>[!warning]
>This work has not been peer reviewed. It is intended as an entry paper and pointer to the full relational vault and codebase.
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# From Relational Resolution to Unified Physics: An Entry Paper
**Author:** Christoph Janke
**Affiliation:** Unaffiliated
**E-mail:**
[email protected]
**URL:** [[Overview — RRT, RST, and applications]]
**Country:** Germany
**Suggested categories:** cs.IT (Information Theory); physics and math groups.
**Version:** Preprint. Full theory, derivations, and applications live in a relational vault and repository (see §7).
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## Abstract
Relational Resolution Theory (RRT) is a meta-ontological framework that extends Landauer's principle to a relational ontology; Relational Substrate Theory (RST) applies it to the physical universe and obtains a Resource Triangle $W^n = \Omega^n + N^n$ and fidelity $\mu(\eta,n) = \eta/(1+\eta^n)^{1/n}$. The main equation is $I = \Omega \cdot \mu$. In the gravity sector the field equation $\mu(q'/a_0,n)\,q' = g_N$ has the form of MOND (Milgrom 1983; Famaey & McGaugh 2012); $a_0 = cH/(2\pi)$ and the transition sharpness $n$ are fixed or constrained by the theory. The gravity sector is compared to SPARC galaxy rotation curves (~86% acceptable fits; median $\Upsilon_{\text{disk}} \approx 0.49$). The same mathematical structure is mapped in 14 foundation topics and in further applications; a Reality Engine (ledger simulator) verifies the formal limits of the ledger. This entry paper summarizes the axioms, equations, derivation chain, and checks; full development is a relational vault and repository.
**Keywords:** Relational Resolution Theory; Relational Substrate Theory; resource allocation; Landauer principle; modified Newtonian dynamics; SPARC; fidelity; information theory.
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## 1. Introduction
Landauer (1961) established that maintaining a bit against thermal noise costs at least $k_B T \ln 2$ per erased bit. Relational Resolution Theory (RRT) extends this to a relational ontology in which **resolution**, **translation**, **proper time**, and **relational distance** are the primitives, and persistence is an active, dissipative process. Relational Substrate Theory (RST) applies RRT to the physical universe and obtains one Resource Triangle and one fidelity function that describe how “signal” (the structure being maintained) and “noise” (the environmental floor) share a finite budget. In the gravity sector the resulting field equation has the form of MOND (Milgrom 1983; Famaey & McGaugh 2012): Newton when acceleration exceeds a threshold, MOND-like below it; the noise floor links to the cosmological constant. The same structure is mapped in 14 foundation topics and in further applications; a simulator (Reality Engine) verifies the formal limits of the ledger.
This document is an **entry paper**: it states the axioms, the main equations, the derivation chain (Resource Triangle → field equation; disformal lensing), the SPARC comparison, and the scope of mappings. Full derivations, code, and exploratory notes are in the vault and repository (§7).
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## 2. Relational Resolution Theory (RRT)
RRT is a meta-ontological framework with five axioms and one principle.
**Axioms (A1–A5):**
| Axiom | Name | Content |
|:---|:---|:---|
| A1 | Format | The independent, dynamic substrate. It sets the local noise level and refresh interval; gravity is a property of the format. |
| A2 | Information (resolution $\sigma$) | The effective bit-count required to maintain distinctions against thermal noise; $\sigma = -\sum p_j \log_2 p_j$ over distinguished states. |
| A3 | Translation | The irreversible mechanism of change. Every change is an event that costs energy; identity is dissipative. |
| A4 | Proper time ($\tau$) | The local refresh interval of a system’s relations. Time is not a background; it is the duration between discrete translations. |
| A5 | Relational distance ($d$) | The number of translation steps to relate one system to another. Cost scales with $d$. |
**Relational Landauer Principle:** The minimum power to maintain resolution against noise is
$\Phi_{\min} = k_B \ln 2 \sum_{i=1}^{n} \frac{d_i \cdot \sigma_i \cdot T_i}{\tau_i}.$
No system can persist below $\Phi_{\min}$; the ratio $K_R = \Phi_{\text{measured}}/\Phi_{\min} \geq 1$ is universal.
**Inherent constraints:** Irreversibility, energy tax per event, causality, entropy production, and relational friction (cost increases with $d$) follow from the format. **Inherent principles:** Everything changes; change is gradual; everything builds on what came before; reality has a reach; existing has a cost. Coarse-graining (reducing $\sigma$ or $d$) lowers $\Phi_{\min}$ and is a thermodynamic survival strategy (zoom logic).
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## 3. Relational Substrate Theory (RST)
RST identifies the RRT concepts with physical quantities and derives the core geometry.
**Identifications:** Format → spacetime + aether field; noise → expansion scalar $\theta = 3H$ (FLRW); gravitational noise floor → $a_0 = c\theta/(6\pi) = cH/(2\pi)$ (e.g. Famaey & McGaugh 2012); resolution → signal; $\Phi_{\min}$ → gravitational workload.
