The Lyapunov Window

Status: proposed analysis and experiment. A forward/backward integration can measure numerical reconstruction error. It does not, by itself, prove an inverse, physical accuracy or an advantage for one representation. No ThreeBody experiment comparing IEEE FP64, posit or b-posit arithmetic has been completed for this design.

The proposed Lyapunov window is the span over which a stated reconstruction procedure meets a stated error tolerance under established assumptions. Its useful length depends on the method, step schedule, state scaling, arithmetic, input errors and dynamics. A Lyapunov estimate can inform that investigation; it is not automatically a certified maximum error bound.

The Reversibility Horizon

Separate three properties:

• A continuous physical model can have time-reversal symmetry.
• A discrete method can be symmetric in exact arithmetic: applying its negative step undoes its positive step under the method’s assumptions.
• A finite-state implementation can have an exact inverse on its admitted states.

Symplecticity preserves a geometric structure; it does not imply either of the last two properties. Symmetric and symplectic methods are related but distinct subjects in geometric numerical integration. Ordinary rounded implementations of a symmetric method need not be bitwise reversible.

A common sensitivity model is

If λ is positive, Cε₀ is nonzero, and tolerance δ exceeds Cε₀, this model suggests

This estimate treats error as an initial perturbation. A simulation also injects error during subsequent operations. A quire may reduce some of those injections; it does not turn every operation’s error into one initial constant or remove integration error. Effective precision is magnitude- and operation-dependent, so substituting a format’s storage width for ε₀ is not generally justified.

A more useful conditional bound tracks every step. Suppose the admitted state domain and a specified scaled norm establish an amplification bound Aᵢ and a local error bound ρᵢ. Then a proposed error analysis can use

For constant bounds A ≥ 1 and ρ ≥ 0, induction gives

and Eₙ ≤ E₀ + Nρ when A = 1. This is a mathematical consequence of the stated bounds, not evidence that those bounds have been established for ThreeBody. Forward and reverse operations both need coverage. If the reference is the continuous trajectory, ρ must also cover discretization error; comparison with a specified discrete reference has a different error budget.

The state norm must account for units and scales. Combining position and momentum errors as bare numbers produces a tolerance whose meaning changes with units. An initial-condition set, a norm, a time interval and a reference procedure are therefore part of the proposed contract.

Three distinct techniques

JANUS combines integer and floating-point arithmetic to construct a bitwise time-reversible N-body integrator. It supplies a different computational property from merely reducing floating-point error. Its exact reversal depends on its discrete construction and admissible execution conditions; it is not a numerical method with just a larger approximate horizon.

Stam’s reversible integrator also uses fixed/integer state with floating-point force calculations to obtain bitwise reconstruction. Exact return does not establish an exact solution of the continuous equations, unlimited range or unconditional stability.

A posit or b-posit quire instead provides exact accumulation of represented products while its finite capacity conditions hold. Inputs, non-accumulation operations and the final rounding still matter. Whether that improves a particular forward/backward residual over FP64 is an experimental question. A comparison should include relevant exact-accumulator or compensated IEEE alternatives, not assume ordinary summation is the only IEEE implementation.

Checkpointing stores states and trades storage against recomputation. Bennett’s reversible simulation is an information-preserving construction with time/space trade-offs. Checkpoint scheduling does not itself supply a physically accurate integrator or a certified sensitivity bound. These techniques can be combined only after their distinct premises are made explicit.

Revolve addresses checkpoint scheduling for reverse/adjoint differentiation. Its scheduling optimality is relative to its cost model and assumptions. It does not certify the error of approximately reversing a chaotic trajectory, and should not be described as assuming that all numerical arithmetic is exact.

Information and saved state

Chaotic sensitivity does not imply that an invertible evolution literally erases its fine-grained state information at a universal bit rate. Statistical entropy results do not, without a specified model and assumptions, provide the minimum number of checkpoint bytes for a finite computation. A bijective finite-state update can retain its predecessor exactly; a rounded many-to-one update generally cannot be inverted from its result alone.

Restoring a checkpoint reproduces its saved bits. It does not recover precision missing from those bits or prove that the preceding approximate reverse calculation was correct. A checkpoint may already differ from the intended physical trajectory. Resetting reconstruction error relative to that saved state therefore does not reset its physical error to zero.

