qftt-wesh

Canonical repository comprising the full manuscript, formal verification (Lean 4), and experimental validation datasets for the framework:

"Quantum Field Theory of Time (QFTT), the Weak Entanglement Symmetry Hypothesis (WESH) and Emergent Spacetime" — L. Casagrande (2026).

The manuscript introduces a Quantum Field Theory of Time where time is promoted to a local quantum field operator T(x). Physical time does not pre‑exist but is generated by eigentime events occurring at an intensity Γ[ρ] > 0, with

dt/ds = Γ[ρ(s)],

so that the observable time t emerges from the dynamics of the state ρ. Spatial geometry is induced indirectly: the pattern of eigentime events, weighted by entanglement and correlated by a causal kernel, seeds the smooth four-dimensional spacetime reconstructed at the infrared fixed point.

A Weak Entanglement Symmetry Hypothesis (WESH) is imposed at the level of completely positive, trace‑preserving (CPTP) maps. A generator–level Noether constraint (WESH–Noether) selects a covariant GKSL structure with

• a quadratic local dissipator D[T^2(x)],
• a bilocal channel L_xy = T^2(x) − T^2(y) gated by a Rényi‑2 entanglement functional and a causal kernel K_ξ(σ),

and defines an emergent time–sector metric

g^(T)_μν = ζ^{-1} ⟨ ∂_μ T ∂_ν T ⟩.

At the stationary WESH bootstrap fixed point the time field aligns with the gravitational potential, ∂_μ T ∝ ∂_μ Φ, yielding Einstein–Hilbert gravity with O(1/N) corrections and reproducing black–hole thermodynamics

S_BH = A/(4 L_P^2) + γ log(A/L_P^2) + …

from the same dynamics, without additional thermal ansatz.

Shot noise from the discrete eigentime substrate propagates to the cosmological constant as δΛ ~ 1/√V₄; at the Hubble scale (V₄ ~ H⁻⁴) this yields

Λ ~ H²

matching the observed order of magnitude without anthropic selection or fine tuning.

---

Formal Verification (Lean 4)

The core mathematical results of QFTT‑WESH have been formally verified in Lean 4 using the Mathlib 4 library. All files compile and type‑check cleanly. Axioms are limited to standard textbook results and to QFTT–WESH-specific content derived elsewhere in the Lean corpus.

File	Content	Key results
formal-verification/Section1.lean	Section 1 (QFTT framework & WESH)	Derivation of the WESH master equation as the unique dynamical–dissipative completion of the Wheeler–DeWitt constraint, selected by CPT symmetry, WESH–Noether conservation, causality, and collective N² stability
formal-verification/Section5.lean	Section 5 (GR emergence)	Derivation of the Einstein field equations from WESH stationarity via gradient alignment and hidden-sector cancellation
formal-verification/Section6_Part1.lean	Section 6 (BH thermodynamics — part I)	Derivation of Bekenstein–Hawking entropy with universal 1/4 prefactor from bipartite pairing and swap-even projection
formal-verification/Section6_Part2.lean	Section 6 (BH thermodynamics — part II)	Derivation of the holographic bound from chronogenetic stability (Γ > 0 ⟹ S ≤ A/4L²_P)
formal-verification/AppendixA.lean	Appendix A	Kernel conventions, Yukawa structure, Planck anchoring
formal-verification/AppendixB.lean	Appendix B	Chronogenesis map, eigentime bootstrap, monotonicity of physical time
formal-verification/AppendixC.lean	Appendix C	WESH–Noether conservation at operator level
formal-verification/AppendixD.lean	Appendix D	Fixed-point uniqueness via Dobrushin contraction, variational alignment
formal-verification/AppendixF.lean	Appendix F	Angular dependence law: lattice averaging, Fourier→cos²θ, final law Γ(θ) = Γ̄(1+ε cos²θ)
formal-verification/AppendixG.lean	Appendix G	Operator-level backbone for WESH–Noether (commutant structure, T-neutrality)
formal-verification/AppendixH.lean	Appendix H	Complete positivity, trace preservation, no-signaling
formal-verification/AppendixI.lean	Appendix I	Quadratic selection from collective stability, CPT-evenness
formal-verification/AppendixJ.lean	Appendix J	Derivation of the cosmological constant as ontological shot noise of eigentime production, yielding Λ ~ H²

What is formally verified:

• The WESH master equation, derived as the unique dynamical–dissipative extension of the Wheeler–DeWitt constraint compatible with complete positivity, operator-level Noether conservation, CPT symmetry, causality, and collective stability. This equation unfreezes the WDW "frozen time" sector.

