systrophe 0.14.1

Co-rotating dust cylinder pair, van Stockum / Tipler exterior with log-periodic sinusoid superposition

Συστροφή — Systrophe

A co-rotating Tipler-cylinder pair as a tunable time-travel harness.

Systrophē (Greek Συστροφή, "twisting-together"): the joint exterior of two co-rotating, dual-positive-mass van Stockum dust cylinders, whose log-periodic Tipler sinusoids superpose with a tunable relative phase offset.

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What this is

A complete numerical and analytic implementation of:

1. The single rotating cylinder — exact van Stockum 1937 interior + analytic Bonnor Case III exterior (closed forms for F, K, L).
2. The Lewis–Papapetrou ODE integrator — Ernst-equation numerical exterior for any rotation parameter, validated to machine precision against the closed forms.
3. The Systrophe pair — two co-axial supercritical cylinders linearly superposed, producing an off-set Tipler sinusoid whose CTC band positions are tunable by the relative phase δ₂ − δ₁.
4. A time-machine harness — locate CTC bands, derive timelike-orbit angular-velocity sectors, tune coordinate-time-per-revolution to a target value, simulate forward and backward time-travel orbits.
5. A bridge to the Δῖνος (Dinos) Dirac–Kerr–Newman framework — exact identification of the Tipler log-grid fundamental eigenvalue with the Z₃ Möbius cover branch=0 fundamental; non-trivial branches realise the off-set sectors of the systrophic pair.

The package ships with a comprehensive whitepaper (paper/systrophe_time_travel.pdf) deriving every formula, all five figures, and reproducing the simulation results end-to-end.

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Installation

pip install -e ".[dev]"        # editable install with dev tools
pytest                         # 406 tests, ~3 minutes

Optional extras:

• .[symbolic] — SymPy for the one-shot derivation script (tools/derive_lewis_papapetrou.py).
• pip install z3-solver — required if you want the Dinos bridge module (systrophe.dinos_bridge).

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Quickstart: build a single time machine

import numpy as np
from systrophe import VanStockumInterior, find_single_cylinder_windows, harness_time_loop

# A supercritical van Stockum cylinder: a = ω R = 1
cyl = VanStockumInterior(omega=1.0, R=1.0)

# Locate CTC bands in r ∈ [1.001, 200]
windows = find_single_cylinder_windows(cyl, r_min=1.001, r_max=200.0)
for w in windows:
    print(f"CTC band:  r ∈ [{w.r_inner:.3f}, {w.r_outer:.3f}]   "
          f"deepest L = {w.L_min:.3f}")

# Tune a backward-time-travel orbit:  Δt per revolution = -1
orbit = harness_time_loop(windows[0], target_dt_per_rev=-1.0, n_revolutions=10)
print(f"Ω = {orbit['Omega']:+.4f}")
print(f"per rev: Δt = {orbit['dt_per_revolution']:+.3f},  Δτ = {orbit['dtau_per_revolution']:.3f}")
print(f"after 10 revs: Δt_total = {orbit['total_coord_time_advance']:+.3f},  "
      f"Δτ_total = {orbit['total_proper_time_advance']:.3f}")

Expected output:

CTC band:  r ∈ [1.001, 6.134]   deepest L = -3.351
CTC band:  r ∈ [37.622, 200.000] deepest L = -126.065
Ω = -6.2832
per rev: Δt = -1.000,  Δτ = 11.354
after 10 revs: Δt_total = -10.000,  Δτ_total = 113.537

The particle moves backwards in coordinate time by 10 units while advancing 113.5 units of its own proper time.

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Quickstart: tune the harness via a co-rotating pair

from systrophe import SystrophePair, VanStockumInterior

cyl = VanStockumInterior(omega=1.5, R=1.0)
pair = SystrophePair.from_cylinders(cyl, cyl, delta_offset=0.7853981633974483)  # π/4

print(f"phase offset = {pair.phase_offset:.4f} rad")
bands = pair.ctc_bands(r_min=1.05, r_max=20.0)
print(f"{len(bands)} CTC bands; first at r ∈ [{bands[0][0]:.3f}, {bands[0][1]:.3f}]")

The phase offset between the two cylinders continuously shifts the CTC band positions. At exact anti-phase (δ = π), all CTC bands extinguish — a topological off-switch.

