1982). - P. G. Doyle and J. L. Snell, *Random Walks and Electric Networks*, AMS (1984). - D. Stauffer and A. Aharony, *Introduction to Percolation Theory*, 2nd ed. (1994). - C. Tsallis, "Possible generalization of Boltzmann-Gibbs statistics," *J. Stat. Phys.* **52**, 479 (1988). - F. Lelli et al., "SPARC: Mass models for 175 disk galaxies," *Astron. J.* **152**, 157 (2016).