Workstreams And API

This page preserves historical agent workstreams, the suggested repository structure, and the minimal computational API from the original formalization. Durable current task status now lives in Beads and the reader-facing proof status lives in the roadmap.

11. Codex agent workstreams

Each workstream should produce code, derivations, tests, and a short report.

Agent A: Symbolic GR recovery

Goal: Verify that the Pulse Model reproduces standard relativistic clock and motion equations.

Tasks:

1. Implement symbolic proper-time functional.
2. Derive SR time dilation from Minkowski metric.
3. Derive Schwarzschild gravitational time dilation.
4. Derive weak-field combined expression:

$$\frac{d\tau}{dt} \approx 1+\frac{\Phi}{c^2}-\frac{v^2}{2c^2}$$

5. Derive Newtonian action from relativistic action.
6. Derive Euler-Lagrange equations and recover:

$$\mathbf{a}=-\nabla\Phi$$

Deliverables:

• derivations/sr_time_dilation.md
• derivations/weak_field_limit.md
• src/pulse_model/relativity.py
• unit tests checking dimensional consistency and known numeric cases

Acceptance criteria:

• GPS-scale correction signs are correct.
• Circular orbit zero-dilation altitude calculation is reproduced.
• Symbolic derivations match standard formulas.

---

Agent B: Quantum phase and interferometry

Goal: Build simulations of phase accumulation along alternative paths.

Tasks:

1. Implement path phase:

$$\Theta = S/\hbar$$

2. Simulate two-path phase difference in uniform gravity.
3. Reproduce COW phase:

$$\Delta\Theta = \frac{mgA}{\hbar v}$$

4. Simulate atom interferometer phases with laser pulses.
5. Compare phase view with proper-time view.

Deliverables:

• src/pulse_model/phase.py
• notebooks/cow_phase.ipynb
• notebooks/atom_interferometer_phase.ipynb
• numerical plots of phase versus area, mass, velocity, and gravity

Acceptance criteria:

• COW scaling matches literature.
• Phase is dimensionless.
• Classical stationary path emerges numerically from phase cancellation.

---

Agent C: Quantum clock superposition simulator

Goal: Model a clock in a superposition of proper times.

Tasks:

1. Represent internal clock Hamiltonian $H_C$.
2. Evolve internal states along two worldlines:

$$\lvert \chi_i \rangle=e^{-iH_C\tau_i/\hbar}\lvert \chi_0 \rangle$$

3. Compute visibility:

$$\mathcal{V} = \left| \mathrm{Tr} \left( \rho_C e^{-iH_C\Delta\tau/\hbar} \right) \right|$$

4. Simulate pure two-level clocks.
5. Simulate thermal internal states.
6. Add gravitational height difference:

$$\Delta\tau \approx \frac{gH}{c^2}T$$

Deliverables:

• src/pulse_model/quantum_clock.py
• notebooks/proper_time_superposition.ipynb
• parameter sweeps for $H,T,\Delta E$

Acceptance criteria:

• Visibility equals 1 when $\Delta\tau=0$.
• Visibility oscillates for pure two-level states.
• Visibility decays for broad energy distributions.

---

Agent D: Pulse universality tests

Goal: Parameterize and constrain deviations from metric universality.

Tasks:

1. Implement deviation model:

$$\frac{dN_i}{dt} = f_i \left[ 1 + (1+\alpha_i)\frac{\Phi}{c^2} - (1+\beta_i)\frac{v^2}{2c^2} \right]$$

2. Collect public clock comparison data.
3. Fit bounds on $\alpha_i,\beta_i$.
4. Compare with existing local position invariance and Lorentz-violation frameworks.

Deliverables:

• src/pulse_model/universality.py
• data/clock_tests/
• reports/pulse_universality_bounds.md

Acceptance criteria:

• Existing null results imply $\alpha_i,\beta_i$ consistent with zero.
• Code can forecast sensitivity of future clocks.

---

Agent E: Pulse-network metric reconstruction

Goal: Infer metric components from synthetic clock/signal data.

Tasks:

1. Generate synthetic clock records in known spacetime.

2. Define pulse-record data structure:

   clock_id
   event_id
   local_pulse_count
   emitted_signal_id
   received_signal_id
   signal_phase
   local_acceleration

3. Fit $g_{00}$ in weak static fields.

4. Fit full weak-field metric:

$$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$$

5. Add loop holonomy analysis.

Deliverables:

• src/pulse_model/network.py
• src/pulse_model/reconstruct_metric.py
• notebooks/metric_reconstruction.ipynb

Acceptance criteria:

• Recover known gravitational potential from clock ratios.
• Recover synthetic gravitational wave perturbation from clock network residuals.
• Provide uncertainty estimates.

---

Agent F: Stress-energy as phase-response

Goal: Formalize the identity

$$T_{\mu\nu} = -\frac{2\hbar}{\sqrt{-g}} \frac{\delta \Theta_{\mathrm{m}}}{\delta g^{\mu\nu}}$$

for common fields.

Tasks:

1. Derive $T_{\mu\nu}$ for scalar field action.
2. Derive $T_{\mu\nu}$ for electromagnetic field action.
3. Derive point-particle stress-energy from worldline action.
4. Express each result in phase-response language.
5. Identify which components correspond to energy density, momentum flux, pressure, and stress.

Deliverables:

• appendix/h4_stress_energy_as_phase_response.md
• tests/test_h4_stress_energy_phase_response.py
• optional symbolic checks only if later work needs machine-checked variation algebra

Acceptance criteria:

• Standard stress-energy tensors are recovered.
• Units and signs are consistent.
• The phase-response interpretation handles pressure, not just mass density.

