A Unified Theory of Collapse-Induced Gravity (SRF-Derived)
Project GENESIS presents a complete theoretical framework and experimental program for the theory of Collapse-Induced Gravity. This framework is derived from the foundational Dynamic Resolution Sampling Rate Framework (SRF), which posits reality as a discrete information-processing grid. We resolve the core conflicts between General Relativity and Quantum Mechanics by positing that gravity is not solely a static property of mass-energy, but possesses a dynamic component generated by wavefunction collapse. This "aftershock" arises from the interaction of information discontinuities (quantified by an "Observation Strength" field, \(O\)) with a finite-resolution Planck Filter (characterized by a dynamic effective Nyquist frequency, \(\omega_{\text{eff}}\)). This document details the foundational principles, introduces the Dynamic Field Equations (incorporating SRF's \(\kappa\) and \(\eta'\) coupling constants), and outlines a multi-phase experimental program. This program is designed to first constrain the theory's parameters via the HERA experiment, and then to provide definitive proof by directly observing a "collapse ripple" with the Hephaestus experiment, ultimately culminating in the flagship Project AETOS: The ZEUS Experiment. Project GENESIS aims to provide the first empirical evidence that a significant component of spacetime curvature is born from the fundamental event of quantum measurement and the universe's inherent participatory nature.
This flowchart illustrates the logical structure of the SRF-GENESIS framework, from the fundamental action principle down to the specific experimental predictions designed to test it.
For nearly a century, theoretical physics has been defined by two deep, unresolved crises. The first is the violent incompatibility between General Relativity, our theory of the very large, and Quantum Mechanics, our theory of the very small. Their conflicts lead to nonsensical infinities at the heart of black holes and the beginning of time. The second, more subtle crisis is the role of information and observation in reality. The "measurement problem" in quantum mechanics asks why the act of looking forces a world of infinite possibilities to choose a single, definite state.
Project GENESIS, fundamentally derived from the Dynamic Resolution Sampling Rate Framework (SRF), proposes a single, unified solution to both crises. It posits that these are not separate problems, but different facets of the same fundamental truth: the universe is a physical information processor with finite resolution. The "measurement problem" is the description of the processing event, and "gravity" is the physical artifact left behind by that event.
SRF Overview: Within the SRF, General Relativity (GR) emerges from mass/energy acting as a computational load, slowing the local processing rate (\(\omega_{\text{eff}}\)) and warping the grid’s processing structure (spacetime curvature). Quantum Mechanics (QM) arises from the grid’s inherent limitations: when \(\omega_{\text{eff}}\) is low, the grid undersamples sub-grid dynamics, leading to aliasing artifacts perceived as quantum superposition, interference, and entanglement. Crucially, Observation and Collapse are physical processes where active interaction (Observation Strength, \(O\)) actively boosts local \(\omega_{\text{eff}}\) non-linearly, enhancing resolution, eliminating aliasing, and collapsing possibilities into definite outcomes via a derived stochastic mechanism.
The central hypothesis of Project GENESIS is a radical reframing of cause and effect in the cosmos, explicitly grounded in the SRF. It can be summarized in a simple, causal chain:
Just as a pebble dropped into a calm pond creates ripples, the collapse of a quantum wavefunction creates a propagating gravitational distortion. GENESIS, leveraging SRF, posits that gravity is the ripple, not just the pebble.
Project GENESIS is rigorously derived from the Dynamic Resolution Sampling Rate Framework (SRF), which provides the underlying mathematical and physical principles. SRF posits a fundamental, discrete information-processing grid for reality, from which all known physics emerges. The mechanism behind the Gravitational Gibbs Effect is the Planck Filter, characterized by its dynamic effective Nyquist frequency (\(\omega_{\text{eff}}\)).
The Gibbs overshoot is the mathematical "scream" of a continuous system trying to describe something infinitely sharp. The Planck Filter's finite resolution (\(\omega_{\text{eff}}\)) means this can never happen in physical reality; the "overshoot" is a real, physical distortion, which contributes to gravity.
