# Fact-Check Notes

**Verification status of every empirical claim in v2**

*These notes document the evidence base for the Oscillator Model and Cascade Hexgrid spec. Every claim that rests on empirical research has been verified against current peer-reviewed literature. The model itself is a working theory — empirical predictions are flagged as such.*

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## Summary Table

| # | Claim | Verification | Notes |
|---|---|---|---|
| 1 | Dunbar's ~150 arises from neocortical cognitive limit | ✅ Verified as original claim, ⚠️ contested as currently valid | Dunbar 1992 (J. Hum. Evol.); multiple replications; Powell et al. critique (PMC3327164) shows relationship doesn't survive rigorous reanalysis |
| 2 | Altman & Taylor social penetration (depth + breadth) | ✅ Verified | Altman & Taylor 1973; widely replicated; onion model is foundational |
| 3 | Bernieri & Rosenthal interactional synchrony | ✅ Verified | Bernieri, Rosenthal, et al. 1991 (Cambridge University Press, "Interpersonal coordination: Behavior matching and interactional synchrony") |
| 4 | Giles Communication Accommodation Theory (convergence/divergence) | ✅ Verified | Giles 1973 onward; multiple meta-analyses confirm effects on relational outcomes |
| 5 | Gottman rupture-repair research | ✅ Verified | Gottman & Silver multiple works; recent replication in J. Marital Fam. Ther. 2023 (3RS system); 5:1 positive-to-negative ratio widely cited |
| 6 | Damped harmonic oscillator: ζ = c/(2√(km)), regimes (under/critically/overdamped), ω_d = ω_n √(1−ζ²) | ✅ Verified | Standard references (ScienceDirect Topics; Nature Comm. Phys. 2025) |
| 7 | Hexagonal close-packing geometry yields 1 + 6 + 12 + 18 + 24 + 30 + 36 = 127 hexes | ✅ Verified mathematically | Sum: 1 + Σ(6·n) for n=1..6 = 1 + 126 = 127 |
| 8 | The 127-hex count is the geometric upper bound of the canonical map | ✅ Verified | Each ring adds 6 hexes in a perfect radial pattern; the count matches the geometric ceiling |
| 9 | The 127-hex count is taken as canonical in the model | ✅ Verified | Matches `sources/cascade-hexgrid.pdf`; treated as a geometric ceiling, not an empirically validated relational limit |
| 10 | Granovetter 1973 | ✅ Verified | Granovetter 1973; foundational work on social networks |

---

## Detailed Verification

### 1. Dunbar's number

**Original claim (verified):** Robin Dunbar's 1992 paper *Neocortex size as a constraint on group size in primates* (J. Hum. Evol.) proposed that primate social group size scales with neocortex volume, predicting ~150 meaningful relationships for humans. The cross-species primate regression is well-documented and cited in thousands of subsequent works.

**Critique (verified):** Powell et al. (PMC3327164, *Embracing covariation in brain evolution*) argue that the brain-group-size relationship may not survive rigorous statistical scrutiny, that developmental and plasticity factors are underrepresented, and that residual brain-size variation attributed to social factors is small relative to overall variation. **v2 accepts the original claim as historical anchor but does not load-bear on it.**

**What v2 does with this:** The Oscillator Model explicitly *replaces* Dunbar's cognitive-ceiling framing with a thermodynamic-ceiling framing. Dunbar's number remains in the canon as a foundational anchor, but the model's primary commitment is to energy, not neocortex.

### 2. Social penetration theory (Altman & Taylor)

**Verified:** Irwin Altman and Dalmas Taylor's 1973 treatise *Social penetration: The development of interpersonal relationships* established the depth-breadth framework and onion model. The theory is foundational in interpersonal communication.

**Multiple replications** exist, including cross-cultural and longitudinal studies. The Gilbert & Whiteneck (1976) extension added the valence (positivity/negativity) dimension. The Dermer & Thiel study shows asymmetric disclosure patterns in actual interactions.

**What v2 does with this:** Maps cleanly onto the **A (amplitude)** parameter. Amplitude is the depth-breadth intensity of disclosure; A increases as social penetration progresses.

### 3. Interactional synchrony (Bernieri & Rosenthal)

**Verified:** Bernieri & Rosenthal (1991), in *Fundamentals of nonverbal behavior* (eds. Feldman & Rimé, Cambridge University Press, pp. 401–432), defined interactional synchrony as "non-random, patterned, and synchronized behavior between two or more people that emerges rapidly and spontaneously during real-time social interaction."

**Extensions:** Bernieri, Reznick & Rosenthal (1988) on mother-infant entrainment; Frontiers in Psychology 2017 review paper confirms the three defining attributes (social interaction with co-presence; rapid spontaneous emergence; demonstrable timing/form synchronization).

**What v2 does with this:** Maps cleanly onto **φ (phase)**. Synchrony is the relational expression of phase alignment.

### 4. Communication Accommodation Theory (Giles)

**Verified:** Howard Giles (1973) introduced Speech Accommodation Theory (later CAT). Convergence (matching) and divergence (distinguishing) are well-documented across contexts. Soliz, Giles & Gasiorek (2021) provide a current synthesis.

**Meta-analysis (2014):** 76 studies (N = 18,382) confirm the relationship between accommodation and relational outcomes (solidarity, quality of contact).

**What v2 does with this:** Reinforces the **φ-alignment** parameter. Accommodation is how φ manifests in communicative behavior.

### 5. Gottman rupture-repair research

**Verified:** John Gottman and colleagues (multiple works including *The Science of Trust* 2011; *What Makes Love Last* 2012) developed the Sound Relationship House framework with rupture-repair as a core mechanism. The **5:1 positive-to-negative interaction ratio** is widely cited.

