>[!warning] >This content has not been peer reviewed. # Expanded Theory Applied — Applications Roadmap Applications of **[[expanded theory/Relational Substrate Theory (RST)|RST]]** (and the axioms of [[Relational Resolution Theory (RRT)|Relational Resolution Theory]]) to open problems. Each application lives in its own subfolder with an application note, optional Python verification, and a Code note where relevant. This document is the **roadmap**: it lists all current and planned applications and reflects their real purpose (index of problems + priority order for solving). --- ## The RST view: one curve In the old view, these problems lived in different departments: - **Black Holes** (astrophysics) - **Hubble Tension** (cosmology) - **Measurement** (quantum physics) - **Metabolism** (biology) RST treats them as the **same problem** — different points on the **Substrate Fidelity Curve**: | Condition | Outcome | Domain | |:---|:---|:---| | **Too small** | You **blur** (quantum / measurement) | Quantum physics | | **Too large** | You **buffer** (dark matter / low SNR) | Galaxies, cosmology | | **Too heavy** | You **saturate** (black hole / resolution collapse) | Astrophysics | | **Alive** | You **optimize** (biology / homeostasis) | Metabolism, life | So: one fidelity curve $\mu(\eta,n)$; scale, mass, and “aliveness” determine where you sit on it. Applications below are ordered by **foundational building blocks first** (the maths and physics essential to map any problem into RRT), then **Millennium**, then **priority for solving** (physics → chemistry → biology). --- ## Foundational maths and physics for RRT mapping Before mapping a given problem to RRT, these are the **essential** areas: they appear directly in the axioms, the Resource Triangle, or the fidelity derivation. Each has an application note under **foundation/**; the **[[foundation/Foundation index]]** is the bridge layer (RRT to Foundations to RST to Applications). **Full ordered list:** **[[foundation/Foundation index]]** (14 topics; the first 10 are the core building blocks with scripts in the combined runner; topics 11–14 also have notes and scripts). | Priority | Topic | Why essential for RRT mapping | RRT/RST link | |:---|:---|:---|:---| | 1 | **[[foundation/Information and Entropy/Information and Entropy (RST)]]** — Information theory and entropy | Resolution $\sigma$, bit-count, signal vs noise; defines "Information" and the split in the triangle. | A2 (Resolution); $I$, $N$, $\Omega$ as request / noise / workload. | | 2 | **[[foundation/Thermodynamics and Landauer/Thermodynamics and Landauer (RRT)]]** — Thermodynamics and Landauer | Cost of maintaining a bit ($k_B T \ln 2$); persistence is dissipative; defines the "noise" cost. | Relational Landauer Principle; $\Phi_{\min}$; A2, A3 (Translation). | | 3 | **[[foundation/Measure and budget allocation/Measure and budget allocation (RST)]]** — Measure, budget allocation, $L^p$ norms | Splitting a finite budget (signal vs noise); $W^n = \Omega^n + N^n$, $\mu^n + \nu^n = 1$. | Resource Triangle; Fidelity Derivation steps 4–6; homogeneity. | | 4 | **[[foundation/Scaling and dimensional analysis/Scaling and dimensional analysis (RST)]]** — Scaling and dimensional analysis | Scale-free response, power laws; derivation of $\mu(\eta,n)$; $n$ as critical/fractal exponent. | Transition sharpness $n$; MOND/Newton limits; turbulence (e.g. $n=4/3$). | | 5 | **[[foundation/Spectral and analytic structure/Spectral and analytic structure (RST)]]** — Spectral and analytic structure | $L$-functions, zeta, zeros; primes as "scales"; noise report and vacated workload. | BSD (rank = order of zero); Riemann (critical line); primes as consistency checks. | | 6 | **[[foundation/Continuum limits and PDEs/Continuum limits and PDEs (RST)]]** — Continuum limits and PDEs | Smoothness, blow-up, singularities; what "solution" means in the classical setting. | Navier–Stokes; substrate capacity as regulator; A1 (finite resolution). | | 7 | **[[foundation/Gauge theory and field theory/Gauge theory and field theory (RST)]]** — Gauge theory and field theory | Relational "glue," invariance, mass gap as resolution floor. | Yang–Mills; $\Lambda_{\mathrm{QCD}}$; A1, A3; four-force mapping. | | 8 | **[[foundation/Cosmology and GR/Cosmology and GR (RST)]]** — Cosmology and GR | Expansion $\theta = 3H$, FLRW; noise