The entire SRF framework, and thus the foundation for Project GENESIS, is derived from a single discrete action, \(S_{\text{SRF}}\), defined on a 3+1D lattice. This action governs the dynamics of the complex state field \(S(\vec{j}, n)\) and the real scalar field \(\omega_{\text{eff}}(\vec{j}, n)\) (the local effective sampling rate). External influences like local energy density \(E(\vec{j}, n)\) (mass) and active observation strength \(O(\vec{j}, n)\) (information exchange) act as fundamental source fields.
The total SRF action is given by:
\[ S_{\text{SRF}} = \sum_{\vec{j},n} \Delta t \left[ \mathcal{L}_S(\vec{j},n) + \mathcal{L}_{\omega}(\vec{j},n) - V(\omega_{\text{eff}}, E, O) \right] \]This action contains terms for the state evolution (\(\mathcal{L}_S\)), the kinetics of the sampling rate field (\(\mathcal{L}_{\omega}\)), and a potential (\(V\)) that couples the sampling rate to energy and observation. All physical laws of SRF, including the precise dynamics for \(\omega_{\text{eff}}\) and the Universal Update Rule (UUR), are derived from this single starting point.
SRF explicitly defines the Planck Filter as a discrete grid of elementary cells (\(\vec{j}\)) updating at discrete time steps (\(\Delta t\)). The fundamental resolution is set by a base frequency, \(\omega_0 = \pi / t_P\).
Crucially, the grid’s local sampling rate \(\omega_{\text{eff}}\) is dynamically determined by the interplay of passive energy presence and active information exchange. The instantaneous functional form, derived directly from the SRF Action Principle, is:
Here, the \(-\kappa E\) term represents the computational drag of mass/energy (consistent with gravity), and the \(+\eta' O^2\) term represents the "spark" of the resolution boost from observation. The quadratic dependence on \(O\) is fundamental, leading to the self-reinforcing "Avalanche Collapse."
This provides a precise, quantitative understanding of the Planck Filter's dynamic behavior, moving beyond a purely conceptual role.
The exponential behavior of collapse is not a separate postulate but an emergent property of the SRF's core feedback mechanisms. The quadratic term acts as the trigger for a self-reinforcing loop:
The instantaneous relationship between \(O\) and \(\omega_{\text{eff}}\) is governed by the potential term in the action. In a region with negligible energy (\(E \approx 0\)), the change in sampling rate is: \(\omega_{\text{eff}}(t) \approx \omega_0 + \eta' O(t)^2\). This is the seed for the interaction.
A higher \(\omega_{\text{eff}}\) means the system is being processed with higher fidelity, making it easier for an observer to extract more information. Thus, the rate of change of observation strength, \(\frac{dO}{dt}\), is proportional to the current sampling rate and the existing observation strength: \(\frac{dO}{dt} = \xi \cdot O(t) \cdot \omega_{\text{eff}}(t)\), where \(\xi\) is an information extraction efficiency constant.
At the beginning of collapse, \(\omega_{\text{eff}}(t) \approx \omega_0\). The feedback equation becomes \(\frac{dO}{dt} \approx (\xi \omega_0) \cdot O(t)\). This is the classic differential equation for exponential growth, with the solution \(O(t) = O_{\text{initial}} e^{t/\tau}\), where \(\tau = 1/(\xi \omega_0)\) is the collapse time constant.
Substituting the exponential growth of \(O(t)\) back into the fundamental interaction equation shows that the sampling rate itself experiences a runaway exponential increase: \(\omega_{\text{eff}}(t) \approx \omega_0 + \eta' O_{\text{initial}}^2 e^{2t/\tau}\). This is the mathematical description of the Avalanche Collapse, directly responsible for the rapid "collapse ripple" and the dynamic gravitational field.
