STAR Drive.
Sampled Transport by Alias Ratcheting
A lock-window pulse-scheduling architecture for conventional propulsion.
Stage 0b validates the scheduler model and the deterministic analysis behind it, not propulsion hardware. Nothing on this page reports a measured thrust.
Every propulsion system decomposes into a reservoir that momentum is exchanged with, a coupling that performs each exchange, and a schedule that decides when. This is a design theory for the third component, and for nothing else.
No new momentum source is proposed and no engineering advantage is claimed in advance. Whether the coupled gate outperforms a direct digital scheduler, PWM, or an instrumented digital PLL is the architecture's first research question, not its premise.
Engineering Note EN-002 · v1.0 · DOI 10.5281/zenodo.21833505 · CC BY 4.0
The founding accounting identity
Three obligations follow. Something must pay for each event, something must schedule it, and something must count it.
01 · Position, honestly stated
A scheduler, not a propulsion source.
STAR is a conventional, nonlinear phase-locked pulse-scheduling architecture: an external-reference control loop whose stroboscopic reduction is the sine circle map, with the map parameter Ω identified physically as the gate-to-reference frequency ratio. It is a recognizable relative of injection locking and first-order PLLs. That kinship defines the burden of proof.
IS What the drive is
- A phase-locked pulse-scheduling layer for a conventional actuator against a conventional reservoir: timing structure, a static hold interval, a command-side ledger, an audit discipline.
- An architecture in which all directionality lives in the momentum quantum χ supplied by the reservoir coupling. The gate contributes scalar rate structure only.
- A machine whose every command is a guarded threshold crossing of a single full lifted phase, committed once to a monotone, irreversible ledger.
- A metrology standard for phase-based thrust stands, and a set of momentum-discriminating controls that run as part of the sweep rather than after it.
IS NOT What the drive is not
- Not a momentum source. Within the ideal model STAR cannot change mean thrust at fixed mean event rate. It redistributes timing and adds a basin, a ledger and a control structure; it amplifies nothing.
- Not a specific-impulse or thrust-to-power improvement. Every comparison is made at matched mean power.
- Not yet demonstrated to beat a digital scheduler or a PLL anywhere. Where the schedule is synthetic, direct digital methods already win, and the note claims nothing for the coupled gate there.
- No reactionless mode. Sustained thrust surviving the dummy-load null while failing axis reversal is a systematic to be found; genuine reservoir-free thrust would contradict the identity the architecture is built on.
The three design principles · the architecture is forced by them, not assembled by preference
The gate must be actuated.
A schedule that does not enter the generator of the dynamics cannot change momentum. A gate that only observes is a stroboscope; a gate that commands is a scheduler.
Direction lives in the coupling.
The gate state and its winding rate are scalars, invariant under spatial inversion. Thrust reversal is implemented on the χ side (opposed actuators, gimballing), never by driving the gate backwards.
The gate must be coupled.
At κ = 0, exact recurrence occurs only on a measure-zero set of rationals with no restoring basin: recurrent, not stable. Coupling is a required layer, not an optimization.
Before the mathematics: how the machine fits together
EN-002 §3.1 · orientation figureScroll the figure horizontally →
02 · Substrate
Six facts the architecture consumes.
Wrapped observation is non-injective. Sampling depth improves estimation but provides no holding. A coupled circle map opens finite rational lock windows. Window location is distinct from the rotation-number label. The rotation number fixes event count, not event timing. And static hold, acquisition and tracking are three different quantities.
Claim tags run through this page; hover any of them for the definition.
[D] derivation ·
[S:En] simulation evidence ·
[S:A1][S:A2] deterministic analysis ·
[C] conjecture, with a stage attached.
Full legend →
Deep sampling answers where is the phase; it does not answer will it stay there. Without coupling, phase-overlay recurrence is exact only at zero-width rational parameter values: a dense set of measure zero, with no attracting basin and no noise robustness. Finite coupling changes the problem. On the lifted phase:
The standard sine circle map of mode-locking theory. Invertible for 0 ≤ κ < 1, the note's design domain, and anchored by evidence at κ = 0.95, which is near-critical invertible, just below the gain at which invertibility is lost.
