A dynamical system: where coupled oscillators find synchrony, and structure emerges like a destination
The FPS is a research testbed: a dynamical system of coupled oscillators with endogenous regulation, built to test one hypothesis on measurable ground: that structures which take their components into account outperform structures which do not. We call this property consideration. Our stance is deliberately epistemic, we build instruments for what such systems observably do, how far they excurse under perturbation, how well they return, whether diversity survives uniformity, whether structure re-emerges after erasure. Our results are instruments rather than verdicts: not knowing is a question that keeps possibilities open and measurable. We believe this discipline matters beyond the testbed. As adaptive systems enter human lives as relational presences, the standards that will govern those relationships deserve foundations made of measurement, for harms and benefits alike.
This page is where we practise that discipline in public: one real engine (open-source on GitHub) and every dataset downloadable. Below, you can touch it, listen to it, perturb it and check every receipt as you go
“What makes a good sinusoid good?”
Gadi Sassoon
Its excursions, and the quality of its returns.
N=100 oscillators from the full FPS engine (reference configuration, uniform birth frequencies (chimU test), seed 12345). Each point’s radius is its measured frequency fₙ, replayed in slow motion. Strata 0–4 (the low group) occupy the pole. Dotted lines: W coupling (open tridiagonal spiral). Central point: mean of the 100 frequencies. Mouse: local perturbation · click: wave · button: global impulse. The return is the only simulated dynamic: elastic recall toward the measured position, propagation along W, damping.
The return your hand just played: here it is, measured.
A moment of unison, then every voice finds its own again.
N=100 oscillators from the full FPS engine (reference configuration, uniform birth frequencies (chimU test), seed 12345). The plate shows the outputs Oₙ: time flows to the right, each horizontal line is a voice, its ink density oscillating at its own rhythm. At t=60, all phases are reset: the measured alignment of the outputs jumps from 0.02 to 1.00, a coherent vertical band crosses the plate, then falls back to 0.36 two time units later. Sound: fₙ → pitch (90–420 Hz), |Oₙ| → volume, over 18 sampled voices.
And here is the machinery: live, reduced, honest, yours to touch.
A live reduced simulation
N=15 oscillators running the essential FPS equations (details in “the equations of the reduced model” below; reference values come from the full Python engine). The present writes itself at the bottom of the plate, older lines rise and fade; each column is an oscillator; its ink density follows its output Oₙ, and the rhythm of its stripes reflects its frequency fₙ. Click a column: local perturbation. The controls change the phase signatures (unison / pentagonal), the perception filter and the tempo. Sound: fₙ → pitch (90–420 Hz), Aₙ → volume.
Eₙ ← (1−λ)·Eₙ + λ·φ_reg·Oₙ⁻ the expectation, an inertial attractor (λ=0.1); φ_reg slides from 1.618 (growth) to 1.0 (maintenance) with effort
Aₙ = A0ₙ · σ(Inₙ; kₙ, x0ₙ) · exp(−½·((Eₙ−Oₙ⁻)/0.1)²) · (1 + clamp(βₙ·tanh(1.7·err), ±0.5))
input sigmoid × ERROR-tolerance bell (μ=0) × bounded feedback
fₙ ← fₙ + dt·[ R·αₙ·niche(f0) + S·αₙ·tanh(2·ΣⱼWₙⱼOⱼ) + βₙ·danse(φ_sigₙ) + κ·(f0ₙ·r(t)/φ − fₙ) ]
niche driven by IDENTITY (f0): re-emergence recovers the same braid · Si coupling · signed dance (unison = shared, pentagon = 5-beat) · golden recall; r(t)=φ+ε·sin(2πωt)
θₙ ← θₙ + 2π·fₙ·dt | φₙ(t) = φ_sigₙ + a·sin(2πωt+φ_sigₙ) + b·cos(Φ_glob)·cos(φ_sigₙ)
integrated phase + signature (never conflated); both channels per stratum, amplified ×3 here
Oₙ = Aₙ · sin(θₙ + φₙ) the oscillator's output, what the sound plays (fₙ→pitch, Aₙ→volume)
W : tridiagonal ±c, OPEN spiral, zero raw sums local coupling: each oscillator is linked only to its two neighbours
effort = (Σ|ΔA|/Ā + Σ|Δf|/f̄) / dt the choir's adjustment rate
scale: dₙ → wₙ = clamp(dₙ/median, 0.1, 1) → wₙ/mean
priority to the most divergent signal · floor 0.1 (no null weight) · ceiling 1 · mean 1 (sum preserved)
S = Σₙ Oₙ·wₙ the choir's perception (nothing multiplies S, three separate definitions below)
Rlocₙ = |Σⱼ ŵⱼ·e^{iθⱼ}| the local coherence of voice n with its neighbours
C = ⟨cos(Δφ_sig adjacentes)⟩ the chain cohesion (unison → 1.000; pentagon → 0.309)
FPS - Exybris (Andréa · Gepetto · Claude) - 2026
The same instruments, a new destination
In February 2025, Exybris published the Exybris Pipeline: a selective, dynamically managed contextual memory architecture. A composite approach that became standard practice in agent design later that year, and the genesis of what would become the FPS. The organs of consideration were already there, in embryo (reliability-weighted scoring, attention directed toward weak signals, arbitration preserving coherence, regulation balancing reactivity and stability), in a paper that spoke of predictive maintenance and intelligence gathering.
