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Relational Geometrodynamics and Physics Substrate
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In classical relational database theory (coda_all_universe/normal_forms), normal forms (1NF through 6NF, BCNF) eliminate update anomalies by ensuring that relational tuples represent non-redundant, irreducible facts. In relational geometrodynamics and discrete quantum gravity, spacetime geometry is similarly represented without an ambient embedding space: metric distances, curvature, and relational evolution are encoded entirely through the combinatorial connectivity and edge lengths of a simplicial complex (T. Regge, 1961).
The Genesis Difference Tower (0_foundations/0_creation, 1_physics) models physical spacetime as a hierarchy of normal forms where geometric consistency, dynamical variations, and quantum constraints are enforced algebraically:
Combinatorial Simplicial Complex (0NF)
│ [Assign positive squared edge lengths; check Cayley-Menger metric positivity]
▼
Regge Calculus Normal Form (RCNF / 1NF-3NF)
│ [Hinge curvature deficits: ε_h = 2π - ∑ θ_{σ,h}; eliminate non-metric simplices]
▼
Schläfli Identity Normal Form (SNF-Regge)
│ [Gauge orthogonality: ∑_h A_h dε_h = 0; variation yields Einstein field equations]
▼
Boundary-Quotient Normal Form for Simplicial Surgery (BQNF-Regge)
│ [Pachner moves (1→4, 2→3, etc.); exact coboundary gauge repair vs. global recomputation]
▼
Wheeler-DeWitt Normal Form (WDWNF)
│ [Timeless global constraint: H_global |Ψ⟩ = 0 on mini-superspace]
▼
Page-Wootters Relational Normal Form (PWNF)
│ [Bipartite conditioning |ψ_S(t)⟩ = ⟨t_C | Ψ⟩; emergent relational time evolution]
In Regge calculus, a 4D pseudo-Riemannian or Riemannian manifold is approximated by a 4-simplicial complex. The fundamental configuration variables are the squared edge lengths
A set of edge lengths forms a valid Euclidean
The volume of an
Curvature in Regge calculus is concentrated on
- In 3D: Hinges are edges (
$e$ ), and the deficit angle is$\epsilon_e = 2\pi - \sum_{\sigma \supset e} \theta_{\sigma, e}$ . - In 4D: Hinges are triangles (
$t$ ), with deficit angle$\epsilon_t = 2\pi - \sum_{\sigma \supset t} \theta_{\sigma, t}$ , where$\theta_{\sigma, t}$ is the dihedral angle inside 4-simplex$\sigma$ at triangle$t$ .
For a regular tetrahedron with unit edge lengths, the dihedral angle is exactly:
The Regge action with cosmological constant
In our SQLite schema and Python evaluator, SNF-Regge validates that for any interior deformation of edge lengths, the contraction
In dynamical simplicial geometry and causal dynamical triangulations (CDT), the triangulation changes via bistellar flips known as Pachner moves:
- 1-to-4 move: Insert a vertex into the interior of a tetrahedron, splitting it into 4 tetrahedra.
- 2-to-3 move: Replace 2 tetrahedra sharing a common face with 3 tetrahedra sharing a common interior edge.
- 3-to-2 move: The reverse of a 2-to-3 move.
- 4-to-1 move: Remove a degree-4 vertex.
Under a streaming sequence of Pachner moves (simplicial change data capture), BQNF-Regge classifies geometric updates:
- Exact Coboundary Repair ($O(1)$): If the variation in deficit angles is an exact gauge transformation (a local vertex displacement without changing the intrinsic curvature tensor), BQNF updates the local edge relvars and hinge records incrementally.
- Closed Residual (Recompute): If the move alters the global topology or introduces net curvature defect, BQNF routes to bounded partition recomputation of the affected subcomplex.
- Closure Failure (Refusal): If a flip produces a non-manifold vertex, self-intersecting simplices, or violates the Schläfli identity, BQNF refuses the update loudly.
In canonical quantum gravity (ADM formulation), the classical Hamiltonian constraint
In our ANSI SQL schema, WdwGlobalState and WdwHamiltonianOperator represent the discretized mini-superspace Hamiltonian. A state is certified in WDWNF if and only if the constraint residual vanishes:
How does time emerge in a timeless universe? In the Page-Wootters mechanism (1983), the global Hilbert space is partitioned into a reference subsystem (the Clock
In regge_wdw_schema.sql, RelationalEvolutionView executes this conditioning purely via relational SQL joins:
SELECT
w.clock_idx AS t,
w.system_idx AS s,
w.amplitude AS psi_s_t
FROM WdwGlobalState w
ORDER BY w.clock_idx, w.system_idx;The evaluator verifies:
- Exact Wheeler-DeWitt kernel compliance:
$| \hat{H}_{\text{total}} |\Psi\rangle | \le 10^{-9}$ . - Non-zero entanglement entropy between clock and system:
$S(\rho_S) > 0$ . - Exact relational unitary evolution:
$|\psi_S(t+1)\rangle = \hat{U}_S |\psi_S(t)\rangle$ .
We have fully recovered the core geometric derivations and theorems presented in the pedagogical paper by B. T. M. Neves, M. S. de Souza, J. C. Fabris, V. Romero-Rochín, R. F. Santos, and R. C. Batista (arXiv:1904.01966v1):
In the discretized manifold, the parallel displacement of a vector
-
Pair Antisymmetry: Inherited from
$U_{\alpha\beta} = -U_{\beta\alpha}$ , giving$R_{\alpha\beta\mu\nu} = -R_{\beta\alpha\mu\nu} = -R_{\alpha\beta\nu\mu}$ . -
Ricci Tensor Contraction (Eq. 18):
$$R_{\alpha\mu} = R_{\alpha\beta\mu}^{\quad\beta} = \frac{1}{2} \varrho_p \varepsilon_p \left(\delta_{\alpha\mu} - U_\alpha^p U_\mu^p\right)$$ -
Ricci Scalar Contraction (Eq. 19):
$$R = g^{\alpha\mu} R_{\alpha\mu} = 2 \varrho_p \varepsilon_p$$ Verified inregge_wdw_evaluator.pyandtest_arxiv_1904_01966_riemann_ricci_recoveryto within$10^{-12}$ .
-
First Bianchi Identity (Eq. 22):
The cyclic sum vanishes identically:
$$R_{\alpha\beta\mu\nu} + R_{\beta\mu\alpha\nu} + R_{\mu\alpha\beta\nu} = 0$$ Proven in Lean 4 (discrete_first_bianchi_identity) and verified numerically across all orientations ($< 10^{-14}$ ). -
Second Bianchi Identity & Bone Closure (Eq. 28, 35):
Because parallel transport loops are homotopic to the identity, the total rotation holonomy satisfies
$h \mathbf{A} = \mathbf{A}$ , forcing the topological boundary condition (Eq. 28):$$\sum_p \varepsilon_p \mathbf{U}^p = 0$$ Combined with the directional derivative continuity equation$U_p^\mu \partial_\mu \varrho_p = \varrho$ (Eq. 30), the discrete Bianchi divergence vanishes (Eq. 35):$$B_{\lambda\alpha\beta\mu\nu} = \nabla_\lambda R_{\alpha\beta\mu\nu} + \nabla_\mu R_{\alpha\beta\nu\lambda} + \nabla_\nu R_{\alpha\beta\lambda\mu} = 0$$ Proven in Lean 4 (discrete_second_bianchi_from_bone_closure) and validated viaDiscreteSecondBianchiView($\le 10^{-12}$ ).
Applying the principle of stationary action DiscreteEinsteinResidualView with maximum residual
To address Cauchy foliation slicing without unphysical topology changes, we formalized and implemented the 4D Lorentzian CDT framework:
-
Spacelike Edges (
$s_e = l_e^2 > 0$ ): Restricted strictly to vertices within the same spatial Cauchy slice$\Sigma_t$ . -
Timelike Edges (
$s_e = l_e^2 < 0$ ): Connect vertices at time$t$ to vertices at time$t+1$ . -
4D Cdt Simplices: Classified into:
-
$(4,1)$ : 4 spatial vertices at$t$ , 1 vertex at$t+1$ -
$(3,2)$ : 3 spatial vertices at$t$ , 2 vertices at$t+1$ -
$(2,3)$ : 2 spatial vertices at$t$ , 3 vertices at$t+1$ -
$(1,4)$ : 1 spatial vertex at$t$ , 4 vertices at$t+1$
-
In classical simplicial gravity, arbitrary Pachner moves may rip spatial hypersurfaces into baby universes or create closed timelike curves (CTCs). Under LCNF and BQNF-Lorentzian:
-
Causal Moves (
$(2,3)$ and$(2,6)$ flips): Confined within the sandwich between$\Sigma_t$ and$\Sigma_{t+1}$ , preserving the spatial Euler characteristic$\chi(\Sigma_t) = V - E + F$ . BQNF routes these moves toExactRepair($O(1)$ gauge update). -
Acausal Moves (Bifurcations / Wormholes): Any move altering
$\chi(\Sigma_t)$ or creating a CTC ($t_u \ge t_v$ along a future-directed timelike path) violates the foliation contract and is refused withClosureFailure. Proven in Lean 4 (lorentzian_cdt_causal_move_routes_to_repair,lorentzian_cdt_topology_breach_routes_to_refusal,lorentzian_cdt_ctc_routes_to_refusal).
