RST v1.0 — Minimal Closure (First Closed Form)
RST Correspondence Principle (v1.0)
Principle: Reactive Substrate Theory (RST) must reproduce the empirically verified predictions of General Relativity (GR) and Quantum Mechanics (QM) within the domains where GR and QM are experimentally confirmed, while allowing a different underlying ontology (substrate dynamics) to generate those effective behaviors.
CP-1: GR Limit (Weak-Field / Low-Frequency / Large-Scale)
In the regime of weak substrate perturbations and slowly varying fields, RST must reduce to the standard post-Newtonian behavior of GR:
- Gravitational redshift and time dilation
- Light bending and lensing to leading order
- Perihelion precession and Shapiro delay
- Near-equivalence of inertial and gravitational response
Operationally: there must exist an effective potential Φ derived from the substrate such that test-particle motion and wave propagation match GR to experimental precision in this limit.
CP-2: QM Limit (Coherent Excitation / Smooth Substrate / Weak Coupling)
In the regime where the substrate varies slowly compared to the coherence scale, the resonance field Ψ must exhibit effective wave dynamics that reproduce:
- interference and diffraction
- bound-state spectra (quantized modes)
- tunneling behavior
- unitary/energy-conserving evolution in isolated systems (at least approximately)
Operationally: measurable observables derived from Ψ (such as |Ψ|², phase gradients, and mode frequencies) must align with standard quantum outcomes in the regime where quantum theory is known to hold.
CP-3: Non-Contradiction + Minimal Extension
RST must not introduce new terms unless required by: (1) a failure to reproduce established GR/QM limits, or (2) a clear empirical anomaly where RST provides a simpler explanation. The core equations define identity; extensions must remain minimal and test-driven.
GR → RST Constraint Table (What GR Forces S to Do)
| Empirical GR Result | What It Means Physically | RST Constraint (on S and its mapping) |
|---|---|---|
| Gravitational redshift | Clock rates differ in gravitational potential wells. | There must exist a substrate→clock mapping such that dτ/dt depends on S (or δS) in the weak-field limit. |
| Time dilation near mass | Proper time slows in deeper potentials. | The effective potential Φ must be a functional of (S − S̄) such that the metric limit is: ds² ≈ (1 + 2Φ)c²dt² − (1 − 2Φ)dx². |
| Light bending / lensing | Null trajectories curve near mass. | Wave propagation must see effective refractive structure from ∇S (or Φ[S]), reproducing lensing angles to post-Newtonian order. |
| Shapiro delay | Signals take longer passing near mass. | The substrate-induced time-rate mapping must generate propagation delay consistent with GR for weak-field solar-system tests. |
| Perihelion precession | Orbits deviate slightly from Newton. | The effective force derived from Φ[S] must contain the same leading corrections as GR in the weak-field expansion. |
| Equivalence principle (approx.) | All test bodies fall the same way. | The substrate coupling that generates “gravity” must be universal (composition-independent) to experimental bounds—i.e., the clock-rate and motion mapping cannot depend on particle identity in the GR limit. |
Key takeaway: GR does not force you to rewrite the substrate PDE. It forces you to specify a mapping from substrate state (S, ∇S, δS) to an effective metric/potential that reproduces GR in the weak-field limit.
QM Observables → RST Quantities (How Ψ Must Behave to Match QM)
| QM Observable | Standard Meaning | RST Mapping (Ψ / S) |
|---|---|---|
| Wavefunction Ψ | Complex amplitude encoding quantum state. | Ψ is a physically real coherence/resonance field: organized excitation modes in the substrate. |
| |Ψ|² | Probability density (Born rule). | |Ψ|² is coherence density / excitation intensity. In the QM limit it must reproduce Born statistics operationally (via ensemble/measurement coupling). |
| Phase arg(Ψ) | Phase controls interference. | Phase is literal medium phase. Gradients ∇θ act like flow; topological phase defects yield nontrivial “magnetic-like” behavior in emergent gauge limits. |
| Energy levels | Quantized eigenvalues in bound systems. | Discrete resonance modes of the coupled (S, Ψ) system. Soliton + trapped modes correspond to particle-like spectra. |
| Tunneling | Nonzero transmission through barriers. | Coherence leakage across substrate barriers: Ψ penetrates regions where effective mass/phase becomes evanescent due to S-coupling. |
| Decoherence | Loss of phase coherence via environment. | Interaction between Ψ and substrate fluctuations/gradients: turbulence, noise, or mode-mixing in S destroys coherent phase relations. |
Key takeaway: QM constrains how Ψ must behave and what Ψ’s observables mean, without forcing you to discard the core PDE. It forces correspondence in the smooth-substrate, coherent-excitation limit.