• The chronogenesis law dt/ds = Γ[ρ] ≥ 0, with eigentime activation, monotonicity, and the emergence of physical time and four-dimensional spacetime geometry from pre-geometric quantum dynamics.

• The Einstein field equations, derived from WESH stationarity via gradient alignment (∂_μT = k∂_μΦ) and hidden-sector cancellation.

• Bekenstein–Hawking black hole entropy S = A/(4L²_P) + γ ln(A/L²_P) + ..., formally verified. The universal 1/4 prefactor emerges from the interplay of bipartite pairing (1/2) and swap-even RAQ projection (1/2). The proof includes the complete linear-algebraic trace calculation (trace_swap: Tr(G_xy) = D via orthonormal basis decomposition), the asymptotic limit (factor_RAQ → 1/2 as D → ∞), and the Big-O bounds for subleading corrections. No thermal ansatz, no Euclidean continuation, no free parameters.

• The cosmological constant as ontological shot noise of eigentime production, with scaling Λ ~ H² derived from CLT on the eigentime counting process.

All derivations are mechanically certified by the Lean 4 type-checker. The formalization comprises 6,700+ lines of verified code (LOC).

To verify locally:

git clone https://github.com/Luca-Casagrande/qftt-wesh.git
cd qftt-wesh
lake update
lake build    

This work was developed with the assistance of a cross-inferencing multi-AI workflow, primarily involving ChatGPT-5.2 Pro and Claude Opus 4.5 for theoretical development, mathematical derivations, and iterative refinement, but also Gemini 3 Pro and other models. Aristotle v.~0.6.0 (Harmonic) was also used, in conjunction with ChatGPT-5.2 Pro and Claude Opus 4.5, as a translation tool in the final stages to convert established mathematical content into Lean 4 syntax. The author intends to further document this multi-AI architecture in future work, hoping it may contribute to AI-assisted theoretical physics research. Throughout the process, the research, conceptual framework and its details, the workflow orchestration, and the verification of every output remained with the author.

---

Experimental Validation (Section 3)

This repository also contains the numerical and experimental material underpinning Section 3 (Experimental Validation):

Highlights

• Collective stability: coherence time scaling

  τ_coh ∝ N^2

  under inverse bilocal couplings, in contrast with standard decoherence where τ_coh remains approximately flat or decreases with N.

• Angular law: an entanglement‑dependent cos²θ angular dependence of the effective decay rate,

  Γ(θ) ∝ 1 + ε cos²θ,

  with W‑state anti‑dependence (ε < 0) indicating state‑dependent behaviour.

• Hardware protection gap: multipartite GHZ states show a persistent parity advantage corresponding to an effective decoherence rate ≈ 2.6× smaller than matched separable controls.

Data

• Classical simulations (pre‑asymptotic collision model, N = 2 … 16) yield an effective scaling

  γ(N) ∝ N^{-1.804}

  with tight confidence intervals, consistent with an approach towards the predicted N^{-2} behaviour within the accessible pre‑asymptotic regime.

• Experiments on IBM Eagle (127‑qubit superconducting device) and Rigetti Ankaa‑3 (82‑qubit) — totaling over 3×10^6 hardware shots — display:

  • a clear cos²θ angular dependence of the decay rate,
  • a robust GHZ vs PRODUCT protection gap,
  • control tests (Fake‑GHZ, W‑state) that substantially constrain gate‑overhead and classical‑noise explanations.