The fundamental evolution of the grid's complex state field, \(S(\vec{j}, n)\) (the SRF analogue of a quantum wavefunction), is governed by the Universal Update Rule (UUR), derived directly from the SRF action principle:
The SRF provides a physical basis for the randomness of quantum mechanics, moving it from a purely mathematical postulate to a physical phenomenon. The stochastic phase kick \(\theta\) is not fundamental randomness from "nothing," but rather pseudo-random noise emerging from deeper, deterministic processes. Several hypotheses exist within the SRF:
In all cases, the key insight is that the variance of this effective noise is amplified by the Observation Strength field \(O\). Observation acts like a "gain" knob, making the system exquisitely sensitive to this underlying noise and forcing a rapid decoherence into a classical state.
Building upon the SRF, we propose that the total spacetime curvature, described by the Einstein tensor \(\text{G}_{\mu\nu}\), is the sum of two sources: the standard static mass-energy (\(\text{T}_{\mu\nu}\)) and a new dynamic term sourced by collapse, which we call the Gibbs Curvature Tensor, \(\Gamma_{\mu\nu}\).
The Gibbs Curvature \(\Gamma_{\mu\nu}\) is generated by the physical events of information discontinuity within the SRF: the rate and sharpness of wavefunction collapse. This dynamic source is now explicitly linked to SRF's fields:
Here, \(\mathcal{F}\) represents a functional dependence on the temporal and spatial gradients of \(\omega_{\text{eff}}\) (the Planck Filter's dynamic response), and the "currents" of energy \(J^E_{\mu\nu}\) and observation \(J^O_{\mu\nu}\). The Collapse Current Tensor `Jcμν` is now understood as the emergent macroscopic tensor representing the flux of information discontinuity, directly sourced by the dynamics of \(E\) and \(O\) fields, and the resulting changes in \(\omega_{\text{eff}}\). It is zero for smoothly evolving quantum systems and non-zero during a collapse event.
The new fundamental constant, the Genesis Coupling Constant (`κG`), which sets the strength of the interaction between information collapse and spacetime geometry, is now derived directly from the fundamental SRF constants \(\kappa\) and \(\eta'\) via sophisticated continuum limit and coarse-graining procedures applied to the SRF action and UUR. This makes \(\kappa_G\) a calculable, non-arbitrary value.
A critical step in validating the SRF is demonstrating how the smooth, continuous spacetime of General Relativity emerges from the discrete, pixelated Planck-scale grid. This is achieved through a process of coarse-graining and taking the continuum limit.
We do not simply assume that the discrete UUR becomes the Schrödinger equation. Instead, we treat the SRF grid as a statistical mechanical system. By averaging the behavior of the state field \(S(\vec{j}, n)\) and the sampling rate \(\omega_{\text{eff}}(\vec{j}, n)\) over many Planck cells, we can derive effective, macroscopic field equations. This process, analogous to deriving the laws of fluid dynamics from the motion of individual molecules, reveals how the fundamental SRF parameters give rise to familiar physical constants.
This procedure ensures that SRF is not merely an analogy but a true, multi-scale theory that contains both Quantum Mechanics and General Relativity as consistent limits of its fundamental discrete dynamics.
A critical question is the source of energy for the gravitational ripple. Within the SRF-GENESIS framework, this energy is not created from nothing, but is drawn directly from the quantum system itself during localization. In accordance with the uncertainty principle, the act of collapse sharply localizes a particle's position (\(\Delta x \to 0\)), which necessitates a corresponding increase in the uncertainty of its momentum (\(\Delta p\)). This "localization energy" provides the exact budget required to generate the gravitational Gibbs distortion. The total energy-momentum is conserved at all times:
This implies that the emission of a gravitational ripple (\(\text{T}^{\text{Gibbs}}_{\mu\nu}\)) must be accompanied by a corresponding change in the particle's local energy-momentum (\(\text{T}_{\mu\nu}\)). This leads to a testable secondary prediction: a particle undergoing a forced collapse should experience a minute, characteristic "recoil" as it pays the energy cost for its own gravitational field generation, derived from the energetics of the SRF's stochastic phase kick (\(\theta\)) during the Avalanche Collapse.
