All Equations Derived in Timeline-Relative Quantum Collapse (TRQC):
Canonical Derivation and Mathematical Foundation (CDMF) Paper found
HERE
1
Collapse Field
Math
\[\Psi_c(x^{\mu})\]
Primary field whose dynamics govern collapse behavior in TRQC. (CDMF Section 1.1 pg. 6)
Glossary
\(\Psi_c\)
Collapse field over spacetime \(x^{\mu}\); the main dynamical variable of TRQC.
\(x^{\mu}\)
Spacetime coordinate with index \(\mu\) running over time and spatial components.
2
Observer Field
Math
\[O^{\mu}(x^{\nu}), \quad O^{\mu}O_{\mu} = 1\]
Unit timelike vector field encoding observer flow and orientation. (CDMF Section 1.2 pg. 6)
Glossary
\(O^{\mu}\)
Observer vector field; unit timelike, meaning \(O^{\mu}O_{\mu}=1\).
\(x^{\nu}\)
Spacetime coordinates with index \(\nu\).
3
Collapse Time Function
Math
\[\tau(x^{\mu}), \quad \frac{d\tau}{ds} > 0\]
Scalar “time terrain” ordering collapse layers in TRQC. (CDMF Section 1.3 pg. 7)
Glossary
\(\tau\)
Scalar field representing structural time in TRQC.
\(\frac{d\tau}{ds} > 0\)
Monotonic increase along proper time \(s\), ensuring forward progression in structural time.
4
Canonical TRQC Lagrangian
Math
\[
\mathcal{L}_{\text{TRQC}}=\tfrac12 \nabla^\mu\Psi_c\nabla_\mu\Psi_c - V(\Psi_c,\tau)
+ \lambda(O^\mu O_\mu - 1) + \gamma O^\mu \nabla_\mu\Psi_c
+ \beta R^\tau_{\mu\nu} + \tfrac12\kappa R[g^{\text{eff}}_{\mu\nu}]
\]
Master density combining collapse, observer, terrain, and effective curvature terms. (CDMF Section 2.2 pg. 13)
Glossary
\(\mathcal{L}_{\text{TRQC}}\)
Lagrangian density for the TRQC framework.
\(\nabla^\mu\)
Covariant derivative.
\(V(\Psi_c,\tau)\)
Collapse potential depending on \(\Psi_c\) and \(\tau\).
\(\lambda, \gamma, \beta, \kappa\)
Coupling constants for constraints, damping, terrain curvature, and effective curvature.
\(R^\tau_{\mu\nu}\)
Terrain-time curvature tensor.
\(R[g^{\text{eff}}_{\mu\nu}]\)
Ricci scalar of the effective metric \(g^{\text{eff}}_{\mu\nu}\).
5
Collapse Potential
Math
\[V(\Psi_c,\tau) = \frac{\alpha}{4}(\Psi_c^2 - \Psi_0^2)^2 + S(\tau)\Psi_c\]
Bistable collapse potential with terrain-induced bias.
Glossary
\(V(\Psi_c,\tau)\)
Collapse potential governing the energetically preferred values of \(\Psi_c\); includes self-interaction and terrain bias.
\(\alpha\)
Collapse stiffness constant; controls the curvature of the bistable potential wells around \(\pm \Psi_0\).
\(\Psi_0\)
Preferred collapse amplitude; defines the minima of the symmetric double-well structure.
\(S(\tau)\)
Terrain-induced bias term; modulates the potential landscape based on the local value of the terrain field \(\tau(x)\).
6
Euler–Lagrange Equation
Math
\[
- \Box \Psi_c + \alpha(\Psi_c^2 - \Psi_0^2)\Psi_c + S(\tau) - \gamma \nabla_\mu O^\mu = 0
\]
Canonical Euler–Lagrange equation for the collapse field \(\Psi_c\), derived from the TRQC Lagrangian under metric signature \(({-}{+}{+}{+})\).
Glossary
\(\Box\)
Covariant d’Alembertian operator: \(\Box = \nabla^\mu \nabla_\mu\)
\(\alpha(\Psi_c^2 - \Psi_0^2)\Psi_c\)
Nonlinear restoring force from bistable collapse potential; pushes \(\Psi_c\) toward ±\(\Psi_0\) minima depending on terrain stability
\(S(\tau)\)
Scalar collapse bias term derived from the terrain time field \(\tau(x)\); encodes asymmetric collapse preference due to stack position
\(\gamma\)
Observer–collapse coupling constant
\(\nabla_\mu O^\mu\)
Divergence of the observer field \(O^\mu\); positive divergence damps collapse, negative divergence amplifies it
7
Collapse PDE (3+1D Form)
Math
\[
\frac{\partial^2 \Psi_c}{\partial t^2}
- \nabla^2 \Psi_c
+ \alpha(\Psi_c^2 - \Psi_0^2)\Psi_c
+ s_0 \tau_0 = 0
\]
Collapse field equation in flat 3+1D spacetime, derived from the canonical Euler–Lagrange equation under comoving observer and uniform terrain assumptions.
Glossary
\(\Psi_c\)
Collapse field over spacetime; primary dynamical variable in TRQC projection dynamics.
\(\nabla^2 \Psi_c\)
Spatial Laplacian of \(\Psi_c\), representing wave-like diffusion in three spatial dimensions.
\(\alpha(\Psi_c^2 - \Psi_0^2)\Psi_c\)
Nonlinear restoring force from the bistable collapse potential; stabilizes the field near \(\pm \Psi_0\).
\(s_0 \tau_0\)
Constant collapse bias from the terrain field \(\tau(x)\), under the assumption that \(\tau = \tau_0 = \text{const}\).
\(\frac{\partial^2}{\partial t^2}, \nabla^2\)
Second-order time derivative and spatial Laplacian in flat Minkowski spacetime (\(\eta_{\mu\nu} = \text{diag}(-1, +1, +1, +1)\)).
8
Collapse Curvature (Scalar)
Math
\[R_\tau = \frac{\partial^2 \tau}{\partial x^2}\]
Scalar curvature associated with terrain-time.
Glossary
\(R_\tau\)
Scalar curvature derived from \(\tau\).
\(\tau\)
Terrain-time scalar field.
9
Projection Operator
Math
\[g^{\text{eff}}_{\mu\nu} = \Pi_{\text{QEC}}[\Psi_c, O^\mu, \tau]\]
Defines effective metric from projection-compatibility conditions.
Glossary
\(g^{\text{eff}}_{\mu\nu}\)
Effective projection-compatible metric.
\(\Pi_{\text{QEC}}\)
Projection operator restricted to the QEC domain.
\(\Psi_c, O^\mu, \tau\)
Collapse field, observer vector, and terrain-time scalar.
10
Effective Metric
Math
\[g^{\text{eff}}_{\mu\nu} = \eta_{\mu\nu} + \alpha_1 \nabla_\mu\Psi_c\nabla_\nu\Psi_c + \alpha_2\nabla_\mu\tau\nabla_\nu\tau\]
Projection-compatible metric derived from collapse field gradients and terrain structure; governs emergent curvature in TRQC.