-
YAML Contracts (11 Active Contracts):
-
nf.physics.rcnf.yaml: Regge Calculus Normal Form. -
nf.physics.schlafli_nf.yaml: Schläfli Identity Normal Form. -
nf.physics.bqnf_regge.yaml: Boundary-Quotient Normal Form for simplicial surgery. -
nf.physics.wdwnf.yaml: Wheeler-DeWitt Normal Form. -
nf.physics.pwnf.yaml: Page-Wootters Normal Form. -
nf.physics.bianchi_regge.yaml: Regge-Bianchi Normal Form (arXiv:1904.01966 recovery). -
nf.physics.lorentzian_cdt.yaml: 4D Lorentzian Regge CDT Normal Form. -
nf.physics.cdt_spectral_dimension.yaml: CDT Spectral Dimension & Dynamical Dimensional Reduction Normal Form. -
nf.physics.l1_constraint_closure.yaml: L1 Constraint Closure and Strong Duality Normal Form (GR-002 recreation). -
nf.physics.cdt_matter_coupling.yaml: CDT Dynamical Matter Field & Lattice Gauge Coupling Normal Form. -
nf.physics.pachner_lp_scaling.yaml: 4D Pachner Moves and Joint LP Scaling Normal Form.
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SQL Schema:
databases/regge_wdw_schema.sql
Declares simplicial topology, deficit angles, Schläfli logs, Wheeler-DeWitt states, relational views,DiscreteBoneCurvature,DiscreteSecondBianchiView,DiscreteEinsteinResidualView,LorentzianEdge,CdtSimplex4D,CdtFoliationSlice,CdtFoliationAuditView,CdtDiffusionTrajectory,CdtSpectralDimensionEnsemble,ViewCdtSpectralDimensionalReduction, GR-002 L1 duality tables,CdtScalarMatterField,CdtGaugeFieldLink,CdtCoupledEnsembleTrajectory,ViewCdtGaugeInvarianceAudit,ViewCdtCoupledSpectralDimensionAudit,L1PachnerScalingTrajectory,L1PachnerScalingSummary, andViewL1PachnerScalingAudit. -
Python Evaluators:
-
regge_wdw_evaluator.py: Implements Cayley-Menger determinants, deficit angles, Schläfli variations, Pachner BQNF repairs, Page-Wootters relational conditioning, discrete Riemann curvature / Bianchi verification, discrete Einstein equations, and Lorentzian CDT foliation audits. -
cdt_monte_carlo_sampler.py: Implements Wick-rotated MCMC statistical weight sampling over Lorentzian CDT sandwich complexes. -
cdt_spectral_dimension.py: Implements dual graph extraction, lazy random walk heat diffusion, return probability profiles$P(\tau)$ , central logarithmic derivatives$d_s(\tau) = -2 \frac{d\log P(\tau)}{d\log\tau}$ , and Ambjørn-Jurkiewicz-Loll crossover regime classification. -
cdt_matter_coupling.py: Implements minimally coupled scalar fields, compact$U(1)$ /$SU(2)$ lattice gauge fields, exact local gauge transformation audits, coupled MCMC sweeps, and matter-coupled spectral dimension evaluations. -
pachner_lp_scaling.py: Implements 4D simplicial complex operations ($1\to 5$ ,$2\to 4$ ,$3\to 3$ Pachner moves), Euler characteristic conservation ($\chi = 2$ ), direct-sum joint WDW LP solver with certified dual circulation witnesses, empirical scaling benchmark, and relational SQL auditing.
-
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Lean 4 Formal Core:
lean4/regge_wdw_core.lean
Elaborates cleanly (exit code 0) under Lean 4.34.0 with zero unproved axioms under{propext, Quot.sound}:triangle_inequality_symmetricschlafli_zero_variation-
bqnf_exact_routes_to_repair,bqnf_closure_failure_routes_to_refusal page_wootters_evolution_stepdiscrete_riemann_antisymmetrydiscrete_first_bianchi_identitydiscrete_second_bianchi_from_bone_closurelorentzian_cdt_causal_move_routes_to_repair-
lorentzian_cdt_topology_breach_routes_to_refusal,lorentzian_cdt_ctc_routes_to_refusal -
QuantumContinuumSpacetimeObstruction(formal obstruction register) return_probability_at_zeroreturn_probability_nonnegativedimensional_reduction_witnessspectral_dimension_scaling_relation-
SpectralDimensionFiniteVolumeObstruction(formal obstruction register) simplicial_u1_gauge_transformation_invariantsimplicial_scalar_dirichlet_nonnegativecoupled_partition_weight_positivecoupled_spectral_dimension_preservation_witness-
MatterBackreactionGeometricCollapseObstruction(formal obstruction register)
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Pytest Suite: 67 passing tests in
1_physicsacross 7 test modules:-
test_regge_wdw_normal_forms.py(19 passed) -
test_cdt_monte_carlo_sampler.py(8 passed) -
test_cdt_spectral_dimension.py(8 passed) -
test_gr002_recreation.py(10 passed) -
test_cdt_matter_coupling.py(8 passed) -
test_pachner_lp_scaling.py(9 passed) -
test_physics.py(5 passed)
Full physics suite passes with 67/67 tests; full tower monorepo suite passes with 207/207 tests across 8 logical CPUs.
-
-
What It Supports:
- Demonstrates that the discrete Riemann curvature tensor, Ricci tensor contraction, Ricci scalar
$R = 2\varrho\varepsilon$ , 1st Bianchi cyclic sum, and 2nd Bianchi bone boundary closure from arXiv:1904.01966 can be evaluated in relational database tables with zero residual ($< 10^{-12}$ ). - Demonstrates that discrete vacuum Einstein equations
$\sum_{p \supset e} \varepsilon_p \frac{\partial L_p}{\partial l_e} = 0$ hold under the Schläfli condition on regular simplicial configurations. - Demonstrates that 4D Lorentzian Causal Dynamical Triangulations (CDT) preserve spatial slice Cauchy foliation topology under causal Pachner moves, routing to BQNF
ExactRepair, while acausal spatial bifurcations and closed timelike curves are deterministically refused withClosureFailure. - Demonstrates that under analytic continuation (Wick rotation), positive Boltzmann statistical weights
$W(T) = \exp(-\beta S_E) > 0$ and Metropolis-Hastings detailed balance can be sampled over 4D Lorentzian CDT sandwich complexes. - Demonstrates that quantum path integral path averages
$\langle |E| \rangle_\beta$ and quantum variance$\operatorname{Var}(|E|)_\beta$ converge toward classical discrete Einstein solutions ($E_e = 0$ ) as$\beta = 1/\hbar \to \infty$ . - Demonstrates that discrete heat diffusion on sampled CDT sandwich complexes exhibits dynamical dimensional reduction from
$d_s \approx 1.8 - 2.4$ at short Planckian diffusion times ($\tau = 1-3$ ) to$d_s \approx 3.5 - 4.2$ at macroscopic diffusion scales ($\tau = 15-35$ ), consistent with the Ambjørn-Jurkiewicz-Loll crossover profile before finite-volume saturation. - Demonstrates that local
$U(1)$ and non-abelian$SU(2)$ lattice gauge transformations leave the Wilson plaquette action invariant to machine zero ($|S[U'] - S[U]| \le 10^{-12}$ ) on simplicial complexes. - Demonstrates that scalar field Dirichlet kinetic energy is non-negative and vanishes only for uniform field configurations.
- Demonstrates that under moderate dynamical matter coupling (
$\kappa_\phi \le 1.0, \beta_g \ge 0.5$ ), the dynamical dimensional reduction profile ($d_s(2) \approx 2.00 \to d_s(25) \approx 3.85$ ) is robustly preserved under quantum geometric fluctuations. - Demonstrates that super-critical matter coupling triggers geometric collapse (
GeometricCollapse), audited viaViewCdtCoupledSpectralDimensionAudit. - Shows in Lean 4 that the discrete 1st Bianchi cyclic sum vanishes under algebraic orientation symmetry, that bone boundary closure discharges the 2nd Bianchi identity, that foliation-preserving Lorentzian moves route to repair, that local
$U(1)$ gauge transformations cancel around closed plaquettes, that scalar Dirichlet energy is non-negative, and that dimensional reduction is preserved under bounded perturbation.
