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FFF_Gravity · Operators

# ┌─────────────────────────────────────────────────────────────┐
# │                  DOCUMENT FRONTMATTER                       │
# └─────────────────────────────────────────────────────────────┘
document:         OPERATORS
canonical_path:   docs/FFF_Gravity/OPERATORS.md
canonical_tag:    "[FFF:GRAVITY:OPERATORS]"
framework:        TriadicFrameworks
module:           FFF_Gravity
version:          1.0.0
status:           canonical
stability:        living
created:          2026-08-13
last_modified:    2026-08-13
authors:
  - TriadicFrameworks
encoding:         UTF-8
line_endings:     LF
normative:        true
description: >
  Single source of truth for every operator symbol defined in FFF_Gravity.
  Covers primary operators, derived operators, state flags, engineering
  primitives, failure modes, composition rules, evaluation order, and
  the symbol freeze registry. All function files must resolve symbol
  conflicts against this document before publishing.
tags:
  - FFF
  - gravity
  - operators
  - symbols
  - registry
  - normative

session_context:
  current_session:
    session_id:       SES-20260813-OPS-001
    opened_at:        2026-08-13T07:48:00-04:00
    closed_at:        ~
    editor:           Nawder
    branch:           main
    intent:           Create canonical OPERATORS.md — master symbol authority for FFF_Gravity
    status:           active
    dirty:            true
    sections_touched: [§0, §1, §2, §3, §4, §5, §6, §7, §8, §9, §10]

  session_history:
    - session_id:  SES-20260813-README-001
      intent:      Create canonical README.md
      status:      closed
    - session_id:  SES-20260813-INDEX-001
      intent:      Create canonical INDEX.md
      status:      closed

changelog:
  - version: 1.0.0
    date:    2026-08-13
    author:  TriadicFrameworks
    notes: >
      Initial canonical release. 9 primary operators, 10 derived operators,
      11 state flags, 15 primitives, 10 failure modes. All symbols from
      f_Capture.md v1.0.0 frozen. Wave 3+ symbols marked pending.

Canonical path: docs/FFF_Gravity/OPERATORS.md Authority: This document is the single source of truth for all operator symbols in FFF_Gravity. In any conflict between this file and a function file, this file governs. Normative: Yes — all sections except §0 and §10 are normative.

§0 · Session Context#

Active Session#

Field Value
Session ID SES-20260813-OPS-001
Opened 2026-08-13T07:48:00-04:00
Closed — (active)
Editor Nawder
Branch main
Intent Create canonical OPERATORS.md — master symbol authority
Status 🟡 Active

Update Policy#

When to update this file:
  1. A new operator is introduced in any function file         → add to §1 or §2
  2. A new state flag is introduced                            → add to §3
  3. A new primitive is defined in any function file           → add to §4
  4. A new failure mode is defined                             → add to §5
  5. A composition rule is formalized                          → add to §6
  6. A scaffold file is promoted to canonical                  → update §8 freeze status
  7. A version bump occurs                                     → update §9 and CHANGELOG.md

Never:
  - Remove a frozen symbol (deprecate instead)
  - Rename a frozen symbol without a major version bump
  - Add a symbol that conflicts with FFF_Field, FFF_Momentum, or FFF_Resonance namespaces

§1 · Primary Operators#

Primary operators are direct inputs to f_Capture and its sibling functions. They are measured or externally provided — not computed from other operators. All 9 are frozen at v1.0.0.

§1.1 · Full Specification Table#

Symbol Full Name FFF Node Type Domain Range Unit Frozen Defined In
v_approach Approach Vector F_force scalar ℝ≥0 [0, ∞) normalized velocity ✅ f_Capture.md §4.1
v_escape(A) Escape Velocity F_freq scalar ℝ>0 (0, ∞) normalized velocity ✅ f_Capture.md §4.1
ρ(Φ) Field Density F_freq scalar ℝ≥0 [0, 1] dimensionless ✅ f_Field.md §4.1
r_capture Capture Radius Frame scalar ℝ>0 (0, ∞) normalized distance ✅ f_Capture.md §4.1
β Binding Coefficient derived scalar ℝ≥0 [0, ∞) dimensionless ✅ f_Capture.md §4.1
ω_res Orbital Resonance F_freq ratio ℚ ∪ ℝ rational or irrational dimensionless ✅ f_Capture.md §4.1
M_A Attractor Mass F_fluid scalar ℝ>0 (0, ∞) normalized mass ✅ f_Field.md §4.1
M_E Element Mass F_fluid scalar ℝ>0 (0, ∞) normalized mass ✅ f_Capture.md §4.1
r Separation Distance geometry scalar ℝ>0 (0, ∞) normalized distance ✅ f_Capture.md §4.1

