# The Oscillator Model: A Thermodynamic Alternative to Dunbar's Number

*Working theory draft — Cascade theoretical foundations*

## 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.**

## 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 | Rough current analogue in Cascade |
|---|---|---|
| **Coherence** | How phase-aligned two people's signals are right now — a state variable | Hearts (direct overlap), Stars (category congruence) |
| **Resonance** | The amplification that occurs when coherent signals reinforce each other over time — a growth process | Existing Resonance system |
| **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 |

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) |
| **φ** (phi) | Phase — timing offset between two oscillators | Synchrony / rhythm alignment | Interactional synchrony (Bernieri & Rosenthal); Communication Accommodation Theory (Giles) |
| **ζ** (zeta) | Damping ratio — how fast oscillation settles after a disturbance | Resilience / conflict-repair capacity | Rupture-and-repair research (Gottman; Tronick) |

## 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.
- **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.

## 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.

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.

## 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** could shift from a quarterly values-quiz score to a running ledger of energy spent vs. available — the actual accounting currency for this model.
- **Ring promotion/demotion logic** gets a mechanism (§4) instead of a flat day-count threshold.
- **The 126 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.

## 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.