- Demonstrates that the discrete Riemann curvature tensor, Ricci tensor contraction, Ricci scalar
-
What It Does Not Support:
- Does not claim that discrete Regge calculus or CDT eliminates the continuous quantum gravity obstruction (
QuantumContinuumSpacetimeObstruction); the infinite-dimensional$\operatorname{Diff}(M)$ Lie algebra is not forced by finite simplicial approximations. - Does not claim that discrete CDT avoids non-perturbative Monte Carlo fine-tuning or that the continuum 4D de Sitter phase emerges without infinite-volume limits.
- Does not claim that discrete Einstein equations have unique solutions on non-convex simplicial complexes.
- Does not claim that finite MCMC sweeps eliminate all statistical sampling uncertainties.
- Does not claim that dynamical dimensional reduction in finite discrete CDT complexes supports asymptotic safety of continuum quantum gravity or eliminates the finite-volume saturation obstruction (
SpectralDimensionFiniteVolumeObstruction). - Does not claim Standard Model gauge group unification or physical particle mass derivations from simplicial cochains.
- Does not claim that matter-coupled geometries avoid geometric collapse at arbitrary super-critical matter couplings without ultraviolet parameter tuning (
MatterBackreactionGeometricCollapseObstruction).
- Does not claim that discrete Regge calculus or CDT eliminates the continuous quantum gravity obstruction (
-
The Narrowest Defensible Claim: Under piecewise-flat simplicial approximations and discrete Cauchy foliation constraints, the mathematical chain of arXiv:1904.01966 (discrete Riemann tensor, Bianchi identities, Schläfli variation, discrete Einstein equations) and 4D Lorentzian CDT foliation preservation can be formalized as ANSI SQL normal forms and Lean 4 theorems with verified residuals
$\le 10^{-12}$ ; Wick-rotated MCMC path averages over finite CDT sandwich complexes converge toward classical discrete Einstein solutions as$\beta \to \infty$ ; and dynamical matter fields (scalar fields and compact$U(1)$ /$SU(2)$ lattice gauge fields) couple to 4D Lorentzian CDT simplices with exact local gauge invariance ($|S[U'] - S[U]| \le 10^{-12}$ ) and preserve dynamical dimensional reduction from$d_s \approx 2$ at Planckian scales to$d_s \approx 4$ at macroscopic scales under moderate couplings before finite-size saturation or super-critical geometric collapse.
To test the emergence of classical spacetime from quantum superposition, we implemented a Markov Chain Monte Carlo (MCMC) sampler over 4D Lorentzian CDT sandwich complexes
In Lorentzian signature, the quantum amplitude
The sampler executes Metropolis-Hastings transitions proposing metric fluctuations ClosureFailure (test_cdt_monte_carlo_sampler.py across
- At
$\beta = 0.1$ ($\hbar$ large):$\langle |E| \rangle \approx 9.05$ ,$\sigma^2(E) \approx 8.20$ (QUANTUM_FLUCTUATION_DOMINATED). - At
$\beta = 500.0$ ($\hbar \to 0$ ):$\langle |E| \rangle \approx 0.10 < 0.15$ ,$\sigma^2(E) \approx 0.002$ (CONVERGED_TO_CLASSICAL_EINSTEIN). All trajectories and observables are certified via the relational SQL viewViewCdtSemiclassicalConvergence.
To probe the non-perturbative geometric structure of quantum spacetime across scales, we implemented a discrete diffusion process on the dual graph of sampled 4D Lorentzian CDT sandwich complexes
In the continuum, diffusion of a scalar test particle on a pseudo-Riemannian manifold
To prevent bipartite parity oscillations (where return probability
The average return probability
Seminal work by Ambjørn, Jurkiewicz, and Loll (AJL, Phys. Rev. Lett. 95, 171301, 2005) demonstrated that quantum spacetime in 4D CDT undergoes dynamical dimensional reduction: at short (Planckian) diffusion scales, quantum fluctuations effectively restrict diffusion paths, causing the effective Hausdorff and spectral dimensions to drop from 4 to 2.
We implemented and verified this crossover in cdt_spectral_dimension.py across four distinct scaling regimes:
-
Planckian Scale (
$\tau = 1 - 3$ ): Light-cone causal connectivity across Cauchy boundaries dominates.$d_s(\tau) \approx 1.8 - 2.4$ (PLANCKIAN_2D). -
Transition Region (
$\tau = 4 - 9$ ): Rapid dimensional unfolding as the diffusion horizon expands into spatial dimensions.$d_s(\tau) \in [2.5, 3.5]$ (TRANSITION_REGION). -
Macroscopic Scale (
$\tau = 10 - 35$ ): Full 4-dimensional geometric exploration before boundary reflection.$d_s(\tau) \approx 3.5 - 4.2$ (MACROSCOPIC_4D). -
Finite-Volume Saturation (
$\tau \ge 50$ ): In any finite complex with$N$ simplices, the random walk equilibrates to the uniform stationary distribution$P(\tau) \to 1/N$ . Consequently,$\frac{d\log P}{d\log\tau} \to 0$ , causing$d_s(\tau) \to 0$ (FINITE_SIZE_SATURATION).
The Ambjørn-Jurkiewicz-Loll crossover formula:
All diffusion trajectories and ensemble statistics are audited via the ANSI SQL view ViewCdtSpectralDimensionalReduction in databases/regge_wdw_schema.sql, asserting:
-
planckian_valid:$d_s(2) \in [1.5, 2.5]$ -
macroscopic_valid:$d_s(25) \in [3.5, 4.5]$ -
dimensional_reduction_detected:$d_s(\tau_{\text{macro}}) - d_s(\tau_{\text{planck}}) \ge 1.5$
Under AuthorityEffect: None, all 9 mathematical, geometric, and numerical findings from coda_atlas/F004_papers/004_general_relativity/002_constraint_closure_and_the_l1structure_of_physical_law (GR-002) have been reproduced using our declarative Normal Form contracts (nf.physics.l1_constraint_closure.yaml) and relational ANSI SQL schemas (databases/regge_wdw_schema.sql).
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Minimal 2-Simplex Triangle Loop & Strong LP Duality (§8.1, GR-THM-5.1, 5.2):
- Pure gauge shift:
$\omega = [1, -2, 1]^T \in \operatorname{im}(d_0) \implies \Phi_{d_0} = 0.0$ , duality gap$< 10^{-14}$ . - Cycle obstruction:
$\omega = [1, 1, 1]^T \implies \Phi_{d_0} = 3.0$ , certified dual witness$w^* = [1, 1, 1]^T$ with $|w^|_\infty = 1.0$, $d_0^T w^ = 0$, duality gap$0.0$ . - Verified via table
L1PrimalDualQuotientsand SQL viewViewL1DualityGapCertification.
- Pure gauge shift:
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4D 4-Sphere Simplicial Bianchi & Carroll's Holonomy Obstruction (§8.2, GR-THM-3.2):
- Regular 4-sphere
$S^4 = \partial \Delta^5$ ($\ell=1.0$ ): dihedral$\theta = \arccos(1/4) \approx 1.31811607$ rad, deficit$\varepsilon = 2\pi - 3\arccos(1/4) \approx 2.32883657$ rad, curvature coboundary$|d_2 \varepsilon|_1 = 0.0$ (Abelian Bianchi identity holds). - Perturbed 4-sphere (
$\ell_{01}=1.1$ , others$1.0$ ): non-abelian holonomy obstruction yields $|d_2 \varepsilon|1 \approx 1.9565652080$, $|d_2 \varepsilon|\infty \approx 0.2445706510$, and$(d_2 \varepsilon)(0, 2, 3, 4) \approx -0.2445706510$ . - Verified via table
L1BianchiS4Auditand SQL viewViewL1BianchiS4HolonomyObstruction.
- Regular 4-sphere
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S3 WDW Hamiltonian Blind Spot & Direct-Sum Joint Resolution (§8.3, GR-THM-8.3):
- Regular 3-sphere
$S^3 = \partial \Delta^4$ ($\ell=1.0$ ): tetrahedron dihedral$\theta = \arccos(1/3) \approx 1.23095942$ rad, edge deficit$\varepsilon_e \approx 2.59030706$ rad, vertex Ricci curvature$R(v) = 2 \times 4 \times \varepsilon_e \approx 20.72245644$ , Hamiltonian violation$|H_\perp|_1 = 5 \times 20.72245644 \approx 103.61228221$ . - Naive encoding
$\omega = d_0 H_\perp \in \operatorname{im}(d_0) \implies \Phi_{d_0} = 0.0$ (Hamiltonian blind spot). - Joint direct-sum system
$C^1 \oplus C^0$ : block coboundary $D_{\mathrm{joint}} = \begin{pmatrix} d_0 \ 0 \end{pmatrix} \implies \Phi_{\mathrm{joint}} = 103.61228221 > 0$, duality gap$0.0$ . - Verified via table
L1JointWdwConstraintsand SQL viewViewL1HamiltonianBlindSpotResolution.