Glossary
\(g^{\text{eff}}_{\mu\nu}\)
Emergent metric defined by collapse and terrain structure; determines projection curvature and replaces background geometry in TRQC.
\(\eta_{\mu\nu}\)
Flat Minkowski background metric; used as seed for projection deformation.
\(\alpha_1, \alpha_2\)
Collapse and terrain coupling constants; control how gradients of \(\Psi_c\) and \(\tau\) modify the effective metric.
11
Termination Surface
Math
\[\Sigma^{\tau}_{(O)}: \ \nabla\Psi_c \to 0 \ \ \text{or} \ \ |\partial^2\tau| > \theta_{\text{shear}}\]
Surface where projection halts or terrain shear becomes too large.
Glossary
\(\Sigma^{\tau}_{(O)}\)
Termination surface at structural time \(\tau\) for observer \(O\).
\(\nabla\Psi_c\)
Gradient of the collapse field; approaching zero indicates projection failure.
\(|\partial^2\tau|\)
Magnitude of terrain-time curvature.
\(\theta_{\text{shear}}\)
Threshold shear parameter for terrain-time.
12
Collapse Horizon Condition
Math
\[\nabla\Psi_c \to 0 \ \Rightarrow \ \text{projection fails}\]
Defines projection failure via vanishing collapse field gradient.
Glossary
\(\nabla\Psi_c\)
Gradient of the collapse field.
Projection failure
Loss of projection compatibility when the collapse field gradient vanishes.
13
Collapse Redshift Mapping
Math
\[\tau(z) = \tau_0 (1 + z)^{-p}\]
Links cosmological redshift to structural time depth in TRQC; predicts redshift flattening as collapse resolves.
Glossary
\(\tau(z)\)
Collapse time at redshift \(z\); represents structural depth in the τ-stack, not coordinate age.
\(\tau_0\)
Local structural time at the projection surface (\(z = 0\)); reference point for collapse depth.
\(p\)
Collapse curvature exponent; determines how steeply projection depth varies with redshift. Empirically fit (typically \(p \approx 1\)).
14
Functional Variations
Math
\[\frac{\delta g^{\text{eff}}_{\mu\nu}}{\delta \Psi_c},\quad \frac{\delta R}{\delta \Psi_c}\]
Variational derivatives of metric and curvature with respect to the collapse field.
Glossary
\(\delta g^{\text{eff}}_{\mu\nu} / \delta \Psi_c\)
Change in effective metric due to variation in collapse field.
\(\delta R / \delta \Psi_c\)
Change in scalar curvature due to variation in collapse field.
15
Hamiltonian Density
Math
\[\mathcal{H} = \tfrac12 (\partial_t\Psi_c)^2 + \tfrac12 (\nabla\Psi_c)^2 + V(\Psi_c,\tau) - \gamma O^\mu \nabla_\mu\Psi_c\]
Energy density of the collapse field including observer coupling.
Glossary
\(\mathcal{H}\)
Hamiltonian (energy) density.
\(\partial_t\Psi_c\)
Time derivative of the collapse field.
\(\nabla\Psi_c\)
Spatial gradient of the collapse field.
\(V(\Psi_c,\tau)\)
Collapse potential.
\(\gamma O^\mu \nabla_\mu\Psi_c\)
Damping term from observer field coupling.
16
Projection-Aligned Frame
Math
\[\chi^\mu(x) = O^\mu(x)\]
Coordinate frame aligned with the observer vector field.
Glossary
\(\chi^\mu(x)\)
Projection-aligned coordinate basis vector.
\(O^\mu(x)\)
Observer vector field at spacetime point \(x\).
17
Smoothed Projection Operator
Math
\[\psi(x) = \int_{\mathcal{N}_x} \Psi_c(x')\, \eta_\varepsilon(\tau(x') - \tau(x))\, \eta_\varepsilon(\chi^\mu(x') - \chi^\mu(x))\, d^3x'\]
Neighborhood-averaged projection of the collapse field.
Glossary
\(\psi(x)\)
Smeared projection function at point \(x\).
\(\mathcal{N}_x\)
Neighborhood region around \(x\).
\(\eta_\varepsilon\)
Smoothing kernel with width \(\varepsilon\).
\(\tau, \chi^\mu\)
Terrain-time scalar and projection-aligned coordinates.
18
Collapse Energy Density
Math
\[\rho(x) = \nabla_\mu\Psi_c \nabla^\mu\Psi_c\]
Local energy density of the collapse field.
Glossary
\(\rho(x)\)
Energy density at point \(x\).
\(\nabla_\mu\Psi_c \nabla^\mu\Psi_c\)
Contraction of the collapse field gradient with itself.
19
Born Deviation Functional
Math
\[\Delta_i = \int_{S_i} (|\psi(x)|^2 - \rho(x))\, d^3x\]
Quantifies terrain-induced deviations from Hilbert-space probability in decoherence-permitted regions.
Glossary
\(\Delta_i\)
Quantifies deviation from Born rule predictions in region \(S_i\); zero when Hilbert-space probability matches collapse energy density.
\(|\psi(x)|^2\)
Squared amplitude of the terrain-smoothed observable field; equivalent to conventional Born probability density under decoherence stability.
\(\rho(x)\)
Collapse energy density: \(\rho(x) = \nabla_\mu \Psi_c \nabla^\mu \Psi_c\); measures structural decoherence support in the terrain.
20
Terrain Curvature Tensor
Math
\[R^\tau_{\mu\nu} = \nabla_\mu \nabla_\nu \tau - \tfrac12 \nabla_\mu\tau\, \nabla_\nu\log(\nabla^\sigma\tau\nabla_\sigma\tau)\]
Second-derivative structure of terrain-time scalar.
Glossary
\(R^\tau_{\mu\nu}\)
Terrain-time curvature tensor.
\(\tau\)
Terrain-time scalar field.
\(\nabla_\mu, \nabla_\nu\)
Covariant derivatives with respect to coordinates \(\mu, \nu\).
21
Collapse Path Integral
Math
\[\int \mathcal{D}\Psi_c\, \mathcal{D}O^\mu\, \mathcal{D}\tau \ e^{iS[\Psi_c, O^\mu, \tau]}\]
Constrained integral over collapse, observer, and terrain fields.
Glossary
\(\mathcal{D}\Psi_c, \mathcal{D}O^\mu, \mathcal{D}\tau\)
Functional measures for the collapse field, observer vector, and terrain scalar.
\(S[\Psi_c, O^\mu, \tau]\)
Action functional for TRQC over the specified fields.
\(e^{iS}\)
Phase factor in the path integral formalism.
22
Terrain-Normalized Probability Integral
Math
\[P_i = \int_S |\Psi_c(x)|^2\, d^3x\]
Probability in a projection-compatible slice normalized to terrain conditions.
Glossary
\(P_i\)
Probability associated with region \(S\).
\(|\Psi_c(x)|^2\)
Probability density from the collapse field amplitude.