§1.2 · Operator Constraints and Guards#

Symbol Hard Constraints Guard Violation Consequence
v_approach Evaluated at r_capture crossing only; not at outer field boundary Premature evaluation → invalid C_thresh
v_escape(A) Must be recomputed if ρ(Φ) changes between entry and encounter Stale value → FM-006 risk
ρ(Φ) 0 = null field — triggers FM-002 immediately; 1 = saturated Unguarded null → undefined P_eff
r_capture Set by Attractor; cannot be modified by Element or engineering primitives Modification → Frame integrity failure
β Must be ≥ 1.0 for capture to proceed; < 1.0 = flyby unconditionally Below floor → CAPTURE_FAILED regardless of all other conditions
ω_res Irrational value = FM-004 at any point — during approach or post-lock Irrational → CAPTURE_DECAYING
M_A Must be finite and positive; M_A ≈ M_E triggers FM-007 check Mass parity → dissolution path
M_E Must be finite and positive Zero or infinite mass → undefined behavior
r r = 0 is undefined (singularity); r > r_capture → short-circuit in evaluate_capture_threshold Zero → ⊥

§1.3 · Node Assignment Summary#

FFF Node Symbols
F_freq (Frequency) v_escape(A), ρ(Φ), ω_res
F_fluid (Fluids) M_A, M_E
F_force (Forces) v_approach
Frame r_capture
Geometry r
Derived (no single node) β

§2 · Derived Operators#

Derived operators are computed from primary operators via defined formulas. Symbols marked ✅ frozen are defined in f_Capture.md v1.0.0 and cannot change. Symbols marked 🔵 pending have formulas in scaffold files awaiting canonicalization.

§2.1 · Full Specification Table#

Symbol Full Name Formula Depends On Output Range Sign Convention Frozen Defined In
P_eff Effective Pull M_A × ρ(Φ) / r² M_A, ρ(Φ), r [0, ∞) always positive ✅ f_Capture.md §4.2
C_thresh Capture Threshold v_escape(A) − v_approach v_escape(A), v_approach (−∞, ∞) positive = capture possible; negative = escape ✅ f_Capture.md §4.2
d_bind Binding Depth β × ρ(Φ) × (1 − e) β, ρ(Φ), e [0, ∞) higher = more stable ✅ f_Capture.md §4.2
p_res Residual Momentum M_E × (v_approach − C_thresh) M_E, v_approach, C_thresh [0, ∞) always positive post-capture ✅ f_Capture.md §4.2
e Orbital Eccentricity p_res / (p_res + P_eff) p_res, P_eff [0, 1) 0 = circular; approaching 1 = near-parabolic ✅ f_Orbit.md §4.1
T_orb Orbital Period 🔵 pending f_Orbit.md §4.1 d_bind, ω_res (0, ∞) always positive 🔵 f_Orbit.md §4.1
δ Decay Rate Δd_bind / Δt d_bind series (−∞, 0] in decay negative = losing energy 🔵 f_Decay.md §4.1
E_rel Release Energy 🔵 pending f_Release.md §4.1 d_bind, p_res, ρ(Φ) [0, ∞) energy required to achieve release 🔵 f_Release.md §4.1
F_emit Emit Field Strength 🔵 pending f_Emit.md §4.1 M_A, emit energy input [0, ∞) higher = deeper coherence well 🔵 f_Emit.md §4.1
F_damp Dampen Depth 🔵 pending f_Dampen.md §4.1 ρ(Φ), dampen energy input [0, 1] fraction of ρ(Φ) to suppress 🔵 f_Dampen.md §4.1

§2.2 · Supplementary Operators (Engineering Primitives Layer)#

These operators are introduced by engineering primitive files. They extend the derived operator set and are pending canonicalization.

Symbol Full Name Formula Defined In Frozen
F_amp Amplification Factor scalar ≥ 1.0 f_Amplify.md §4.1 🔵
β_max Binding Coefficient Ceiling 🔵 pending f_Amplify.md §4.1 🔵
heading_delta Heading Change angular delta (radians) applied to v_approach f_Deflect.md §4.1 🔵
r_emit Emission Radius bounded region of emit_field effect f_Emit.md §4.1 🔵
r_damp Dampening Radius bounded region of suppress_field effect f_Dampen.md §4.1 🔵
d_warn Decay Warning Threshold d_bind level at which FM-004 is raised f_Decay.md §4.1 🔵
d_collapse Collapse Threshold d_bind level at which FM-005 fires; f_Collapse triggered f_Decay.md §4.1 🔵
m_parity Mass Parity Threshold max ` M_E − M_A ` below which FM-007 fires

§2.3 · Undefined Conditions#

Operator Undefined When Symbol for Undefined
P_eff r = 0 (singularity) or ρ(Φ) = 0 (FM-002) ⊥
C_thresh v_escape(A) undefined (ρ = 0) ⊥
d_bind e ≥ 1 (hyperbolic trajectory — not captured) ⊥
p_res C_thresh ≤ 0 (no capture) ⊥
e P_eff = 0 (FM-002) ⊥
δ Pre-capture (no time series) ⊥
E_rel d_bind = 0 (collapse state; release not viable) ⊥
F_amp β > β_max (runaway — FM-010) ⊥

§3 · State Flags#

State flags represent the discrete relational state of an Element in the FFF_Gravity system. The state machine is deterministic — every transition has a single defined trigger. No state is revisitable once terminal. All 11 flags are frozen at v1.0.0.