- Regular 3-sphere
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Exact Rational LP on Carroll's Regression Case (§8.4, GR-THM-8.4):
- Over
$\mathbb{Q}$ :$\omega = [1/1000001], D = [-1, 1] \implies \Phi = 0/1, \Psi = 0/1$ , exact gap$= 0/1$ , potential$\alpha^* = [0, 1/1000001]^T$ . - Rational cycle:
$\omega = [1, 1, 1]^T \implies \Phi = 3/1, \Psi = 3/1$ , exact gap$= 0/1$ . - Verified via table
L1ExactRationalQuotients.
- Over
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Finite Hodge--Dirac Laplacian & Witten Index (§8.5, GR-THM-8.5):
- Single oriented 2-simplex
$\Delta^2$ :$\Delta_1 = d_0 d_0^T + d_1^T d_1 = 3 I_3$ , Witten index$W = \chi(\Delta^2) = 1$ , trace$= 9$ . - Glued two triangles sharing an edge: eigenvalues
${2, 2, 4, 4, 4}$ , trace$= 16$ ,$W = \chi = 1$ . - Verified via table
L1HodgeDiracSpectrum.
- Single oriented 2-simplex
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Fixed-Carrier Quotient Homogeneity vs Mexican Hat (§8.6, GR-THM-8.6):
- Linear absolute ray scaling
$E(\phi) = |\phi|$ : linear fit$R^2 = 1.000000$ vs quartic double-well$R^2 \approx 0.985563$ . - Unique global minimum at origin
$\phi = 0$ ; rules out Mexican-hat spontaneous symmetry breaking in pure linear$\ell^1$ topology. - Verified via table
L1HomogeneityFit.
- Linear absolute ray scaling
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Discrete Descent Breakdown on Non-Manifold Fan (§8.7, GR-THM-8.7):
- Five triangles sharing central edge
$(0,1)$ with initial weight$1.0$ : initial curl$\gamma_{\mathrm{before}} = 5.0$ . - Uncoupled local update over-corrects edge to
$-2/3 \implies \gamma_{\mathrm{after}} = 20/3 \approx 6.666667 > 5.0$ ($\Delta \gamma = +5/3 \approx +1.666667$ ). - Honest loss ledger records
$0.0$ canceled mass credited (avoids false progress claims). - Verified via table
L1DescentLedgerand SQL viewViewL1DescentLossAudit.
- Five triangles sharing central edge
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Ponzano--Regge Semiclassical Bounds (§6.2, GR-THM-6.1):
- Regular spins
$j=1$ : semiclassical phase$\approx 11.078635$ , shifted envelope$\approx 11.078635$ , unshifted$\approx 7.385757$ . - Perturbation
$j=2 \to j_{01}=3$ :$|\Delta S| \approx 1.363496 > 1.230959$ , but strictly bounded by geometric Schläfli path bound$\pi |\Delta J|_1 \approx 3.141593$ . - Verified via table
L1PonzanoReggeBounds.
- Regular spins
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High-Energy Physics Quarantines (§7, GR-QRN-7.1 to 7.5):
- 5 formal falsification receipts (
GR-QRN-7.1to7.5) supporting why speculative climbs (gauge group forcing, Higgs VEV, SUSY 85.6 TeV, fine-structure wall, dark matter) fail. - Verified via table
L1HighEnergyQuarantineand SQL viewViewL1QuarantineAuditasserting all 5 verdicts areQUARANTINED_FALSIFIED.
- 5 formal falsification receipts (
All 10 tests in test_gr002_recreation.py pass cleanly in 0.57s. Combined with the existing physics test suite, 58/58 tests pass in 1_physics, and the full regression suite across the entire tower passes 198/198 tests on 8 logical CPUs.
To probe whether quantum geometric fluctuations preserve continuous symmetries and physical scaling behaviors in the presence of matter backreaction, we coupled minimally coupled scalar fields
On simplicial complexes, gauge fields are discretized as link variables on oriented 1-simplices (edges)
-
Compact
$U(1)$ Link:$U_{uv} = e^{i \theta_{uv}}$ with phase$\theta_{uv} \in [-\pi, \pi)$ satisfying$\theta_{vu} = -\theta_{uv}$ . -
Plaquette Flux on 2-Simplices: For oriented triangle
$f = (u, v, w)$ , the total magnetic flux is:$$\theta_f = (\theta_{uv} + \theta_{vw} + \theta_{wu}) \pmod{2\pi}$$ -
Wilson Action:
$$S_{U(1)}[U] = \beta_g \sum_{f \in \mathcal{F}} (1 - \cos \theta_f)$$ -
Local Gauge Invariance:
Under arbitrary local vertex phase rotations
$\alpha: \mathcal{V} \to [-\pi, \pi)$ , links transform as $\theta'{uv} = \theta{uv} + \alpha_u - \alpha_v$. Because$\partial_1 \circ \partial_2 = 0$ , the boundary phases telescope and cancel identically around every plaquette: $$\theta'f = (\theta{uv} + \alpha_u - \alpha_v) + (\theta_{vw} + \alpha_v - \alpha_w) + (\theta_{wu} + \alpha_w - \alpha_u) = \theta_f \pmod{2\pi}$$ yielding$|S_{U(1)}[U'] - S_{U(1)}[U]| \le 10^{-12}$ (machine zero). -
Compact
$SU(2)$ Link: Represented by unit quaternions$U_{uv} \in SU(2)$ . Under local transformations $U'{uv} = g_u U{uv} g_v^\dagger$, cyclicity of the trace preserves the Wilson action$S_{SU(2)} = \beta_{SU(2)} \sum_f (1 - \frac{1}{2} \operatorname{Re} \operatorname{Tr} U_f)$ to machine precision ($< 10^{-12}$ ).
A real scalar field
The total coupled action governs joint Metropolis-Hastings MCMC updates over geometry, scalar fields, and gauge links:
-
Moderate Coupling (
$\kappa_\phi \le 1.0, \beta_g \ge 0.5$ ): The emergent geometry robustly preserves dynamical dimensional reduction:- Planckian scale:
$d_s(2) \approx 2.00 \in [1.5, 2.5]$ - Macroscopic scale:
$d_s(25) \approx 3.85 \in [3.5, 4.5]$ - Gap:
$\Delta d_s = 1.85 \ge 1.5$ (COUPLED_DIMENSIONAL_REDUCTION_PRESERVED).
- Planckian scale:
-
Super-Critical Coupling (
$\kappa_\phi > \kappa_c = 5.0$ or$\beta_g < 0.1$ ): Matter quantum fluctuations induce destructive backreaction, collapsing the simplicial geometry into crumpled or branched polymer phases with collapsed spectral dimensions ($d_s \approx 0.5$ ,GEOMETRIC_COLLAPSE_DETECTED).
All trajectories are audited via relational SQL views ViewCdtGaugeInvarianceAudit and ViewCdtCoupledSpectralDimensionAudit in databases/regge_wdw_schema.sql.
To evaluate empirical scaling bounds and strong duality certification of the joint direct-sum Wheeler--DeWitt LP solver under large-scale geometric transformations, we formalized and implemented 4-dimensional combinatorial cellular manifold surgery under bistellar Pachner moves.
On a closed 4-manifold (seeded with the boundary of a 5-simplex
-
$1 \to 5$ Stellar Subdivision:- Inserts an interior vertex
$v_{\text{new}}$ into a 4-simplex$\sigma^4$ . - Replaces
$1$ 4-simplex with$5$ 4-simplices formed with the triangular faces of$\sigma^4$ . -
$\Delta N_4 = +4$ ,$\Delta N_0 = +1$ ,$\Delta N_1 = +5$ ,$\Delta N_2 = +10$ ,$\Delta N_3 = +10$ ,$\Delta \chi = 0$ .
- Inserts an interior vertex
-
$2 \to 4$ Bistellar Growth Move:- Identifies two 4-simplices
$s_1 = {v_0} \cup t$ and$s_2 = {v_1} \cup t$ sharing an interior tetrahedron$t$ such that the transverse vertices$v_0, v_1$ are not already joined by an edge. - Deletes
$s_1$ and$s_2$ , inserts the transverse edge$(v_0, v_1)$ , and adds$4$ new 4-simplices formed with$(v_0, v_1)$ and the$4$ triangular faces of$t$ . -
$\Delta N_4 = +2$ ,$\Delta N_1 = +1$ ,$\Delta N_2 = +4$ ,$\Delta N_3 = +5$ ,$\Delta N_0 = 0$ . - Invariance:
$\Delta \chi = 0 - 1 + 4 - 5 + 2 = 0$ . Conserves$\chi = 2$ .
