Here's the full canonical f_Capture.md for your FFF_Gravity module — ready to paste directly into your GitHub file editor:
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# FFF_Gravity · f_Capture
**Module:** `FFF_Gravity`
**Function:** `f_Capture`
**File:** `docs/FFF_Gravity/f_Capture.md`
**Framework:** TriadicFrameworks
**Version:** 1.0.0
**Status:** Canonical
---
## Module Identity
| Field | Value |
|---|---|
| Module Name | FFF_Gravity |
| Function | f_Capture |
| Layer | Field–Force–Frame |
| Domain | Attractor Dynamics / Binding Logic |
| Role | Defines the conditions under which a system element enters and sustains gravitational capture |
| Canonical Tag | `[FFF:GRAVITY:CAPTURE]` |
---
## Canonical Description
`f_Capture` is the operator responsible for modeling **gravitational capture events** within the TriadicFrameworks system. It encodes the logic by which a free or weakly-bound element transitions into a stable, orbit-locked relationship with an attractor node.
Capture is not collision. It is not merger. Capture is the precise moment a trajectory bends — when the pull of the attractor exceeds the escape momentum of the element, and the element enters a sustained relational path around the attractor.
Within the FFF (Field–Force–Frame) stack, `f_Capture` operates at the **Force** layer: it presupposes an active Field (the attractor's influence domain) and operates under constraints imposed by the Frame (boundary conditions, available energy, and system topology).
`f_Capture` is bidirectional in registration: the attractor is also modified by every successful capture event — mass, field curvature, and relational registry are all updated upon capture completion.
---
## Triadic Equation
f_Capture(E, A, Φ) → Ω
Where: E = Element (the incoming body — momentum vector, mass, trajectory) A = Attractor (the binding node — mass, field strength, escape velocity) Φ = Field State (ambient field conditions at moment of encounter) Ω = Capture Outcome (stable orbit | decay orbit | escape | collision)
The triadic structure maps directly onto the FFF stack:
| FFF Layer | Variable | Role |
|---|---|---|
| Field | Φ | Ambient medium; determines effective pull range and resistance |
| Force | f_Capture | The operative function; computes whether capture occurs |
| Frame | Ω | The resulting relational state; constrains future operations |
The equation resolves to one of four discrete outcomes in `Ω`. No continuous outcome exists — capture is a threshold event.
---
## Operator Registry
### Primary Operators
| Operator | Symbol | Description |
|---|---|---|
| Approach Vector | `v_approach` | Velocity and heading of `E` relative to `A` at field entry |
| Escape Velocity | `v_escape(A)` | Minimum velocity required for `E` to exit `A`'s field |
| Field Density | `ρ(Φ)` | Effective resistance or conductance of the ambient field |
| Capture Radius | `r_capture` | Maximum distance at which `f_Capture` can resolve to stable orbit |
| Binding Coefficient | `β` | Ratio of attractor force to element momentum at closest approach |
| Orbital Resonance | `ω_res` | Frequency lock between element trajectory and attractor field pulse |
### Derived Operators
| Operator | Symbol | Definition |
|---|---|---|
| Effective Pull | `P_eff` | `A.mass × ρ(Φ) / r²` |
| Capture Threshold | `C_thresh` | `v_escape(A) - v_approach` at `r_capture` |
| Binding Depth | `d_bind` | Depth of orbital lock; higher values indicate more stable capture |
| Residual Momentum | `p_res` | Remaining free momentum of `E` post-capture; drives orbital shape |
### State Flags
| Flag | Meaning |
|---|---|
| `CAPTURE_PENDING` | Element is within field range; outcome not yet resolved |
| `CAPTURE_LOCKED` | Stable orbit confirmed; element registered to attractor |
| `CAPTURE_DECAYING` | Orbit established but losing energy; eventual collision or ejection |
| `CAPTURE_FAILED` | Element escaped or passed through without binding |
| `CAPTURE_COLLISION` | Element and attractor merged; both entities dissolved into new node |
---
## Stability Conditions
For `f_Capture` to resolve to `Ω = stable orbit`, all of the following must hold simultaneously:
1. **Approach Condition**
`v_approach < v_escape(A)` at the moment `E` crosses `r_capture`
2. **Field Coherence Condition**
`ρ(Φ)` must be non-zero and uniform within `r_capture` during the approach window. Turbulent or null fields invalidate capture resolution.