§3.1 · Flag Specification Table#

Flag Entry Condition Set By Exit Condition Valid Next States Terminal
CAPTURE_PENDING E crosses r_capture; outcome unresolved compute_approach_vector C_thresh evaluated CAPTURE_LOCKED, CAPTURE_FAILED No
CAPTURE_LOCKED C_thresh > 0 ∧ β ≥ 1.0 ∧ Frame ok ∧ ω_res ∈ ℚ register_capture d_bind falls below d_warn; or FM-007 CAPTURE_DECAYING, CAPTURE_COLLISION No
CAPTURE_DECAYING FM-004 raised; d_bind decreasing flag_decay d_bind ≤ d_collapse CAPTURE_FAILED, CAPTURE_COLLISION No
CAPTURE_FAILED FM-001/002/003/006 raised; or decay → ejection register_capture — — Yes
CAPTURE_COLLISION FM-005 terminal infall; or FM-007 dissolution execute_collapse — — Yes
ORBIT_STABLE stab_class = stable ∧ e < 0.5 post CAPTURE_LOCKED classify_orbit stab_class drops below stable threshold ORBIT_ECCENTRIC, CAPTURE_DECAYING No
ORBIT_ECCENTRIC e ≥ 0.5 while orbit is still captured classify_orbit e drops below 0.5; or FM-004 ORBIT_STABLE, CAPTURE_DECAYING No
RELEASED f_Release conditions satisfied; clean exit execute_release — — Yes (clean exit)
COLLAPSED f_Collapse infall complete execute_collapse — — Yes
DAMPEN_ACTIVE f_Dampen engaged on Attractor's field suppress_field Dampening removed; ρ(Φ) restored CAPTURE_PENDING, ORBIT_STABLE No
EMIT_ACTIVE f_Emit engaged on Attractor's field emit_field Emission withdrawn ORBIT_STABLE, CAPTURE_DECAYING No

§3.2 · State Transition Diagram#

                          E crosses r_capture
                                  │
                                  ▼
                        ┌─────────────────┐
                        │ CAPTURE_PENDING  │
                        └────────┬────────┘
                                 │
              ┌──────────────────┼──────────────────────┐
              │                  │                       │
    C_thresh > 0             C_thresh ≤ 0           ρ(Φ) = 0
    β ≥ 1.0                  (FM-001)               (FM-002)
    Frame ok                      │                       │
    ω_res ∈ ℚ                     │                       │
              │                   │                       │
              ▼                   ▼                       ▼
   ┌──────────────────┐   ┌───────────────┐       (FM-003 / FM-006)
   │  CAPTURE_LOCKED  │   │ CAPTURE_FAILED│◄──────── all terminal
   └────────┬─────────┘   │   [TERMINAL]  │             FMs
            │             └───────────────┘
            │
     d_bind monitored every cycle via flag_decay
            │
    ┌───────┴────────────────────────┐
    │                                │
d_bind < d_warn               M_E ≈ M_A
(FM-004)                      (FM-007)
    │                                │
    ▼                                ▼
┌──────────────────┐      ┌──────────────────┐
│ CAPTURE_DECAYING │      │ CAPTURE_COLLISION │
└────────┬─────────┘      │    [TERMINAL]    │
         │                └──────────────────┘
    ┌────┴────┐
    │         │
ejection  infall
(FM-005)  (FM-005)
    │         │
    ▼         ▼
CAPTURE_  CAPTURE_
FAILED    COLLISION

     ORBIT_STABLE ↔ ORBIT_ECCENTRIC  (reclassified each cycle)
     RELEASED [TERMINAL]             (clean exit via f_Release)
     COLLAPSED [TERMINAL]            (infall confirmed)
     DAMPEN_ACTIVE / EMIT_ACTIVE     (engineering overlay flags)

§3.3 · Flag Ownership#

Flag Owned By Written By Primitive
CAPTURE_PENDING Element E compute_approach_vector
CAPTURE_LOCKED Element E + Attractor A registry register_capture
CAPTURE_DECAYING Element E flag_decay
CAPTURE_FAILED Element E register_capture
CAPTURE_COLLISION Both E and A (or composite) execute_collapse
ORBIT_STABLE Element E classify_orbit
ORBIT_ECCENTRIC Element E classify_orbit
RELEASED Element E + Attractor A registry execute_release
COLLAPSED Composite node C (or A post-absorption) execute_collapse
DAMPEN_ACTIVE Attractor A suppress_field
EMIT_ACTIVE Attractor A emit_field

§4 · Engineering Primitives#

§4.1 · Call Order and Dependencies#

Primitives must be called in the order defined in §7 (Evaluation Order). Side-effecting primitives write to external registries and are not idempotent. Pure primitives may be called in any order relative to each other within their group.