**Recent replication:** Application of the Rupture Resolution Rating System (3RS) to emotion-focused couple therapy (J. Marital Fam. Ther. 2023) confirms ruptures and repairs are common; poor-outcome couples show more ruptures; injured partners tend to generate more ruptures than injurers.

**What v2 does with this:** Maps cleanly onto **ζ (damping)**. Damping is the relational expression of repair capacity. The asymmetric repair pattern (one person doing more repair work) is acknowledged as a limitation of the symmetric-ζ simplification (§5 of `oscillator-model.md`).

### 6. Damped harmonic oscillator physics

**Verified — formulas:**
- Equation of motion: ẍ + 2ζω₀ ẋ + ω₀² x = 0
- Damping ratio: ζ = c / (2√(km)) = c / (2mω₀)
- Damped natural frequency: ω_d = ω₀ √(1 − ζ²)
- Underdamped (ζ < 1): oscillatory with decaying amplitude
- Critically damped (ζ = 1): fastest return without oscillation
- Overdamped (ζ > 1): no oscillation, exponential decay

**Sources:** ScienceDirect Topics (Engineering: Damping Ratio); Nature Communications Physics 2025 (Table 1: Characteristics and solutions of damped harmonic oscillators); Springer chapter (Oscillatory Motion).

**What v2 does with this:** The physics is taken verbatim. The relational mapping (ω → interaction frequency, A → amplitude, φ → phase, ζ → damping) is the model's novel contribution; the underlying physics is standard.

### 7. Hexagonal close-packing geometry

**Verified mathematically:**
```
R0: 1 hex
R1: 6 hexes (ring 1)
R2: 12 hexes (ring 2)
R3: 18 hexes (ring 3)
R4: 24 hexes (ring 4)
R5: 30 hexes (ring 5)
R6: 36 hexes (ring 6)
Total: 1 + 6 + 12 + 18 + 24 + 30 + 36 = 127
```

This is hexagonal close-packing: each ring adds 6 hexes in a perfect radial pattern. The count is the geometric upper bound of the canonical map.

**What v2 does with this:** Takes the 127-hex count as canonical (matches `sources/cascade-hexgrid.pdf`). Notes that this is a geometric ceiling, not an empirically validated relational limit.

---

## Empirical Claims That Are NOT Yet Established

The Oscillator Model commits to several empirical predictions that are NOT yet verified:

### Prediction A: Energy ceiling shrinks under depletion / expands under surplus

This is a structural prediction. We expect grief, burnout, or illness to reduce the operator's effective `E_available`. We expect joy, rest, or surplus to expand it. **Not yet measured.**

**Test:** Longitudinal cohort study tracking `E_available` against life events.

### Prediction B: Ring promotion follows ζ-drop mechanism

We expect promotion from a higher-ω ring to a lower-ω ring to follow from accumulated resonance lowering ζ, not from an arbitrary day-count threshold. **Not yet tested in v1 deployment.**

**Test:** A/B test with two cohorts — one using the v1 day-count threshold, one using the ζ-drop criterion. Compare promotion rates and relationship outcomes.

### Prediction C: Symmetric ζ is a usable approximation

We claim that symmetric ζ (one shared damping ratio per dyad) is structurally useful even though empirically asymmetric ζ is more accurate. **Not yet validated.**

**Test:** Cohort member self-reports on whether the symmetric approximation feels honest. Identify cases where the asymmetry breaks the model.

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## What We Are NOT Claiming

To be explicit:

1. **We are not claiming the Oscillator Model is correct.** It is a working theory with structural predictions that could be wrong.
2. **We are not claiming Dunbar's number is wrong.** It is a useful historical anchor. The Oscillator Model is a *complement*, not a *replacement*.
3. **We are not claiming social penetration theory, interactional synchrony, CAT, or Gottman research is complete or uncriticizable.** These are research-grounded anchors; the mapping onto oscillator parameters is our own contribution and is open to revision.
4. **We are not claiming the energy budget is the right unit.** Time, attention, and self-reported scale are alternatives (§8 of `oscillator-model.md`). The protocol uses self-reported scale as the practical default.
5. **We are not claiming "energy" is a real physiological substrate.** The Oscillator Model does not inherit empirical legitimacy from ego-depletion research. Hagger et al.'s 23-lab preregistered replication (2016) found d ≈ 0.04, essentially null; the multi-site preregistered paradigmatic test (Vohs et al., 2021, N ≈ 3,500) reinforced the same finding. The glucose-as-substrate claim is no longer supported. The Oscillator Model uses "energy" as a metaphorical scaffold and a protocol commitment, not as a literal claim.
6. **We are not claiming the model captures Granovetter's strong/weak ties axis.** Granovetter (1973) and subsequent work show that weak ties (bridging) deliver novel information, while strong ties (bonding) provide redundant but supportive content. The Oscillator Model tracks frequency, depth, synchrony, and repair — it does NOT separately track whether a tie is structurally bridging or bonding. This is an acknowledged gap; see §5.5 of `oscillator-model.md`.

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## Open Empirical Questions

1. Does ζ get recalculated per-interaction, or only at decision points?
2. What's the actual unit for "available energy"?
3. Should Resonance be built from Coherence, or are they independent?
4. Does symmetric ζ make the model blind to one-sided-repair patterns that predict failure?
5. How does the Oscillator Model generalize beyond dyads to groups, cells, cohorts?

These are listed in §8 of `oscillator-model.md` and are the v3 research agenda.

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*Every claim here was verified as of June 25, 2026. Future updates should flag any new replications or refutations that change the verification status.*