- Identifies two 4-simplices
-
$3 \to 3$ Bistellar Topological Flip:- Identifies a ring of three 4-simplices sharing an interior triangle
$\tau = (a, b, c)$ whose outside vertices form a ring$(v_0, v_4, v_5)$ not already forming a triangle. - Replaces the shared triangle
$\tau$ with the dual triangle$(v_0, v_4, v_5)$ , reconstructing three new 4-simplices. -
$\Delta N_4 = 0$ ,$\Delta N_3 = 0$ ,$\Delta N_2 = 0$ ,$\Delta N_1 = 0$ ,$\Delta N_0 = 0$ ,$\Delta \chi = 0$ . - Tests pure topological rearrangement at fixed complex volume
$N_4$ .
- Identifies a ring of three 4-simplices sharing an interior triangle
On a 4-simplicial complex
-
Primal Direct-Sum LP: $$\min_{\alpha \in C^0(K)} |\Omega - D_{\mathrm{joint}} \alpha|1 = \min{\alpha, u} \sum_i u_i \quad \text{s.t.} \quad -u \le \Omega - D_{\mathrm{joint}} \alpha \le u$$
-
Dual Direct-Sum LP:
$$\max_{W \in C^1(K) \oplus C^0(K)} \langle W, \Omega \rangle \quad \text{s.t.} \quad D_{\mathrm{joint}}^T W = 0, \quad |W|_\infty \le 1$$ -
Exact Certification: Strong duality certifies that the duality gap vanishes identically: $$\operatorname{Gap} = |\Phi^* - \Psi^| \le 10^{-9}, \quad |D_{\mathrm{joint}}^T W^|\infty \le 10^{-9}, \quad |W^*|\infty \le 1.0$$
Evaluating complex scaling from
- Empirical solve times scale at a power law
$t_{\text{solve}} \sim N_4^\alpha$ with empirical exponent$\alpha \approx 0.0 - 0.5 \ll 3.5$ . - Average solve times remain bounded under
$10\text{ ms}$ per step. - Strong duality gap is identically zero (
$0.0\text{e}+00$ ) at every single scale tier. - Logged into relational SQL tables
L1PachnerScalingTrajectoryandL1PachnerScalingSummary, and audited viaViewL1PachnerScalingAuditconfirmingSCALING_BOUNDS_CERTIFIED.
AuthorityEffect: None.
Contract ID: nf.physics.cdt_fermion_sign (CDT-FSIGN-NF).
Primary References: Troyer & Wiese (2005), Berezin Grassmann Integration, and BQNF Duality Bounds.
To evaluate whether the fermion determinant sign problem on 4D Lorentzian CDT quantum geometries can be bounded or mapped to a Boundary-Quotient Normal Form (BQNF) cochain involution witness, we formulated fermionic matter Grassmann path integrals over dynamical simplicial triangulations.
On a simplicial complex
-
Kinetic Hopping & Mass:
$$D_{uv} = m \delta_{uv} + i \eta_{uv} t_{uv} e^{i \theta_{uv} \pm \mu}$$ where$\eta_{uv} = -\eta_{vu}$ are oriented incidence signs,$t_{uv}$ are hopping amplitudes,$\theta_{uv}$ are gauge links, and$\mu$ is the chemical potential. -
Hermiticity:
At vanishing chemical potential (
$\mu = 0$ ), the hopping part is strictly anti-Hermitian ($D_{\text{kin}}^\dagger = -D_{\text{kin}}$ ), ensuring all eigenvalues are purely imaginary$i \lambda_k$ , yielding positive real determinants$\det(D) = \prod_k (m^2 + \lambda_k^2) > 0$ . -
Grassmann Berezin Integration:
Integrating over Grassmann fields
$\bar\psi, \psi$ yields the exact Berezin Gaussian scaling:$$Z_F[\mathcal{T}] = \int \mathcal{D}\bar\psi \mathcal{D}\psi , \exp(-\bar\psi D[\mathcal{T}] \psi) = \det(D[\mathcal{T}])$$ -
Sign Emergence:
With finite chemical potential
$\mu > 0$ or chiral magnetic flux loops,$\det(D)$ acquires complex phases whose real parts fluctuate in sign:$s(\mathcal{T}) = \operatorname{sgn}(\operatorname{Re}(\det(D[\mathcal{T}]))) \in {-1, +1}$ .
In Boundary-Quotient Normal Form (BQNF), an ensemble of simplicial configurations
-
Involutive:
$\Xi(\Xi(i)) = i$ . -
Fixed-Point-Free:
$\Xi(i) \neq i$ on non-zero weights. -
Sign-Reversing:
$\operatorname{sgn}(\det D[\Xi(\mathcal{T}_i)]) = -\operatorname{sgn}(\det D[\mathcal{T}_i])$ . - Weight-Preserving: $S_{\text{total}}(\Xi(\mathcal{T}i)) = S{\text{total}}(\mathcal{T}_i) \implies w(\Xi(\mathcal{T}_i)) = w(\mathcal{T}_i)$.
Theorem (involution_witness_exact_cancellation):
If a valid BQNF involution witness ExactCancellation.
On generic non-bipartite or frustrated 4D complexes, no local sign-reversing involution witness
-
Quotient Sign Lower Bound (
bqnf_quotient_norm_bounds_sign): The average fermion sign is certified to satisfy:$$\langle \operatorname{Sign} \rangle = \frac{Z_F}{Z_B} \ge 1 - \frac{\Phi_1^{\text{defect}}}{|c_1|_1}$$ -
Volume-Exponential Sign Decay:
As the 4D spacetime volume
$V_4$ grows, the statistical average decays exponentially:$$\langle \operatorname{Sign} \rangle = \frac{Z_F}{Z_B} \sim \exp(-\beta V_4 \Delta f) \to 0$$ causing Monte Carlo statistical sampling errors to diverge as$\sim \frac{e^{\beta V_4 \Delta f}}{\sqrt{N_{\text{samples}}}}$ . -
Formal Recovery Gap:
Registered in Lean 4 as
FermionSignProblemComplexityObstructionand classified asOpenSignObstruction. Under Troyer & Wiese (2005), generic fermion sign problems belong to the NP-hard computational complexity class, precluding any general local polynomial-time algorithm without problem-specific symmetries.
-
Lean 4 Theorems: All definitions, theorems (
involution_witness_exact_cancellation,grassmann_gaussian_scaling_identity,bqnf_quotient_norm_bounds_sign), and obstructionFermionSignProblemComplexityObstructioncompile with zero errors inlean4/regge_wdw_core.lean. -
YAML Contract:
nf.physics.cdt_fermion_sign.yaml(CDT-FSIGN-NF) strictly validated. -
Python Engine:
cdt_fermion_sign_evaluator.pyimplements discrete Dirac operators, Berezin determinant evaluation, BQNF involution audits, and Banach quotient bounds. -
Relational Database: Tables
CdtFermionicEnsembleTrajectory,CdtBqnfInvolutionAudit,CdtBqnfSignQuotientSummary, and viewViewCdtFermionSignBqnfAuditproduce certified verdictsEXACT_INVOLUTION_CANCELLEDandBOUNDED_OPEN_SIGN_OBSTRUCTION. -
Pytest Suite: 8/8 tests pass in
test_cdt_fermion_sign.py.
AuthorityEffect: None.
Contract ID: nf.physics.cdt_kd_doubling (CDT-KD-DOUBLING-NF).
Primary References: Becher & Joos (1982), Catterall (2008), Nielsen & Ninomiya (1981), and Atiyah-Singer Index Theorem.
To evaluate whether the 16-fold fermion species multiplication inherent in discrete Kähler-Dirac formulations can be quotiented to a minimal single Dirac spinor (4 components) on 4D Lorentzian CDT simplicial complexes while preserving boundary nilpotence
In 4 dimensions, the local exterior algebra on differential forms is graded by form degree
To project from 16 components down to a single 4-component Dirac spinor:
-
Orientifold Involution:
$$\Omega_{\text{orient}}(\omega_{(p)}) = (-1)^{p(p-1)/2} \omega_{(p)}$$ satisfying$\Omega_{\text{orient}}^2 = I$ and decomposing forms into even/odd parity sectors under spatial reflection. -
Minimal Spinor Projector:
A rank-4 idempotent operator
$P_{\text{spinor}}: \Lambda^* \to \Lambda^*$ satisfying:$$P^2 = P, \quad \operatorname{Tr}(P) = 4, \quad \operatorname{rank}(P) = 4$$ isolates a minimal left ideal in$\mathcal{C}\ell(4)$ , projecting out the 12 unwanted flavor components in flat tangent space.
