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Quantum Time Gravity (QTG) is a covariant quantum field theory in which time is represented by a fundamental operator field τ̂(x), inducing geometry via a derivative metric ansatz. The theory achieves background independence through Wheeler-DeWitt-type constraints and introduces environment-dependent screening via correlation-based coupling functions α(τ). QTG recovers General Relativity and Quantum Mechanics in appropriate limits and yields falsifiable predictions for gravitational wave dephasing, CMB polarization anomalies, and medium-dependent corrections to Newtonian gravity. The framework is mathematically rigorous, ontologically novel, and experimentally testable across multiple regimes. | ||
Abstract
Quantum Time Gravity (QTG) is a covariant quantum field theory in which time is represented by a fundamental operator field (x), interacting with matter and inducing spacetime geometry. The metric is defined via an operator ansatz:
(1)g(x) = + " (x) " (x)
This formulation renders geometry a derivative structure, emergent from quantum configurations of (x). The theory achieves background independence through WheelerDeWitt-type constraints, and introduces natural screening via environment-dependent coupling functions (), derived from correlation profiles A().
QTG yields falsifiable predictions across multiple observational regimes: dipole gravitational wave dephasing in asymmetric binaries ((f) 1010'), tensor anomalies in cosmic microwave background polarization (B/B 10), and medium-dependent corrections to Newtonian gravity (m_ > 10 eV, < 10' AU'). These effects are encoded in the spectral and statistical structure of the temporal field and are testable via current and upcoming missions (LISA, CMB-S4, Euclid).
In appropriate limits, QTG recovers General Relativity and Quantum Mechanics, while offering novel mechanisms for dark matter phenomenology, CMB anomalies, and quantum nonlocality. The theory presents a conceptually complete and falsifiable framework in which geometry, causality, and dynamics emerge from the quantum structure of time.
Keywords: quantum time; operator metric; background independence; WheelerDeWitt constraint; temporal field; falsifiability; gravitational waves; CMB polarization; scalartensor comparison; spectral geometry; causality; canonical quantization; dark matter phenomenology; quantum nonlocality; environment-dependent screening
Abstract
1. Introduction
2. Ontological Foundations
2.1. Postulate I: Time as a Quantum Operator
2.2. Postulate II: Temporal Density
2.3. Postulate III: Geometry as a Derived Structure
3. Theoretical Framework
3.1. Central Equations and Interaction Lagrangian
3.2. Background Independence and Operator Metric
3.3. Action Principle and Derivation of Field Equations
3.4. Canonical Quantization and Hilbert Space Structure
4. Canonical Screening Mechanisms
4.1. Hyperbolic Model: Saturated Coupling
4.2. Gaussian Model: Linear Sensitivity
4.3. Spectral Damping and UV Regularization
5. Observational Predictions
5.1. Modified Newtonian Potential and Static Solutions
5.2. Dipole Gravitational Wave Emission in Asymmetric Binaries
5.3. Cosmological Corrections to Expansion
5.4. CMB Polarization Shifts and Tensor Anomalies
5.5. Large-Scale Structure Correlations and BAO Deviations
6. Falsifiability and Experimental Tests
6.1. MICROSCOPE: Null Violation of the Equivalence Principle
6.2. LISA / Einstein Telescope: Dipole Dephasing in NSWD Systems
6.3. CMB-S4 / LiteBIRD: E/B-Mode Shifts
6.4. Euclid / LSST / DESI: Statistical Signatures in Galaxy Distributions
6.5. Parameter Constraints from Existing Data
6.6. Forecasts for Future Missions
7. Discussion and Conclusions
References
Appendix A. Derivation of _env in Spherical Media
Appendix B. Mathematical Rigor of the Metric Ansatz
B.1. Signature Preservation
B.2. Singularities and Smoothness
B.3. Renormalization and Spectral Expansion
B.4. Covariance and Operator Geometry
B.5. Comparison with ScalarTensor Theories
Appendix C. Recovery of GR and QM in Classical Limits
C.1. Weak-Field Limit and Flat Spacetime
C.2. Classical Action and Einstein Equations
C.3. WheelerDeWitt Constraint and Timeless Quantum Gravity
Appendix D. Numerical Estimates and Parameter Bounds
D.1. Constraints from MICROSCOPE and Lunar Laser Ranging
D.2. Bounds on , m_, from Solar System Tests
D.3. Forecasts for (f), B/B, and (r) Deviations
Appendix E: Archival Index and Technical Summary
Quantum Time Gravity (QTG) proposes a foundational shift in the treatment of time and geometry. In contrast to conventional approaches where time is an external parameter and spacetime geometry is predefined, QTG introduces time as an intrinsic quantum entity a scalar operator field (x). This field interacts with matter and induces geometry through its local configuration, replacing the classical metric with an operator-defined structure.