\(d^3x\)
Volume element in 3D space.
23
Collapse Probability Operator
Math
\[\hat{P}_{\text{collapse}} = \int_D w(x;O^\mu,\tau)\, |\Psi_c(x)|^2\, d^3x\]
Observer-weighted projection probability operator defined over terrain-compatible domains.
Glossary
\(\hat{P}_{\text{collapse}}\)
Observer- and terrain-weighted projection operator yielding structural collapse probability over domain \(D \subset D_{\text{QEC}}\).
\(w(x;O^\mu,\tau)\)
Weighting function encoding alignment with the observer field \(O^\mu\), terrain slope \(\tau(x)\), and local collapse coherence; normalized over \(D_{\text{QEC}}\).
\(|\Psi_c(x)|^2\)
Squared amplitude of the collapse field; represents raw projection strength prior to observer smoothing or kernel filtering.
24
Collapse Brightness Function
Math
\[\mathcal{B}(\tau) = \int_{\Sigma_\tau} L(x)\, d^3x\]
Brightness across a structural-time slice.
Glossary
\(\mathcal{B}(\tau)\)
Brightness function parameterized by structural time \(\tau\).
\(\Sigma_\tau\)
Hypersurface of constant structural time \(\tau\).
\(L(x)\)
Local luminosity or emission density at \(x\).
25
Collapse Luminosity Rate
Math
\[\frac{d\mathcal{B}}{d\tau} = -\kappa\, \frac{dM}{d\tau}\]
Links change in brightness to change in mass over structural time.
Glossary
\(\frac{d\mathcal{B}}{d\tau}\)
Rate of change of brightness with structural time.
\(\kappa\)
Proportionality constant relating mass change to brightness change.
\(\frac{dM}{d\tau}\)
Rate of change of mass with structural time.
26
Photon–Mass Collapse Coupling
Math
\[\frac{dN_\gamma}{d\tau} = -\alpha\, \frac{dM}{d\tau}\]
Relates photon emission rate to collapse-driven mass change.
Glossary
\(\frac{dN_\gamma}{d\tau}\)
Rate of photon number change over structural time.
\(\alpha\)
Proportionality constant between photon and mass change rates.
\(\frac{dM}{d\tau}\)
Rate of change of mass with structural time.
27
Collapse Termination Condition
Math
\[\lim_{\tau \to \tau_\infty} \mathcal{B}(\tau) = 0 \ \Rightarrow\ \left.\frac{d\tau}{ds}\right|_{\mathcal{B}=0} = 0\]
When brightness vanishes, structural time flow ceases.
Glossary
\(\tau_\infty\)
Asymptotic structural time where collapse ends.
\(\mathcal{B}(\tau)\)
Brightness function.
\(\frac{d\tau}{ds}\)
Rate of change of structural time along proper time \(s\).
28
Entropy–Brightness Divergence
Math
\[\dot{S} > 0, \quad \dot{\mathcal{B}} < 0, \quad |\dot{\mathcal{B}}| \propto (\nabla_\mu \tau \nabla^\mu \tau)^{-1}\]
Entropy rises as brightness falls, with rate set by terrain-time gradient magnitude.
Glossary
\(\dot{S}\)
Rate of change of entropy.
\(\dot{\mathcal{B}}\)
Rate of change of brightness.
\(\nabla_\mu \tau \nabla^\mu \tau\)
Norm squared of the terrain-time gradient.
29
Collapse Termination Theorem (Repeat)
Math
\[\int^\infty \frac{dM}{d\tau}\, d\tau = M_0 < \infty \ \Rightarrow\ \mathcal{B}(\tau) \to 0\]
Finite total mass loss implies brightness vanishes at termination.
Glossary
\(M_0\)
Initial mass available for collapse.
\(\frac{dM}{d\tau}\)
Rate of change of mass over structural time.
\(\mathcal{B}(\tau)\)
Brightness function.
30
Black Hole Feedback Functional
Math
\[\mathcal{F}_{\mathrm{BH}}(x) = \delta R^{\tau}_{\mu\nu}\]
Feedback from curvature variations near collapse cores.
Glossary
\(\mathcal{F}_{\mathrm{BH}}(x)\)
Black hole feedback functional at point \(x\).
\(\delta R^{\tau}_{\mu\nu}\)
Variation in terrain-time curvature tensor.
31
Brightness Gradient Field
Math
\[\nabla_\mu \mathcal{B}(x^\mu)\]
Spatial variation of brightness on terrain-time slices.
Glossary
\(\nabla_\mu\)
Covariant derivative with respect to coordinate \(\mu\).
\(\mathcal{B}(x^\mu)\)
Brightness function as a field over spacetime coordinates \(x^\mu\).
32
Brightness-Modulated Projection
Math
\[\Pi_{\text{QEC}}[\Psi_c, O^\mu, \tau] \to \Pi_{\text{QEC}}[\Psi_c, O^\mu, \tau, \mathcal{B}(\tau)]\]
Projection operator extended by dependence on brightness function.
Glossary
\(\Pi_{\text{QEC}}\)
Projection operator restricted to the quantum event collapse domain.
\(\mathcal{B}(\tau)\)
Brightness function at structural time \(\tau\).
33
Collapse Clock Functional
Math
\[T_{\text{structural}}=\int_{\tau_0}^{\tau}\mathcal{B}(\tau')\,d\tau'\]
Defines an internal clock from accumulated brightness over structural time.
Glossary
\(T_{\text{structural}}\)
Accumulated brightness measure used as a structural time clock.
\(\tau_0, \tau\)
Starting and current structural times.
\(\mathcal{B}(\tau')\)
Brightness at intermediate structural time \(\tau'\).
34
Collapse Layer Entropy
Math
\[S_{BH} = k \,\log\!\left(\frac{\tau_{\text{core}} - \tau_0}{\Delta \tau}\right)\ \ \text{or}\ \ \sim\ k \,\log\!\left(\frac{A}{\Delta \tau_{\min}}\right)\]
Entropy scaling for collapse layers and boundaries.
Glossary
\(S_{BH}\)
Entropy of the collapse (black hole) layer.
\(k\)
Boltzmann constant (sets entropy scale).
\(\tau_{\text{core}}, \tau_0\)
Core and reference structural times.
\(\Delta \tau\)
Structural time thickness of the layer.
35
Collapse Curvature Emission
Math
\[\mathcal{G}(x) = \nabla^{\mu} R^{\tau}_{\mu\nu}\]
Emission-like term from gradients of terrain-time curvature.
Glossary
\(\mathcal{G}(x)\)
Collapse curvature emission density at position \(x\).
\(R^{\tau}_{\mu\nu}\)
Terrain-time curvature tensor.
\(\nabla^{\mu}\)
Covariant derivative with raised index \(\mu\).
36
Collapse Entropy Boundary
Math
\[S_{BH} = \int_{\partial D_{BH}} k \,\log\!\left(\frac{\tau_{\text{core}} - \tau_0}{\Delta \tau}\right)\, d^3 x\]
Boundary integral form of collapse entropy.