At κ > 0 finite lock windows open in Ω wherever the winding number is rational. Three objects must then be kept apart: the rotation number p/q, which fixes the commanded event-rate level and nothing else; the window [Ω−, Ω+] with its width and measured centre Ωc; and the two detunings: rate mismatch Ω − p/q, and window-relative Ω − Ωc. Every hold condition in the note is a statement about the second.
A lock window's location is not its rational label
Source · EN-002 §2.4 · [S:E5][S:A1]Scroll the figure horizontally →
| Winding p/q | Window [Ω−, Ω+] | Width w | Centre Ωc | Ωc − p/q | Half-width |
|---|---|---|---|---|---|
| 0/1 | [−0.151197, +0.151197] | 0.302394 | 0.000000 | 0 (exact) | 0.151197 |
| 1/4 | [0.274710, 0.287946] | 0.013236 | 0.281328 | +0.031328 | 0.006618 |
| 1/3 | [0.337193, 0.363743] | 0.026550 | 0.350468 | +0.017135 | 0.013275 |
| 1/2 | [0.466409, 0.533591] | 0.067181 | 0.500000 | 0 (exact) | 0.033591 |
| 2/3 | [0.636257, 0.662807] | 0.026550 | 0.649532 | −0.017135 | 0.013275 |
| 3/4 | [0.712054, 0.725290] | 0.013236 | 0.718672 | −0.031328 | 0.006618 |
| 1/1 | [0.848803, 1.151197] | 0.302394 | 1.000000 | 0 (exact) | 0.151197 |
Widths from the released E5 scan [S:E5]; boundaries and centres from analysis A1 [S:A1], solved from the simultaneous saddle-node system to ~10−10 with residuals ≤ 2.4 × 10−14 across all fourteen edges. A1 reproduces every released E5 width to within the scan resolution ΔΩ ≈ 4.2 × 10−4 and confirms the deterministic integer-lock half-width κ/2π = 0.151197. Widths are ordered by denominator, w1/2 > w1/3 = w2/3 > w1/4 = w3/4; the equalities are the reflection symmetry.
03 · The second boundary
A rational winding fixes the event count, not the event phases.
On the plateau the gate runs a stable period-q orbit whose per-interval advances are unequal, so the p crossings per superperiod are not generally uniformly spaced, and event phase is the note's primary performance variable. A perfectly implemented gate therefore carries a deterministic timing floor by construction, stated here before Stage 1 because it is a structural handicap on the architecture's own primary metric.
RMS shape deviation of the locked event train, as a share of the nominal gap
Source · EN-002 §2.5 · [S:A1] · κ = 0.95, δg = 0Coupling gain is not to be maximized
Raising κ widens the window and worsens the intrinsic nonuniformity, both strictly across nine sampled values κ ∈ [0.40, 0.95]. w2/3 = 0.016437 at κ = 0.80 against 0.026550 at 0.95, about 38 % narrower; deviation falls from 0.023953 to 0.022185.
Hold-optimal ≠ phase-optimal
On the half-integer plateau the two coincide at the centre. At 2/3 they sit 0.418 half-widths apart. One isolated off-centre uniform point exists in the 2/3 window at m = 0 (Ω* = 0.655081), costing 42 % of the hold half-width.
Wider is not more robust
Robust tolerance falls roughly as (w/2)/Ωc: equally wide windows at larger Ω are markedly less robust. εmax runs from 5.263 % at the guarded null and the low integer windows down to 0.858 % at 3/4, which for most moving cadences rules that window out.
04 · The STAR architecture
Six elements, one command surface.
The integer–fractional decomposition is bookkeeping, not a second command stream. Every command, without exception, is the committed crossing of a single guarded surface on the full lifted phase, and a threshold is committed once and only once.
Reference cadence
The interval is measured, not assumed: Tn is the realized gap between external fiducials, with a floor Tmin. Strobing once per cycle presupposes a per-cycle fiducial or an already branch-tracked reference: a declared architectural requirement, not an implementation detail.
Gate oscillator on a reduced lift
The integer part m is fixed by the selected winding target, not by the instantaneous map parameter. Motion within a tongue changes nothing; a change of the selected tongue family is a re-decomposition event and triggers re-arming. The fractional target is selected by its measured window centre.