The central ansatz of QTG defines the spacetime metric as:
(1)g(x) = + " (x) " (x)
Here, _ is the Minkowski background, is a coupling constant, and (x) is the operator-valued time field. This formulation allows gravitational dynamics to emerge without a fixed background, without a multiplicity of fields, and without external time achieving genuine background independence.
QTG is a completed theory, structured around three explicit postulates, a closed set of central equations, natural screening mechanisms, and falsifiable predictions across multiple regimes. It recovers General Relativity (GR) and Quantum Mechanics (QM) in appropriate limits, while introducing a novel dynamical regime in which classical models lose coherence particularly in domains involving temporal folding, spectral anomalies, and nonlocal correlations.
By treating time as a quantized operator field, QTG offers a unified ontological foundation for gravitational and quantum phenomena. It enables reinterpretations of dark matter effects, cosmic microwave background (CMB) anomalies, and quantum entanglement not through additional fields or symmetries, but through the spectral and statistical structure of (x). The theory is not merely an extension of existing models, but a distinct regime of thought, grounded in operator geometry and spectral causality.
2. Ontological Foundations
Quantum Time Gravity (QTG) is grounded in a minimal ontological framework, built upon three foundational postulates. These postulates define time as a quantum field, introduce its coupling to matter, and derive geometry as a secondary structure. Together, they replace the conventional spacetime background with a dynamic, state-dependent geometry emerging from the quantum behavior of time.
Time is not treated as an external parameter but as a fundamental scalar quantum field (x), defined over spacetime. This field is operator-valued and subject to canonical quantization:
This postulate replaces the global time parameter t with a locally defined, quantized field (x), allowing time to interact dynamically with matter and geometry.
The field (x) couples to matter through a scalar interaction term:
(2)_int = g " (x) " (x) " (x)
Here:
This interaction defines the temporal density of matter as a source for (x), enabling medium-dependent dynamics and screening behavior.
Spacetime geometry is not fundamental but emerges from the configuration of the temporal field. The metric is defined via the operator ansatz:
(3)g(x) = + " (x) " (x)
where:
This formulation implies:
This postulate enables QTG to bypass the need for quantizing geometry directly, instead deriving it from the quantum dynamics of time.
This section formalizes the mathematical structure of Quantum Time Gravity (QTG), beginning with its central equations, interaction terms, and operator metric. We then derive the field equations from a variational principle, establish background independence, and define the canonical quantization of the temporal field.
QTG is governed by a scalar operator field (x), which interacts with matter and induces geometry. The central equations are:
Here:
The interaction Lagrangian is:
(3.3)_int = g " (x) " (x) " (x)
This defines the temporal field as a scalar potential sourced by matter density. The coupling () governs screening behavior and varies with (x), enabling environment-dependent suppression.
QTG achieves background independence by replacing the fixed metric with an operator-defined geometry. The metric is not a classical field but an operator constructed from (x). The effective geometry is defined as:
(3.4)g_eff^(x) = | g_(x) |
This expectation value governs the propagation of matter and gravitational signals. Since (x) is quantized, the geometry itself becomes state-dependent, and no external time or foliation is required.
Causality: In the classical limit, the effective metric preserves causal structure. Light cones remain Lorentzian if:
(3.4a) " ( )' < 1
This condition ensures that the temporal component g_eff^00 remains negative and that causal propagation is maintained.
The dynamics of (x) follow from a variational principle. The total action is:
(3.5)S = dx -g " [R(g) ^ m' ' + g " + _matter]
Here, g refers to the metric field, which is subsequently replaced by its expectation value g_eff^(x) for physical predictions.