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End-to-end demonstration

python examples/time_travel_simulation.py

Runs the full numerical experiment from the whitepaper: identifies CTC bands, sweeps offset, computes time-travel orbits, writes machine-readable JSON results to examples/time_travel_simulation_results.json.

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Mathematical highlights

Tipler log-frequency. For a = ω R > 1/2 (supercritical), the exterior metric components oscillate as functions of u = ln(r/R) with frequency

α = √(4 a² − 1).

Closed forms (all three Bonnor regimes).

Supercritical (a > 1/2): with γ = π − arctan α,

F(r) = (r/R) · sin(α u + γ) / sin γ
K(r) = (r/α)  · [ ((α² − 1)/2) sin(α u + γ) − α cos(α u + γ) ]
L(r) = (r R sin γ / α²) · [ Q sin(α u + γ) + α(α² − 1) cos(α u + γ) ]

with Q = α² − (α² − 1)² / 4.

Critical (a = 1/2):

F(r) = (r/R)(1 − u)        K(r) = (r/2)(1 + u)        L(r) = (rR/4)(3 + u)

Subcritical (a < 1/2): with β = √(1 − 4a²) and S± = cosh(βu) ± sinh(βu)/β,

F(r) = (r/R) S₋(u)         K(r) = a r S₊(u)           L(r) = rR(1 − a²S₊²)/S₋

The constraint F·L + K² = r² holds identically in every regime; verified to machine precision in the test suite.

Pair superposition. For matched α, the joint envelope is a single sinusoid whose amplitude and phase come from the phasor sum

A_eff · exp(i δ_eff) = A₁ exp(i δ₁) + A₂ exp(i δ₂).

Time-travel orbit. A circular orbit at fixed r in a CTC band has

Δt    = 2 π / Ω                              (coordinate time per rev)
Δτ    = √(F − 2 K Ω − L Ω²) · |Δt|           (proper time per rev)

The full derivation, with Lewis–Papapetrou Ernst-equation reduction and Bonnor's Case classification, is in the whitepaper.

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Architecture

src/systrophe/                  Classical-GR backbone (v0.1-v0.6)
  vanstockum.py                 — Interior metric + analytic Case III exterior
  lewis_papapetrou.py           — Numerical Ernst-equation integrator
  lp_robust.py                  — Regime-dispatching robust solver
  sinusoid.py                   — TiplerSinusoid log-periodic envelope
  pair.py                       — SystrophePair co-axial superposition
  off_axis.py                   — OffAxisPair parallel-axis
  ctc.py                        — CTC band detector
  geodesic.py                   — Circular orbits, integrate_geodesic
  time_machine.py               — TimeMachineWindow + harness
  dinos_bridge.py               — Optional Dinos-DKN interop

src/systrophe/                  Quantum / QFT layer (v0.7-v0.13; see paper II)
  dirac.py, dirac_spectrum.py   — Radial Dirac operator + bound-state spectrum
  dirac_sea.py                  — Dirac-sea pressure, horizon divergence
  particle_creation.py          — Bogoliubov-style horizon emission
  qftcs_backreaction.py         — QFTCS curvature back-reaction trace
  quantum_diagnostics.py        — Ricci, surface gravity, Hawking T, Tolman
  point_splitting.py            — 4D Riemann/Kretschmann/trace anomaly
  hadamard_offtrace.py          — Full <T_munu>_ren tensor
  floquet.py                    — Adiabatic Floquet on radial Dirac
  floquet_mobius.py             — Joint Floquet on (time x Z_3 branch)
  casimir.py                    — Topological Casimir / Z_3 mode sums
  casimir_throat.py             — Brown-Maclay <T_munu> at cavity
  anomaly_inflow.py             — APS eta + Callan-Harvey Z_3 closure
  tipler_fractal.py             — DSI + cascade-DSI extension
  horned_torus.py               — Regular + inverted horn modes
  acoustic_metric.py            — Unruh acoustic-metric mapping
  newton_kantorovich.py         — NK solver + Picard comparison
  back_reaction.py              — Self-consistency composite residual
  floquet_engineering.py        — CTC stability map (drive_amp, omega)
  dsi_observables.py            — Log-periodic precursor fits
  adm_export.py                 — ADM 3+1 hand-off for NR codes
  d_ctc.py                      — Deutsch CTC fixed-point on Z_3 cover