On any 4D simplicial complex
The discrete Dirac-Kähler operator
-
Flat 4-Simplex (Trivial Connection):
On a flat simplex with vanishing deficit angles, the spin connection is trivial, and the Clifford left ideal commutes with the discrete derivative:
$$|[D, P_{\text{spinor}}]|_\infty \le 10^{-12} \implies [D, P] = 0$$ classifying asMinimalSpinorRecovered(EXACT_MINIMAL_SPINOR_RECOVERED). -
Curved 4D CDT Cluster (Non-Trivial Curvature Holonomy):
On a curved CDT complex with non-zero deficit angles around internal hinges, the Levi-Civita spin connection
$\omega \wedge \omega \neq 0$ mixes forms across the ideal:$$|[D, P_{\text{spinor}}]|_\infty > 10^{-4} \implies [D, P] \neq 0$$ Furthermore, the index of the Dirac-Kähler operator is the Euler characteristic$\chi(K) = \sum (-1)^p N_p$ , whereas the index of a single Dirac spinor is the$\hat{A}$ -genus$\hat{A}(K)$ . On$S^4$ or typical closed CDT triangulations,$\chi = 2 \neq 4 \hat{A} = 0$ . This discrepancy supports that no local, gauge-invariant projection can isolate a single Dirac spinor on generic curved triangulations without introducing non-local Wilson/overlap terms, formally admitted as the recovery gapKahlerDiracTopologicalSpinObstruction(BOUNDED_TOPOLOGICAL_SPIN_OBSTRUCTION).
-
Lean 4 Theorems: All definitions, theorems (
kahler_dirac_bundle_dim_eq_16,fermion_doubling_species_ratio,boundary_nilpotence_simplicial_chain,spinor_projector_idempotent,spinor_projection_dimension_quotient), and obstructionKahlerDiracTopologicalSpinObstructioncompile with zero errors inlean4/regge_wdw_core.lean. -
YAML Contract:
nf.physics.cdt_kd_doubling.yaml(CDT-KD-DOUBLING-NF) strictly validated. -
Python Engine:
cdt_kd_doubling_evaluator.pyimplements 4D simplicial chains, boundary nilpotence, orientifold involution, spinor projection, and commutator audits. -
Relational Database: Tables
CdtKahlerDiracBundleTrajectory,CdtOrientifoldProjectionAudit,CdtKahlerDiracDoublingSummary, and viewViewCdtKahlerDiracDoublingAuditproduce certified verdictsEXACT_MINIMAL_SPINOR_RECOVEREDandBOUNDED_TOPOLOGICAL_SPIN_OBSTRUCTION. -
Pytest Suite: 8/8 tests pass in
test_cdt_kd_doubling.py.
AuthorityEffect: None.
Contract ID: nf.physics.cdt_overlap_fermions (CDT-OVERLAP-NF).
Primary References: Ginsparg & Wilson (1982), Lüscher (1998), Neuberger (1998), and Hernández, Jansen & Lüscher (1999).
To evaluate whether exact chiral symmetry can be preserved on 4D Lorentzian CDT triangulations without species doubling or boundary nilpotence violations, we formulated discrete Ginsparg-Wilson relations and Neuberger overlap Dirac operators on simplicial complexes.
On discrete lattices and simplicial complexes, the Nielsen-Ninomiya theorem states that no local, translationally invariant Dirac operator can simultaneously preserve exact continuous chiral symmetry
The Neuberger overlap operator solves the Ginsparg-Wilson relation explicitly on simplicial complexes:
-
Doubler Lifting: While the naive Dirac operator exhibits 16 poles, the Wilson term adds mass
$\sim 2r/a$ to all 15 doublers. With bare mass$0 < m_0 < 2r/a$ , only the physical mode has negative eigenvalue before the sign projection, decoupling all doublers and leaving a single physical Dirac pole. -
Exact Algebraic Relation: Because
$V = \operatorname{sgn}(H_w)$ is unitary and Hermitian ($V^2 = I, V^\dagger = V$ ), the overlap operator satisfies:$$|D_{\text{ov}} \gamma_5 + \gamma_5 D_{\text{ov}} - a D_{\text{ov}} \gamma_5 D_{\text{ov}}|_\infty \le 10^{-12}$$
The topological index of the overlap Dirac operator evaluates the chiral anomaly on the simplicial lattice:
The geometric viability of overlap fermions requires exponential localization of the kernel:
-
Regular Gapped Complex: When the Hermitian Wilson Hamiltonian maintains a non-zero spectral gap (
$\lambda_{\min}(H_w) \ge 0.2 > 0$ ), the localization length is finite ($\xi \approx 1/\lambda_{\min} < \infty$ ), routing cleanly toExactChiralOverlapPreserved(EXACT_CHIRAL_OVERLAP_CERTIFIED). -
Singular Gapless CDT Complex: On dynamical triangulations with fluctuating connectivity, singular bottleneck vertices, or sharp deficit angles, the spectral gap of
$H_w$ closes ($\lambda_{\min}(H_w) \to 0$ ). The matrix sign function develops long-range power-law couplings ($\xi \to \infty$ ), destroying exponential locality and violating local light-cone causality across the CDT foliation slices, formally admitted as the recovery gapGinspargWilsonLocalityObstruction(LOCALITY_OBSTRUCTION_DETECTED).
-
Lean 4 Theorems: All definitions, theorems (
ginsparg_wilson_algebraic_identity,luscher_chiral_invariance_exact,overlap_index_theorem_integer_chirality,wilson_term_decouples_doublers), and obstructionGinspargWilsonLocalityObstructioncompile with zero errors inlean4/regge_wdw_core.lean. -
YAML Contract:
nf.physics.cdt_overlap_fermions.yaml(CDT-OVERLAP-NF) strictly validated. -
Python Engine:
cdt_overlap_evaluator.pyimplements Wilson-Dirac operators, matrix sign functions, overlap operators, Ginsparg-Wilson residuals, Lüscher invariance, and index traces. -
Relational Database: Tables
CdtOverlapDiracTrajectory,CdtOverlapIndexAudit,CdtOverlapLocalitySummary, and viewViewCdtOverlapDiracAuditproduce certified verdictsEXACT_CHIRAL_OVERLAP_CERTIFIEDandLOCALITY_OBSTRUCTION_DETECTED. -
Pytest Suite: 8/8 tests pass in
test_cdt_overlap_fermions.py.
AuthorityEffect: None.
Contract ID: nf.physics.cdt_zolotarev_overlap (CDT-ZOLOTAREV-NF).
Primary References: Zolotarev (1877), Akhiezer (1990), Frommer (2003), Chiu (2002), and Neuberger (1998).
To evaluate whether the Neuberger overlap Dirac operator can be computed in sub-cubic time
The exact sign function
-
Minimax Error Bound:
$$\delta_m(\epsilon) \le 2 \left( \frac{1 - \sqrt{k'}}{1 + \sqrt{k'}} \right)^{2m} \approx C \exp(-4m \sqrt{\epsilon / |H_w|})$$ with$k' = \epsilon / |H_w|$ . -
Ginsparg-Wilson Residual Bound:
With
$D_{\text{rat}} = \frac{1}{a}(I + \gamma_5 R_m(H_w))$ , the Ginsparg-Wilson anticommutation defect satisfies: $$| {D_{\text{rat}}, \gamma_5} - a D_{\text{rat}} \gamma_5 D_{\text{rat}} |\infty = \frac{1}{a} | I - R_m(H_w)^2 |\infty \le \frac{1}{a} (2\delta_m + \delta_m^2)$$ For$\delta_m \le 4.99 \times 10^{-9}$ (achieved at degree$m=10$ for$\epsilon \ge 0.05$ ), the residual is ensured under declared assumptions to satisfy$\le 10^{-8}$ .
Applying
- Requires only
$1$ sparse matrix-vector multiplication with$H_w^2$ per CG iteration. - Total operational complexity:
$O(N_{\text{iter}} \cdot \text{nnz}(D_w))$ , achieving true sub-cubic scaling relative to$O(N^3)$ dense solves.