Glossary
\(\partial D_{BH}\)
Boundary of the black hole (collapse) domain.
\(k, \tau_{\text{core}}, \tau_0, \Delta\tau\)
As in Collapse Layer Entropy.
37
Collapse Feedback Integral
Math
\[\Psi_c^{(n+1)}(x) = \Psi_c^{(n)}(x) + \int_{\Sigma_{BH}} K(x,x')\, \mathcal{F}_{BH}(x') \, d^3 x'\]
Iterative update from black-hole feedback over a surface.
Glossary
\(\Psi_c^{(n)}\)
Collapse field at iteration \(n\).
\(K(x,x')\)
Kernel function coupling points \(x\) and \(x'\).
\(\mathcal{F}_{BH}(x')\)
Black hole feedback functional at point \(x'\).
38
Collapse Recursion Operator
Math
\[\mathcal{R}\,\Psi_c^{(n)}(x) = \Psi_c^{(n)}(x) + \int K(x,x')\left(\delta R^{\tau}(x') - \alpha\,\mathcal{B}(\tau)\right)\,\Psi_c^{(n)}(x)\, d^3 x'\]
Operator form of feedback update including curvature and brightness terms.
Glossary
\(\mathcal{R}\)
Recursion operator.
\(\delta R^{\tau}(x')\)
Variation of terrain-time curvature at point \(x'\).
\(\alpha\)
Brightness coupling constant.
39
Restart Threshold
Math
\[\mathcal{F}_{BH}(x)\cdot \|\nabla \tau\| > \lambda \,\log\!\left(1+\Delta S\right)\]
Condition to restart recursion based on feedback and entropy change.
Glossary
\(\mathcal{F}_{BH}(x)\)
Black hole feedback functional at point \(x\).
\(\|\nabla \tau\|\)
Norm of terrain-time gradient.
\(\lambda\)
Proportionality constant in restart condition.
\(\Delta S\)
Change in entropy.
40
Collapse Entropy Budget
Math
\[S_{\text{usable}} = \int_{D_{\text{QEC}}} \log\!\left(\frac{\tau_{\text{core}} - \tau_0}{\Delta \tau(x)}\right)\, d^3 x\]
Integrated usable entropy over the projection-compatible domain.
Glossary
\(S_{\text{usable}}\)
Usable entropy within the QEC domain.
\(D_{\text{QEC}}\)
Quantum event collapse domain.
\(\tau_{\text{core}}, \tau_0, \Delta\tau(x)\)
Core time, reference time, and local structural time thickness.
41
Recursion Convergence
Math
\[\Delta_{\text{rec}} = \int_{\Sigma_{\tau}} \left|\Psi_c^{(n+1)}(x) - \Psi_c^{(n)}(x)\right|^2\, d^3 x\]
Measures convergence of successive recursion steps in collapse field updates.
Glossary
\(\Delta_{\text{rec}}\)
Quantitative measure of change between iterations.
\(\Sigma_{\tau}\)
Constant structural-time hypersurface.
\(\Psi_c^{(n)}(x)\)
Collapse field at iteration \(n\) at point \(x\).
42
Curvature Smoothing
Math
\[\overline{R}^{\tau}_{\mu\nu}(x) = \int G(x,x')\, R^{\tau}_{\mu\nu}(x')\, d^3 x'\]
Smooths terrain-time curvature using a kernel convolution.
Glossary
\(\overline{R}^{\tau}_{\mu\nu}(x)\)
Smoothed curvature tensor at \(x\).
\(G(x,x')\)
Smoothing kernel function between points \(x\) and \(x'\).
\(R^{\tau}_{\mu\nu}(x')\)
Terrain-time curvature tensor at \(x'\).
43
Time Dilation Lag
Math
\[\left(\frac{d\tau}{ds}\right)_{\text{lagged}} = \frac{\sqrt{\nabla^{\mu}\tau\,\nabla_{\mu}\tau}}{1 + \beta\, R(x)}\]
Structural time flow slowed by curvature, producing lag relative to proper time.
Glossary
\(\frac{d\tau}{ds}\)
Rate of change of structural time along proper time \(s\).
\(\nabla^{\mu}\tau\,\nabla_{\mu}\tau\)
Norm squared of terrain-time gradient.
\(\beta\)
Curvature coupling constant.
\(R(x)\)
Curvature scalar at point \(x\).
44
Collapse Emission Spectrum
Math
\[\mathcal{S}(\omega,x) = \frac{\omega^{3}}{\exp\!\left(\frac{\omega}{\gamma\,\mathcal{B}(\tau) + \delta\,R(x)}\right) - 1}\]
Spectral distribution of collapse emission influenced by brightness and curvature.
Glossary
\(\mathcal{S}(\omega,x)\)
Emission spectrum at frequency \(\omega\) and position \(x\).
\(\gamma\)
Brightness coupling constant.
\(\mathcal{B}(\tau)\)
Brightness function at structural time \(\tau\).
\(\delta\)
Curvature-emission coupling constant.
\(R(x)\)
Curvature scalar at position \(x\).
45
Born Deviation (Energy Density)
Math
\[\Delta_i = \int_{S_i} \left(|\psi(x)|^2 - \nabla_{\mu}\Psi_c\,\nabla^{\mu}\Psi_c\right)\, d^3 x\]
Deviation from Born rule using energy density as comparator.
Glossary
\(\Delta_i\)
Deviation measure for subset \(S_i\) of space.
\(|\psi(x)|^2\)
Projected probability density.
\(\nabla_{\mu}\Psi_c\,\nabla^{\mu}\Psi_c\)
Energy density from collapse field gradients.
46
Collapse–Coupled Fermion Lagrangian
Math
\[\mathcal{L}_f = \bar{\psi}_f \left(i\gamma^{\mu} D_{\mu} - Y_f \,\|\nabla \tau\|\right)\psi_f\]
Fermion sector Lagrangian with terrain-time gradient coupling.
Glossary
\(\mathcal{L}_f\)
Fermion sector Lagrangian density.
\(\bar{\psi}_f, \psi_f\)
Dirac adjoint and fermion field of flavor \(f\).
\(i\gamma^{\mu} D_{\mu}\)
Dirac kinetic term with gauge-covariant derivative.
\(Y_f\)
Effective Yukawa coupling for fermion flavor \(f\).
\(\|\nabla \tau\|\)
Norm of terrain-time gradient.
47
Yukawa Overlap Functional
Math
\[Y_f = \frac{2\pi\, A_f A_n\, \sigma_f \sigma_n}{\sigma_f^{2} + \sigma_n^{2}} \,\exp\!\left[-\frac{(x_f - x_n)^{2}}{2(\sigma_f^{2} + \sigma_n^{2})}\right]\]
Defines effective Yukawa coupling from mode overlaps.
Glossary
\(Y_f\)
Effective Yukawa coupling for fermion flavor \(f\).
\(A_f, A_n\)
Amplitudes of fermion and normal modes.
\(\sigma_f, \sigma_n\)
Widths of the fermion and normal mode distributions.