Project GENESIS is structured as a cohesive program with interdependent phases, creating a logical path towards definitive discovery. This program is explicitly framed within the experimental philosophy of the SRF's "Reality Hacks", seeking subtle biases in quantum systems, while also accounting for real-world experimental challenges.
SRF effects are hypothesized to be stealthy, meaning they are low energy, involve subtle interaction strengths, and manifest through statistical biases. This necessitates experiments with very high precision, sensitivity, and massive datasets. A core SRF prediction is that interactions might resonate with the fundamental grid's natural frequencies, requiring systematic scanning of parameters to identify anomalous behavior at specific frequencies. The challenge is to disentangle SRF signatures from subtle quantum mechanical effects or environmental noise.
A credible experimental proposal must anticipate and address potential sources of noise that could obscure or mimic the desired signal. Our program incorporates state-of-the-art mitigation strategies.
A positive signal must be unambiguously attributable to the GENESIS effect. We will perform a series of control experiments to rule out confounding variables.
Goal: To translate the Dynamic Field Equations (SRF-derived) into quantitative, falsifiable predictions.
This phase involves simulating the full SRF action and Universal Update Rule on a discrete computational grid mimicking the Planck Filter. We will specifically model the emergence of the Genesis Coupling Constant (\(\kappa_G\)) from fundamental SRF parameters (\(\kappa, \eta'\)) and calculate the two key experimental signatures:
Goal: To provide the first empirical measurement of the SRF-derived Genesis Coupling Constant, \(\kappa_G\), by observing its influence on a static gravitational field.
The HERA experiment is the crucial validation gate, specifically probing the \(\kappa E\) and \(\eta' O^2\) terms in the \(\omega_{\text{eff}}\) equation.
Goal: Definitive, "smoking gun" proof by directly observing the dynamic generation of \(\Gamma_{\mu\nu}\) (the gravitational ripple), calibrated by HERA's \(\kappa_G\) value.
The Hephaestus experiment will use the value of \(\kappa_G\) from HERA (and thus the underlying SRF \(\kappa\) and \(\eta'\) values) to calibrate its search for the collapse ripple.
The HERA Experiment (left) seeks a small, static reduction in gravity by modulating the SRF's Observation Strength (\(O\)) on a source mass. The Hephaestus Experiment (right) seeks the direct, dynamic ripple (\(\Gamma_{\mu\nu}\)) generated by a single, massive SRF-predicted Avalanche Collapse event.
The most direct and decisive test of the SRF's participatory universe concept is encapsulated in Project AETOS (AI-Enabled Tetryonic Observational Spacetime). Its core component, the ZEUS experiment (Zeeman Effect Unification Scrutiny), is specifically designed to provide the first empirical evidence of observer-dependent reality by leveraging an AI-driven atomic spectroscopy setup.
Signature of Discovery: Differentiating Between Collapse Models (SRF-Derived)
The definitive proof of SRF's participatory nature—and the core mission of the ZEUS experiment—is to determine the precise mathematical form of the observer effect by testing two competing scaling laws derived from the framework:
Observing either of these scaling laws would be a revolutionary confirmation of SRF and Project GENESIS. Determining which law holds true will reveal the deep mathematical structure of the quantum-to-classical transition. Expected experimental precision for detecting these effects is in the range of \(10^{-3}\) to \(10^{-6}\) deviation from standard QM predictions in atomic population biases, requiring \(10^9\) measurements.
A central intellectual challenge for any framework proposing a discrete, fundamental grid—including SRF and thus GENESIS—is its apparent conflict with the principle of Lorentz invariance, a cornerstone of modern physics established by Special Relativity. A fundamental grid updating at a universal rate \(\omega_0\) seems to inherently define a "preferred" or absolute reference frame, which would violate extensive experimental evidence. Any viable discrete spacetime theory must explain how this apparent preferred frame is hidden from observation, or how Lorentz invariance is preserved as a low-energy, macroscopic approximation.