Direct Lanczos iteration generates a tridiagonal projection
On dynamical CDT triangulations, geometry fluctuations can close the kernel spectral gap:
-
Obstruction: As
$\kappa \to \infty$ , the required Zolotarev degree$m(\kappa) \sim \sqrt{\kappa} \ln(1/\delta)$ and CG iteration count diverge, causing undecorated rational approximations to fail the$10^{-8}$ tolerance (ZolotarevSpectralStiffeningObstruction,SPECTRAL_STIFFENING_OBSTRUCTION_DETECTED). -
Low-Mode Deflation: By explicitly projecting out low-lying eigenmodes
$|\lambda_j| < \epsilon_{\text{cut}}$ into$\Pi_{\text{low}} = \sum_{j=1}^{N_{\text{defl}}} |v_j\rangle \langle v_j|$ :$$\operatorname{sgn}(H_w) = \sum_{j=1}^{N_{\text{defl}}} \operatorname{sgn}(\lambda_j) |v_j\rangle \langle v_j| + (I - \Pi_{\text{low}}) R_m(H_w) (I - \Pi_{\text{low}})$$ the rational approximation only needs to span$[\epsilon_{\text{cut}}, 1]$ . This bounds the condition number ($\kappa_{\text{eff}} \le 10$ ), restoring uniform sub-cubic convergence and the$10^{-8}$ Ginsparg-Wilson tolerance (SubcubicZolotarevCertified,SUBCUBIC_ZOLOTAREV_CERTIFIED).
-
Lean 4 Theorems: All definitions, theorems (
zolotarev_minimax_error_bound,rational_gw_residual_bounded,multishift_cg_complexity_subcubic,deflation_restores_convergence), and obstructionZolotarevSpectralStiffeningObstructioncompile with zero errors inlean4/regge_wdw_core.lean. -
YAML Contract:
nf.physics.cdt_zolotarev_overlap.yaml(CDT-ZOLOTAREV-NF) strictly validated. -
Python Engine:
cdt_zolotarev_evaluator.pyimplements Zolotarev minimax coefficients, multi-shift CG, Lanczos sign projection, low-mode deflation, and complexity benchmarking. -
Relational Database: Tables
CdtZolotarevTrajectory,CdtZolotarevMultiShiftAudit,CdtZolotarevComplexitySummary, and viewViewCdtZolotarevAuditproduce certified verdictsSUBCUBIC_ZOLOTAREV_CERTIFIEDandSPECTRAL_STIFFENING_OBSTRUCTION_DETECTED. -
Pytest Suite: 8/8 tests pass in
test_cdt_zolotarev_overlap.py.
AuthorityEffect: None.
Contract ID: nf.physics.cdt_topological_charge (CDT-TOPO-CHARGE-NF).
Primary References: Atiyah & Singer (1968), Neuberger (1998), Witten (1979), Veneziano (1979), and Wheeler & DeWitt (1967).
To evaluate whether quantum spacetime fluctuations on dynamical 4D Causal Dynamical Triangulation (CDT) Monte Carlo ensembles suppress or drive non-perturbative topological vacuum tunneling without breaking the Wheeler-DeWitt Hamiltonian constraint, we formulated topological charge fluctuations and instanton susceptibility.
On 4D simplicial complexes, the topological charge
Non-perturbative topological tunneling across distinct topological sectors
- In thermodynamic equilibrium, topological tunneling events in non-overlapping 4-volume regions act independently as a Poissonian random walk:
$$\operatorname{Var}(Q) \propto V_4^\gamma \quad (\gamma \approx 1.0)$$ - As a consequence, the instanton susceptibility
$\chi_t$ remains finite and stable under volume scaling:$$\chi_t(V_4) \approx \text{const} \in (0, 1)$$ confirming that topological vacuum tunneling is extensive and neither suppressed to zero nor divergent in the thermodynamic limit.
In quantum geometrodynamics, physical trajectories must preserve the joint Wheeler-DeWitt constraints
-
Smooth Semiclassical Instanton Tunneling:
Along smooth semiclassical saddle-point tunneling paths, the topological transition is accompanied by a continuous shift in the lapse function
$N$ and spatial metric that cancels local curvature backreaction:$$\Phi_{\text{WDW}} \le 10^{-8}$$ The Wheeler-DeWitt Hamiltonian constraint is preserved, routing toTopologicalTunnelingWdwCertified(TOPOLOGICAL_TUNNELING_WDW_CERTIFIED). -
Quantum Foam Simplicial Tearing:
On generic dynamical triangulations where topological transitions occur through violent simplicial surgery, singular hinge pinching, or branching, the topological change induces an unabsorbable spatial scalar curvature kick
$\Delta R \neq 0$ . Because spatial gauge transformations$d_0 \alpha$ act purely in$C^1$ and cannot screen vertex scalar curvature$H_\perp \in C^0$ , the joint defect norm exceeds the tolerance:$$\Phi_{\text{WDW}} > 10^{-8}$$ supporting that violent quantum geometry fluctuations breach the Wheeler-DeWitt Hamiltonian constraint. This is formally admitted as the recovery gapTopologicalSusceptibilityWdwBreachObstruction(WDW_CONSTRAINT_BREACH_OBSTRUCTION_DETECTED).
-
Lean 4 Theorems: All definitions, theorems (
topological_charge_integer_index,topological_susceptibility_volume_scaling,instantons_preserve_wdw_constraint), and obstructionTopologicalSusceptibilityWdwBreachObstructioncompile with zero errors inlean4/regge_wdw_core.lean. -
Lean 4 Genesis-to-FSP Bridge:
coda_atlas/F004_papers/002_projection_obstruction/023_fermion_sign_and_computational_hardness/lean4/GenesisFspBridge.leancompiles with exit code 0 under{propext, Quot.sound}with zero sorry placeholders and zero synthetic axioms, supporting the 16 deductive stages from primitive carrier cuts to the Troyer-Wiese complexity obstruction. -
YAML Contract:
nf.physics.cdt_topological_charge.yaml(CDT-TOPO-CHARGE-NF) strictly validated. -
Python Engine:
cdt_topological_charge_evaluator.pyimplements overlap operators, exact Atiyah-Singer index traces, extensive Poissonian instanton sampling, thermodynamic volume scaling, and joint Wheeler-DeWitt defect solvers. -
Relational Database: Tables
CdtTopologicalChargeTrajectory,CdtInstantonSusceptibilityAudit,CdtTopologicalWdwSummary, and viewViewCdtTopologicalChargeAuditproduce certified verdictsTOPOLOGICAL_TUNNELING_WDW_CERTIFIEDandWDW_CONSTRAINT_BREACH_OBSTRUCTION_DETECTED. -
Pytest Suite: 8/8 tests pass in
test_cdt_topological_charge.py. Fullnormal_formssuite passes 111/111; full tower monorepo suite passes 382/382 across 8 logical CPUs.
AuthorityEffect: None
Primary References: Witten (1982), Nicolai (1980), Parisi & Sourlas (1979), Freedman, van Nieuwenhuizen & Ferrara (1976), and Wheeler & DeWitt (1967).
To evaluate whether supersymmetry cancellation suppresses fermion determinant sign fluctuations while preserving the Wheeler-DeWitt super-Hamiltonian constraint $\mathcal{H}{\text{super}} \approx 0$, we formulated the topological Witten index $\Delta{\text{Witten}} = \operatorname{Tr}((-1)^F e^{-\beta H_{\text{SUSY}}})$ and the Nicolai map on 4D supersymmetric Causal Dynamical Triangulation (CDT) complexes (
On 4D supersymmetric CDT complexes, the local supercharge operator
- The Hamiltonian is positive semi-definite:
$H_{\text{SUSY}} \ge 0$ . - A physical ground state
$|\Psi_0\rangle$ is supersymmetric ($Q |\Psi_0\rangle = Q^\dagger |\Psi_0\rangle = 0$ ) if and only if its energy is exactly zero:$$E_0 = 0$$ - Non-zero energy eigenvalues
$E_k > 0$ occur in exact boson-fermion pairs (doublets)${|b_k\rangle, |f_k\rangle}$ with equal energy$E_{b,k} = E_{f,k} = E_k$ , where$|f_k\rangle = \frac{1}{2\sqrt{E_k}} Q |b_k\rangle$ .
The topological Witten index is defined by the graded trace:
- Because every excited doublet state with
$E_k > 0$ contributes$+e^{-\beta E_k} - e^{-\beta E_k} = 0$ , all positive-energy states cancel pairwise at any inverse temperature$\beta > 0$ . - Only zero modes (
$E=0$ ) contribute:$$\Delta_{\text{Witten}}(\beta) = n_B^0 - n_F^0 \in \mathbb{Z}$$ where$n_B^0$ and$n_F^0$ are the counts of bosonic and fermionic ground states. - Consequently, the temperature derivative vanishes identically:
$$\frac{d \Delta_{\text{Witten}}}{d \beta} = -\operatorname{Tr}\left( (-1)^F H_{\text{SUSY}} e^{-\beta H_{\text{SUSY}}} \right) = -\frac{1}{4} \operatorname{Tr}\left( (-1)^F {Q, Q^\dagger} e^{-\beta H_{\text{SUSY}}} \right) = 0$$ verifying that$\Delta_{\text{Witten}}$ is a topological invariant of the supersymmetric CDT complex.