tests/                          — 367 passing tests across 30 modules
paper/
  systrophe_time_travel.tex/pdf   Whitepaper I (classical, v0.1-v0.6)
  systrophe_qft_on_ctc.tex/pdf    Whitepaper II (QFT, v0.7-v0.13)
docs/
  INTERPRETATIONS.md            — 6 open ansatz claims, with required input
  EXPERIMENTAL_ACOUSTIC_ANALOG.md — BEC-vortex design proposal
examples/                       — Verification batteries + simulation scripts
tools/                          — One-shot SymPy derivations

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Quantum layer (v0.13)

Beyond the classical-GR core, Systrophe v0.7–v0.13 adds:

• Renormalised stress tensor on the LP background, with off-trace components <T_{μν}>_ren = (1/2880π²) R_{μρστ} R_ν^{ρστ}, whose trace recovers the conformal anomaly exactly.
• Anomaly inflow on the Z₃ Möbius cover: APS η-invariants give (0, 1/3, −1/3) summing to zero; nonzero gauge twist is closed by Chern-Simons coefficient 1/(24π²).
• Acoustic-metric mapping: identification c² − v² = F makes the chronology horizon an acoustic horizon. Gravitational and acoustic Hawking temperatures agree to machine precision.
• Joint Floquet on (time-circle × Z₃-branch), with cyclic- permutation symmetry verified and the (e_b − e_b') resonance identified.
• Brown-Maclay flat-space Casimir at the cavity, with LP curvature-correction scale K · d⁴ (small means flat-space approximation valid).
• Newton-Kantorovich back-reaction solver demonstrating that naive Picard iterations converge linearly, not quadratically.
• Cascade discrete-scale invariance (tipler_fractal.py): proves the base Tipler sinusoid is not fractal (dim 0) but the multi-cylinder cascade is (dim > 0.3).
• Horned torus topology with regular (pinch) and inverted (bulge) variants.
• ADM 3+1 export for hand-off to Einstein Toolkit / similar NR codes.
• D-CTC Deutsch-CTC fixed-point solver on the Z₃ cover.

Full derivations are in paper/systrophe_qft_on_ctc.pdf (Whitepaper II). Open ansatz-level interpretations are documented in docs/INTERPRETATIONS.md. A BEC-vortex experimental-analog design proposal is in docs/EXPERIMENTAL_ACOUSTIC_ANALOG.md.

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The Δῖνος bridge

The package optionally interoperates with Dinos-DKN, exposing a striking structural correspondence:

Theorem (numerical, exact to machine precision). Sample the supercritical Tipler exterior at N nodes per log-period 2π/α. The fundamental discrete-Laplacian eigenvalue equals exactly the branch = 0 mode-1 eigenvalue of the Dinos Z₃ Möbius cover at the same N. The non-trivial branches b ∈ {1, 2} (complex conjugate pair, 2π/3 phase advance per node) sit at strictly lower eigenvalue and are the discrete signature of the off-set sectors of the systrophic pair (δ = ±2π/3).

from systrophe import VanStockumInterior
from systrophe.dinos_bridge import z3_branch_match_to_tipler_alpha

vs = VanStockumInterior(omega=1.0, R=1.0)
out = z3_branch_match_to_tipler_alpha(vs, N=24)
# {'tipler_eigenvalue': 0.06814834..., 'z3_eigenvalues': (0.06815, 0.00761, 0.00761),
#  'best_branch_match': 0, 'relative_residual': 0.0}

Requires pip install z3-solver and Dinos-DKN on PYTHONPATH. The test suite skips silently if either is unavailable.