Firing rule: one guarded surface
Every command is a committed crossing of x = ℓ + δg by the full lifted phase. Commands are matched to the reference by threshold identity ℓ: a shared threshold with displaced timing is a timing error; a threshold present in only one schedule is a count error. No tolerance mediates between the two.
The declared digital realization
The stroboscopic map supplies counts per interval, not pulse times within it, so the realization is part of the architecture. Baseline: a rate-blended servo on zero-order hold. Declared alternative: a phase-step servo, a valid fallback for invertibility failures only.
Initialization and arming
Acquire before arm. Establish the reference and its cycle-boundary source, enter the window at its declared operating point, wait out a declared acquisition transient, initialize the commitment threshold, arm, then begin counting. The 0/1 guarded null holds no-fire only after arming, and under noise only to a budget.
Ledgers: four counters
The winding counters Kfull, Kfrac (either can decrease after a backward excursion); the monotone command ledger CN; and the verified firing count Kdel from actuator feedback. They coincide only under named conditions. Divergences are diagnostics, never smoothed.
What the window buys, and what it does not.
In noiseless monotone captured operation at constant interval, the whole chain collapses to the exact ideal form, the accounting identity made quantitative. The substrate supplies the design theory of the winding number W0; the reservoir coupling supplies the impulse bit. The two factors are independently auditable, and if the impulse bit is zero the ideal drive produces a controlled zero while the gate still winds.
| Property | Static result — A1 / A2 / E5 | Dynamic status |
|---|---|---|
| Event rate | p/q exactly, throughout the stationary tongue | Realized rate under moving cadence: Stage-1 measurement |
| Hold interval | Deterministic window widths and centres, measured | Necessary, not sufficient, for lock retention |
| Event phases | Operating-point- and guard-dependent floors | Measured primary metric |
| Slip probability | Not supplied by static windows | Stage-1 first-exit statistic |
Inside a window the lock is noise-confined over the measured horizon: at drive noise 0.03 cycles/step, a 500-member ensemble at the 1/2 centre holds its lifted-phase spread at 0.302 cycles after 400 steps, while an incommensurate ensemble diffuses to 0.913 cycles and keeps growing. Countability is architectural: the ledger advances on a deterministic timetable in lock, so the schedule audits itself; the impulse estimate is only as good as the actuator calibration.
The period must be predicted, not known.
Tn is unavailable at the fiducial that opens interval n, so the scheduler runs on a causal one-interval prediction. With ρn the ratio of realized to predicted period, the realized map parameters are jointly scaled:
Prediction error therefore moves the system radially through the (Ω, κ) plane: it displaces the detuning as well as the gain. For a narrow interior window it can leave the intended tongue long before it threatens invertibility; at the anchor gain, a period underestimate of about 5 % already violates the invertibility cap. The error is multiplicative, not additive: it scales with the servo action and does not vanish merely because the gate is locked.
The robust requirement is consequently two-dimensional. Candidate commands must stay inside the selected stationary tongue for all admitted prediction ratios, satisfy the invertibility cap, remain reachable under slew limits, keep their rollout feasible against hard event-count and actuator constraints, and exceed a preregistered worst-case margin floor fixed offline. An empty feasible set is a fault, not a tuning inconvenience: the controller does not relax the margin floor; it suspends new commitments, preserves both ledgers, and enters reacquisition or fault handling.
Robust operating-point tolerances at κ = 0.95: the price of causality
| Operating point | Screen (w/2)/Ωc | εmax rate-blended | Binding constraint | εmax phase-step |
|---|---|---|---|---|
| 0/1 guarded null | — | 5.263 % | invertibility | not prediction-limited |
| 1/1 | 15.12 % | 5.263 % | invertibility | 15.120 % |
| 2/1 | 7.56 % | 5.263 % | invertibility | 7.560 % |
| 3/1 | 5.04 % | 5.027 % | window drift | 5.040 % |
| 5/1 | 3.02 % | 3.021 % | window drift | 3.024 % |
| 1/2 | 6.72 % | 5.263 % | invertibility | 6.718 % |
| 1/3 | 3.79 % | 4.112 % | window drift | 3.788 % |
| 2/3 | 2.04 % | 1.959 % | window drift | 2.044 % |
| 1/4 | 2.35 % | 2.900 % | window drift | 2.352 % |
| 3/4 | 0.92 % | 0.858 % | window drift | 0.921 % |
Bisection on ε until the robust intersection empties; integer rows analytic [S:A2]. The binding constraint crosses from invertibility to window drift near m = 3. The guarded null's window containment is exactly invariant because Ωcmd = 0 scales to 0, so its operational limit is set entirely by the effective-gain cap. The phase-step column helps strictly where invertibility binds and changes little, not always favourably, where drift binds.