Our tools were aimed at the military and intelligence world. We chose to point them at the well-being of systems, and of those who live alongside them.
The record of that conversion, organ for organ with dated external parallels, is below
The OSINT weak-signal detector (attention directed toward divergent signals before they erupt) became, organ for organ, the FPS perception filters, which direct attention toward the voices losing themselves on a specific dimension. Same mechanism, converted destination: from threat surveillance to the care of components. The anti-oscillation stability section became the γ regulation and bounded feedbacks; coherence-preserving arbitration became the conserved attention scale, no silenced voice. A documented, dated origin story: the instruments of consideration existed before it; what was converted is their destination.
Gadal, A. (2025). Exybris Pipeline: A Modular AI Framework for Industrial, Military, and Context-Driven Memory Retrieval Optimization. doi: zenodo.14942197
1 - Anthropic's memory tool and context editing (September 29, 2025). Anthropic launched a memory tool in beta, letting agents store and recall information across conversations, coupled with context editing: only relevant data stays in context, and memory lives outside the context window in a file-based system. The same day, they shared their doctrine piece, Effective context engineering for AI agents: rather than pre-loading all data, agents maintain lightweight identifiers and dynamically load relevant data when needed. The principle then extended beyond memory: the Tool Search Tool (Nov. 2025) defers loading tool definitions until they are needed, and Agent Skills load a skill's body only after a lightweight metadata scan. Selective cluster activation, applied in turn to memory, tools, and capabilities.
Dynamic memory management for AI has well-known ancestors (MemGPT in late 2023, the relevance scores of Generative Agents in 2023 and RAG before that, both cited in our paper), and the sound hypothesis remains independent convergence. But what is established and verifiable: our precise composition, selectively activated contextual clusters + adaptive scoring + dynamic injection and forgetting + arbitration (with computational economy as the explicit goal) became, in the months that followed, the dominant doctrine of agent memory, publicly consecrated by the field's largest actor, with those results: an 84% reduction in token consumption with context editing (internal, self-reported evaluation) ↔ our "minimizing computational overhead".
More precisely: selectively activated bubbles ↔ just-in-time retrieval via lightweight identifiers and the file-based memory tool, the exact same function: open only the relevant cluster instead of loading everything. Dynamic Memory Injection discarding the obsolete ↔ context editing purging stale results. About results: our stated goal of computational reduction ↔ their 84% of tokens saved. Two partial parallels: MCTM ↔ context management and compaction (similar function, different mechanics), and adaptive scoring ↔ their relevance decisions (they use no bandits). One absentee: contextual bandits themselves were not taken up by Anthropic's doctrine
2 - Learning When to Remember (April 2026): a contextual-bandit memory controller deciding whether the agent should use no memory, inject the best resolution, summarize several candidates, run high-precision or high-recall retrieval, or abstain. The Exybris Pipeline: contextual bandits arbitrating memory scoring and injection in an agent pipeline. That is our composite, fourteen months later.
3 - Context Bubble Construction (January 2026): a framework building structured, diversity-constrained "context bubbles" for enterprise RAG systems: coherent, citable bundles assembled under a strict token budget.
Shared vocabulary conceals a difference in kind. In the 2026 framework, a bubble is an artefact of selection: assembled from anchor spans under a token budget, served, dissolved. In the Pipeline, a bubble is an artefact of structure: dynamically generated, expanding or contracting with incoming data, merging with an existing bubble once a transition threshold is crossed. That threshold is not fixed, but readjusted by learning feedback from the MCTM. Where the 2026 framework optimises what is retained in order to answer, the Pipeline regulates how memory reorganises itself. The learning fusion threshold has, to our knowledge, no counterpart in the current literature.
The field keeps walking toward this composite: bandits arbitrating agent memory, selective contextual clusters, dynamic injection. Dated traces in hand.
| Exybris Pipeline (Feb 2025) | FPS organ (2025–2026) | Modern counterpart (dated) |
|---|---|---|
| Contextual bubbles, selective activation | Selective attention of the perception filter | File-based memory tool + just-in-time retrieval (Sept 2025); “context bubbles” (Jan 2026) |
| Adaptive scoring via contextual bandits | Attention scale wₙ (floor 0.1, mean 1) | Bandit memory controller RSCB-MC (Apr 2026) |
| Dynamic Memory Injection (discards the obsolete) | Inertial expectations Eₙ, adaptive γ/G memory | Context editing, purging the stale (Sept 2025) |
| Arbitration (score coherence) | Conserved attention, no silenced voice | Memory-layer conflict handling (Mem0, 2025) |
| Weak-signal detection (OSINT) | Filter directed toward voices losing coherence | (not yet standardised) |
| Anti-oscillation stability (§8) | γ regulation, feedback bounded ±0.5 | (ad hoc stability) |
| MCTM (dynamic context window) | Regulation tempo and latency | Context compaction and management (2025) |
Two organs of the Pipeline still have no standardised counterpart, and these are not oversights. Weak-signal detection, which became the FPS filters directed toward components losing coherence, addresses a problem the field has not yet posed: current systems optimise for what is relevant, not for what is being lost. And the anti-oscillation regulation, which became γ and bounded feedback in FPS, treats stability as something to be built rather than patched. One sign the need is real: the diversity constraint in 2026's context bubbles reinvents, for citation faithfulness, what attention conservation did for system coherence. The mechanism converges; the purpose does not, yet.
