# Geometry, retained history and response: a bounded model

Version 0.2 · 18 October 2026

A driven oscillator with an explicitly imposed, fading internal record. Compare it with a matched baseline, inspect the geometry law and close the oscillator energy ledger. This model demonstrates mathematical consequences, not universal learning or newly observed physics.

## One bounded model, one inspectable claim

Version 0.2 turns one part of Christopher Hill’s framework into an explicit mathematical exercise: a driven oscillator retains a fading record of input intensity, and that record changes its later mechanical response. This is a dimensionless, phenomenological toy model. It is not a unified theory, a derivation of new fundamental dynamics, or a model of Alpha 1. Every plotted trajectory is a calculation from selected equations, not experimental data. The contribution is a model that readers can inspect, compare with a baseline and challenge.

Its question is narrow: can a specified history variable alter the response to the same later probe, and can the oscillator’s energy balance be closed when its stiffness changes? Classical dynamics supplies the oscillator framework; material-memory research supplies examples of history encoded in physical states. Neither source derives the equations chosen here. A successful simulation demonstrates consequences of those choices, while a physical realization would require an identified substrate and measurements.

Sources: geometry-g-tong-dynamics, information-info-keim2019

## Define the state before interpreting it

The state is displacement q, velocity p and a scalar record m. Time, coordinates, forces and energy are expressed in normalized units, with mass set to one. L is a fixed relative length for each run. The parameters satisfy L > 0, ζ ≥ 0, α ≥ 0, β ≥ 0 and τ > 0. Both candidate and baseline begin at q = p = 0; the candidate also begins at m = 0. These choices ensure positive stiffness and keep the continuous-time record between zero and one.

The word record is deliberately limited. m summarizes a fading, saturating influence of past squared input; it does not store the entire waveform or a meaningful message. Different histories can produce the same m. Nothing in the model represents subjective experience, biological memory, an inferred external environment or a target task. It is a structural-memory demonstration, not a task-learning system.

Sources: geometry-g-buckingham, information-info-keim2019, information-info-shannon1948

## The proposed equations

The oscillator receives input u(t), loses energy through damping and responds to stiffness k. Geometry enters through the selected L⁻² dependence; retained history enters through the selected multiplier 1 + βm. The record increases with squared input, saturates as m approaches one and relaxes on timescale τ. These laws are imposed for this demonstration. They have not been derived from a material microstructure, measured in a device or inferred as a universal relationship.

Keeping the damping ratio ζ fixed means the viscous coefficient 2ζ√k changes when stiffness changes. The candidate therefore modifies both stiffness and the damping coefficient, in a linked way. An observed change in the simulated probe should not be attributed solely to stiffness. A different model holding the viscous coefficient fixed would be a useful competing architecture, but it must have a separate equation and version.

q̇ = p; ṗ = u − 2ζ√k p − kq; k = L⁻²(1 + βm); ṁ = αu²(1 − m) − m/τ

Dots denote derivatives with respect to normalized time. L is constant during a run. The equations define this toy model; cited literature supplies background rather than empirical validation of this particular system.

Sources: geometry-g-tong-dynamics, information-info-keim2019

## Compare the same training, rest and probe

The input schedule has three declared phases: training, a rest interval with u = 0, and a probe. Use the same complete seeded waveform, time grid and phase boundaries for candidate and baseline. A seed alone is insufficient if different solver calls draw different random samples; preserve or deterministically reconstruct the actual input as a function of time. The baseline enforces m ≡ 0 and therefore has k = L⁻² throughout. It receives the same input and shares every other parameter.

Probe forcing also updates m under the stated law. A weak probe may reduce that disturbance, but the model does not assume readout is nondestructive. Rest can reduce residual displacement and velocity without making them identically equal between models. Report q, p and m at probe entry. If testing memory alone, use a separately declared state-matched intervention; resetting q and p requires accounting for the removed oscillator energy.

Sources: methods-method-prereg, information-info-keim2019, methods-method-repro

## What the record actually retains

During an unforced rest interval, m decays exponentially. For constant input amplitude A, it approaches a plateau set by competition between acquisition and forgetting. These exact relationships are verification targets: the numerical solution should reproduce them as the time step is refined. They also show that m tracks weighted exposure rather than every detail of history. Histories with identical u²(t) and initial m have identical record trajectories, even if the force signs differ.

A changed probe response is possible, not guaranteed for every parameter choice or observable. It may disappear after a long rest, when coupling is zero, or when the chosen probe is insensitive. An increase in stiffness does not guarantee a larger response amplitude. Frequency, damping, phase and residual motion all matter. The useful output is a specified response difference with its conditions, not a universal claim that training improves performance.

The bounded interval for m follows from the equation, not a data fit: at m = 0 its derivative is αu² ≥ 0, and at m = 1 its derivative is −1/τ < 0. Within the interval, a record can persist when the input is removed without making the equations depend on an inaccessible past. Once q, p and m are specified, the next state follows from the current input. History dependence in a reduced description is therefore compatible with ordinary state-based dynamics.

u = 0: m(t₀ + Δt) = m(t₀)e^(−Δt/τ).  u = A: m* = αA²/(αA² + 1/τ)

For constant A, m approaches m* exponentially at rate αA² + 1/τ. These are mathematical consequences of the selected memory law, not measured material constants.