RST v1.0 — Minimal Closure (First Closed Form)
This section freezes a minimal, testable closure of Reactive Substrate Theory (RST). The goal is not to claim the final form of nature, but to define the smallest complete system that: (1) remains faithful to the RST ontology, (2) preserves GR/QM correspondence in their validated limits, and (3) is implementable in simulations and falsifiable in observations.
1) Closure Choice A: Coherence Functional C[Ψ]
We choose the simplest physically meaningful coherence measure:
C[Ψ] ≡ ρ ≡ Ψ*Ψ = |Ψ|²
Interpretation: ρ is the local coherence density (excitation intensity). In the QM limit, this quantity corresponds operationally to the Born-density observable.
2) Closure Choice B: Resonance Coupling Function FR
We choose a minimal resonance driver that explicitly encodes phase alignment between the coherence field and a local natural oscillation rate:
F_R(x,t) ≡ ρ(x,t) cos( ω₀ t )
Interpretation: Resonance is maximized when coherence density is high and oscillations remain phase-aligned with the substrate’s local natural mode.
Optional time-averaged (simulation-friendly) form:
⟨F_R⟩_T(x,t) ≡ (1/T) ∫[t−T/2 to t+T/2] ρ(x,t') cos(ω₀ t') dt'
This removes rapid oscillatory sign-flips while preserving the physical content: sustained resonance requires persistent alignment over a finite window.
3) Closure Choice C: Weak-Field Map (Substrate → Effective Potential)
We choose the minimal weak-field identification of an effective gravitational potential Φ with substrate deviation from a background mean:
Φ(x,t) ≡ A ( S(x,t) − S̄(t) )
Interpretation: Φ is not assumed fundamental. It is the leading-order effective potential in the GR correspondence limit, where gradients in substrate state produce the same observable weak-field phenomena as GR.
4) The Closed RST v1.0 Field System
Substrate field (closed form):
∂²ₜ S − c²∇²S + βS³ = σ(x,t) · ρ(x,t) cos(ω₀ t)
Resonance / coherence field (unchanged core):
∂²ₜ Ψ − v²∇²Ψ + μΨ + λ|Ψ|²Ψ = κ S Ψ
with:
ρ(x,t) = |Ψ(x,t)|² Φ(x,t) = A( S(x,t) − S̄(t) )
5) What This Closure Buys You Immediately
- Closed system: nothing undefined remains (C[Ψ], F_R, Φ are explicit).
- Simulation-ready: can be evolved numerically in 1D/2D/3D with standard methods.
- Correspondence-compatible: weak-field observables can be mapped through Φ.
- Resonance physics is explicit: regions of sustained coherence alignment become active drivers rather than passive bystanders.
6) Minimal Honesty Label (Important)
RST v1.0 Minimal Closure is a deliberately conservative choice: it is not claimed to be unique. It is chosen because it is the smallest closure that preserves the core ontology and enables concrete falsifiable modeling. If empirical mismatch appears, the first place to refine is the functional form of FR (not the entire framework).
RST v1.0: Variables, Units, and Interpretation
This table defines the minimal set of variables appearing in the RST v1.0 equations, together with their physical dimensions, interpretation, and correspondence to familiar quantities in General Relativity (GR) and Quantum Mechanics (QM). The goal is not to introduce new speculative objects, but to close the theory operationally so that simulations, comparisons, and tests can be implemented without guessing.
Core Fields
| Symbol | Units | Interpretation (RST) | GR / QM Correspondence |
|---|---|---|---|
| S(x,t) | dimensionless or energy-density–scaled | Substrate state / tension field. Encodes emergent geometry, time-rate, and large-scale gravitational behavior. |
GR: Newtonian potential Φ (weak-field limit) GR: gtt component via Φ ∝ (S − S̄) |
| Ψ(x,t) | √(density) | Coherence (soliton) field representing matter-like excitations. Supports localized stable solutions and phase dynamics. | QM: Wavefunction ψ (reinterpretation, not replacement) |
Derived Quantities
| Symbol | Definition | Units | Physical Meaning |
|---|---|---|---|
| ρ | |Ψ|² | density | Coherence density. Acts as effective matter density and resonance strength for substrate sourcing. |
| S̄(t) | spatial average of S | same as S | Cosmological background substrate state controlling expansion and average clock rate. |
| Φ | A(S − S̄) | dimensionless | Effective gravitational potential in weak-field limit. |
Coupling and Parameters
| Symbol | Units | Role | Interpretation |
|---|---|---|---|
| c | length / time | Substrate signal speed | Maximum propagation speed of substrate disturbances (reduces to speed of light observationally). |
| v | length / time | Coherence propagation speed | Effective wave speed in the Ψ sector (≤ c). |
| β | 1 / S² | Substrate self-interaction | Controls stiffness, saturation, and avoidance of singularities. |
| μ | 1 / time² | Linear mass term | Sets natural oscillation scale of Ψ excitations. |
| λ | 1 / (density·time²) | Nonlinear coherence coupling | Enables soliton stability and quantized modes. |
| κ | 1 / S | Substrate–coherence coupling | Determines how matter-like excitations backreact on the substrate. |
Source and Resonance Closure (v1.0)
RST v1.0 adopts a minimal closure consistent with existing GR and QM results:
C[Ψ] = |Ψ|² = ρ
FR[ρ] = ⟨ρ⟩T (time-averaged resonance strength)
This choice ensures:
- No hidden variables
- No explicit nonlocal signaling
- Compatibility with quantum probability density
- Controlled sourcing of the substrate by coherent matter
Interpretive Summary
In RST v1.0, nothing new is added arbitrarily. General Relativity constrains the weak-field mapping of S to Φ, while Quantum Mechanics constrains the interpretation of Ψ and ρ. The substrate does not replace existing theories; it acts as the mechanical medium whose macroscopic and microscopic limits reproduce them.
