# The Oscillator Model **The thermodynamic re-foundation of Cascade's ring system** *This document replaces Dunbar's-number framing in the v1 Cascade integration with a four-parameter damped-oscillator model. The arguments and parameter assignments are verified against current research (see `sources/fact-check-notes.md`).* --- ## 1. The Core Claim Cascade's ring system was originally framed around Dunbar's number — the idea that ~150 relationships is a *cognitive* ceiling, fixed by neocortical capacity for tracking relationship state. This document proposes a different foundation: the limit on a personal Cascade is not cognitive but **thermodynamic**. Relationships are not slots in memory; they are **open systems that require continuous energy input to stay coherent**, and the true ceiling is wherever total energy expenditure across all active relationships meets a person's available energy budget. Under this model, ring position is not "social distance." It's an **inverse measure of how much energy a relationship currently needs to stay alive.** ### Why this matters The Dunbar framing has three weaknesses: 1. **Empirically contested.** Modern reanalysis (Powell et al.) shows the neocortex-group-size relationship does not survive rigorous statistical scrutiny. Dunbar's number is a useful historical anchor, not a load-bearing premise. 2. **Fixed per person.** Dunbar's ~150 doesn't shrink under grief or expand under joy. Real relational capacity is dynamic. 3. **Not actionable.** A cognitive ceiling tells you nothing about what to do today. A thermodynamic ceiling does — it tells you where the energy is going. The Oscillator Model is a working theory. Its empirical predictions are flagged in §8 as open work. --- ## 2. The Three Macro Traits Three observable, relationship-level quantities sit above the physics layer. These are what you'd actually notice about a relationship without measuring anything formally. | Trait | Definition | Cascade grounding | Research anchor | |---|---|---|---| | **Coherence** | How phase-aligned two people's signals are right now — a state variable | Hearts (direct overlap), Stars (category congruence) | Interactional synchrony (Bernieri & Rosenthal 1991); Communication Accommodation Theory (Giles 1973) | | **Resonance** | The amplification that occurs when coherent signals reinforce each other over time — a growth process | Existing Resonance system | Self-disclosure reciprocity (Altman & Taylor 1973) | | **Dissipation** | The rate at which alignment decays without fresh input — the thermodynamic cost of *not* maintaining a connection | Currently informal "inactivity decay" in ring-demotion logic | Damped-oscillator physics (standard references) | These three are not independent. They form a causal chain (see §4). --- ## 3. The Four Micro-Parameters Borrowed from damped-oscillator physics, these are the underlying dials that *produce* the three macro traits above. They are not a parallel system to map 1:1 against Coherence/Resonance/Dissipation — they operate one level down. | Symbol | Physics meaning | Relational equivalent | Comm-theory grounding | |---|---|---|---| | **ω** (omega) | Angular frequency — how often the system cycles | Interaction frequency / contact rate | Already built: the hourly→yearly tempo ladder (R6→R1) | | **A** | Amplitude — magnitude of the oscillation | Intensity / depth of self-disclosure | Social penetration theory (Altman & Taylor 1973) | | **φ** (phi) | Phase — timing offset between two oscillators | Synchrony / rhythm alignment | Interactional synchrony (Bernieri & Rosenthal 1991); Communication Accommodation Theory (Giles 1973) | | **ζ** (zeta) | Damping ratio — how fast oscillation settles after a disturbance | Resilience / conflict-repair capacity | Rupture-and-repair research (Gottman; Tronick) | ### Prerequisite for φ-alignment φ-alignment (phase-lock) requires *compatible* ω. Two people whose natural rhythms are incompatible — say, one a daily-pulse person and one a yearly-pulse person — cannot achieve phase-lock at the higher tempo. The system will lock at the slower tempo, or not at all. This is why the Cascade ring system has discrete tempos (hourly, daily, weekly, monthly, quarterly, yearly): each ring represents a *phase-lockable* ω, and the rings do not interlace. A relationship runs at one tempo, and that tempo is a structural choice. --- ## 4. How the Layers Connect The micro-parameters combine to produce the macro traits in a specific causal order: ``` φ-alignment (requires compatible ω) ↓ COHERENCE ↓ Coherence + sustained A (disclosure/intensity) ↓ RESONANCE ↓ Sustained resonance → lowers ζ over time (repair gets easier with trust) ↓ Lower ζ → less ω required to sustain the relationship ↓ RING PROMOTION ``` In plain terms: two people get in sync (φ), which only works if their natural rhythms are compatible (ω). Sustained sync plus real depth of exchange (A) builds resonance. Resonance, sustained long enough, makes the relationship more resilient to disturbance (lowers ζ) — repair gets easier because there's a track record. And a relationship that's gotten easier to repair needs *less frequent contact* (lower ω) to stay warm — which is mechanically what ring promotion means. This gives ring promotion a **causal mechanism** rather than an arbitrary threshold ("30 days of daily pulses crosses a line"). The relationship doesn't get promoted because a counter hit a number — it gets promoted because ζ has actually dropped as a structural consequence of accumulated resonance, and the system needed less ω as a result. --- ## 5. The Symmetric-ζ Decision A real fork: should ζ be **asymmetric** (each person has their own repair tendency, combined somehow) or **symmetric** (one shared damping ratio for the dyad)? - **Empirically**, asymmetric is more accurate. Repair research consistently finds one person doing more repair work than the other, and that imbalance is itself diagnostic of relationship health. The prescriber/reciprocator pattern in social penetration (Dermer & Thiel) is one example; Gottman's 5:1 positive-to-negative ratio research shows similar asymmetry. - **Structurally**, symmetric is what licenses treating the relationship as a single physical system with one energy budget, rather than two separate internal states forced into a number. This model takes the symmetric path **as a deliberate simplification**, not an oversight, in order to make the next move possible: a real energy accounting. **Cost of the simplification:** we lose the ability to detect the exact one-sided-repair pattern that is most predictive of relationship failure in the underlying research. This should be flagged in any future spec, not buried. --- ## 5.5 The Honest-vs-Functional Tension The model needs to be useful at the practice level (the Daily Three must run, cohort members must make decisions) AND honest at the theory level (we can't claim physiological substrates we don't have). ### What honesty costs - The "energy" metaphor has no proven physiological substrate (see §6.1 below for full caveat) - The symmetric-ζ simplification erases the one-sided-repair pattern Gottman identifies as the strongest single predictor of relationship failure - Granovetter's strong/weak ties distinction (1973, Strength of Weak Ties) is a SEPARATE dimension the model does not capture — the model tracks frequency/depth/sync/repair, not whether a tie is structurally bridging or bonding. Acknowledging that weak ties often deliver more novel information than strong ties is a missing axis. ### What functional compression gives us - The Daily Three is runnable in 3 minutes by a non-expert (it requires no measurement apparatus, no relational-psychology training, no granular tracking) - Decisions are made at decision points (one per day), not continuously — this matches the rate at which structural decisions actually matter - The vocabulary is small (ω, A, φ, ζ, E_available, E_total) — a cohort member can hold it in working memory ### How we resolve the tension - At the protocol level: optimize for functional compression. Run the Daily Three. Make decisions. Track the budget. - At the theory level: maintain radical honesty. Flag what's metaphorical vs. what's structural. Don't claim physiological substrate. Don't paper over Granovetter's axis. Don't pretend symmetric ζ is empirically accurate. - At the upgrade path: every honest concession is a v3 research question, not a hidden assumption. The Oscillator Model is a working theory that knows what it doesn't know. The honest-vs-functional tension is not resolved by being more honest OR more functional — it is resolved by being honest about which level we are speaking at. --- ## 6. From Damping to Energy If ζ is shared by the dyad, the relationship behaves as one coherent oscillating system — which means it can have a real energy cost, not just a metaphorical one. This reframes the whole model as an energy-budget problem: - Every active relationship has an ongoing energy cost to keep above the noise floor, determined by its ω, A, and shared ζ. - A person's total energy available for relationships at any given time is finite and shared across all active connections. - **The ceiling on Cascade's size is wherever total energy expenditure