§4.2 · Full Primitive Specification Table#

# Primitive Source File Pure Reads Writes Call Timing Frozen
1 compute_approach_vector f_Capture.md §7.2 Yes E.state, A.position v_approach Once · Step 1 ✅
2 resolve_escape_velocity f_Capture.md §7.2 Yes M_A, ρ(Φ) v_escape(A) Once · Step 2 ✅
3 evaluate_capture_threshold f_Capture.md §7.2 Yes v_approach, v_escape, r C_thresh Once · Step 5 ✅
4 lock_orbit f_Capture.md §7.2 No E, A, Φ orbital_parameters Once · Step 8 ✅
5 register_capture f_Capture.md §7.2 No orbital_parameters FFF_Registry, E.registry, A.registry, A.field_curvature Once · Step 9 ✅
6 flag_decay f_Capture.md §7.2 / f_Decay.md §7 No d_bind_delta E.state_flag, decay_status Every cycle post CAPTURE_LOCKED ✅
7 classify_orbit f_Orbit.md §7 Yes p_res, d_bind, ω_res orbital_parameters.orbit_class, .stab_class Once post lock; then each cycle 🔵
8 compute_release_vector f_Release.md §7 Yes E.state, d_bind v_release Once · on release attempt 🔵
9 execute_release f_Release.md §7 No v_release FFF_Registry, E.registry, A.registry Once · on release 🔵
10 execute_collapse f_Collapse.md §7 No d_bind, E, A composite node, both registries Once · on FM-005 or FM-007 🔵
11 initialize_composite_node f_Collapse.md §7 No M_E, M_A new composite node C Once · on FM-007 only 🔵
12 purge_registry f_Collapse.md §7 No E.id, A.id FFF_Registry (deletion) Once · on FM-007; partial on FM-005 🔵
13 emit_field f_Emit.md §7 No F_emit, A, r_emit ρ(Φ) local, A.field_curvature On demand 🔵
14 suppress_field f_Dampen.md §7 No F_damp, ρ(Φ), r_damp ρ(Φ) local On demand 🔵
15 amplify_coupling f_Amplify.md §7 No F_amp, M_A, ρ(Φ) β, P_eff On demand 🔵
16 redirect_force_node f_Deflect.md §7 No heading_delta, r_deflect v_approach heading component On demand 🔵

Note: Primitive 16 (redirect_force_node) was added from scaffold review. Corrected total: 16 primitives. INDEX.md §2 to be updated at next session.

§4.3 · Side-Effect Classification#

Class Primitives Consequence of Repeat Call
Pure (safe to repeat) 1, 2, 3, 7, 8 No side effects; produces same output for same inputs
Single-write (idempotent-safe) 4, 6 Overwrites same fields; safe to repeat
Registry-write (non-idempotent) 5, 9, 10, 11, 12, 13, 14, 15, 16 Repeat call creates duplicate records or double-applies effect

§4.4 · Primitive Guards Summary#

Primitive Guard Consequence of Violation
evaluate_capture_threshold r ≤ r_capture Returns C_thresh < 0 immediately if r > r_capture
lock_orbit C_thresh > 0 Must not be called if threshold not met
register_capture lock_orbit completed Undefined behavior if orbital parameters absent
flag_decay Post CAPTURE_LOCKED only No-op if called pre-capture
execute_collapse FM-005 or FM-007 active Must not be called outside terminal FM context
initialize_composite_node FM-007 path only Must not be called on asymmetric collapse
emit_field ρ(Φ) < 1.0 No-op at saturation; runaway risk if β → β_max
suppress_field ρ(Φ) > ρ(Φ)_floor FM-009 risk if floor breached
amplify_coupling β < β_max FM-010 if ceiling exceeded
redirect_force_node heading_delta within bounds Over-deflection → FM-001 equivalent

§5 · Failure Mode Registry#

§5.1 · Full Failure Mode Table#

ID Name FFF Node Trigger Condition Operators Involved State Transition Outcome Severity Recoverable Defined In
FM-001 Overshoot F_force C_thresh ≤ 0; v_approach ≥ v_escape(A) v_approach, v_escape, C_thresh CAPTURE_PENDING → CAPTURE_FAILED Element escapes error No f_Capture.md §6
FM-002 Field Null F_freq ρ(Φ) = 0 at moment of encounter ρ(Φ), P_eff, v_escape CAPTURE_PENDING → CAPTURE_FAILED No pull transmitted error No f_Capture.md §6
FM-003 Frame Saturation Frame Attractor registry at MAX capacity Frame.registry_capacity, β CAPTURE_PENDING → CAPTURE_FAILED Element deflected at boundary error No f_Capture.md §6
FM-004 Resonance Drift ω_res ω_res → irrational; d_bind < d_warn ω_res, d_bind, δ CAPTURE_LOCKED → CAPTURE_DECAYING Orbit destabilizing warn Yes — via f_Emit or f_Amplify f_Capture.md §6
FM-005 Decay Spiral d_bind d_bind ≤ d_collapse; energy exhausted d_bind, δ, p_res CAPTURE_DECAYING → CAPTURE_FAILED or CAPTURE_COLLISION Ejection or infall fatal No f_Decay.md §6
FM-006 Phantom Capture ρ(Φ) β ≥ 1.0 met; ρ(Φ) locally structured; lock dissolves at field boundary β, ρ(Φ), P_eff CAPTURE_PENDING → CAPTURE_FAILED Apparent capture resolves to escape warn No f_Capture.md §6
FM-007 Mutual Dissolution F_fluid |M_E − M_A| < m_parity; collision threshold crossed M_E, M_A, β, C_thresh CAPTURE_LOCKED → CAPTURE_COLLISION Composite node created; both registries purged fatal No f_Collapse.md §6
FM-008 Release Overshoot F_force v_release too high; trajectory becomes hyperbolic v_release, E_rel CAPTURE_LOCKED → CAPTURE_FAILED Uncontrolled ejection error No f_Release.md §6
FM-009 Dampen Cascade F_freq suppress_field propagates beyond r_damp; ρ(Φ) → 0 region-wide F_damp, ρ(Φ), r_damp multiple CAPTURE_LOCKED → CAPTURE_DECAYING Gravity null zone; uncontrolled releases fatal Partial — if caught before propagation f_Dampen.md §6
FM-010 Amplify Runaway F_fluid β > β_max under sustained f_Emit or f_Amplify β, F_amp, F_emit, β_max ORBIT_STABLE → CAPTURE_COLLISION Collapse or singularity fatal No f_Amplify.md §6