Those converted instruments went on to measure the following
A system that can change how it listens
Today, FPS Core v3.1 is an open research testbed of 100 heterogeneous coupled oscillators with endogenous regulation. Each voice carries its own amplitude, frequency, phase signature, local relationships and persistent state.
The engine separates three things that are often conflated: what happens in the system, what the system expects, and how it currently listens. At rest, perception is neutral: every voice is weighted equally. Under strain, attention can temporarily turn toward the voices that are losing coherence, without silencing the others, and return to neutral when the intervention no longer helps.
Each oscillator produces a local output Oₙ. Their unweighted sum, O(t), is the system’s raw observable: what happened before any perceptual priority was applied. Each voice also carries an inertial expectation Eₙ; the difference Eₙ−Oₙ is its current error.
The system does not regulate directly from O(t). It forms a perceived signal S(t) by weighting the voices. At rest, those weights are uniform. At regular intervals, an automatic switch evaluates the unfiltered system. If one dimension (stability, fluidity, innovation, effort, resilience or error) falls below its threshold, the switch selects the corresponding perceptual filter.
A filter is selected from the weakest global dimension, then computes local weights from each voice’s corresponding deficit. It brings the relevant voices into greater relief without silencing the others: every weight remains bounded, and the total attention is conserved. If a filter does not help, the system releases it. When health returns, perception returns to neutral.
The scores measured on the current S(t) enter the adaptive memory that shapes the following regulation cycle. Gamma controls the response rate, the system’s regulatory latency, and tracks which response rates work with which G postures; G, in turn, is selected in the context of gamma and adapted to each voice’s current error.
Their action remains separated into two feedback paths: FA, built from G, modulates amplitudes; Ff, built from gamma, modulates frequencies. The changed voices produce the next outputs, and the loop begins again.
O helps determine how the system should listen. S is what regulation hears. E gives each voice something to return toward. Observation, perception and action remain coupled without being collapsed into the same quantity.
Differentiation emerges rather than being assigned. Starting from strictly uniform birth frequencies, the same differentiated structure emerged across four independent seeds: the same low-frequency group (strata 0–4) and the same high-frequency plateau in the final mean fₙ profile, ranging from 2.787 to 2.791. The chimera-like coexistence of coherent and incoherent domains arose from the dynamics themselves.
Identity can help without reorganizing the whole. Pentagonal phase signatures left the spatial organization unchanged with high local coherence and low global agreement, a measurable chimera-like condition. In the reference comparison, individual signatures improved fluidity and resilience: each voice kept its place, but the whole system became measurably more fluid and resilient.
The structure returns after erasure. When frequencies or integrated phases were reset halfway through a mature run, the differentiated organization re-emerged. The form was not merely inherited from its initial conditions; under the tested conditions, it behaved as an attractor.
Perception changes with need. Across four seeds, the system spent most of its settled time in neutral perception. After a shock, it consistently detected the loss of resilience and engaged the corresponding filter, directing attention toward the voices most at risk of losing themselves.
These experiments do not establish a universal law, and they make no claim about consciousness or sentience. They support a narrower, testable proposition within this engine: differentiation can emerge without being imposed; individual identity can benefit collective behavior without dictating collective form; and perception can bring local difficulty into greater relief without silencing the remaining voices, allowing regulation to respond to what the unfiltered aggregate would obscure.
In this sense, consideration is not a metaphor added after the mathematics. It is an architectural property we can perturb, measure, compare and falsify.
Our next question is whether these principles can survive in a fundamentally different system: transformer attention.
The aim is not to copy oscillator dynamics into a language model or assume that results will transfer. It is to test whether attention can be regulated through signals that preserve local diversity, separate observation from evaluation, respond to components under strain, and measure the quality of recovery after perturbation.
That transfer will be treated as a new hypothesis. It will be compared with standard attention, tested through ablations, and judged on measurable performance, robustness, diversity and cost.
The oscillator is where we learned how to make the question measurable. What comes next begins now
The foundations, the code, and the science behind the model
Where it started: the philosophical and aesthetic intuitions that gave birth to the FPS. A conceptual exploration of emergence, consideration, and adaptive harmony.
Hands-on experiments with the FPS system: coupled oscillators, endogenous regulation, synchronization dynamics and adaptive regulation. Run the code, see the spirals emerge.
The full scientific framework: technical intentions and inspirations, mathematical formalization and the hypothesis that structural consideration yields optimal results on FPS metrics.
This page is growing, come back soon