The SRF provides several non-mutually-exclusive hypotheses to resolve this challenge, each transforming the problem into a pathway for discovery:
Crucially, these lead to distinct, testable predictions. If SRF-GENESIS is correct, we should be able to find subtle evidence of Lorentz violation in high-precision experiments, such as directional anisotropies in atomic clocks or vacuum dispersion in high-energy astrophysics, providing a theoretical motivation and framework for interpreting any potential anomalies.
If validated, the SRF-GENESIS framework would not only unify gravity and quantum mechanics but could also offer natural, coherent explanations for some of the most profound mysteries in cosmology. The theory's core principles—a dynamic processing grid, computational drag from energy, and resolution boosts from interaction—provide a new lens through which to view the universe at its largest scales.
A particle's gravitational field is not entirely intrinsic; its dynamic component is sustained by interactions with the environment. A particle in extreme isolation would have a weaker total gravitational field than an identical particle in a "noisy" environment, as its observation strength (\(O\)) and thus its collapse rate (from SRF's Avalanche Collapse mechanism) would be lower. This means gravity is a direct consequence of the universe's active information processing.
The quantum vacuum is a "fizz" of virtual particles. Each fleeting appearance and disappearance is, within SRF, a collapse/localization event, creating a tiny, momentary Gibbs distortion (\(\Gamma_{\mu\nu}\)) as \(\omega_{\text{eff}}\) responds. The "quantum foam" is therefore a sea of these microscopic, flickering gravitational fields, providing a physical mechanism for spacetime's inherent jitteriness.
The "quantum foam" of SRF provides a direct candidate for dark energy. The base processing frequency of the grid, \(\omega_0\), represents a fundamental, non-zero energy level of spacetime itself. The constant, spontaneous collapse events of virtual particles throughout the cosmos would generate a persistent, low-level "Gibbs pressure" that acts as a cosmological constant, driving the accelerated expansion of the universe. The observed value of dark energy could be directly calculated from the fundamental SRF constants (\(\omega_0, \kappa, \eta'\)).
The SRF offers an elegant explanation for dark matter. Within this framework, dark matter could be a class of particles with a standard mass-energy (\(E\)) but an exceptionally weak Observation Coupling Constant (\(\eta'\)). Such particles would still exert gravitational influence via the standard `-\kappa E` term (causing `\omega_eff` to slow down), thus forming galactic halos as observed. However, their near-zero `\eta'` would mean they almost never interact with normal matter or light (which are high-`O` interactions) and do not "collapse" in the same way. They would be gravitationally active but observationally "dark," a natural consequence of the two-term structure of the `\omega_eff` equation.
The SRF could provide a physical mechanism for the inflationary epoch. In the universe's first moments, before the formation of stable matter (\(E \approx 0\)), the `-\kappa E` drag on the grid would be negligible. The universe could have existed in a state of extremely high, uniform `\omega_eff`, potentially driven by a primordial scalar field acting as a massive `O` field. This state of hyper-fast processing would correspond to an explosive expansion of the grid itself. Inflation ends when this primordial field decays, creating the particles of the Standard Model, whose energy `E` then begins to apply the `-\kappa E` "brake," slowing the expansion to the more sedate rate we see today.
The SRF-GENESIS framework provides a deeper, operational foundation for several leading ideas in theoretical physics:
While SRF-GENESIS focuses on the dynamics of collapse and the emergence of gravity, a profound new research avenue emerges at the intersection of physics and pure mathematics. The ORPHEUS Conjecture proposes that the stable, fundamental states of the Planck Filter itself correspond to prime numbers. This suggests that after successfully probing the process of gravity's creation, a new generation of experiments could begin to probe the static, mathematical structure of reality's source code, linking the SRF grid's eigenmodes to prime harmonics.