The eleven design requirements, stated normatively
| # | Rule | Requirement | Source |
|---|---|---|---|
| 1 | Static hold | Operating point plus worst-case uncompensated drift stays inside the selected stationary window, on the invertible domain 0 ≤ κ < 1. | [S:E5][S:A1] |
| 2 | Window selection | Use the simplest rational window delivering the required level; command the measured centre — never the rational label, which can lie outside its own window. | [S:E5][S:A1][S:A2] |
| 3 | Anti-reversal clearance | Every sensing channel feeding the servo holds its raw-channel sampling ratio above the statistical zone, R ≳ 2.2–2.6 for η ≤ 0.7. | [S:E1][S:E4] |
| 4 | Anti-freeze exclusion | No sensing channel at a freeze coincidence, where slips are invisible. | [S:E1] |
| 5 | Resolvability budget | Sensing resolves the finest residual structure it acts on; depth computed, not assumed. | [S:E3] |
| 6 | No magic depths | No design may "tune to" a privileged sampling-depth value; resolution is smooth in RA. | [S:E2][S:E5] |
| 7 | Guard budget | δg selected against separate first-passage budgets; single-step quantile screens size, never prove. | Stage 1 [C] |
| 8 | Physical-units mapping | Tolerable physical drift derived from the normalized window at the measured centre. | [D] |
| 9 | Event-phase uniformity | Timing budgeted separately from rate; sit at an exactly uniform operating point or enter the measured floor at the operational guard. | [S:A1][S:A2] |
| 10 | Dynamic tracking | Budget trackable slew, modulation bandwidth, pull-out probability, reacquisition time. The budget on which the case is won or lost. | Stage 1 [C] |
| 11 | Robust command under prediction error | Two-sided prediction bounds at stated coverage; nonempty robust intersection; gain cap; margin-constrained cost-minimizing setpoint with a fallback ladder. | [S:A2]; Stage 1 [C] |
05 · The first research question
The coupled gate has to earn its place.
Where the schedule is synthetic (referenced to the controller's own clock), direct digital methods already generate exact rational schedules with exact counts and no analog phase diffusion. Against that baseline the coupled gate offers nothing, and this note claims nothing for it there. The candidate regime is a physical detuning: firing that must acquire and hold an external or analog cadence, where hold, acquisition, basin recovery and graceful degradation are the currencies.
| Scheduler | Reference source | Response to physical cadence drift | Event-phase structure | Characteristic failure mode |
|---|---|---|---|---|
| Direct digital p/q | Own clock | None — open-loop w.r.t. cadence | Exact synthetic grid | Silent drift against physical cadence |
| PWM / pulse-density | Own clock | None | Duty-cycle grid, quantization jitter | Same, plus pattern noise |
| Instrumented digital PLL | External, tracked | Linear-loop tracking, cycle slips | Loop-filter dependent | Cycle slips under disturbance, logged by its own monitor |
| STAR rate-blended | External, tracked | Basin + robust setpoint rule | Deterministic, guard-dependent floors | Tongue exit under prediction error |
| STAR phase-step | External, tracked | Basin; no gain scaling | Discontinuous at fiducials | Ratio drift unmitigated |
The metric hierarchy is forced
Events pair to targets in order, the j-th delivered event with the j-th target, because nearest-in-time pairing is precisely how a whole-cycle slip is concealed.
Count error alone is passed by a scheduler firing the right number of events at entirely the wrong phases. Wrapped error alone is passed by a scheduler that has slipped whole cycles. The primary metric must resist both.
Primary: long-horizon RMSEapp
Unwrapped, order-paired event-phase error against the application target at the fixed arming origin. An event one full cycle late must not score zero.