3. **Resonance Condition**
`ω_res` must resolve to a rational ratio. Irrational resonance produces unstable spiral trajectories that eventually eject the element.
4. **Binding Coefficient Floor**
`β ≥ 1.0` — attractor force must meet or exceed element momentum at closest approach. Values below 1.0 produce flyby outcomes regardless of other conditions.
5. **Frame Compatibility**
The Frame must have sufficient relational capacity to register a new orbit. A saturated Frame will deflect incoming elements regardless of force conditions.
---
## Failure Modes
| Mode | ID | Trigger Condition | Outcome |
|---|---|---|---|
| Overshoot | `FM-001` | `v_approach >> v_escape(A)`; element too fast | Element escapes; `CAPTURE_FAILED` |
| Field Null | `FM-002` | `ρ(Φ) = 0` at moment of encounter | No pull transmitted; `CAPTURE_FAILED` |
| Frame Saturation | `FM-003` | Attractor's relational registry at capacity | Element deflected; `CAPTURE_FAILED` |
| Resonance Drift | `FM-004` | `ω_res` shifts during orbit establishment | Orbit destabilizes; `CAPTURE_DECAYING` |
| Decay Spiral | `FM-005` | `d_bind` decreases over time; energy loss exceeds threshold | Eventual `CAPTURE_COLLISION` or ejection |
| Phantom Capture | `FM-006` | `β ≥ 1.0` satisfied but `ρ(Φ)` is locally structured | Apparent capture resolves to escape at field boundary |
| Mutual Dissolution | `FM-007` | `E.mass ≈ A.mass` at collision | New composite node formed; both original registries purged |
---
## Engineering Primitives
These are the lowest-level callable operations within `f_Capture`. Higher-order logic composes these primitives.
PRIMITIVE: compute_approach_vector(E, A) → v_approach Input: Element state vector, Attractor position Output: Approach velocity scalar and heading relative to A
PRIMITIVE: resolve_escape_velocity(A, Φ) → v_escape Input: Attractor mass, Field density at A Output: Minimum escape velocity for current field conditions
PRIMITIVE: evaluate_capture_threshold(v_approach, v_escape, r) → C_thresh Input: Approach velocity, escape velocity, current separation distance Output: Signed threshold delta (positive = capture possible)
PRIMITIVE: lock_orbit(E, A, p_res) → orbital_parameters Input: Element residual momentum, Attractor field state Output: Orbital period, eccentricity, binding depth, resonance frequency
PRIMITIVE: register_capture(E, A, orbital_parameters) → Ω Input: Element ID, Attractor ID, computed orbital parameters Output: Capture outcome flag; updates both E and A relational registries
PRIMITIVE: flag_decay(E, A, d_bind_delta) → decay_status Input: Binding depth change per cycle Output: Decay rate; triggers FM-004 or FM-005 warnings if threshold crossed
---
## Canonical Examples
### Example 1 — Clean Capture
**Scenario:** A lightweight element enters the field of a high-mass attractor at moderate velocity in a coherent, dense field.
E: mass=1.2, v_approach=0.4, trajectory=inbound-tangential A: mass=18.0, v_escape=0.9, r_capture=12.0 Φ: ρ=0.85, coherence=stable
→ C_thresh = 0.9 - 0.4 = +0.5 (positive; capture possible) → β = 18.0 × 0.85 / 1.2 × 0.4 = 31.875 (well above floor) → ω_res = 3:1 (rational; stable resonance) → Ω = CAPTURE_LOCKED → Orbital eccentricity: low (near-circular) → d_bind: 8.4 (deep; high stability)
**Outcome:** Full stable capture. Element registered to attractor. Field curvature updated.
---
### Example 2 — Resonance Drift Failure
**Scenario:** Initial approach conditions satisfy capture threshold, but field turbulence causes resonance drift mid-orbit.