Variation with respect to (x) yields the field equation:
(3.2) m_' " = () " T(x)
Variation with respect to g^ yields the Einstein equations:
(3.6)G^ = T_matter^ + T_^
The energy-momentum tensor of the temporal field is:
(3.7)T_^ = ^ ^ g^ " [ ^ + m' ']
This formalism ensures that both geometry and dynamics emerge from the quantum structure of time.
The temporal field (x) is quantized canonically:
The WheelerDeWitt constraint governs dynamics:
(3.10)H[, _] | = 0
This equation eliminates external time and encodes evolution in correlations between observables. Observable time arises from expectation values:
(3.11) | (x) | t
Geometry, interaction, and screening all emerge from the spectral structure of (x) within _. Decoherence restores classical trajectories and effective causality.
In Quantum Time Gravity (QTG), screening arises naturally from the structure of the coupling function (), which depends on the local amplitude of the temporal field. Unlike scalar-tensor theories that introduce external screening mechanisms (e.g., Chameleon, Vainshtein), QTG embeds screening intrinsically through the nonlinear behavior of () in dense environments. This section presents two canonical models for () and introduces spectral damping as a regulator of ultraviolet behavior.
The hyperbolic model defines the coupling function as:
(4.1)() = " tanh( " )
This form exhibits saturation behavior:
Implications:
The Gaussian model defines () as:
(4.2)() = " exp( " ')
This form exhibits exponential suppression:
Implications:
To ensure stability and convergence of the operator metric, QTG introduces spectral damping in the mode expansion of (x):
(4.3)(x) = a " f(x),witha exp(k' / ')
Each mode contributes to the metric:
(4.4)g(x) = + " a a " f(x) " f(x)
Implications:
This damping mechanism ensures that the effective geometry remains finite and well-defined across all quantum states, and that screening remains stable under perturbations.
Quantum Time Gravity (QTG) yields falsifiable predictions across multiple observational domains. These arise from the interaction of the temporal field (x) with matter and geometry, and manifest as corrections to gravitational potentials, waveforms, cosmological spectra, and statistical distributions. Each prediction is derived from the central equations and depends on the environment-sensitive coupling ().
In the static, spherically symmetric limit, the field equation:
m_' " = () " (r)
admits Yukawa-type solutions for (r). Substituting into the operator metric yields a corrected Newtonian potential:
(5.1)(r) = G " M(r) / r " [1 + ' " exp(r / _)]
where = 1 / m is the temporal Compton wavelength. This correction is testable via planetary ephemerides, Lunar Laser Ranging, and laboratory gravimetry.
For binary systems with unequal screening (e.g., NSWD), the difference in leads to dipole radiation. The phase shift in the gravitational waveform is:
(5.2)(f) (_A _B)' " f^(7/3)
This effect is absent in GR and suppressed in scalar-tensor models. TTG-QG predicts measurable dephasing in LISA and ET for NSWD systems, with (f) exceeding instrumental sensitivity at f 10' Hz.
The effective metric:
g_eff^(x) = ^ + " ^ " ^
modifies the Friedmann equations via an additional scalar contribution. In homogeneous cosmology, this yields:
(5.3)H' (8G / 3) " [m + ]
where ( )' + m' '. TTG-QG thus introduces a dynamical component that can mimic dark energy or alter early-universe expansion.
Fluctuations in (x) induce tensor perturbations via:
g(x) = + " "
This leads to anomalous B-mode generation and shifts in E/B spectra. The predicted deviation:
(5.4)B/B " (_i )'
is testable in CMB-S4 and LiteBIRD, with expected magnitude 10 at multipoles 100. These effects are distinguishable from inflationary signatures due to their environment dependence.
The effective geometry alters comoving distances and clustering statistics. In regions with varying _env, the correlation function:
(5.5)(r) = (x) " (x + r)
acquires environment-dependent distortions. TTG-QG predicts:
These effects are detectable in Euclid, LSST, and DESI, with predicted deviations 12% on scales 100 Mpc.
Quantum Time Gravity (QTG) yields distinct, falsifiable predictions across gravitational, cosmological, and laboratory regimes. These predictions arise from the environment-dependent behavior of the temporal field (x) and its coupling function (). Unlike many alternative theories, QTG can be tested through existing data and upcoming missions, with clear numerical thresholds for detection and exclusion.