This table defines the theory tightly enough for numerical implementation, experimental comparison, and falsification — while remaining deliberately minimal.
RST v1.0: Assumptions and Domain of Validity
Reactive Substrate Theory (RST) v1.0 is intentionally minimal. Its assumptions are explicitly stated to define the range of regimes in which the theory is expected to agree with observation and where it may safely be applied without extrapolation.
Explicit Assumptions
- Substrate Continuity: The substrate field S(x,t) is continuous and differentiable on scales larger than the Planck length. No discrete spacetime structure is assumed.
- Weak-Field Correspondence: In regimes where gravitational fields are weak and velocities are non-relativistic, RST must reduce to the predictions of General Relativity.
- Quantum Correspondence: The coherence field Ψ reproduces standard quantum-mechanical observables (probability density, interference, conservation laws) when interpreted through |Ψ|².
- No Exotic Matter Assumed: RST v1.0 does not require negative energy densities, exotic fields, or ad hoc dark-sector particles.
- Minimal Closure: Resonance sourcing is time-averaged and local in v1.0 to avoid untested nonlocal dynamics.
Domain of Validity
RST v1.0 is expected to be valid under the following conditions:
- Substrate gradients |∇S| are small compared to background values
- Time-rate modulation remains perturbative (|α·δS| ≪ 1)
- Ψ maintains phase coherence over the region of interest
- No attempt is made to model Planck-scale or trans-Planckian physics
Claims outside these regimes are explicitly deferred to later versions of the theory.
RST v1.0: What Would Falsify This Immediately
RST is framed as a falsifiable physical theory. The following observations or experimental results would directly contradict RST v1.0 in its current form.
Immediate Falsifiers
- No Weak-Field Correspondence: If substrate-based models cannot reproduce Newtonian gravity and post-Newtonian corrections already confirmed by experiment, RST fails.
- No Quantum Correspondence: If mapping |Ψ|² → probability density produces measurable deviations from quantum statistics already observed, RST is ruled out.
- Absence of Substrate Signatures: If clock-comparison, interferometric, and multi-band observations show no correlated residuals beyond GR expectations at increasing precision, RST loses empirical motivation.
- Violation of Causality: If the coupled system permits information transfer faster than substrate signal speed c, the model is rejected.
- Uncontrolled Energy Growth: If simulations show runaway substrate energy or unstable soliton collapse inconsistent with observed stability of matter and spacetime, RST v1.0 is invalid.
These criteria are intentionally strict. RST v1.0 is not protected by interpretive flexibility against direct empirical conflict.
RST v1.0 → v1.1: What Changes, and Why
RST v1.0 is not intended as a final theory. It serves as a closed, testable baseline. Version 1.1 introduces changes only where empirical necessity or internal consistency requires them.
Planned v1.1 Extensions
- Dynamic Resonance Function: Replace time-averaged FR[ρ] with a frequency-selective or phase-sensitive resonance term once stability is demonstrated.
- Anisotropic Substrate Response: Allow direction-dependent substrate stiffness to model polarization, birefringence, or preferred-frame effects if observed.
- Nonlinear Backreaction: Introduce controlled nonlinear feedback from Ψ to S beyond |Ψ|² when soliton–substrate coupling becomes strong.
- Strong-Field Regime: Extend beyond perturbative Φ mapping to explore compact objects and extreme environments.
- Experimental Calibration: Fix coupling constants using observational data rather than theoretical normalization.
What Does Not Change
- The substrate ontology
- The reinterpretation (not rejection) of GR and QM
- The rejection of ad hoc dark-sector components
- The requirement of falsifiability
RST v1.1 is therefore an evolution, not a rewrite. Any extension must preserve agreement with v1.0 in its validated domain while expanding predictive reach.