\(x_f, x_n\)
Centers of the fermion and normal mode distributions.
48
Master Lagrangian — \( \mathcal{L}_{\mathrm{Master}} \)
Math
\[
\mathcal{L}_{\mathrm{Master}}
= \mathcal{L}_{\mathrm{collapse}}
+ \mathcal{L}_{\mathrm{observer}}
+ \mathcal{L}_{\mathrm{terrain}}
+ \mathcal{L}_{\mathrm{fermion}}
+ \mathcal{L}_{\mathrm{gauge}}
+ \mathcal{L}_{\mathrm{feedback}}
\]
Complete TRQC Lagrangian; sum of all sector contributions. Expand sub-items for definitions.
Glossary
\(\mathcal{L}_{\mathrm{collapse}}\)
Collapse field kinetic and potential terms.
\(\mathcal{L}_{\mathrm{observer}}\)
Observer field coupling and constraints.
\(\mathcal{L}_{\mathrm{terrain}}\)
Terrain-time curvature terms.
\(\mathcal{L}_{\mathrm{fermion}}\)
Fermion sector dynamics with terrain coupling.
\(\mathcal{L}_{\mathrm{gauge}}\)
Gauge sector dynamics and alignment terms.
\(\mathcal{L}_{\mathrm{feedback}}\)
Curvature feedback and effective-metric variations.
48a
Collapse Core Terms — \( \mathcal{L}_{\mathrm{collapse}} \)
Math
\[\mathcal{L}_{\mathrm{collapse}} = \tfrac{1}{2}\,\nabla^{\mu}\Psi_c\,\nabla_{\mu}\Psi_c - V(\Psi_c,\tau)\]
Core kinetic term for the collapse field plus its potential.
Glossary
\(\nabla^{\mu}\Psi_c\,\nabla_{\mu}\Psi_c\)
Kinetic term from collapse field gradients.
\(V(\Psi_c,\tau)\)
Collapse potential depending on field and terrain-time.
48b
Observer Terms — \( \mathcal{L}_{\mathrm{observer}} \)
Math
\[\mathcal{L}_{\mathrm{observer}} = \gamma\,O^{\mu}\nabla_{\mu}\Psi_c + \lambda\,(O^{\mu}O_{\mu} - 1)\]
Couples observer field to collapse dynamics and enforces unit norm constraint.
Glossary
\(\gamma\)
Observer–collapse coupling constant.
\(O^{\mu}\nabla_{\mu}\Psi_c\)
Directional derivative of collapse field along observer vector.
\(\lambda\,(O^{\mu}O_{\mu} - 1)\)
Constraint enforcing \(O^\mu\) as a unit timelike vector.
48c
Terrain Curvature Term — \( \mathcal{L}_{\mathrm{terrain}} \)
Math
\[\mathcal{L}_{\mathrm{terrain}} = \beta\,R^{\tau}_{\mu\nu}\]
Contribution from terrain-time curvature tensor.
Glossary
\(\beta\)
Coupling constant for terrain curvature.
\(R^{\tau}_{\mu\nu}\)
Terrain-time curvature tensor derived from \(\tau\).
48d
Fermion Sector — \( \mathcal{L}_{\mathrm{fermion}} \)
Math
\[\mathcal{L}_{\mathrm{fermion}} = \bar{\psi}_f\!\left(i\gamma^{\mu}D_{\mu} - Y_f\,\|\nabla\tau\|\right)\!\psi_f\]
Fermion dynamics with terrain-time gradient coupling.
Glossary
\(\bar{\psi}_f, \psi_f\)
Dirac adjoint and fermion field of flavor \(f\).
\(i\gamma^{\mu}D_{\mu}\)
Dirac kinetic term with gauge-covariant derivative.
\(Y_f\,\|\nabla\tau\|\)
Yukawa coupling modulated by terrain-time gradient norm.
48e
Gauge Sector — \( \mathcal{L}_{\mathrm{gauge}} \)
Math
\[\mathcal{L}_{\mathrm{gauge}} = -\tfrac{1}{4}F^{a}_{\mu\nu}F^{a\mu\nu} + \xi\,\mathrm{Tr}\!\big([O^{\mu},A^{a}_{\mu}]^2\big)\]
Gauge field kinetic term plus observer–gauge alignment penalty.
Glossary
\(-\tfrac{1}{4}F^{a}_{\mu\nu}F^{a\mu\nu}\)
Yang–Mills kinetic term for gauge fields.
\(\xi\)
Observer–gauge alignment penalty constant.
\(\mathrm{Tr}([O^{\mu},A^{a}_{\mu}]^2)\)
Trace over squared commutator between observer and gauge fields.
48f
Curvature Feedback Terms — \( \mathcal{L}_{\mathrm{feedback}} \)
Math
\[\mathcal{L}_{\mathrm{feedback}} = \delta\,R[g^{\mathrm{eff}}] + \eta\,\nabla^{\mu}R^{\tau}_{\mu\nu} + \tfrac{1}{2}\,\kappa\,R\!\left[g^{\mathrm{eff}}_{\mu\nu}\right]\]
Effective-curvature variations and terrain-curvature gradient terms.
Glossary
\(\delta\)
Coupling to effective metric curvature variation.
\(\eta\)
Coupling to divergence of terrain-time curvature tensor.
\(\kappa\)
Coupling to Ricci scalar of effective metric.
49
Collapse Mode Overlap Matrix
Math
\[Y_{ij} = \int \bar{\psi}_i(x)\, \Psi_c(x)\, \psi_j(x)\, d^{4}x\]
Overlap of fermion modes with the collapse field mode functions.
Glossary
\(Y_{ij}\)
Matrix element of mode overlap between fermion states \(i\) and \(j\).
\(\bar{\psi}_i, \psi_j\)
Dirac adjoint and fermion mode functions.
\(\Psi_c(x)\)
Collapse field at spacetime point \(x\).
50
Gauge Transport Condition
Math
\[[O^{\mu}, A^{a}_{\mu}] = 0,\qquad \nabla^{\mu} A^{a}_{\mu} \ll \nabla^{\mu} \Psi_c \,\nabla_{\mu} \Psi_c\]
Alignment of gauge fields with observer field; small divergence relative to collapse gradients.
Glossary
\(O^{\mu}\)
Observer vector field.
\(A^{a}_{\mu}\)
Gauge field with group index \(a\).
\(\nabla^{\mu} A^{a}_{\mu}\)
Covariant divergence of gauge field.
\(\nabla^{\mu} \Psi_c \nabla_{\mu} \Psi_c\)
Norm squared of collapse field gradient.
51
Collapse-Layer RG Flow
Math
\[\alpha_i(\tau) = \left\langle F^{a}_{\mu\nu} F^{a\mu\nu} \right\rangle_{\Sigma_{\tau}}\]
Renormalization group flow of collapse-layer parameters via gauge field invariants.
Glossary
\(\alpha_i(\tau)\)
Scale-dependent coupling in the \(i\)-th collapse layer.