---

Agent G: Geometry phase functional

Goal: Explore whether the Einstein-Hilbert action can be interpreted or derived as a pulse-consistency cost.

Tasks:

1. Start from standard:

$$S_{\mathrm{EH}} = \frac{c^3}{16\pi G} \int(R-2\Lambda)\sqrt{-g}\,d^4x$$

2. Express as phase:

$$\Theta_{\mathrm{EH}}=S_{\mathrm{EH}}/\hbar$$

3. Study dimensions in Planck units.
4. Investigate curvature as loop pulse-comparison holonomy.
5. Attempt derivation of $R\sqrt{-g}$ from local holonomy density.
6. Compare with Regge calculus and causal set discretizations.

Deliverables:

• research/geometric_phase_cost.md
• notebooks/regge_pulse_cost.ipynb
• candidate discrete action

Acceptance criteria:

• Reproduce Einstein-Hilbert action in continuum limit or clearly identify failure.
• Preserve diffeomorphism invariance or explain replacement symmetry.
• Avoid introducing preferred frame.

---

Agent H: Quantum source / metric superposition

Goal: Model the relation between superposed matter phase histories and geometry.

Tasks:

1. Model matter source state:

$$\lvert \psi \rangle=\alpha\lvert L \rangle+\beta\lvert R \rangle$$

2. Compare three models:

   • semiclassical metric sourced by $\langle T_{\mu\nu}\rangle$
   • branch metric $\alpha\lvert g_L \rangle\lvert L \rangle+\beta\lvert g_R \rangle\lvert R \rangle$
   • collapse/decoherence model

3. Track pulse histories of probe clocks.

4. Predict entanglement/decoherence signatures.

Deliverables:

• src/pulse_model/metric_superposition.py
• reports/superposed_source_models.md

Acceptance criteria:

• Distinguish predictions between models.
• Identify experiments that can falsify each class.
• Keep assumptions explicit.

---

Agent I: Black hole pulse model

Goal: Translate black hole clock behavior and horizon structure into pulse-history language.

Tasks:

1. Compute proper time for infalling observer in Schwarzschild spacetime.
2. Compute redshift for signals emitted near horizon.
3. Track pulse counts for static, orbiting, and infalling clocks.
4. Analyze horizon as breakdown of external pulse comparison.
5. Explore phase records and information flow.

Deliverables:

• notebooks/black_hole_pulse_counts.ipynb
• reports/horizon_pulse_comparison.md

Acceptance criteria:

• Correctly distinguish local finite proper time from distant infinite redshift.
• Avoid saying "time stops" as an absolute statement.
• Handle null geodesics and massless phase separately.

---

Agent J: Literature and benchmark map

Goal: Keep the research grounded.

Tasks:

1. Build a bibliography of:
   • quantum clocks
   • problem of time
   • atom interferometry
   • gravitational redshift
   • quantum reference frames
   • semiclassical gravity
   • equivalence principle tests
   • clock networks
2. Map each paper to Pulse Model concepts.
3. Identify known no-go theorems and constraints.
4. Maintain a benchmark list of equations and experiments the model must reproduce.

Deliverables:

• references/pulse_model_bibliography.bib
• reports/literature_map.md
• tests/benchmarks.md

Acceptance criteria:

• Every speculative claim is tagged and sourced.
• Benchmarks are converted into executable tests where possible.

---

12. Suggested repository structure

pulse-model/
  README.md
  docs/
    pulse_model_formalization.md
    assumptions.md
    glossary.md
  appendix/
    h1_time_is_relational_pulse_count.md
    h2_metric_reconstruction_from_pulse_comparisons.md
    h3_pulse_comparison_holonomy.md
    h4_stress_energy_as_phase_response.md
  src/
    pulse_model/
      __init__.py
      constants.py
      relativity.py
      phase.py
      quantum_clock.py
      universality.py
      network.py
      reconstruct_metric.py
      metric_superposition.py
  tests/
    test_h4_stress_energy_phase_response.py
  notebooks/
    cow_phase.ipynb
    atom_interferometer_phase.ipynb
    proper_time_superposition.ipynb
    metric_reconstruction.ipynb
    black_hole_pulse_counts.ipynb
  data/
    clock_tests/
    experiments/
  reports/
    pulse_universality_bounds.md
    superposed_source_models.md
    horizon_pulse_comparison.md
    literature_map.md
  tests/
    test_units.py
    test_sr.py
    test_weak_field.py
    test_phase.py
    test_clock_visibility.py
    benchmarks.md
  references/
    pulse_model_bibliography.bib

---

13. Minimal computational API

The first implementation should expose these pure functions.

def proper_time_flat(dt: float, v: float, c: float) -> float:
    """Return proper time for inertial motion in flat spacetime."""
    ...

def weak_field_dtaudt(phi: float, v: float, c: float) -> float:
    """Return dτ/dt in weak gravity and low velocity."""
    ...

def pulse_count(frequency: float, proper_time: float) -> float:
    """Return accumulated pulse count."""
    ...

def free_massive_phase(mass: float, proper_time: float, c: float, hbar: float) -> float:
    """Return free massive action phase."""
    ...

def gravitational_redshift(phi_emit: float, phi_recv: float, c: float) -> float:
    """Return weak-field fractional frequency shift."""
    ...

def cow_phase_shift(mass: float, gravity: float, area: float, velocity: float, hbar: float) -> float:
    """Return COW gravitational phase shift."""
    ...

def clock_visibility(delta_tau: float, energy_levels: list[float], probabilities: list[float], hbar: float) -> float:
    """Return internal-clock path visibility."""
    ...

---