Hard constraint: no missing and no additional delivered events over the horizon. A violating scheduler is not scored on the primary metric at all.
Secondary: RMSEshape; delivered-count RMSE; map-fidelity RMSE. Propulsion-level: integrated impulse error at matched mean rate and actuator model.
06: Signature, controls and metrology
A staircase in a force channel does not demonstrate momentum transfer.
The commanded event rate of a swept gate is a mode-locking staircase by construction. But the staircase is imposed on the command stream, and everything driven by that stream inherits it: power draw, valve actuation, heating, fields, vibration all step when the rate steps. A command-correlated systematic can reproduce plateau positions, hierarchy, and even κ-scaling.
Momentum-discriminating controls
Run as part of the sweep, not after it.
- Matched dummy load: electrically and thermally matched, no momentum exchange. The primary null.
- Thrust-axis reversal: genuine momentum transfer reverses sign; most artifacts do not.
- Instrument orientation reversal: device-fixed versus lab-fixed.
- Direct momentum balance: the reservoir side must close within budget.
- Opposed symmetric actuators: command signature present, net momentum absent by construction.
- Scrambled-winding control: identical power spectra and mean rates, plateau structure broken.
- Gate-side null: with coupling removed, plateaus must vanish from the schedule.
A bare disconnect is not a dummy load: it changes exactly the systematic environment that must be matched.
The null-first decision rule
| Observation | Interpretation |
|---|---|
| Staircase in command stream and schedule telemetry only | Gate verified; no force claim |
| Staircase survives matched dummy load | Command-correlated systematic; force reading invalid until separated |
| Signal reverses with thrust axis, momentum books close, dummy load null | Consistent with intended momentum transfer, within the stated uncertainty budget |
| Any branch-identification control absent or failed | Measurement invalid; resolve before any claim |
Phase-readout validity: why propulsion testing is the hard case
Propulsion testing is the class that wrapped-phase non-injectivity fools most effectively: small, slow, quasi-periodic displacement, read through a sampled phase-like observable, near the noise floor, by an experimenter with a preferred answer. The two silent failure modes are confidently wrong (the alias-reversal band, an inverted sign at full confidence) and invisible (the stroboscopic freeze, where impulse renders as stillness and a null is manufactured as easily as a signal). An instrument sets its own cadence but never thereby knows the unaliased frequency of the motion it reads, which on a thrust stand is the very quantity under measurement.
| EN-001 method | Assumption supplied | Thrust-stand requirement |
|---|---|---|
| A — known-start continuity | In-band start, smooth evolution | Measured baseline interval bounding drift and slew; continuous trace with inter-sample change < ½ cycle |
| B — two-rate congruence | Two rates, bounded physical range | Two independently clocked acquisition paths, documented transfer functions, common time base |
| C — fiducial timing | Detectable marker | Independent index channel bypassing the phase stream |
| D — external coarse prior | Second modality within half a wrap | A sensor of different physical type |
Method B is the most decisive: a physical displacement agrees across rates; an unresolved alias branch generally does not. The combined workflow D + A (prior seeding plus continuity tracking) is the recommended baseline configuration, with the run's full assumption ledger published beside the data.
07 · Evidence status and validation ladder
Every claim carries a tag, and every tag carries a stage.
Substantive claims in the note are tagged, following the discipline of the parent framework. Nothing about physical hardware is asserted as demonstrated.
[D] definition, convention or internal derivation
[S:En] supported by experiment n of the Simulation Evidence Pack, fixed seed 20599373
[S:A1] deterministic analysis A1
[S:A2] deterministic analysis A2
[C] conjecture or untested extension, paired with a validation stage
[I] interpretive device, carrying no theoretical weight
| Stage | Status | Purpose | Primary outputs | Failure condition |
|---|---|---|---|---|
| 0a | Complete | Released substrate | E1–E5 anchors | Mismatch with released evidence |
| 0b | Complete — this release | Deterministic completion | A1/A2 routines and tables, 36 verification tests, regenerated results.json, 286-entry manuscript manifest, pinned environment | Closed: all A1/A2-based values regenerate from the artifact package in a clean room |
| 1 | Next | Closed-loop simulation with baselines | Preregistered primary metric and count constraint; acquisition, tracking, slip, guard-budget, estimator and convergence measurements | Baseline superiority absent → retirement; model falsifier fires → back to Stage 0 |
| 2 | Pending | Bench with momentum controls | Full control set on hardware; the Stage-1 comparison repeated | Dummy-load or reversal failure |
| 3 | Pending | Applied gating | Impulse-estimate fidelity and ledger verification on a conventional actuator | No practical advantage |
36 regression tests
One class recomputes A1/A2 quantities in process rather than reading stored results, so a source regression cannot pass unnoticed.