\(F^{a}_{\mu\nu}F^{a\mu\nu}\)
Gauge field strength contraction.
\(\Sigma_{\tau}\)
Surface of constant structural time \(\tau\).
52
Fermion Mass via Overlap
Math
\[m_f = \lambda \int \Psi_c(x) \, |\psi_f(x)|^{2} \, d^{4}x\]
Fermion mass generated from overlap with collapse field distribution.
Glossary
\(m_f\)
Mass of fermion flavor \(f\).
\(\lambda\)
Coupling constant relating overlap to mass.
\(\Psi_c(x)\)
Collapse field.
\(|\psi_f(x)|^2\)
Probability density for fermion field \(f\).
53
Collapse-Time Quantization Rule
Math
\[[\hat{\tau}(x), \hat{H}] = i\hbar\]
Canonical commutation relation between structural time operator and collapse Hamiltonian.
Glossary
\(\hat{\tau}(x)\)
Structural time operator at point \(x\).
\(\hat{H}\)
Collapse Hamiltonian operator.
\(i\hbar\)
Imaginary unit times Planck’s constant, from quantum commutation relations.
54
Collapse-Sourced Stress-Energy Tensor
Math
\[T^{\mu\nu}_{\text{eff}} = -\frac{2}{\sqrt{-g}}\, \frac{\delta \mathcal{L}_{\text{TRQC}}}{\delta g^{\text{eff}}_{\mu\nu}}\]
Effective stress-energy tensor sourced by collapse-sector Lagrangian.
Glossary
\(T^{\mu\nu}_{\text{eff}}\)
Effective stress-energy tensor from collapse dynamics.
\(g\)
Determinant of the metric tensor.
\(\mathcal{L}_{\text{TRQC}}\)
TRQC Lagrangian density.
55
Collapse-Time Commutation Structure
Math
\[[\hat{\tau}(x), \hat{\Pi}_{\tau}(y)] = i\hbar\, \delta^{3}(x-y)\]
Canonical commutation relation between structural time and its conjugate momentum.
Glossary
\(\hat{\tau}(x)\)
Structural time operator.
\(\hat{\Pi}_{\tau}(y)\)
Conjugate momentum to structural time at point \(y\).
\(\delta^{3}(x-y)\)
3D Dirac delta function ensuring locality of commutation.
56
Collapse-Time Functional Schrödinger Equation
Math
\[i\hbar\, \frac{\delta}{\delta \tau(x)}\, \Psi[\Psi_c,\tau,O^{\mu}] = \hat{H}_{\text{collapse}}(x)\, \Psi[\cdot]\]
Functional Schrödinger equation in structural time representation.
Glossary
\(\frac{\delta}{\delta \tau(x)}\)
Functional derivative with respect to \(\tau\) at point \(x\).
\(\hat{H}_{\text{collapse}}(x)\)
Local collapse Hamiltonian density.
57
Collapse-Time Green’s Equation
Math
\[\mathcal{D}_{\tau}\, G(x,x') = \delta^{4}(x-x'), \quad \mathcal{D}_{\tau} \equiv \Box + \frac{\partial V}{\partial \Psi_c} - \gamma\, \nabla_{\mu} O^{\mu}\]
Green’s function equation for collapse dynamics in structural time formalism.
Glossary
\(G(x,x')\)
Green’s function connecting points \(x\) and \(x'\).
\(\delta^{4}(x-x')\)
4D Dirac delta function.
\(\Box\)
D’Alembertian operator.
\(\frac{\partial V}{\partial \Psi_c}\)
Derivative of collapse potential with respect to collapse field.
58
Collapse-Time Quantization Theorem
Math
\[\tau_n(x) = n\,\Delta\tau + \varepsilon(x), \quad [\hat{\tau}(x), \hat{H}] = i\hbar, \quad [\hat{\tau}(x), \hat{\Pi}_{\tau}(y)] = i\hbar\,\delta^3(x-y)\]
Eigenstate quantization of structural time with canonical commutators.
Glossary
\(\tau_n(x)\)
n-th eigenvalue of structural time at point \(x\).
\(\Delta\tau\)
Quantization spacing in structural time.
\(\varepsilon(x)\)
Small position-dependent offset.
59
Observer Field Commutation Relation
Math
\[[\hat{O}^{\mu}(x), \hat{\Pi}^{(O)}_{\nu}(y)] = i\hbar\, \delta^{\mu}_{\nu}\, \delta^{3}(x-y), \quad \hat{O}^{\mu}\hat{O}_{\mu} = 1\]
Canonical commutation relation for observer field and its conjugate momentum with unit norm constraint.
Glossary
\(\hat{O}^{\mu}(x)\)
Observer field operator at \(x\).
\(\hat{\Pi}^{(O)}_{\nu}(y)\)
Conjugate momentum to observer field at \(y\).
\(\delta^{\mu}_{\nu}\)
Kronecker delta.
60
Projection-Compatible Gauge Commutator
Math
\[[\hat{A}^{a}_{\mu}(x), \hat{\Pi}^{b\nu}(y)] = i\hbar\, \delta^{ab} \delta^{\nu}_{\mu} \delta^{3}(x-y)\]
Canonical commutation relation for gauge fields in a projection-compatible frame.
Glossary
\(\hat{A}^{a}_{\mu}(x)\)
Gauge field operator for group index \(a\) at position \(x\).
\(\hat{\Pi}^{b\nu}(y)\)
Conjugate momentum to gauge field component \(b,\nu\) at \(y\).
\(\delta^{ab}\)
Kronecker delta over group indices.
61
TRQC Hilbert Space Tensor Structure
Math
\[\mathcal{H}_{\mathrm{TRQC}} = \mathcal{H}_{\Psi_c} \otimes \mathcal{H}_{\tau} \otimes \mathcal{H}_{O^{\mu}} \otimes \mathcal{H}_{\mathrm{gauge}} \otimes \mathcal{H}_{\mathrm{fermion}}\]
Full Hilbert space as a tensor product of collapse, terrain-time, observer, gauge, and fermion sectors.
Glossary
\(\mathcal{H}\)
Hilbert space for a given sector.
\(\otimes\)
Tensor product operation.
62
TRQC Path Integral over Constrained Fields
Math
\[Z_{\mathrm{TRQC}} = \int_{\mathcal{C}_{\mathrm{TRQC}}} \mathcal{D}\Psi_c\, \mathcal{D}\tau\, \mathcal{D}O^{\mu}\, \mathcal{D}A_{\mu}\, \mathcal{D}\psi_f \ \exp\left[i \int d^{4}x\, \mathcal{L}_{\mathrm{Master}}\right]\]
Full path integral of TRQC over all constrained fields and sectors.
Glossary
\(Z_{\mathrm{TRQC}}\)
Partition function of TRQC.
\(\mathcal{C}_{\mathrm{TRQC}}\)
Constraint set defining allowed field configurations.
\(\mathcal{D}\)
Functional integration measure.