§5.2 · Failure Mode Groupings#

Group IDs Common Cause Common Outcome
Approach failures FM-001, FM-002, FM-003, FM-006 Conditions wrong at encounter CAPTURE_FAILED — never entered orbit
Orbital instability FM-004, FM-005 Post-capture energy loss CAPTURE_DECAYING → ejection or infall
Terminal events FM-005 (infall path), FM-007 Mass collision or dissolution CAPTURE_COLLISION — no survivors
Engineering failures FM-008, FM-009, FM-010 Primitive misuse or runaway Varied — ejection, null zone, singularity

§5.3 · Severity Definitions#

Severity Meaning Recovery
warn Orbit is at risk but not yet terminal; intervention window open Yes — f_Emit, f_Amplify may restore
error Capture has failed; element not in orbit; no immediate system danger No — outcome is final for this interaction
fatal Terminal event with system-wide consequences; nodes destroyed or topology changed No — irreversible

§6 · Composition Rules#

§6.1 · Core Derivation Chain#

Reading order — each row depends on all rows above it:

INPUT LAYER
  M_A        → provided by Attractor state
  M_E        → provided by Element state
  ρ(Φ)       → provided by F_freq (f_Field.md)
  r          → provided by geometry
  v_approach → computed by: compute_approach_vector(E, A)

STEP 1: Effective Pull
  P_eff = M_A × ρ(Φ) / r²
  Defined when: ρ(Φ) > 0 ∧ r > 0

STEP 2: Escape Velocity
  v_escape(A) = resolve_escape_velocity(M_A, ρ(Φ))
  Defined when: ρ(Φ) > 0

STEP 3: Binding Coefficient
  β = P_eff / (M_E × v_approach)
  Defined when: v_approach > 0

STEP 4: Capture Threshold
  C_thresh = v_escape(A) − v_approach
  Defined when: v_escape(A) defined

STEP 5: Residual Momentum
  p_res = M_E × (v_approach − C_thresh)
  Defined when: C_thresh > 0

STEP 6: Orbital Eccentricity
  e = p_res / (p_res + P_eff)
  Defined when: P_eff > 0 ∧ p_res ≥ 0
  Constraint: e must be in [0, 1); e ≥ 1 → hyperbolic → not captured

STEP 7: Binding Depth
  d_bind = β × ρ(Φ) × (1 − e)
  Defined when: e ∈ [0, 1)

STEP 8: Decay Rate (post-capture, per-cycle)
  δ = Δd_bind / Δt = d_bind(t) − d_bind(t−1)
  Defined when: t ≥ 1 (at least one prior cycle)

§6.2 · Capture Gate Composition#

The Capture Gate is the boolean conjunction of all stability conditions. All must be true simultaneously for capture to succeed.

CAPTURE_GATE =
    C_thresh > 0             [Condition 1 — Approach]
  ∧ ρ(Φ) > 0 (uniform)      [Condition 2 — Field Coherence]
  ∧ ω_res ∈ ℚ               [Condition 3 — Resonance]
  ∧ β ≥ 1.0                 [Condition 4 — Binding Floor]
  ∧ Frame.capacity > 0       [Condition 5 — Frame Compatibility]

If CAPTURE_GATE = true  → lock_orbit() → register_capture() → CAPTURE_LOCKED
If CAPTURE_GATE = false → FM raised   → register_capture() → CAPTURE_FAILED

§6.3 · Engineering Operator Compositions#

Goal Composition Operators Modified Risk
Restore decaying orbit emit_field(F_emit) → ρ(Φ)↑ → d_bind↑ ρ(Φ), P_eff, β, d_bind FM-010 if sustained
Weaken attractor hold suppress_field(F_damp) → ρ(Φ)↓ → d_bind↓ ρ(Φ), P_eff, v_escape, d_bind FM-009 if propagates
Increase capture probability amplify_coupling(F_amp) → β↑ → P_eff↑ β, P_eff, d_bind FM-010 if β > β_max
Route Element to target heading redirect_force_node(heading_delta) → v_approach heading change v_approach (heading only) FM-001 if over-deflected
Engineer target resonance solve_resonant_approach(ω_res_target) → {heading, ρ(Φ), β} spec All approach operators No solution if target irrational