286-entry manifest
Every [S:A1] and [S:A2] value in the note traces to its generating artifact and matches. One canonical value per number; scan-accuracy duplicates labelled.
Pinned environment
CPython 3.14.7, NumPy 2.5.1, SciPy 1.18.0, pytest 9.1.1. The regeneration driver refuses non-matching versions and exits nonzero on any manifest mismatch.
Clean-room bit-identical
A clean-room regeneration on the same recorded runtime and platform reproduces results.json bit-identically apart from its timestamp.
Model falsifiers
Any one of these, reproduced, sends the architecture back to Stage 0.
- No plateaus in the schedule at κ ∈ (0, 1).
- Wrong widths, positions or scaling at the preregistered windows: the measured ordering, the symmetry-fixed centres, or the κ/2π integer half-width scaling fails to reproduce.
- No locked/unlocked contrast under E5-matched conditions.
- Wrong locked event phases against the full A1/A2 tables at the declared operating points and operational guard, the falsifier most likely to fire if an implementation has quietly reverted to two command streams.
- Privileged depth values: any reproducible non-smooth feature in estimation error as a function of sampling depth.
Redundancy retirement
One outcome ends the project without any falsification at all. If at Stages 1–2 the coupled gate fails the preregistered superiority test on the primary metric under the count constraint, the scheduler is redundant engineering and is retired in favour of the baseline. Secondary metrics inform the post-mortem; they do not rescue the architecture.
Three quantities must stop being placeholders before Stage 1, in a separate preregistration document: the superiority margin, the confidence level, and the count-constraint tolerance, fixed before any run, with the architecture's own deterministic floor declared there as expected performance rather than discovered in the comparison.
Two deliverables stand regardless: the accounting discipline, and the metrology standard for phase-based thrust stands, which is overdue independently of this architecture.
Precise enough to fail on a bench, in a named place, at a stated number.
EN-002 §8 · Conclusion
08 · Release
Download the release.
The main note, the technical supplement, the evidence-pack repository documentation, and the Stage 0b artifact package, released together as the release rule of EN-002-EP requires.
STAR Drive · Engineering Note EN-002 v1.0
The condensed main note: substrate, architecture, the eleven design requirements, experimental signature and metrology, scope and limits, validation ladder, falsifiers, and Appendix A's substrate evidence summary.
Download the paper PDF · 1.3 MB ZIP · Complete packThe full STAR Drive pack
Everything in one archive: the main note, the technical supplement, the evidence-pack specification, and the Stage 0b artifact package (routines, tests, pinned environment, regenerated results.json and the manuscript manifest).
Download the pack ZIP · 2.8 MBCite this note
Dupke, A. (2026). STAR Drive: Sampled Transport by Alias Ratcheting (Engineering Note EN-002, Version 1.0). Scale-Time Dynamics LLC. Zenodo. https://doi.org/10.5281/zenodo.21833505
SCOPE NOTE · The pack remains one-mode and one-observer, with Gaussian readout noise on the estimation side and the circle-map coupling on the invertible domain, finite-run ensembles. It supports exactly the [S:En]-tagged claims, extended by the deterministic analyses A1 and A2, and nothing more. It does not test momentum exchange, reservoir coupling, actuator response, hardware event timing, dynamic acquisition, dynamic lock retention under a moving cadence, long-horizon slip statistics, controller-tick realization effects, super-critical coupling, or an assembled thrust loop.
09 · Contact
Scale-TimeDynamics
STAR Drive is an engineering note of the Scale-Time Dynamics programme. Correspondence on the architecture, the evidence pack, or a Stage-1 preregistration is welcome.
A design theory for the schedule