**Resource Triangle:**
$W^n = \Omega^n + N^n$
- $\Omega$ = workload (actual effort);
- $N$ = noise floor (effort against environment);
- $W$ = total budget;
- $n \geq 1$ = transition sharpness (from axioms; empirically $n_0 \approx 1.25$).
**Fidelity** (rendering quality of the format):
$\mu = \frac{\Omega}{W} = \frac{\eta}{(1+\eta^n)^{1/n}}, \qquad \eta = \frac{\Omega}{N}.$
Boundary conditions: $\eta \gg 1 \Rightarrow \mu \to 1$; $\eta \ll 1 \Rightarrow \mu \to \eta$ (Landauer forces linearity in noise). The budget identity $\mu^n + \nu^n = 1$ (with $\nu = N/W$) follows from the triangle. The form $\mu(\eta,n) = \eta/(1+\eta^n)^{1/n}$ is a standard MOND interpolation family; in RST it is obtained from the axioms (vault: Fidelity Derivation).
**Derivation sketch (Triangle → field equation):** (1) Landauer: cost per maintained bit ∝ noise $T$; hence high SNR ⇒ $\mu \to 1$, low SNR ⇒ $\mu \to \eta$. (2) Translation steps (A3, A4) allocate exclusively to signal or noise (**phase fractions** $\varphi_s + \varphi_n = 1$). (3) Scale-free response of the Format (A1): $g(\varphi) = \varphi^{1/n}$, so $\mu^n + \nu^n = 1$, hence the Resource Triangle. (4) Fidelity: $\mu = \Omega/W = \eta/(1+\eta^n)^{1/n}$. (5) Main equation: $I = \Omega \cdot \mu$. (6) Gravity identification: source $I = g_N$ (Newtonian from baryons), workload $\Omega = q'$ (RST acceleration), noise $N = a_0$; thus $g_N = q' \cdot \mu(q'/a_0,\, n)$, i.e. $\mu(q'/a_0,\, n)\, q' = g_N$.
**Main equation (Resource Allocation Equation):**
$I = \Omega \cdot \mu$
where $I$ is the source/request (signal). Equivalently, $I = \Omega^2/W$ (projection of the signal leg onto the hypotenuse). The triangle and the main equation are algebraically identical; a script confirms this identity as an internal consistency check—it verifies the formalism, not physical truth (vault: Super-Relational Mapping Verification; Python: `verify_super_relational_mapping.py`).
**Gravity:** The field equation $\mu(q'/a_0,\, n)\, q' = g_N$ has the form of MOND: $q'$ is the RST acceleration, $g_N$ the Newtonian acceleration from baryons. It gives Newton when $\eta \gg 1$, MOND-like when $\eta \lesssim 1$, and links to $\Lambda$ via the noise floor. In the gravity sector $a_0$ and $n$ are fixed or constrained by the theory (SPARC calibration gives $n_0 \approx 1.25$). Four projections of the triangle are mapped to force-like domains in the vault; these are structural correspondences, not a derivation of the Standard Model.
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## 4. Empirical and numerical verification
**SPARC:** The RST gravity sector is compared to the SPARC database (Lelli et al. 2016, AJ 152, 157). The transition is set by $\eta = g_N/a_0$ with $a_0 = cH/(2\pi)$. Pipeline: rawt package and notebooks; $\mu$ and solver live in `rst_engine`. Reported outcome: ~86% of galaxies with acceptable fits; median stellar mass-to-light $\Upsilon_{\text{disk}} \approx 0.49$ (within SPS range). Vault and code: SPARC evaluation under `expanded theory/sparc evaluation/`.
**Super-relational mapping:** A script verifies the identity between the triangle and the main equation, $\mu^n + \nu^n = 1$, and the roundtrip $I \to \Omega \to I$. This is an algebraic consistency check (vault: Super-Relational Mapping).
**Reproducibility:** Scripts and SPARC pipeline use `rst_engine`; one formula set (clone repo, run scripts to reproduce figures and numbers).
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## 5. Applications and mappings
**Foundations:** 14 foundation topics map classical maths and physics onto RRT/RST (information theory, Landauer, measure, scaling, spectral, continuum, gauge, cosmology, complexity, algebraic structure, and four more; vault Foundation index). The first 10 have notes and scripts that plot $\mu(\eta,n)$ and the budget split in each context; runner `run_all_foundation_scripts.py` produces a combined figure.