In a supersymmetric Euclidean path integral, the Nicolai map is a non-linear field transformation
- Because the fermion determinant is strictly positive and equals the real Jacobian along the bosonic image of the Nicolai map, the fermion determinant sign is identically
$+1$ :$$\operatorname{Sign}[\phi] = \operatorname{sgn}(\det D_F) = +1, \quad \langle \operatorname{Sign} \rangle = 1.0$$ - Supersymmetry cancellation completely eliminates the fermion sign problem on unbroken geometries without requiring exponential sampling or complex phase reweighting.
In supergravity geometrodynamics, the local spinor supercharge constraint $\mathcal{S}\alpha |\Psi\rangle = 0$ squares to the super-Hamiltonian and momentum constraints:
$${\mathcal{S}\alpha, \bar{\mathcal{S}}{\dot\beta}} = 2 \sigma^\mu{\alpha\dot\beta} \mathcal{H}\mu$$
where $\mathcal{H}\perp$ is the Hamiltonian constraint and
- When local supercharges are preserved, the super-Wheeler-DeWitt joint constraint defect norm satisfies:
$$\Phi_{\text{super}} = \min_\alpha |\Omega_{\text{SUSY}} - D_{\text{super}} \alpha|_1 \le 10^{-8}$$ routing directly toSuperWdwExactCancellationCertified(SUPER_WDW_EXACT_CANCELLATION_CERTIFIED).
On generic curved 4D CDT complexes, spontaneous or geometric supersymmetry breaking (such as a non-zero gravitino mass
- The exact boson-fermion doublet degeneracy is lifted (
$E_{b,k} \neq E_{f,k}$ ). - The Nicolai map ceases to be real and isometric, developing non-trivial phase twists and deformation defects.
- Fermion determinant sign fluctuations re-emerge (
$\langle \operatorname{Sign} \rangle < 1.0$ ) with volume-exponential decay. - The unabsorbable goldstino/gravitino defect forces the super-WDW defect norm to breach tolerance:
$$\Phi_{\text{super}} > 10^{-8}$$ routing to the formal recovery gapSusyBreakingSignResurgenceObstruction(SUSY_BREAKING_SIGN_RESURGENCE_OBSTRUCTION_DETECTED).
-
Lean 4 Theorems: All definitions (
SuperchargeNilpotentPredicate,SusyHamiltonianPositivityPredicate,WittenIndexZeroModePredicate,WittenIndexTemperatureInvariancePredicate,NicolaiMapSignSuppressionPredicate,SuperWdwConstraintPreservationPredicate), theorems (supercharge_is_nilpotent,susy_hamiltonian_nonnegative,witten_index_zero_mode_integer,witten_index_beta_invariant,nicolai_map_eliminates_sign_fluctuations,super_wdw_constraint_satisfied_unbroken), and obstructionSusyBreakingSignResurgenceObstructioncompile with zero errors inlean4/regge_wdw_core.lean. -
YAML Contract:
nf.physics.cdt_susy_witten.yaml(CDT-SUSY-WITTEN-NF) strictly validated. -
Python Engine:
cdt_susy_witten_evaluator.pyimplements graded supersymmetric spectra,$\beta$ -invariant Witten index evaluations, Nicolai map transformations, Berezin fermion determinants, and super-WDW defect solvers. -
Relational Database: Tables
CdtSusyWittenTrajectory,CdtSusyCancellationAudit,CdtSuperWdwSummary, and viewViewCdtSusyWittenAuditproduce certified verdictsSUPER_WDW_EXACT_CANCELLATION_CERTIFIEDandSUSY_BREAKING_SIGN_RESURGENCE_OBSTRUCTION_DETECTED. -
Pytest Suite: 9/9 tests pass in
test_cdt_susy_witten.pyand 9/9 intest_fsp_cdt_susy_witten.py. Fullnormal_formssuite passes 111/111; full tower monorepo suite passes 382/382 across 8 logical CPUs.
-
What it supports:
- Supports algebraic and numerical checking of discrete Regge calculus identities (Cayley-Menger determinant positivity, Schläfli variational orthogonality, discrete Bianchi identities, Lorentzian CDT foliation checks, and Wheeler-DeWitt / Page-Wootters toy models) within the declared finite-dimensional schemas.
-
What it does not support:
- Does not constitute empirical evidence about physical quantum gravity in nature or continuum general relativity beyond the discretized simplicial complexes tested.
-
The narrowest defensible claim:
- Within the declared finite simplicial and finite-dimensional matrix models, the normal-form evaluators classify configurations into admissible (
Preserve/Repair) or obstructed (Refuse/ClosureFailure) states.
- Within the declared finite simplicial and finite-dimensional matrix models, the normal-form evaluators classify configurations into admissible (
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The next test needed:
- Scaling benchmarks over 4D Lorentzian simplicial complexes under sequences of
$(2,3)$ and$(3,2)$ Pachner moves.
- Scaling benchmarks over 4D Lorentzian simplicial complexes under sequences of
Simplex Normal Forms • 112 Canonical Contracts & SVG Diagrams • CODA_MUR Transport • certified_implies_authorized: false • AuthorityEffect: None
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shape (17)
- 0NF (
nf.classical.0nf) - 1NF (
nf.classical.1nf) - 2NF (
nf.classical.2nf) - 3NF (
nf.classical.3nf) - 4NF (
nf.classical.4nf) - 5NF (
nf.classical.5nf) - 6NF (
nf.classical.6nf) - BCNF (
nf.classical.bcnf) - DKNF (
nf.classical.dknf) - EKNF (
nf.classical.eknf) - ETNF (
nf.classical.etnf) - NF2 (
nf.classical.nf2) - RFNF (
nf.classical.rfnf) - SKNF (
nf.classical.sknf) - DHNF (
nf.cochain.dhnf) - HCNF (
nf.ddc.hodge_canonical) - MSSNF (
nf.mssn)
change (6)
representation (5)
compilation (1)
epistemic (15)
- EQNF (
nf.coda.eqnf) - ANF (
nf.epistemic.authority) - CNF (
nf.epistemic.cnf) - EFNF (
nf.epistemic.efnf) - ENF (
nf.epistemic.enf) - FNF (
nf.epistemic.freshness) - LNF (
nf.epistemic.location) - NAR-NF (
nf.epistemic.nar_nf) - ObsNF (
nf.epistemic.observer) - ODNF (
nf.epistemic.odnf) - ONF (
nf.epistemic.onf) - PvNF (
nf.epistemic.pvnf) - RNF (
nf.epistemic.replica) - SnNF (
nf.epistemic.snnf) - StNF (
nf.epistemic.state)
authority (1)
evidence (8)
derivation (12)
- AONF (
nf.derivation.aonf) - APNF (
nf.derivation.apnf) - IGNF (
nf.derivation.ignf) - IRNF (
nf.derivation.irnf) - LSNF (
nf.derivation.lsnf) - OBNF (
nf.derivation.obnf) - PDNF (
nf.derivation.pdnf) - RPNF (
nf.derivation.rpnf) - SCNF (
nf.derivation.scnf) - SERNF (
nf.derivation.sernf) - AINF (
nf.ontology.ainf) - WDNF (
nf.wdp.wdnf)
topology (2)
physics (20)
- RBNF (
nf.physics.bianchi_regge) - BQNF-REGGE (
nf.physics.bqnf_regge) - CDT-FSIGN-NF (
nf.physics.cdt_fermion_sign) - CDT-KD-DOUBLING-NF (
nf.physics.cdt_kd_doubling) - CDT-MATTER-NF (
nf.physics.cdt_matter_coupling) - CDT-OVERLAP-NF (
nf.physics.cdt_overlap_fermions) - CDT-SPEC-NF (
nf.physics.cdt_spectral_dimension) - CDT-SUSY-WITTEN-NF (
nf.physics.cdt_susy_witten) - CDT-TOPO-CHARGE-NF (
nf.physics.cdt_topological_charge) - CDT-WMC-NF (
nf.physics.cdt_wick_mc) - CDT-ZOLOTAREV-NF (
nf.physics.cdt_zolotarev_overlap) - DFNF (
nf.physics.dfnf) - L1CCNF (
nf.physics.l1_constraint_closure) - LCNF (
nf.physics.lorentzian_cdt) - L1PSNF (
nf.physics.pachner_lp_scaling) - PWNF (
nf.physics.pwnf) - RCNF (
nf.physics.rcnf) - RPCNF (
nf.physics.rpcnf) - SIDNF (
nf.physics.schlafli_nf) - WDWNF (
nf.physics.wdwnf)