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Tests

pytest                 # 65 tests, ~5 seconds
pytest -v              # verbose
pytest --cov=systrophe # with coverage (requires pytest-cov)

Test suite breakdown:

Module	Tests	Coverage
test_vanstockum.py	7	Minkowski limit, invariants, threshold, exterior rejection
test_sinusoid.py	6	α formula, log-periodicity, fit recovery
test_pair.py	7	Phasor collapse, anti-phase cancellation, principal-range wrap
test_ctc.py	6	Sign-band detection, destructive interference, modulated bands
test_lewis_papapetrou.py	12	Continuity, supercritical numerical-vs-analytic, constraint
test_lp_robust.py	8	Regime dispatch, machine-precision supercritical, F-zero formula
test_off_axis.py	7	Construction guards, reflection symmetries, 2D CTC map
test_dinos_bridge.py	6	Kerr mapping, Z₃ branch correspondence (skipped if z3 absent)
test_offset_sweep.py	6	Convenience constructor, offset-π minimum, type checks
test_geodesic.py	6	Minkowski circular orbit, Ω-bounds, target-time inverse
test_time_machine.py	9	Band detection, target-Δt matching, spacelike rejection

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Whitepaper

paper/systrophe_time_travel.pdf (11 pages, includes 5 figures and 3 tables) covers:

1. Mathematical framework — van Stockum, Lewis–Papapetrou, Ernst, Bonnor cases.
2. Co-rotating cylinder pair — linearised superposition, off-set Tipler sinusoid.
3. Closed timelike curves and time-travel orbits — geodesic structure, coordinate vs proper time.
4. Time-machine harness — single-cylinder and pair-tuned numerical results.
5. Connection to Dinos via Z₃ Möbius eigenvalue correspondence.
6. Implementation, tests, evaluation.
7. Limitations and open questions (chronology protection, asymptotic non-flatness, idealised source).

To regenerate the PDF:

python paper/generate_figures.py     # produces paper/figures/*.pdf
cd paper && pdflatex systrophe_time_travel.tex && pdflatex systrophe_time_travel.tex

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Limitations

• Linearised pair. Two-cylinder Einstein vacuum has no closed form; both SystrophePair (co-axial) and OffAxisPair (parallel-axis) treat the second source as a linearised perturbation. Cross-terms h^(1) · h^(2) are formally O(G²) and not modelled.
• Off-axis quantitative limits. In the off-axis case, the Case III exterior is not asymptotically flat, so both single-cylinder perturbations are simultaneously "large" at most points; the linearised superposition is best read as a qualitative tool for identifying CTC regions rather than as a quantitative orbital framework.
• Idealised source. Infinite, rigid, perfectly axisymmetric dust column. No known matter form realises this; Tipler-cylinder time-travel scenarios are theoretical exercises in the structure of GR vacuum solutions.
• Asymptotic non-flatness. The Case III exterior oscillates indefinitely; there is no privileged "observer at infinity" against whom coordinate time can be synchronised.
• No chronology protection. This is a pre-quantum, classical-GR construction. The chronology-protection conjecture (Hawking 1992) is not addressed.

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Citation

@misc{Knopp2026Systrophe,
  author = {Knopp, Christian},
  title  = {{Systrophē}: A co-rotating Tipler-cylinder pair as a tunable
            time-travel harness},
  year   = {2026},
  note   = {Python implementation of van Stockum interior, Lewis--Papapetrou
            exterior, off-set Tipler sinusoid, and Z\_3 M\"obius cover
            correspondence; 65 passing tests.},
  url    = {https://github.com/Zynerji/systrophe}
}

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License

MIT. See LICENSE.

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Contact

[e-mail]

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References

• W. J. van Stockum, The gravitational field of a distribution of particles rotating about an axis of symmetry, Proc. Roy. Soc. Edin. 57 (1937) 135.
• T. Lewis, Some special solutions of the equations of axially symmetric gravitational fields, Proc. Roy. Soc. London A 136 (1932) 176.
• F. J. Tipler, Rotating cylinders and the possibility of global causality violation, Phys. Rev. D 9 (1974) 2203.
• W. B. Bonnor, The exterior gravitational field of a rotating cylinder of dust, J. Phys. A 13 (1980) 2121.
• S. W. Hawking, The chronology protection conjecture, Phys. Rev. D 46 (1992) 603.