Sources: information-info-keim2019, information-info-sethna1993

## Close the oscillator’s energy ledger

Define the oscillator energy E as kinetic plus instantaneous elastic energy. Differentiating it gives three terms: work from the applied drive, dissipation through damping and work associated with changing stiffness. The last term is essential. Counting the drive alone would produce an apparent imbalance whenever the stored record changes k. For fixed L, k̇ = βL⁻²ṁ. Changing L during a trajectory would require the additional geometric term −2(1 + βm)L⁻³L̇; this version instead holds L fixed.

Integrate the signed drive work Wdrive = ∫up dt and signed parametric work Wparam = ∫q²k̇/2 dt. Integrate dissipation D = ∫2ζ√k p² dt, which is nonnegative in the stated parameter range. The balance is E(t) − E(0) = Wdrive + Wparam − D. Either work channel can be negative. Label the residual and inspect its convergence rather than presenting a small numerical discrepancy as energy production.

E = ½p² + ½kq²; dE/dt = up − 2ζ√k p² + ½q²k̇

This is an exact oscillator-subsystem identity for the stated differential equations. Its numerical residual measures integration and accounting error; it is not a new conservation law.

Sources: geometry-g-tong-dynamics, geometry-g-feynman-energy

## The ledger is not a full-device efficiency

The model has no physical mechanism for writing, maintaining or erasing m, or for actuating the required stiffness. Parametric work identifies the energy exchange with the oscillator, but does not specify all energy consumed by the apparatus that would cause it. Storage, sensing, control, losses and reservoirs could add substantial costs. Consequently, the oscillator ledger cannot establish total-system efficiency, energy savings, net power generation or a thermodynamic advantage of memory.

Landauer’s bound applies to particular logical erasure processes under stated conditions; it cannot be assigned automatically to each numerical update of m. Likewise, a normalized energy is not a value in joules without a calibrated physical mapping. A future device proposal must add its substrate, controls and reservoirs, then close their combined balance. Until then, the honest result is accounted oscillator work and dissipation with an unspecified implementation cost.

Sources: information-info-landauer1961, information-info-berut2012, information-info-seifert2012, information-info-sagawa2010

## Checks that can expose a faulty implementation

When α = 0 and m starts at zero, m remains zero and the candidate must reproduce the baseline. When β = 0, m may evolve, but it cannot affect q or p; oscillator trajectories must again agree. With u = 0 from the initial zero state, the whole state remains zero. These equivalence checks are stronger than a plausible-looking plot because they compare independently specified limiting cases of the same equations.

Also verify the rest-decay solution, constant-drive plateau, nonnegative dissipation and energy-balance convergence. Refine the step size relative to both oscillator and memory timescales. A discrete solver can overshoot m’s continuous bounds; silently clipping it may conceal inadequate integration and alter the energy relation. Disclose solver settings and detect failed convergence. Passing these checks validates computation within tolerance, not the truth or novelty of the chosen physical interpretation.

Sources: methods-method-repro, methods-method-uncertainty, geometry-g-tong-dynamics

## Define a comparison before exploring the sliders

Choose probe observables before examining the output: a displacement trajectory difference, a specified-window RMS amplitude, phase relative to the input, or an independently defined fitted response. State normalization and time windows. A larger difference is not better performance unless a task and a desired outcome were specified independently. Report initial probe states and the input schedule beside each comparison so residual oscillator motion is not mistaken for a pure stored-state effect.

A geometry study should vary L across separate runs and keep other normalized parameters explicit. At fixed m, the undamped natural frequency is √k = √(1 + βm)/L. This is not automatically the damped oscillation frequency or driven resonance peak. Nor does it make full trajectories self-similar when drive frequency and memory timescale remain fixed. The selected scaling law supplies a benchmark whose physical range would need separate testing.

Parameter identification is a separate problem from drawing different curves. If the only measurement is displacement, several combinations of input calibration, damping and stiffness history may explain a similar trace. In particular, an inferred influence through βm does not independently measure β and the physical record. A future study should include additional observables or interventions and report which parameters remain degenerate. Use held-out runs to compare explanations; do not call an adjustable fit a detection of a new memory carrier.

Sources: geometry-g-buckingham, geometry-g-tong-dynamics, methods-method-loo

## What would turn the model into research evidence?

The next empirical step is to identify a material or device whose independently measured internal state could play m’s role, then estimate parameters from one dataset and predict another. Compare the candidate with ordinary linear dynamics, a temperature-dependent stiffness model, passive relaxation and other plausible internal-state laws. A successful fit alone cannot identify the carrier; interventions and new conditions should distinguish explanations. A failed prediction should narrow or replace this model rather than be absorbed by unrestricted retuning.

The wider Hill claims remain separate. This exercise neither derives general relativity nor supports universal fractality, cosmic learning, consciousness, a parent universe or geometric travel. Its value is more specific: it makes a geometry–history proposal inspectable and exposes the work terms, controls and missing mechanism. Readers can contribute a better constitutive law, a numerical counterexample, a reproducible dataset or a failed benchmark without accepting the larger worldview.

Sources: methods-method-loo, methods-method-prereg, information-info-keim2019

## Complete model record

See model-v0.2.json for glossary, comparisons, predictions, limitations and eight hypothesis records.

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