More checkpoints can extend practical replay or navigation coverage, given enough storage or recomputation. They do not create unlimited numerical precision or unbounded guarantees at fixed resource cost. Recording a restored state as a successful inverse would confuse storage recovery with arithmetic reversibility.

The Window as a Coeffect

The proposed graph evidence would relate:

• the admitted initial states, units and norm;
• range and representation requirements from Numeric Selection;
• a specific integrator, force evaluation and step schedule;
• sensitivity and local-error bounds, including their provenance and validity domain;
• a required reconstruction tolerance and resource budget.

Composer could then compare candidate representations and schedules when those premises are available. A measured finite-time exponent would remain estimated evidence; a verified bound would retain its justification and applicable domain. The result should identify which kind of evidence supports it. An unresolved numerical obligation must not become a guaranteed horizon through a default constant or an assumed effective bit count.

This proposed analysis would need numerical transfer functions and consumers that establish the sensitivity and local-error evidence. The arithmetic construction and placement design identifies how that evidence would constrain eligible realizations.

The relation can describe a whole computation region. Its inverse recipe and numerical evidence do not imply a saved dual beside every intermediate value. An admitted exact inverse can reconstruct from current state; an approximate inverse needs its error envelope; replay needs its inputs; a checkpoint policy needs its retention and recomputation budget. The selected realization determines which execution values remain live. A Path Less Traveled connects that distinction to native bidirectional composition.

Forward-mode differentiation adds another, separate choice. A fixed number of current tangents can move through a streaming calculation without retaining all earlier tangents. Their storage follows the live schedule and tangent count. They carry sensitivity information, not a predecessor history or an inverse certificate. Selecting a recovery strategy therefore needs the primal dynamics and arithmetic contract even when its gradient computation has no reverse tape.

Local Exponents and Global Claims

An asymptotic Lyapunov exponent describes long-time perturbation behavior under its defining assumptions. A finite-time estimate samples a trajectory and interval. Neither automatically bounds every perturbation admitted by a program’s input domain. Close encounters and near-singular force evaluation require explicit separation or other domain constraints, and can invalidate a previously adequate step/error model.

A runtime estimator may inform an adaptive schedule. A guaranteed schedule would also need a monitor or validated domain condition, a safe response before a bound is exceeded, and evidence that adaptation preserves the chosen numerical method’s requirements. Merely observing a larger local exponent after an encounter does not validate the earlier interval retroactively.

Approximate forward/backward agreement is a test result for the tested states. It does not prove injectivity or surjectivity over the admitted state space. Exact finite-state inversion requires its own argument; closeness to the physical trajectory, conservation behavior and shadowing require different evidence. Consequently, a reversibility type or structural composition rule cannot by itself certify the numerical residual of a rounded implementation.

Checkpoint Economies

A usable checkpoint must contain enough state for the selected recovery behavior: positions and momenta may be insufficient if replay also depends on an adaptive step controller, random state, cached history or external inputs. Deterministic recomputation needs the same arithmetic semantics and relevant execution choices. Checkpoint integrity, lifetime and retention must be accounted for separately from reconstruction accuracy.

The execution design assigns analysis and selection to Composer, arena ownership and orchestration to Prospero, work to Olivier actors and ready-turn scheduling to Ariel. Those components could carry a selected checkpoint policy; they do not make its numerical assumptions true. This organization does not presume a managed runtime or garbage collector.

An initial ThreeBody experiment should fix its initial-condition set, force law, collision/separation treatment and step policy. Record forward error against an appropriate reference, forward/backward residual, invariant drift, boundary events, execution time, transfer costs and checkpoint storage separately. Compare arithmetic choices at stated storage, accuracy or throughput budgets, and report failures as well as successful cases. Restored checkpoints must be distinguished from computed reverse states in both data and visualization.

The result would test whether a useful reconstruction window exists under those conditions and whether an adaptive policy helps. It would not establish an ordering in which posit always outlasts FP64. See Rounding on Real Hardware for operation-level obligations and Pondering Fearless Parallelism for the broader design discussion.