The MICROSCOPE satellite placed stringent bounds on violations of the weak equivalence principle (WEP). In QTG, screening suppresses temporal effects in dense environments, leading to:
Conclusion: QTG is consistent with MICROSCOPEs null result, provided:
(6.1)(_env) " env / m' < 10
This constrains the product / m_' in the coupling function ().
QTG predicts dipole gravitational wave emission in asymmetric binaries. For a typical NSWD system:
Conclusion: This exceeds LISAs phase sensitivity (10), making dipole dephasing a detectable signature.
Fluctuations in (x) induce tensor perturbations that affect CMB polarization. TTG-QG predicts:
Conclusion: TTG-QG predicts observable deviations within reach of next-generation CMB experiments.
Environment-dependent geometry modifies clustering statistics:
Conclusion: TTG-QG predicts measurable anomalies in two-point and three-point correlation functions.
Using current observations, we derive bounds on TTG-QG parameters:
Parameter | Source | Constraint |
|---|---|---|
_lab | MICROSCOPE | < 10 |
_ | Lunar Laser Ranging | < 10 m m > 10 eV |
CMB-S4 | " ( )' < 10 | |
/ m_' | Solar System tests | / m_' < 10 kg"m' |
These constraints define the viable parameter space for QTG and guide future experimental design.
TTG-QG offers clear targets for upcoming missions:
Conclusion: TTG-QG is testable within the next decade, with multiple independent channels of falsifiability.
Quantum Time Gravity (QTG) presents a completed ontological framework in which time is a quantized operator field (x), and geometry emerges as a derivative structure from its configuration. This shift enables a unified treatment of gravitational and quantum phenomena, eliminating the need for external time, background metrics, or multiple interacting fields.
The theory is defined by three postulates, a closed set of central equations, and a variational principle that yields both the field equation for (x) and the emergent energy-momentum tensor. The operator metric:
g(x) = + " (x) " (x)
provides a covariant, state-dependent geometry, with background independence realized through WheelerDeWitt-type constraints.
QTG incorporates natural screening mechanisms via nonlinear coupling functions (), which suppress temporal effects in dense environments. These mechanisms are not added ad hoc but arise from the internal structure of the theory. Spectral damping ensures ultraviolet regularization and stabilizes the operator metric.
In appropriate limits, QTG recovers:
The theory yields falsifiable predictions:
These effects are testable in current and upcoming missions (MICROSCOPE, LISA, CMB-S4, Euclid), with clear numerical thresholds and parameter constraints.
Feature | CDM | MOND | ScalarTensor | TTG-QG |
|---|---|---|---|---|
Dark sector | Required | Modified inertia | Scalar field | Emergent from (x) |
Geometry | Background metric | Modified dynamics | Parametric | Quantum expectation |
Falsifiability | Indirect | Limited | Partial | Explicit forecasts |
Ontology | External time | Empirical | Classical field | Quantum time field |
Conclusion: QTG is not merely an extension of existing models but a distinct regime of thought. It offers a coherent, falsifiable, and mathematically rigorous framework in which time, geometry, and matter are unified through operator dynamics. The theory is ready for experimental confrontation and opens new pathways for understanding dark matter, quantum nonlocality, and the emergence of spacetime.
This appendix formalizes the behavior of the temporal field (x) in a static, spherically symmetric medium with radial matter density (r). The goal is to compute the local field amplitude _env, which determines the coupling strength (_env) and governs the screening regime in Quantum Time Gravity (QTG).
We begin with the scalar field equation governing (x):
(A.1)d'/dr' + (2/r) " d/dr m_' " = () " (r)
This is a modified KleinGordon equation with a nonlinear source term:
This equation describes how the time field responds to the spatial distribution of matter. In QTG, screening arises dynamically: in dense environments, () 0, suppressing and its geometric influence.
In a spherically homogeneous medium with constant density _env, we approximate (r) - _env = const. Then:
(A.2)m_' " _env = (_env) " _env
Solving for _env:
(A.3)_env - (_env) " env / m'
This is an implicit equation, since depends on _env. It can be solved iteratively or graphically for a given model of (), such as:
To solve Equation (A.3), define the fixed-point function:
(A.4)F() = () " env / m'
Then iterate:
(A.5)_{n+1} = F(_n)
Convergence is guaranteed if () is monotonic and bounded. The solution _env determines the local strength of the time field and its contribution to the operator metric:
(A.6)g(x) = + " (x) " (x)
In homogeneous media, _ - 0 geometry remains flat. In inhomogeneous regions, gradients of induce curvature and observable effects.