63
Collapse-Compatible Mode Expansion
Math
\[\hat{\Psi}_c(x) = \int \frac{d^{3}k}{(2\pi)^{3}} \left[ a_k\, u_k(x) + a_k^{\dagger} \, u_k^{*}(x) \right]\]
Mode expansion of the collapse field operator in a projection-compatible basis.
Glossary
\(a_k, a_k^{\dagger}\)
Mode annihilation and creation operators.
\(u_k(x)\)
Mode function with momentum \(k\) at position \(x\).
64
Fermionic Field Anticommutation Structure
Math
\[\{\hat{\psi}^{\alpha}_f(x), \hat{\psi}^{\beta\dagger}_f(y)\} = \delta^{\alpha\beta} \delta^{3}(x-y)\]
Canonical anticommutation relations for fermion fields of flavor \(f\).
Glossary
\(\hat{\psi}^{\alpha}_f(x)\)
Fermion field operator of spinor component \(\alpha\).
\(\hat{\psi}^{\beta\dagger}_f(y)\)
Creation operator for spinor component \(\beta\).
65
Gauge Field Commutation Structure
Math
\[[\hat{A}^{a}_{\mu}(x), \hat{\Pi}^{b\nu}(y)] = i\hbar\, \delta^{ab} \delta^{\nu}_{\mu} \delta^{3}(x-y)\]
Canonical commutation relations for gauge fields in TRQC.
Glossary
\(\hat{A}^{a}_{\mu}(x)\)
Gauge field operator with group index \(a\).
\(\hat{\Pi}^{b\nu}(y)\)
Conjugate momentum to gauge field component \(b,\nu\).
66
Curvature Quantization Functional
Math
\[\hat{R}^{\tau}_{\mu\nu}(x) = \hat{F}[\Psi_c(x), \tau(x)]\]
Operator form of the terrain-time curvature tensor as a functional of collapse field and terrain-time scalar.
Glossary
\(\hat{R}^{\tau}_{\mu\nu}(x)\)
Quantized terrain-time curvature tensor operator at \(x\).
\(\hat{F}[\Psi_c,\tau]\)
Functional form defining curvature in terms of collapse and terrain fields.
67
Observer Path Integral Structure
Math
\[Z_{O^{\mu}} = \int_{\mathcal{C}_{O}} \mathcal{D}O^{\mu}(x)\, \delta(O^{\mu}O_{\mu} - 1)\, \exp\left(i\int d^{4}x\, \mathcal{L}_{\text{observer}}\right)\]
Path integral over observer field configurations constrained to unit norm.
Glossary
\(Z_{O^{\mu}}\)
Partition function for observer field sector.
\(\delta(O^{\mu}O_{\mu} - 1)\)
Delta functional enforcing unit norm for observer vector.
68
Unitarity Constraint on Collapse Layers
Math
\[\frac{d}{d\tau} \int_{D_{\mathrm{QEC}}} |\Psi[\Psi_c, \tau, O^{\mu}]|^{2} \ \mathcal{D}\Psi_c = 0\]
Probability conservation condition over projection-compatible domain.
Glossary
\(D_{\mathrm{QEC}}\)
Projection-compatible domain.
\(|\Psi|^{2}\)
Probability density functional over collapse field configurations.
69
Collapse-Time Heisenberg Evolution
Math
\[\frac{\delta \hat{A}(x)}{\delta \tau(x)} = \frac{i}{\hbar} \left[ \hat{H}_{\text{collapse}}(x), \hat{A}(x) \right]\]
Heisenberg-picture evolution of operator \(\hat{A}\) in structural time representation.
Glossary
\(\hat{A}(x)\)
Generic operator at point \(x\).
\(\hat{H}_{\text{collapse}}(x)\)
Local collapse Hamiltonian operator.
1
Projection Convergence Theorem
Theorem
\[
\lim_{\varepsilon \to 0} (\nabla_\mu \Psi_c)(\nabla^\mu \Psi_c) = |\psi(x)|^2
\]
In the zero-smoothing limit, collapse field gradients converge to the Born probability density.
Glossary
\(\varepsilon\) Smoothing parameter.
\(\nabla_\mu\) Covariant derivative.
\(\Psi_c\) Collapse field.
\(\psi(x)\) Projected wavefunction.
2
Collapse Kernel Solution Lemma
Theorem
\[
\Psi_c(x) = A\,e^{-r^2/2\sigma^2} \ \Rightarrow\ \Box\Psi_c = -m^2\Psi_c - 2\lambda \Psi_c^3 \quad \text{iff} \quad 3\sigma^2 = m^2 + 2\lambda A^2
\]
Gaussian collapse kernels solve the nonlinear wave equation under the given width–mass–coupling relation.
Glossary
\(A\) Amplitude of the Gaussian kernel.
\(r\) Radial coordinate.
\(\sigma\) Width of the Gaussian kernel.
\(\Box\) D’Alembertian operator.
\(m\) Mass parameter.
\(\lambda\) Self-interaction constant.
3
Terrain Stability Theorem
Theorem
\[
\mathcal{L}_\tau = -\alpha (\nabla_\mu\tau \nabla^\mu\tau)^2 - \beta (\Box\tau)^2
\]
Stable terrain-time dynamics require quartic gradient and curvature-squared terms.
Glossary
\(\mathcal{L}_\tau\) Terrain-time Lagrangian density.
\(\alpha, \beta\) Coupling constants.
\(\tau\) Terrain-time scalar.
\(\nabla_\mu\) Covariant derivative.
\(\Box\) D’Alembertian operator.
4
Threshold Dynamics Lemma
Theorem
\[
\theta_{\mathrm{collapse}} = m^2 + 2\lambda \psi^2, \quad \theta_{\mathrm{shear}} = \frac{3}{L^2}
\]
Collapse and shear thresholds in terms of field amplitude and characteristic length.
Glossary
\(\theta_{\mathrm{collapse}}\) Collapse threshold value.
\(\theta_{\mathrm{shear}}\) Shear threshold value.
\(m\) Mass parameter.
\(\lambda\) Self-interaction constant.
\(\psi\) Projected wavefunction amplitude.
\(L\) Characteristic length scale.
5
Collapse Gradient Arrow Theorem
Theorem
\[
\nabla^\mu \Psi_c \nabla_\mu \Psi_c > 0 \ \Rightarrow\ \frac{d\tau}{ds} > 0
\]
Positive collapse gradient norm enforces forward progression in structural time.
Glossary
\(\nabla^\mu \Psi_c \nabla_\mu \Psi_c\) Norm squared of collapse field gradient.
\(\tau\) Terrain-time scalar field.
\(s\) Proper time parameter.
6
Collapse–Curvature Feedback Identity
Theorem
\[
\frac{\delta R[g^{\mathrm{eff}}_{\mu\nu}]}{\delta \Psi_c} \neq 0
\]
Effective curvature depends functionally on the collapse field.
Glossary
\(R[g^{\mathrm{eff}}_{\mu\nu}]\) Ricci scalar from effective metric.
\(\Psi_c\) Collapse field.