---

Repository layout

qftt-wesh/
├── paper/                       # Manuscript source
│   ├── QFTT-WESH.tex
│   ├── QFTT-WESH.pdf
│   └── figures/
│       ├── Picture1.png         # Figure 1: Conceptual overview
│       ├── Picture2.png         # Figure 2: WESH as WDW extension
│       ├── Picture3.png         # Figure 3: Spacetime bootstrap
│       ├── Picture4.png         # Figure 4: GR emergence
│       └── Picture4b.png        # Figure 5: BH thermodynamics
├── formal-verification/         # Lean 4 proofs (6,700+ LOC)
│   ├── Section1.lean
│   ├── Section5.lean
│   ├── Section6_Part1.lean
│   ├── Section6_Part2.lean
│   ├── AppendixA.lean
│   ├── AppendixB.lean
│   ├── AppendixC.lean
│   ├── AppendixD.lean
│   ├── AppendixF.lean
│   ├── AppendixG.lean
│   ├── AppendixH.lean
│   ├── AppendixI.lean
│   └── AppendixJ.lean
├── experiments/                 # Data and scripts
│   ├── 3.1/
│   ├── 3.2/
│   ├── 3.3-3.4/
│   ├── 3.5/
│   ├── 3.6/
│   ├── 3.7-3.8/
│   └── 3.9/
└── README.md

Paper folder

The paper/ folder contains the complete LaTeX source and compiled PDF of the manuscript, along with the synoptic figures (Figures 1–5) illustrating the QFTT–WESH conceptual architecture.

Experiments folder

Folder	Manuscript figure(s)	Description
3.1/	Fig. 3.1	WESH vs standard local decoherence (purity & coherence scaling, CPU simulations).
3.2/	Fig. 3.2	Coherence under dephasing / projective / amplitude damping with different N‑scaling of the coupling.
3.3-3.4/	Figs. 3.3–3.4	Pre‑asymptotic collision model vs first‑order local model: γ(N) scaling (α ≈ −1.80 vs −1).
3.5/	Fig. 3.5	IBM Eagle QPU: cos²θ angular dependence (N = 3), mini‑scaling at θ ≈ 30°, and W‑state control.
3.6/	Fig. 3.6	Fake‑GHZ control: gate‑matched PRODUCT / FAKE_GHZ / GHZ sequences to separate gate overhead from genuine entanglement effects.
3.7-3.8/	Figs. 3.7–3.8	GHZ vs PRODUCT parity distributions and decay statistics (IBM Eagle Experiment 6): ECDFs, violin plots, significance analysis (≈ 21σ).
3.9/	Fig. 3.9	Cross‑platform countercheck on Rigetti Ankaa‑3: cos²θ angular dependence for N = 3 … 6, 400k shots.

Each folder is self‑contained: running the main script in that directory regenerates the corresponding plots from the included data.

---

Quick start

Reproducing figures (experiments)

1. Clone or download the repository.
2. Navigate to the desired figure folder, e.g. experiments/3.5/.
3. Ensure Python 3.x and the required libraries (numpy, pandas, matplotlib, qiskit, qiskit-ibm-runtime) are installed.
4. Run the main analysis script to regenerate the plots from the provided data.

Hardware access (IBM Quantum, AWS Braket) is not required for reproducing the figures: all QPU results are available as CSV/JSON files.

Verifying formal proofs

1. Install Lean 4 and Mathlib 4.
2. Navigate to formal-verification/.
3. Run lake build to verify all proofs compile.

---

Resources

• Lean 4: https://leanprover-community.github.io/
• Mathlib 4: https://github.com/leanprover-community/mathlib4
• Aristotle (Lean 4 AI assistant by Harmonic): https://aristotle.harmonic.fun/

---

Citation

If you use this repository, its datasets, analysis scripts, or formal proofs, please cite the QFTT–WESH manuscript:

@article{casagrande2026qftt_wesh,
  title   = {Quantum Field Theory of Time (QFTT), the Weak Entanglement Symmetry Hypothesis (WESH) and Emergent Spacetime},
  author  = {Casagrande, Luca},
  year    = {2026},
}

Changelog

• v1.5 (2026-02-09): Table 3 aligned with source data; added Sec 2.3 (discrimination from conventional models)
• v1.4 (2026-02-05): Added lakefile.toml for reproducible one-command builds
• v1.3 (2026-02-03): Added Appendix F formal verification (angular dependence law, 0 sorry, 0 axioms)
• v1.2 (2026-02-01): Fixed Appendix F figure numbering (F.1); added cross-platform outlook for trapped-ion validation
• v1.1 (2026-01-27): Clarified falsifiability hierarchy (core vs diagnostic); removed speculative Kerr paragraph
• v1.0 (2026-01-24): Initial release

---

License

This work is released under the MIT License.