floor $a_0 = cH/(2\pi)$; dark energy as baseline. | Noise Floor; RST regimes (Newton / MOND / $\Lambda$); A4, A5. | | 9 | **[[foundation/Complexity and computation/Complexity and computation (RRT)]]** — Complexity and computation | Polynomial scaling, verification vs search, step-count as cost. | P vs NP; Landauer and resolution cost; A2, A3. | | 10 | **[[foundation/Algebraic and geometric structure/Algebraic and geometric structure (RST)]]** — Algebraic and geometric structure | Cohomology, algebraic cycles, invariants; "bookkeeping" of the format. | Hodge (relational loops, workload accounting); A1, A5. | The first 10 (priority 1–10) are implemented as foundation notes with illustrative scripts and run results; topics 11–14 also have notes and scripts. See **[[foundation/Foundation index]]** and **[[foundation/Run all foundation scripts - Code]]**; **`run_all_foundation_scripts.py`** runs the foundation scripts and builds the combined image. --- ## Roadmap Millennium and other non-foundation applications live under **further applications/**; see **[[further applications/Further applications index]]** for the full list. **Full list of further applications** (including Reality Engine, Path Integral Optimization, Ryu–Takayanagi, Bremermann Governor, Wilson RG Sensor, and plug-ins): **[[further applications/Further applications index]]**. **Derivation source:** Applications that use the **graph engine** (Pure Axiom Substrate, `rst_axiom_pure.py`) operate directly on the relational graph — e.g. **Periodic Table**, Nucleon Ground State. Applications that use the **fidelity engine** (`rst_engine`) apply $\mu(\eta,n)$ and the Resource Triangle — e.g. Navier–Stokes, BSD, MOND, Homeostasis. **[[Graph Size Hierarchy (RST)]]** unifies forces → matter → chemistry → cosmology from graph size $N$ ($K_2$, $K_3$, $K_4$, packing, $N \to \infty$). ### Millennium Prize problems | # | Problem | Subfolder | Status | RST angle (one-line) | |:---|:---|:---|:---|:---| | 1 | **[[further applications/Navier-Stokes Smoothness/Navier-Stokes Smoothness (RST)]]** — Navier–Stokes existence and smoothness | further applications/Navier-Stokes Smoothness | In progress | Substrate capacity + fidelity regulator prevent blow-ups; smoothness from finite resolution (A1, A2) and Resource Triangle. | | 2 | **[[further applications/BSD Conjecture/BSD Conjecture (RST)]]** — Birch–Swinnerton-Dyer conjecture | further applications/BSD Conjecture | In progress | Spectral duality: rank = order of zero at $s=1$ as conservation of workload; fidelity peaks where $L$-function (noise) vanishes. | | 3 | **[[further applications/Hodge Conjecture/Hodge Conjecture (RST)]]** — Hodge conjecture | further applications/Hodge Conjecture | In progress (conceptual) | Algebraic cycles as “relational loops” in the format; cohomology as workload accounting. | | 4 | **[[further applications/P vs NP/P vs NP (RST)]]** — P vs NP | further applications/P vs NP | In progress (conceptual) | Resolution cost (A2, Landauer) vs verification cost; polynomial as substrate scaling. | | 5 | **[[further applications/Riemann Hypothesis/Riemann Hypothesis (RST)]]** — Riemann hypothesis | further applications/Riemann Hypothesis | In progress (conceptual) | Zeros of $\zeta$ as noise-vacated scales; prime distribution as substrate consistency checks. | | 6 | **[[further applications/Yang-Mills/Yang-Mills (RST)]]** — Yang–Mills existence and mass gap | further applications/Yang-Mills | In progress (conceptual) | Gauge sector as relational maintenance; mass gap as resolution floor (cf. Lambda_QCD, Depth Identity). Illustrative script: `rst_yangmills_fractions.py`. | ### Other mathematics | # | Problem | Subfolder | Status | RST angle (one-line) | |:---|:---|:---|:---|:---| | 7 | **[[further applications/E8 Compatibility/E8 Compatibility (RST)]]** — E₈ compatibility | further applications/E8 Compatibility | In progress (conceptual + script) | $E_8$ as saturation limit of format; $n$ from 240 roots and $\alpha^{-1}$; forces as root slices; script derives $n \approx \ln(240)/\ln(137)$. | | 8 | **Levinthal paradox** | (todo) | **Todo** | Folding as search on the fidelity landscape; persistence (A2) and refresh (A4) select viable paths. | ### Total ontology (Great Map) The **Great Map** ties five pillars. The **complexity-scale plot** (μ from Planck to biological) is an RST application: **[[further applications/Grand Map/Grand Map (RST)]]** (script in that folder). The **Great Map** ties five pillars: (1) Thermodynamics (Landauer/Jacobson) — cost of existence; (2) Information (Shannon/Solomonoff) — logic of selection; (3) Fractal geometry (percolation/Bejan) — shape of the wires; (4) Dynamics (MOND/Navier–Stokes) — flow of the render; (5) Intelligence (Friston) — optimisation of the signal. Each pillar below is mapped to RST and has a script that **shows** the mapping in Python. | # | Problem | Subfolder | Status | RST angle (one-line) | |:---|:---|:---|:---|:---| | — | **[[further applications/Solomonoff Induction/Solomonoff Induction (RST)]]** — Solomonoff / Kolmogorov / Occam | further applications/Solomonoff Induction | In progress (script) | Shortest description = most probable; substrate favours compressed states (high $\eta$, high $\mu$); Occam from the ledger; script: $\mu(\eta)$ with regimes. | | — | **[[further applications/Free Energy Principle/Free Energy Principle (RST)]]** — Free Energy Principle (Friston) | further applications/Free Energy Principle | In progress (script) | Agents minimise prediction error $\Rightarrow$ minimise $N$ $\Rightarrow$ raise $\eta$ and $\mu$; Biological Substrate Triangle; script: prediction error vs $\mu$. | | — | **[[further applications/Constructal Law/Constructal Law (RST)]]** — Constructal law (Bejan) | further applications/Constructal Law | In progress (script) | $n \approx 1.25$ as least-resistance dimension; geometry of bandwidth; rivers, vasculature, backbone; script: $\mu(\eta,n)$ and flow-efficiency vs $n$. | | — | **[[further applications/Quantum Error Correction/Quantum Error Correction (RST)]]** — Quantum error correction (surface code) | further applications/Quantum Error Correction | In progress (script) | Entanglement protects information; hardware for A3; logical $\eta$ boosted by code $\Rightarrow$ higher $\mu$; script: logical vs physical $\mu$. | | — | **[[further applications/Bell Nonlocality and Measurement Dependence/Bell Nonlocality and Measurement Dependence (RST)]]** — Bell nonlocality, MI, signalling | further applications/Bell Nonlocality and Measurement Dependence | In progress (script) | Equiprobability vs signalling regimes on $\mu(\eta)$; RST extends Bacciagaluppi et al.; no hidden variables. | | — | **[[further applications/Grand Map/Grand Map (RST)]]** — Grand Map (complexity scale) | further applications/Grand Map | In progress (script) | Synthesis: $\mu$ from Planck to biological; $\eta_{\mathrm{eff}}(\ell)$ with life boost (Friston); script in this folder. | ### Chemistry | # | Problem | Subfolder | Status | RST angle (one-line) | |:---|:---|:---|:---|:---| | 8 | **The Nature of the Chemical Bond** | (todo) | **Todo** | Bond as relational maintenance (shared workload); covalent/ionic as SNR regimes on the same curve. | | — | **[[further applications/Periodic Table (RST)/Periodic Table (RST)]]** — Periodic Table **(graph)** | further applications/Periodic Table (RST) | In progress | Relational Liquid Drop: Semi-Empirical Mass Formula from A2/A3/A5, zero fitted parameters; thermodynamic equilibrium Strong/Pauli; Valley of Stability; discrete crystallography at A<100 | | 9 | **Origin of Life (prebiotic chemistry)** | (todo) | **Todo** | Emergence of persistent replicators as crossing the persistence threshold (A2); first “optimizers” on the fidelity curve. | *Chemistry and biology make sense in RST:* bonding is relational maintenance at molecular scale; origin of life is the first crossing of the persistence threshold; metabolism and homeostasis are identity loops (A2, A4) — same axioms, different scale. ### Biology and scaling | # | Problem | Subfolder | Status | RST angle (one-line) | |:---|:---|:---|:---|:---| | 10 | **Kleiber’s Law (3/4 power law)** | (todo) | **Todo** | Metabolic scaling as network dimension of the substrate (cf. $n \approx 4/3$, fractal backbone). | | 11 | **[[further applications/Homeostasis/Homeostasis (RST)]]** — Homeostasis (the identity loop) | further applications/Homeostasis | In progress (verified) | Identity loop I = Omega*mu; script verifies round-trip (source_from_rst, solve_omega). | | — | **[[further applications/Nuclear pore complex (RST)/Nuclear pore complex (RST)]]** — Nuclear pore complex (selectivity, transit) | further applications/Nuclear pore complex (RST) | In progress (script + data) | Selectivity as $\mu(\eta,n)$; disorder as substrate; transit/passage vs RST curve; relational path (A5). | | 12 | **Consciousness (recursive self-maintenance)** | (todo) | **Todo** | Higher-order persistence: system maintains the *process* of maintaining; recursive workload. | ### Physics and cosmology | # | Problem | Subfolder | Status | RST angle (one-line) | |:---|:---|:---|:---|:---| | 13 | **Black Hole Information Paradox** | (todo) | **Todo** | Saturation regime: resolution collapse at horizon; information as workload that cannot be sustained (same curve, “too heavy”). | | 14 | **Quantum measurement problem** | (todo) | **Todo** | “Too small” regime: blur between signal and noise; measurement as resolution threshold (A1). | | 15 | **Hubble Tension** | (todo) | **Todo** | Inconsistent $H_0$ from different probes; RST: different effective $a_0$/noise on different scales or epochs (fidelity curve calibration). | | 16 | **High-temperature superconductivity** | (todo) | **Todo** | Coherent phase as high-fidelity regime; transition as crossing $\mu(\eta,n)$; pairing from relational cost. | | 17 | **Dark energy / cosmological constant** | (todo) | **Todo** | Baseline workload $W \to N$ when $I \to 0$; $\Lambda$ as standby cost of the substrate (RST Part II). | | 18 | **Hierarchy problem (electroweak vs Planck)** | (todo) | **Todo** | Scale separation as different SNR regimes on the same curve; Substrate Eigenvalues ($G$, $v_\text{EW}$, etc.) as geometric ratios. | | 19 | **[[further applications/Quantum Geometry/Quantum Geometry (RST)]]** — Quantum geometry (Berry curvature, quantum metric) | further applications/Quantum Geometry | In progress (conceptual); calibration in Phase 2 | Conceptual bridge to holonomy ($\theta_W$); Berry phase ↔ EW twist; ghost field ↔ noise projection; Bloch-sector $\mu(\eta,n)$ mapping. | --- ## Implementation status and problems See **[[Implementation status and problems]]** for: which applications have scripts and have been run; which are conceptual-only; known limitations (engine path, two codebases, no Millennium proofs); and how to re-verify scripts. ### Computational capstone: Graph → Sovereign Chain The **Pure Axiom Substrate** (`rst_axiom_pure.py`) derives $n \approx 1.24$ and $d_B \approx 1.22$ (backbone dimension) from five parameter-free axioms — hot genesis, local Landauer cost, relational cooling. The Sovereign Chain is now **derived from first principles**, not empirically matched. Ledger/genesis (`rst_genesis.py`) provides an alternative bridge. See **[[expanded theory applied/Derivation Chain Overview]]**, **[[further applications/Micro-Graph Generator/Micro-Graph Generator (RST)]]**, **[[expanded theory/The Sovereign Chain]]**. --- ## How applications are structured - **Application note:** Problem statement in classical form, then RST reframe (which axioms, which identities), and a short conclusion. - **Code (optional):** Python script that **plugs into the general RST engine** (fidelity $\mu(\eta,n)$, Resource Triangle $W^n = \Omega^n + N^n$, etc.); documented in a `* - Code.md` file in the same subfolder. - **Results (optional):** If a script produces figures or tables, a short results note; include images in the Results note (and optionally in the application note). **Full guide:** [[How we do applications]] — structure, engine integration, including images, internal links, and relational documentation. --- ## Links - **Foundation index** (bridge layer): **[[foundation/Foundation index]]** - **Further applications index:** **[[further applications/Further applications index]]** (run all: `run_all_further_scripts.py`) - **How we do applications:** [[How we do applications]] (code, scripts, images, links, relational docs) - **RST Python engine:** `rst_engine/` subfolder at workspace root (see `rst_engine/README.md`) — single implementation; all application scripts and the core verification script import from this package. - **Core theory:** [[expanded theory/Relational Substrate Theory (RST)]] - **Core equation:** [[expanded theory/RST Core Reduction]] - **Axioms:** [[Relational Resolution Theory (RRT)]] - **Symbol index:** [[expanded theory/Symbol Index]] - **Structural foundations:** [[expanded theory/Structural Foundations]] — resistance, fidelity, walk dimension, isotropic projection, graph attractor