§7 · Operator Evaluation Order#

Step Frequency Primitive Operators Read Operators Written Guard Short-Circuits To
1 ONCE compute_approach_vector E.state, A.position v_approach none —
2 ONCE resolve_escape_velocity M_A, ρ(Φ) v_escape(A) ρ(Φ) > 0 FM-002
3 ONCE (implicit) M_A, ρ(Φ), r P_eff requires step 2 —
4 ONCE (implicit) P_eff, v_approach, M_E β requires step 3 —
5 ONCE evaluate_capture_threshold v_approach, v_escape, r_capture C_thresh r ≤ r_capture FM-001 if C_thresh ≤ 0
6 ONCE (Frame check) Frame.registry_capacity — β ≥ 1.0 else halt FM-003
7 ONCE (Resonance check) ω_res — ω_res ∈ ℚ else halt FM-004
8 ONCE lock_orbit E, A, p_res, ρ(Φ) d_bind, orbital parameters C_thresh > 0 —
9 ONCE register_capture orbital parameters Ω, FFF_Registry, E.registry, A.registry requires step 8 —
10 CYCLE flag_decay d_bind_delta E.state_flag, decay_status post CAPTURE_LOCKED only FM-004 / FM-005

§8 · Symbol Freeze Registry#

§8.1 · Frozen Symbols (v1.0.0)#

All frozen by f_Capture.md v1.0.0:

Symbol Type Frozen In Frozen At
v_approach Primary f_Capture.md v1.0.0
v_escape(A) Primary f_Capture.md v1.0.0
ρ(Φ) Primary f_Capture.md v1.0.0
r_capture Primary f_Capture.md v1.0.0
β Primary f_Capture.md v1.0.0
ω_res Primary f_Capture.md v1.0.0
M_A Primary f_Capture.md v1.0.0
M_E Primary f_Capture.md v1.0.0
r Primary f_Capture.md v1.0.0
P_eff Derived f_Capture.md v1.0.0
C_thresh Derived f_Capture.md v1.0.0
d_bind Derived f_Capture.md v1.0.0
p_res Derived f_Capture.md v1.0.0
CAPTURE_PENDING Flag f_Capture.md v1.0.0
CAPTURE_LOCKED Flag f_Capture.md v1.0.0
CAPTURE_DECAYING Flag f_Capture.md v1.0.0
CAPTURE_FAILED Flag f_Capture.md v1.0.0
CAPTURE_COLLISION Flag f_Capture.md v1.0.0
compute_approach_vector Primitive f_Capture.md v1.0.0
resolve_escape_velocity Primitive f_Capture.md v1.0.0
evaluate_capture_threshold Primitive f_Capture.md v1.0.0
lock_orbit Primitive f_Capture.md v1.0.0
register_capture Primitive f_Capture.md v1.0.0
flag_decay Primitive f_Capture.md v1.0.0
FM-001 through FM-007 Failure Mode f_Capture.md v1.0.0

§8.2 · Pending Freeze (promoted when source file goes canonical)#

Symbol Type Pending In Freeze Trigger
e Derived f_Orbit.md f_Orbit.md → canonical
T_orb Derived f_Orbit.md f_Orbit.md → canonical
δ Derived f_Decay.md f_Decay.md → canonical
d_warn Derived f_Decay.md f_Decay.md → canonical
d_collapse Derived f_Decay.md f_Decay.md → canonical
E_rel Derived f_Release.md f_Release.md → canonical
m_parity Derived f_Collapse.md f_Collapse.md → canonical
F_emit Derived f_Emit.md f_Emit.md → canonical
F_damp Derived f_Dampen.md f_Dampen.md → canonical
F_amp Supplementary f_Amplify.md f_Amplify.md → canonical
β_max Supplementary f_Amplify.md f_Amplify.md → canonical
heading_delta Supplementary f_Deflect.md f_Deflect.md → canonical
ORBIT_STABLE Flag f_Orbit.md f_Orbit.md → canonical
ORBIT_ECCENTRIC Flag f_Orbit.md f_Orbit.md → canonical
RELEASED Flag f_Release.md f_Release.md → canonical
COLLAPSED Flag f_Collapse.md f_Collapse.md → canonical
DAMPEN_ACTIVE Flag f_Dampen.md f_Dampen.md → canonical
EMIT_ACTIVE Flag f_Emit.md f_Emit.md → canonical
FM-008 Failure Mode f_Release.md f_Release.md → canonical
FM-009 Failure Mode f_Dampen.md f_Dampen.md → canonical
FM-010 Failure Mode f_Amplify.md f_Amplify.md → canonical
Primitives 7–16 Primitive Wave 3 files each source file → canonical

§9 · Versioning Policy#

§9.1 · Version Bump Rules#

Change Type Bump Required Example
Add a new symbol (not conflicting) minor — e.g. v1.0.0 → v1.1.0 Adding T_orb when f_Orbit.md canonicalizes
Change a frozen symbol's formula major — e.g. v1.1.0 → v2.0.0 Changing P_eff formula
Rename a frozen symbol major Renaming d_bind to d_orbital
Remove a frozen symbol major + deprecation notice Removing ω_res
Add a failure mode minor Adding FM-011
Change a failure mode's severity minor Upgrading FM-004 from warn to error
Fix a typo in a non-normative section patch — e.g. v1.0.0 → v1.0.1 Fixing a table cell
Add a note or example patch Adding a new canonical example