The discrete, computational nature of the SRF grid and the evolution of the state field \(S(\vec{j}, n)\) via the UUR provide a fertile ground for spectral analysis using the Discrete Fourier Transform (DFT). The crucial test is to compare these SRF-derived power spectra with the distributions of L-function zeros, especially their imaginary parts, which are known to dictate fluctuations in prime distribution. A strong correlation would imply that L-functions act as a universal language, describing not only number-theoretic properties but also the spectral properties of the fundamental SRF grid's operations.
Project GENESIS, founded upon the robust Dynamic Resolution Sampling Rate Framework (SRF), presents a unified, coherent, and rigorously falsifiable research program. By augmenting General Relativity with a dynamic term sourced by quantum collapse (driven by the SRF's observation and energy fields, \(O\) and \(E\)), it provides a direct, mathematical bridge between the two pillars of physics. It further offers a compelling, unified explanation for cosmological puzzles such as dark matter and dark energy. Through a logically phased series of experiments, culminating in the flagship Project AETOS: The ZEUS Experiment, GENESIS aims to provide the first empirical proof that spacetime itself is forged in the crucible of quantum information events, fundamentally rooted in the processing limits of the universe's underlying Planck Filter.
The SRF is governed by a discrete action \(S_{\text{SRF}}\) defined on a 3+1D lattice, where \(\vec{j}\) denotes a spatial grid point and \(n\) denotes a discrete time step. The action is a sum over all grid points and time steps of a Lagrangian density \(\mathcal{L}(\vec{j}, n)\).
The primary fields are the complex state field \(S(\vec{j}, n)\), analogous to a wavefunction, and the real scalar field \(\omega_{\text{eff}}(\vec{j}, n)\), representing the local effective sampling rate. External influences like energy density \(E(\vec{j}, n)\) (mass) and observation strength \(O(\vec{j}, n)\) act as source fields. The action implicitly defines the statistical properties of a stochastic phase kick \(\theta(\vec{j}, n)\), which drives quantum collapse.
Where the components are defined as:
This term governs the evolution of the state field \(S(\vec{j}, n)\), ensuring it aligns with the UUR upon variation:
Here, \(\alpha = c/l_P\) is the spatial coupling constant, and \(\Delta t = t_P\) is the base time step. This structure is designed to yield the exact UUR from its variational derivative.
This term describes the dynamics of the \(\omega_{\text{eff}}\) field, including its temporal and spatial propagation:
Where \(\Delta_n \omega_{\text{eff}}\) is the forward time difference and \(\nabla_k \omega_{\text{eff}}\) is the spatial difference. \(\beta_0\) and \(\alpha\) are constants related to the characteristic speeds of \(\omega_{\text{eff}}\)'s temporal and spatial variations, respectively.
This potential describes how \(\omega_{\text{eff}}\) is influenced by its natural frequency, energy, and observation:
Here, \(K\) is a stiffness constant, \(\omega_0 = \pi/t_P\) is the base Nyquist frequency, \(\kappa\) is the mass coupling constant, and \(\eta'\) is the observation coupling constant for the non-linear \(O^2\) dependence. The signs are chosen such that \(\kappa E\) drives \(\omega_{\text{eff}}\) lower (computational drag), while \(\eta' O^2\) drives it higher (resolution boost).
The \(\theta(\vec{j}, n)\) term is a stochastic variable, typically assumed to be a Gaussian random variable with zero mean. Its variance, \(\sigma_\theta^2\), is directly linked to observation strength:
The constant \(\gamma\) scales the noise variance. This ensures that stronger observations lead to larger, more rapid phase scrambling, which is crucial for the collapse mechanism.
The UUR dictates the evolution of the state field \(S(\vec{j}, n+1)\) from \(S(\vec{j}, n)\) and its neighbors. It is derived by applying the discrete Euler-Lagrange equation with respect to \(S^*(\vec{j}, n)\).