across all active relationships meets that available budget** — not a fixed cognitive number. ### A working cost formula For operators, a simple order-of-magnitude approximation: ``` E_rel ≈ k · A · ω · ζ ``` Where: - `k` is a constant of proportionality (start with `k = 0.01`, calibrate per cohort) - `A` is amplitude on a 1–10 scale - `ω` is interaction frequency in interactions per week - `ζ` is damping on a 1–10 scale (lower ζ = less costly per unit A and ω) This is intentionally rough. See `energy-budget-protocol.md` for the daily operating layer. ### Dynamic ceiling This is the structural difference from Dunbar's ~150: Dunbar's number is roughly fixed per person. This model's ceiling is **dynamic** — it shrinks under depletion (grief, burnout, illness) and expands under surplus, because it's tracking energy, not memory slots. *Note:* "shrinks under depletion / expands under surplus" is a structural prediction of the model. It is not yet empirically established. Treat it as the model's commitment, not a known fact. --- ### Honest caveat on 'energy' 'Energy' is metaphorical scaffolding, not a literal physiological claim. The Hagger 23-lab ego-depletion replication (2016, d≈0.04) and the 2021 multi-site preregistered paradigmatic test (N≈3500) show the literal substrate claim doesn't survive. The Oscillator Model does NOT inherit empirical legitimacy from ego-depletion research. The energy budget is actually doing a protocol commitment and structural metaphor. Functional at practice level, open at theory level, and the protocol is honest if it does not claim physiological reality it doesn't have. --- ## 7. What Changes Mechanically If This Goes In - **Ring tempo (ω)** stays largely as already built — it was already correct under this model, just not yet understood as "minimum forcing frequency needed to overcome ζ." - **Hourglasses** shift from a quarterly values-quiz score to a running ledger of energy spent vs. available — the actual accounting currency for this model. See `energy-budget-protocol.md`. - **Ring promotion/demotion logic** gets a mechanism (§4) instead of a flat day-count threshold. - **The 127-hex cap** becomes a derived quantity, not an assumed constant — and could legitimately vary by person and by life season. --- ## 8. Open Questions - Does ζ get recalculated per-interaction, or only re-assessed at decision points (ring transitions)? - What's the actual unit for "available energy" — time, attention, a self-reported scale, or something inferred from app behavior? - Should Resonance be *built from* Coherence (Coherence as raw material, Resonance as what happens when it's acted on over time), or should they be modeled as fully independent variables? Current lean: built-from. - Is there a risk that symmetric ζ, even as a deliberate simplification, will make the model blind to exactly the one-sided-repair pattern that's most predictive of relationship failure in the underlying research? Worth a note in any future spec, not just here. - **Multi-party extension:** the Oscillator Model is currently dyadic. What happens when one hex represents a group, a cell, or a cohort? The energy budget would have to become a shared pool, and ζ would have to generalize to n-party systems. --- ## 9. Relationship to Adjacent Energy-Based Models Worth flagging explicitly: this model is **not** an attempt to plug into Sovereignosis-style energy economics (Ergon, C/T/M frameworks, society-scale currency replacement). That project operates at the scale of macro-economic exchange and treats energy as fungible currency. This model operates at the scale of a single dyad and treats energy as a non-transferable maintenance cost — closer to a thermodynamic *constraint* than a *currency*. Any resemblance is thematic (physics-grounded alternative to a conventional metric) rather than mechanical, and the two should stay structurally separate. --- ## 10. Reading Order for Cohort Members If you're new to the Oscillator Model: 1. Read §1 (the core claim) and §2 (the three macro traits) — this is the intuition 2. Read §3 (the four micro-parameters) and §4 (the causal chain) — this is the mechanism 3. Skim §5 (symmetric-ζ) and §6 (energy) — the honest simplifications 4. Read `energy-budget-protocol.md` for the daily practice 5. Return to §7–§9 for what changes mechanically and what this isn't If you're a reviewer: read §1, §4, §5, and §8 in that order. The four micro-parameters in §3 are research-grounded and don't need deep review. --- *This is the theoretical heart of v2. The energy-budget-protocol is its daily practice; the cascade-hexgrid-spec is its visual substrate.*