§9.2 · Namespace Reservation#

The following symbol prefixes are reserved for future use and must not be used by any function file without first registering here:

Prefix Reserved For Example
v_ velocity operators v_approach, v_escape, v_release
ρ field density operators ρ(Φ), ρ(Φ,θ), ρ(Φ,t)
ω_ resonance and frequency operators ω_res, ω_target
d_ depth and distance operators d_bind, d_warn, d_collapse
F_ engineering force/field operators F_emit, F_damp, F_amp
M_ mass operators M_A, M_E
G_ GravityGraph operators G_edge, G_degree, G_stability
FM- failure mode IDs FM-001 through FM-999
EX- canonical example IDs EX-001, EX-RES-001
SES- session IDs SES-20260813-OPS-001

§10 · Document Metadata#

Field Value
Canonical Path docs/FFF_Gravity/OPERATORS.md
Version 1.0.0
Status Canonical
Stability Living — update when any symbol is added, frozen, or deprecated
Normative Yes
Created 2026-08-13
Last Modified 2026-08-13
Authors TriadicFrameworks
License See /LICENSE at repository root
Encoding UTF-8 · LF
Supersedes Operator tables in f_Capture.md §4 (those remain normative; this file is the consolidated authority)

Single source of truth for all FFF_Gravity operator symbols. All function files derive their symbol definitions from this document. Update at the close of any session that introduces or freezes a symbol.


---

**Commit message:**

feat(FFF_Gravity): add canonical OPERATORS.md — master symbol authority, 9P/10D operators, 11 flags, 16 primitives, 10 FMs, composition rules, freeze registry [SES-20260813-OPS-001]

Two corrections surfaced while writing this — both logged in the file, both need propagating:

Discovery Location to update
Primitive count is 16, not 13 (INDEX said 13, scaffolds said 15) — redirect_force_node was the missing one INDEX.md §2 master registry row for primitives; FFF_Gravity_module.json
Primitive count correction note in §4.2 is self-documenting No separate action needed

Wave 3 Addendum — Core Function Operators (PRIM:007–024)#

Session: SES-20260813-W3 | Sealed: 2026-08-13

f_Orbit Operators (Wave 3 · PRIM:007, 012)#

Symbol Type Domain Description Frozen in
T_orb float > 0 Orbital period; time for one complete orbit around source node PRIM:012
orbit_class enum {STABLE, DECAYING, ESCAPE} Orbit classification derived from binding strength and δ PRIM:007
stab_class enum {HIGH, MED, LOW, CRITICAL} Stability classification derived from T_orb and decay trajectory PRIM:012

f_Release Operators (Wave 3 · PRIM:008–009)#

Symbol Type Domain Description Frozen in
v_release float ≥ 0 Scalar release velocity at moment of unbinding PRIM:009
r_release float > 0 Release radius; distance at which unbinding condition is met PRIM:008

f_Decay Operators (Wave 3 · PRIM:010–011)#

Symbol Type Domain Description Frozen in
δ float (0, 1) Decay rate coefficient; fractional binding-distance loss per cycle PRIM:010
d_warn float > 0, < d_bind Binding distance level at which decay warning is raised PRIM:010
d_collapse float > 0, ≤ d_warn Binding distance threshold triggering collapse evaluation PRIM:011

f_Collapse Operators (Wave 3 · PRIM:013–014)#

Symbol Type Domain Description Frozen in
m_parity float > 0 Mass parity ratio between collapsing nodes PRIM:013
C_node node_id — Identifier of the surviving/dominant node after collapse resolves PRIM:014

f_Emit Operators (Wave 3 · PRIM:015–017)#

Symbol Type Domain Description Frozen in
F_emit float ≥ 0 Emission force magnitude applied outward into the field PRIM:015
ρ(Φ)_delta float any Change in field density ρ(Φ) caused by a single emission event PRIM:016
r_emit float > 0 Emission radius; spatial extent of emitted influence PRIM:017

f_Dampen Operators (Wave 3 · PRIM:018–020)#

Symbol Type Domain Description Frozen in
F_damp float ≤ 0 Dampening force; negative magnitude opposing field escalation PRIM:018
ρ(Φ)_floor float ≥ 0 Minimum field density enforced when dampening is active PRIM:019
cascade_guard bool {0, 1} Flag active when cascade dampening guard is engaged PRIM:020

f_Amplify Operators (Wave 3 · PRIM:021–022)#

Symbol Type Domain Description Frozen in
F_amp float ≥ 0 Amplification force magnitude applied to field PRIM:021
β_max float > 1 Maximum amplification coefficient; hard ceiling on β PRIM:021
amp_cost float ≥ 0 Energy cost of amplification; deducted from ρ(Φ) each cycle PRIM:022

f_Deflect Operators (Wave 3 · PRIM:023–024)#

Stub resolution note: heading_delta was declared as a forward stub in f_Force.md §4.3 (PRIM:005). That stub is fully resolved here. f_Force.md §4.3 remains the declaration site; f_Deflect.md (PRIM:023) is the authoritative definition and domain authority.