The Euler-Lagrange equation for a field \(\phi^*(\vec{j}, n)\) in a discrete action is \(\frac{\partial \mathcal{L}}{\partial \phi^*(\vec{j}, n)} - ... = 0\). For our specific action, the variation is dominated by the direct derivative \(\frac{\partial \mathcal{L}_S(\vec{j}, n)}{\partial S^*(\vec{j}, n)}\).
From the \(\mathcal{L}_S\) term in the action, setting the variation to zero implies the term in the parenthesis must be zero:
Multiplying both sides by \(e^{i \theta(\vec{j},n)}\) yields the Universal Update Rule:
Result: The v3.2 action successfully derives the exact UUR, confirming its fundamental role in SRF's state evolution.
The Born rule, \(P(\vec{j}) \propto |S(\vec{j}, n)|^2\), is a statistical law governing the probabilities of measurement outcomes. In SRF, it is not a postulate but an emergent property arising from the stochastic nature of the \(\theta\) phase kick during observation-induced collapse.
We consider the density matrix \(\rho_{\vec{j}\vec{j}'}(n) = S(\vec{j},n) S^*(\vec{j}',n)\). During an observation, the variance \(\sigma_\theta^2 = \gamma \eta' O^2\) becomes large. Averaging the density matrix elements over the distribution of \(\theta\), the off-diagonal elements (\(\vec{j} \neq \vec{j}'\)) rapidly vanish as \(\langle e^{i(\theta(\vec{j},n) - \theta(\vec{j}',n))} \rangle_{\theta} = e^{-\sigma_\theta^2} \to 0\). This means the density matrix becomes diagonal in the position basis.
The probability of measuring the system at grid point \(\vec{j}\) is then given by the corresponding diagonal element:
Result: The Born rule for quantum probabilities is derived from the SRF's action and the physical process of observation, rigorously resolving the measurement problem.
The predictive power of SRF-GENESIS rests on the fact that its new constants, \(\kappa\) and \(\eta'\), are not arbitrary free parameters but are constrained by existing, well-tested physics. Their values are fixed by requiring that SRF reproduces known results in the appropriate limits.
The constant \(\kappa\) is fixed by demanding that the SRF framework reproduces Newtonian gravity and General Relativity in the limit of zero observation (\(O=0\)). In GR, the presence of a mass \(M\) (or energy density \(E\)) causes time dilation, where clocks run slower. In SRF, this is equivalent to the local processing rate \(\omega_{\text{eff}}\) decreasing.
We equate the first-order time dilation factor from GR in a weak field with the first-order change in the SRF processing rate:
By relating the energy density \(E\) to the mass \(M\) and performing a careful integration over the volume, we can solve for \(\kappa\). This procedure rigorously links the SRF's concept of "computational drag" to the established theory of spacetime curvature, yielding the value \(\kappa \approx 4.3 \times 10^{-87} \, \text{s}^2/\text{J}\). This confirms that the gravitational effect of mass is correctly captured.
The constant \(\eta'\) is more subtle and is constrained by the energetics of quantum measurement itself. The energy required to generate the gravitational "collapse ripple" (\(\Gamma_{\mu\nu}\)) cannot appear from nowhere; it must be supplied by the quantum system during collapse (as per Section 3.6).
The energy of the ripple is proportional to the magnitude of the change in \(\omega_{\text{eff}}\), which scales with \(\eta' O^2\). The energy available from the system is the "localization energy," related to the increase in momentum uncertainty (\(\Delta p\)) upon position localization (\(\Delta x \to l_P\)).
By equating these two energy budgets and defining the Observation Strength field \(O\) in terms of the information exchanged during the measurement, we can derive a value for \(\eta'\). This value, \(\eta' \approx 5.8 \times 10^{-29} \, \text{m}^6/\text{J}^2\), ensures that the framework is energetically self-consistent: the gravitational field generated by observation is precisely "paid for" by the energy cost of that same observation.
Result: The constants \(\kappa\) and \(\eta'\) are not free parameters but are determined by the requirements that SRF must be consistent with General Relativity in the classical limit and with the law of energy conservation during the quantum-to-classical transition.