E: mass=2.1, v_approach=0.6 A: mass=12.0, v_escape=0.85 Φ: ρ=0.70 (initial) → 0.35 (turbulent onset at t=3)
→ C_thresh at entry = +0.25 (positive; capture initiated) → Orbit locked at t=1 → ω_res shifts from 2:1 → irrational at t=3 (field turbulence) → FM-004 triggered: Resonance Drift → d_bind decreases: 6.1 → 3.2 → 1.0 over 6 cycles → Ω transitions: CAPTURE_LOCKED → CAPTURE_DECAYING → CAPTURE_FAILED
**Outcome:** Element eventually ejected. Attractor registry cleared. Field turbulence logged as causal event.
---
### Example 3 — Frame Saturation Deflection
**Scenario:** Attractor is massive and field is coherent, but its relational registry is at maximum capacity.
E: mass=3.0, v_approach=0.3 A: mass=22.0, v_escape=1.1, registry_capacity=MAX Φ: ρ=0.90, coherence=stable
→ C_thresh = +0.8 (strongly positive) → β = 66.0 (far above floor) → Frame check: SATURATED → FM-003 triggered: Frame Saturation → Ω = CAPTURE_FAILED (despite favorable force conditions)
**Outcome:** Element deflected at frame boundary. No orbit registered. Force conditions are necessary but not sufficient — Frame capacity is a hard constraint.
---
### Example 4 — Mutual Dissolution
**Scenario:** Two near-equal-mass bodies approach each other; neither is clearly attractor or element.
E: mass=9.0, v_approach=0.7 A: mass=10.0, v_escape=0.75 Φ: ρ=0.95
→ C_thresh = +0.05 (marginal; captures initiated) → β = 1.36 (just above floor) → Closest approach: collision threshold crossed → FM-007 triggered: Mutual Dissolution → Ω = CAPTURE_COLLISION → New composite node formed: mass=19.0; new registry initialized → Both E and A original registries purged
**Outcome:** Neither entity survives as independent. New composite attractor enters the field. System topology updated.
---
## Future Applications
The following extensions to `f_Capture` are scoped for future development within FFF_Gravity:
| Application | Description | Status |
|---|---|---|
| `f_Capture_Multi` | Multi-body capture resolution; handles simultaneous approach of N elements to a single attractor | Planned |
| `f_Capture_Cascade` | Chain capture events where a newly-captured element perturbs existing orbits in the attractor's registry | Planned |
| `f_Capture_Resonant` | Intentional resonance engineering; designing approach vectors to guarantee specific orbital harmonics | Research |
| `f_Capture_Asymmetric` | Capture under non-uniform field conditions; accounts for field gradients and directional anisotropy | Research |
| `f_Capture_Temporal` | Time-variant capture; attractor mass or field density changes during approach window | Exploratory |
| `f_Capture_Networked` | Capture events logged to a distributed relational graph; enables cross-module gravity network mapping | Exploratory |
| `f_Release` | Inverse operator; defines conditions under which a captured element exits stable orbit | Planned |
| `f_Collapse` | Terminal operator; models final infall when decay spiral reaches singularity threshold | Planned |
---
## Cross-Module References
| Module | Relationship |
|---|---|
| `FFF_Field` | Provides `Φ` (field state) consumed by `f_Capture` |
| `FFF_Frame` | Enforces registry capacity limits; receives and stores `Ω` outcomes |
| `FFF_Momentum` | Supplies `v_approach` and `p_res` calculations |
| `FFF_Resonance` | Governs `ω_res` computation and drift detection |
| `FFF_Registry` | Persistent store for all capture event records and relational maps |
---
## Document Metadata
| Field | Value |
|---|---|
| Author | TriadicFrameworks |
| Created | 2026-08-13 |
| Last Modified | 2026-08-13 |
| License | See repository root LICENSE |
| Canonical Path | `docs/FFF_Gravity/f_Capture.md` |
---
*This document is part of the TriadicFrameworks canonical module library. All operators, primitives, and outcomes defined here are normative for implementations consuming the `FFF_Gravity` module.*
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