**Engine:** A **Reality Engine** (ledger simulator) implements a discrete MCMC path-integral over the A5 relational friction cost. Without hardcoding kinematic force laws or interpolation functions, the engine structurally reproduces: (1) relational scale invariance (Wilsonian coarse-graining passing with a 1.0 ratio); (2) the low-SNR acceleration boost (the MOND regime); and (3) the External Field Effect (where an external background signal $\Omega_{\text{ext}}$ analytically suppresses the local fidelity boost). It also demonstrates that a higher noise floor ($a_0 \propto H$) naturally accelerates structural collapse, establishing a formal mechanism for rapid early-epoch formation. Vault: Reality Engine.
**Further applications:** Homeostasis (identity loop verified by script), Millennium-style reframes (Navier–Stokes, BSD, Hodge, etc.), and other mappings are in the vault; runner `run_all_further_scripts.py`.
**Lensing:** Conformal coupling alone does not bend light; disformal terms are used in relativistic MOND (Bekenstein 2004). In the vault, a disformal photon metric is derived from A1–A4 and a congestion identity (**where high local workload restricts the update rate for passing signals**); numerical implementation in `build_all_calculated_proofs.py`.
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## 6. Limitations and scope
This is an entry paper and pointer to the vault and repository. **Gravity:** compared to SPARC; results reported as above. **Other sectors (four-force, foundations, further applications):** exploratory mappings and structural correspondences; not derivations of the Standard Model or proofs of the mapped conjectures. Other emergent-gravity approaches (e.g. Verlinde 2016) address dark phenomena from different premises. Full formalism, derivations, and caveats are in the vault.
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## 7. Conclusion and availability
RRT and RST provide one axiom set and one physical formalism. In the gravity sector one curve $\mu(\eta,n)$ and the equation $I = \Omega \cdot \mu$ give a MOND-form field equation; $a_0$ and $n$ are fixed or constrained by the theory. SPARC is used as an empirical comparison; algebraic consistency is checked by script. Predictions and the data each is matched or matchable against (SPARC, Gates 1–3, lensing) are summarised in the vault note **Predictions and data matching**. The same mathematical structure is mapped in 14 foundation topics and in further applications; a Reality Engine implements the ledger. The full development is a **relational vault** (linked notes) and a **repository** with one Python engine (`rst_engine`) and documented scripts.
**Full content:**
- **Vault (relational web):** Overview and full graph: <https://tripstoph.org/about+systems/Overview+%E2%80%94+RRT%2C+RST%2C+and+applications>
- **Code and figures:** [Repository](https://codeberg.org/ChristophTripstoph/RRT_RRS) (clone or download). Python 3; `rst_engine` at repo root; scripts under `expanded theory applied/foundation/`, `expanded theory applied/further applications/`, `expanded theory/python calculations/`, `expanded theory/sparc evaluation/`. Figures are generated by running the scripts; the vault references them. No `.py` files are served by Obsidian Publish; the repo holds the code and the commands to reproduce every figure and number.
- **Author:** https://www.tripstoph.org
**Suggested citation (entry paper):**
C. Janke, “From Relational Resolution to Unified Physics: An Entry Paper,” preprint, 2025. Full theory and applications: relational vault and repository (tripstoph.org).
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## References
1. R. Landauer, “Irreversibility and heat generation in the computing process,” *IBM J. Res. Dev.* **5**, 183 (1961).
2. M. Milgrom, “A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis,” *Astrophys. J.* **270**, 365 (1983).
3. B. Famaey and S. S. McGaugh, “Modified Newtonian dynamics (MOND): Observational phenomenology and relativistic extensions,” *Living Rev. Relativ.* **15**, 10 (2012); [arXiv:1112.3960](https://arxiv.org/abs/1112.3960).
4. J. Bekenstein, "Relativistic gravitation theory for the modified Newtonian dynamics paradigm," *Phys. Rev. D* **70**, 083509 (2004); [arXiv:astro-ph/0403694](https://arxiv.org/abs/astro-ph/0403694).
5. E. Verlinde, "Emergent gravity and the dark universe," *SciPost Phys.* **2**, 016 (2017); [arXiv:1611.02269](https://arxiv.org/abs/1611.02269).
6. F. Lelli, S. S. McGaugh, and J. M. Schombert, "SPARC: Mass models for 175 disk galaxies with Spitzer photometry and accurate rotation curves," *Astron. J.* **152**, 157 (2016); [arXiv:1606.09251](https://arxiv.org/abs/1606.09251).
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*This work has not been peer reviewed. It is an entry paper and pointer to the full relational development (vault + repository).*
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## Version history
| Version | File | Changes |
|:---|:---|:---|
| v1 (2025) | `entry_paper.tex` | Original: gravity sector only (SPARC). Published on Zenodo. Archived. |
| **v2 (2026)** | `theory_paper.tex` | **Canonical Zenodo version.** Self-contained; no internal citation of v1. Adds Dimensional Ladder (14 sectors), five $R^2=1$ identities (Lorentz, particle-in-box, hydrogen, Michaelis-Menten, Hill equation), Sovereign Chain (9 SM constants from $n=1.25$ at