7
Collapse Redshift Limit Theorem
Theorem
\[
z_{\mathrm{threshold}} = \left(\frac{\tau_0}{\tau_{\mathrm{crit}}}\right)^{1/p} - 1, \quad z > z_{\mathrm{threshold}} \ \Rightarrow\ D_{\mathrm{QEC}} = \varnothing
\]
Above a redshift threshold set by terrain time, no projection-compatible domains exist.
Glossary
\(z_{\mathrm{threshold}}\) Maximum redshift for projections.
\(\tau_0\) Reference terrain time.
\(\tau_{\mathrm{crit}}\) Critical terrain time.
\(p\) Power-law index.
\(D_{\mathrm{QEC}}\) Projection-compatible domain.
8
Observer Horizon Theorem
Theorem
\[
\nabla_\mu O^\mu \le \theta_{\mathrm{collapse}} \ \text{or}\ |\partial^2\tau| > \theta_{\mathrm{shear}} \ \Rightarrow\ x \in \Sigma^\tau_{(O)}
\]
Observer horizons occur when collapse divergence is below threshold or terrain shear exceeds limits.
Glossary
\(O^\mu\) Observer vector field.
\(\theta_{\mathrm{collapse}}\) Collapse threshold value.
\(\theta_{\mathrm{shear}}\) Shear threshold value.
\(\Sigma^\tau_{(O)}\) Observer horizon surface at time \(\tau\).
9
Decoherence Domain Lemma
Theorem
\[
x \in D_{\mathrm{QEC}} \ \Leftrightarrow\ \|\nabla\Psi_c\| \ge \theta_{\mathrm{collapse}},\ \ |\partial^2\tau| \le \theta_{\mathrm{shear}}
\]
Characterizes projection-compatible regions by collapse gradient and terrain shear limits.
Glossary
\(D_{\mathrm{QEC}}\) Projection-compatible domain.
\(\|\nabla\Psi_c\|\) Magnitude of collapse field gradient.
\(\theta_{\mathrm{collapse}}\) Collapse threshold.
\(\theta_{\mathrm{shear}}\) Shear threshold.
10
Collapse Projection Orthogonality Lemma
Theorem
\[
O^\mu \nabla_\mu \Psi_c = 0
\]
Collapse field is orthogonal to observer field flow in projection-compatible domains.
Glossary
\(O^\mu\) Observer vector field.
\(\nabla_\mu\) Covariant derivative.
\(\Psi_c\) Collapse field.
11
Collapse Path Integral Well-Formedness
Theorem
\[
\int \mathcal{D}\Psi_c\, \mathcal{D}O^\mu\, \mathcal{D}\tau\ e^{iS[\Psi_c, O^\mu, \tau]} < \infty
\]
The TRQC path integral converges over the collapse, observer, and terrain fields.
Glossary
\(\mathcal{D}\Psi_c, \mathcal{D}O^\mu, \mathcal{D}\tau\) Functional integration measures.
\(S[\Psi_c, O^\mu, \tau]\) Action functional for TRQC fields.
12
Structural Time Theorem
Theorem
\[
\frac{d\tau}{ds} > 0 \ \text{ along integral curves of } O^\mu
\]
Structural time increases monotonically along observer worldlines.
Glossary
\(\tau\) Terrain-time scalar.
\(s\) Proper time.
\(O^\mu\) Observer vector field.
13
Collapse Termination Theorem
Theorem
\[
\text{If } \int \frac{dM}{d\tau} \, d\tau = M_0 < \infty,\ \text{then}\ \mathcal{B}(\tau) \to 0 \ \text{as} \ \tau \to \tau_\infty
\]
Finite total mass loss implies brightness vanishes as structural time approaches its limit.
Glossary
\(M(\tau)\) Mass function over structural time.
\(\mathcal{B}(\tau)\) Brightness function.
\(\tau_\infty\) Termination structural time.
14
Projection Collapse Limit Theorem
Theorem
\[
\nabla^\mu\tau \nabla_\mu\tau \to \infty \ \Rightarrow\ D_{\mathrm{QEC}} = \varnothing
\]
Infinite terrain-time gradient eliminates projection-compatible domains.
Glossary
\(\nabla^\mu\tau \nabla_\mu\tau\) Norm squared of terrain-time gradient.
\(D_{\mathrm{QEC}}\) Projection-compatible domain.
15
Observer Horizon Theorem (Dual Condition Form)
Theorem
\[
\nabla_\mu O^\mu > \theta_{\mathrm{collapse}} \ \text{or} \ |\partial^2\tau| > \theta_{\mathrm{shear}} \ \Rightarrow\ x \in \Sigma^\tau_{(O)}
\]
Observer horizons also occur when collapse divergence exceeds threshold or terrain shear exceeds limits.
Glossary
\(O^\mu\) Observer vector field.
\(\theta_{\mathrm{collapse}}\) Collapse threshold.
\(\theta_{\mathrm{shear}}\) Shear threshold.
\(\Sigma^\tau_{(O)}\) Observer horizon surface.
16
Gauge Group Selection Theorem
Theorem
\[
G_{\mathrm{TRQC}} = SU(3)_C \times SU(2)_L \times U(1)_Y
\]
The TRQC framework selects the Standard Model gauge group.
Glossary
\(G_{\mathrm{TRQC}}\) Gauge group in TRQC.
\(SU(3)_C, SU(2)_L, U(1)_Y\) Standard Model gauge symmetries for color, weak, and hypercharge interactions.
17
Collapse-Generated Chirality Theorem
Theorem
\[
\text{Asymmetry from }\nabla^\mu\tau \ \text{splits projection bandwidths for } \psi_L \ \text{and} \ \psi_R
\]
Collapse-induced terrain-time gradients cause left/right-handed mode asymmetry.
Glossary
\(\nabla^\mu\tau\) Terrain-time gradient.
\(\psi_L, \psi_R\) Left- and right-handed fermion modes.
18
Flavor Multiplicity Theorem
Theorem
\[
N_{\mathrm{flavor}} = 3 \quad\text{given}\quad \mathcal{T}_{\mathrm{Yukawa}} \ \text{stable under}\ D_{\mathrm{QEC}}
\]
Projected Yukawa topology admits exactly three stable decoherence-preserving fermion classes.
Glossary
\(N_{\mathrm{flavor}}\) Number of fermion generations.
\(\mathcal{T}_{\mathrm{Yukawa}}\) Topology of Yukawa coupling network.
\(D_{\mathrm{QEC}}\) Projection-compatible domain.
19
Born Rule Emergence Theorem
Theorem
\[
P = \int_D w(x; O^\mu, \tau)\,|\Psi_c(x)|^2\, d^3x, \quad \Delta_i = \int_{S_i} \left(|\psi(x)|^2 - \rho(x)\right) d^3x
\]
TRQC derives Born probabilities and deviation measures from first principles.
Glossary
\(P\) Born probability for outcome region.
\(w(x; O^\mu, \tau)\) Weighting function.
\(\Psi_c(x)\) Collapse field.
\(\Delta_i\) Deviation functional.
\(\rho(x)\) Energy density from collapse field gradients.