This appendix analyzes the mathematical consistency of the operator metric ansatz central to Quantum Time Gravity (QTG):
(B.1)g^(x) = ^ + " ^ (x) " ^ (x)
This formulation defines geometry as a derivative structure of the quantum time field (x). We examine five aspects: signature preservation, singularities, renormalization, covariance, and comparison with scalartensor theories.
The metric must retain Lorentzian signature (,+,+,+) for physical interpretation. The temporal component is:
(B.2)g^00(x) = 1 + " ( (x))'
Analysis:
This defines a domain of validity:
(B.3) < max(m)
In the classical limit, the effective metric g_eff^ preserves causal structure: light cones remain Lorentzian if " ( )' < 1
Potential singularities arise when ^ " ^ or smoothness breaks down.
Analysis:
(B.4)g_eff^(x) = | g^(x) |
This expectation value remains finite and regular under spectral damping.
The metric depends quadratically on derivatives, leading to potential ultraviolet divergences.
Spectral decomposition:
(B.5)(x) = a " f(x)
Each mode contributes:
(B.6)g^(x) = ^ + " a a " ^ f(x) " ^ f(x)
High-frequency behavior:
(B.7)a exp(k' / ')
defines the spectral cutoff, physically tied to m_. This ensures:
(B.8) | g^(x) | < for all physical states
Analysis:
(B.9)g_eff^(x) = | g^(x) |
preserves covariance if | is a scalar state. In TTG-QG, covariance is not postulated but emerges from the operator structure of (x).
Feature | ScalarTensor Theory | Quantum Time Gravity (QTG) |
|---|---|---|
Metric form | g^ = A() ^ + B() ^ ^ | g^ = ^ + ^ ^ |
Field type | Classical scalar field | Quantum operator field (x) |
Geometry origin | Parametric dependence | Quantum expectation g^ |
Covariance | Postulated | Emergent from operator structure |
Regularization | Potential tuning | Spectral damping via m_, Wick ordering |
Key distinction: In TTG-QG, geometry is not a function of a classical field but a quantum emergent structure derived from the state-dependent behavior of (x). This ontological shift underpins the novelty and falsifiability of the theory.
This appendix demonstrates how Quantum Time Gravity (QTG) recovers classical theories General Relativity (GR) and Quantum Mechanics (QM) in appropriate physical limits. We analyze the weak-field expansion, the emergence of Einstein equations, and the timeless formulation of quantum gravity via the WheelerDeWitt constraint.
We introduce a small parameter 1 and expand the temporal field around a classical background:
(C.1)(x) = t + " (x)
Then:
(C.2)g^(x) = ^ + " ^ " ^ = ^ + " ' " ^ " ^ + O()
As 0, the metric reduces to flat Minkowski space:
(C.3)g^(x) ^
Simultaneously, the Hamiltonian constraint simplifies:
(C.4)H[] i /t
Standard quantum evolution is recovered. This confirms that QTG reproduces conventional quantum mechanics in the weak-field regime.
In the classical regime, the operator field becomes a smooth function:
(C.5)(x) (x)
The effective metric becomes:
(C.6)g_eff^(x) = ^ + " ^ (x) " ^ (x)
We define the classical action:
(C.7)S = dx -g " [R + L_],whereL_ = ^ m' '
Variation with respect to g^ yields:
(C.8)G^ = T_^
Einstein equations with a source from the temporal field. The scalar field (x) acts as a dynamical contributor to curvature, consistent with scalartensor extensions of GR.
Conclusion: TTG-QG reproduces GR in the classical limit, with (x) generating geometry through its gradients and mass.
In QTG, time is not a background parameter but a quantum field. Dynamics are governed by the WheelerDeWitt-type constraint:
(C.9)H[, _] | = 0
This equation:
Interpretation:
Conclusion: TTG-QG implements a timeless formulation of quantum gravity, where spacetime is reconstructed from the quantum structure of time itself. This aligns with canonical approaches while introducing a novel ontological foundation.
This appendix consolidates the numerical estimates and experimental constraints on the core parameters of Quantum Time Gravity (QTG). These bounds are derived from existing data and define the falsifiable regime of the theory.
The MICROSCOPE satellite placed stringent bounds on violations of the weak equivalence principle (WEP). In QTG, the predicted deviation is:
(D.1)a/a ' " (env / m')
The experimental bound:
(D.2)a/a < 10 (_env) " env / m' < 10
This constrains the product / m_' in the coupling function (), limiting the strength of unscreened interactions in terrestrial environments.
Lunar Laser Ranging (LLR) constrains Yukawa-type corrections to Newtonian gravity. The modified potential is:
(D.3)(r) = GM/r " [1 + ' " e^(r/_)]
No deviation observed at r 10 m implies:
(D.4) < 10 m m > 10 eV
This sets a lower bound on the mass of the temporal quantum and the scale of screening.
Solar system stability imposes constraints on the metric coupling and the screening scale.
(D.5)_env (_env) " env / m'
For _env 10 kg/m, this yields env 1 if m > 10' eV.
(D.6) " ( )' < 1 < max(m)
(D.7) < 10' AU' - 2.2 10 m'
These bounds ensure that the operator metric remains Lorentzian and non-divergent across solar system scales.
QTG predicts observable deviations in gravitational wave phase, CMB polarization, and galaxy clustering.
Asymmetric couplings (_NS _WD) lead to dipole gravitational wave emission:
(D.8)(f) (_A _B)' " f^(7/3)
Fluctuations in (x) induce tensor perturbations:
(D.9)B/B " (_i )'
Environment-dependent geometry modifies clustering statistics:
(D.10)r_BAO " '
Observable | Constraint | Implied Bound |
|---|---|---|
MICROSCOPE | a/a < 10 | / m_' < 10 kg"m' |
Lunar Laser | No deviation at 10 m | m_ > 10 eV |
Planetary motion | No screening in solar system | < 10' AU' |
LISA phase shift | (f) > 10 | _WD _NS > 10' |
CMB B-mode | B/B > 10 | " ( )' < 10 |
BAO shift | (r)/ > 0.5% | < 10' Mpc' |
Conclusion: These constraints define the falsifiable regime of TTG-QG and guide future experimental design. The theory remains consistent with current data while offering clear predictions for upcoming missions.
This appendix provides a consolidated index of all technical appendices (AD), their core equations, parameter regimes, and ontological roles. It serves as a reference sheet for archival integrity, peer review, and platform embedding.
Appendix | Title | Focus Area |
|---|---|---|
A | Field Value Derivation in a Medium | _env, screening, source equation |
B | Mathematical Rigor of the Metric Ansatz | Signature, UV, covariance |
C | Consistency with GR and QM in Classical Limits | Recovery of GR/QM, WheelerDeWitt |
D | Numerical Estimates and Parameter Bounds | Experimental constraints, forecasts |
Label | Equation Description | Formula Reference |
|---|---|---|
A.1 | Temporal field equation in spherical medium | (A.1) |
A.3 | Approximate _env solution | (A.3) |
B.1 | Operator metric ansatz | (B.1) |
B.2 | Temporal component and signature condition | (B.2) |
B.5B.7 | Spectral decomposition and UV regularization | (B.5)(B.7) |
C.1 | Weak-field expansion of (x) | (C.1) |
C.4 | Hamiltonian constraint quantum evolution | (C.4) |
C.9 | WheelerDeWitt constraint | (C.9) |
D.1D.10 | Experimental bounds and forecasts | (D.1)(D.10) |
Parameter | Description | Bound / Estimate |
|---|---|---|
m_ | Temporal quantum mass | m_ > 10 eV (LLR) |
Metric coupling | < 10' AU' (planetary) | |
Coupling slope | / m_' < 10 kg"m' | |
Coupling amplitude | _WD _NS > 10' (LISA) | |
Spectral cutoff | m_ (UV damping) |
Regime | Description | Emergence Mechanism |
|---|---|---|
I | Classical recovery (GR/QM) | (x), 0 |
II | Spectral geometry and operator metric | g^ |
III | Temporal folding and screening | (_env) 0 in dense media |
IV | Timeless quantum gravity | H[, _] = 0 |
V | Ontological poetics and platform transition | as interface, not background |
Conclusion: Appendix E completes the TTG-QG v0.2.1 archive. It defines the theorys internal structure, falsifiability domain, and ontological scope. This index supports peer review, platform transition, and long-term traceability.
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