Symbol Type Domain Description Frozen in
heading_delta float [−π, π] Angular deflection applied to the approach vector PRIM:023
r_deflect float > 0 Deflection radius; distance at which deflection force acts PRIM:023
deflect_cost float ≥ 0 Energy cost of deflection; deducted from ρ(Φ) PRIM:024

Wave 4 Addendum — Capture Variant Operators (PRIM:025–040)#

Session: SES-20260813-W4 | Sealed: 2026-08-13

f_Capture_Multi Operators (Wave 4 · PRIM:025–026)#

Symbol Type Domain Description Frozen in
N int ≥ 2 Count of simultaneous capture targets in multi-capture event PRIM:025
eval_order list ordered Evaluation priority sequence for multi-target binding attempts PRIM:025
Φ_perturbed float [0, 1] Field value after perturbation from multi-target interference PRIM:026
δ_perturb float ≥ 0 Perturbation magnitude applied to Φ during multi-capture PRIM:026
k_perturb float > 0 Perturbation scaling coefficient PRIM:026

f_Capture_Cascade Operators (Wave 4 · PRIM:027–028)#

Symbol Type Domain Description Frozen in
cascade_depth int ≥ 0 Current recursion depth within the cascade capture chain PRIM:027
k_max int ≥ 1 Maximum permitted cascade depth; CAS-1 guard enforced inside PRIM:027 PRIM:027
γ float (0, 1) Cascade gain; fraction of parent Ω_cascade transmitted to child step PRIM:028
Ω_cascade float ≥ 0 Cascade transmission energy available at the current depth step PRIM:028

f_Capture_Soft Operators (Wave 4 · PRIM:029–030)#

Symbol Type Domain Description Frozen in
d_soft float > 0 Soft capture binding distance (relaxed threshold vs. d_bind) PRIM:029
soft_threshold float (0, 1) Minimum ρ(Φ) required to sustain soft capture PRIM:029
grace_period int ≥ 0 Cycles permitted below soft_threshold before release triggers PRIM:030
k_grace float > 0 Grace period scaling coefficient PRIM:030

f_Capture_Hard Operators (Wave 4 · PRIM:031–032)#

Symbol Type Domain Description Frozen in
d_hard float > 0 Hard capture binding distance; strict threshold for lock entry PRIM:031
α_hard float > 1 Hard capture alpha; binding distance multiplier PRIM:031
β_hard float > 0 Hard capture beta; force amplification coefficient PRIM:031
β_min_hard float > 0, ≤ β_hard Minimum beta required to achieve and sustain hard lock PRIM:032
lock_cost float ≥ 0 Energy cost deducted from ρ(Φ) to establish hard lock PRIM:032
k_lock float > 0 Lock cost scaling coefficient PRIM:032

f_Capture_Resonant Operators (Wave 4 · PRIM:033–034)#

Symbol Type Domain Description Frozen in
ω_res float > 0 Resonant angular frequency of the source emission cycle PRIM:033
T_res float > 0 Resonant period; T_res = 2π / ω_res PRIM:033
φ_A(t) float [0, 2π) Approach node phase angle at time t PRIM:033
φ_E float [0, 2π) Emission phase angle of source at peak resonance PRIM:033
φ_open float [0, 2π) Phase angle at which resonance acceptance window opens PRIM:033
φ_close float [0, 2π) Phase angle at which resonance acceptance window closes PRIM:033
window_width float (0, 2π) Angular width of resonance acceptance window PRIM:033
p_ratio int ≥ 1 Numerator of the p:q phase-lock ratio PRIM:033
q_ratio int ≥ 1 Denominator of the p:q phase-lock ratio PRIM:033
ρ_res_gain float > 0 Field density gain multiplier active during resonance window PRIM:034
ρ_eff float ≥ 0 Effective field density under resonance conditions PRIM:034
ρ_res_floor float ≥ 0 Minimum field density floor enforced in resonant state PRIM:034
d_bind_res float > 0 Resonance-adjusted binding distance PRIM:034
T_orb_res float > 0 Resonance-adjusted orbital period PRIM:034

f_Capture_Asymmetric Operators (Wave 4 · PRIM:035–036)#

Symbol Type Domain Description Frozen in
mass_ratio float > 0 Ratio of source node mass to approach node mass PRIM:035
asymmetry_factor float ≥ 0 Degree of mass asymmetry (0 = fully symmetric) PRIM:035
parity_warn_threshold float > 0 Asymmetry level at which parity warning is raised PRIM:035
d_bind_asym float > 0 Asymmetry-corrected binding distance PRIM:036
heading_delta_asym float [−π, π] Asymmetry-adjusted heading deflection angle PRIM:036
deflect_tolerance float ≥ 0 Maximum heading_delta_asym before deflection is rejected PRIM:036
asym_decay_risk bool {0, 1} Flag: elevated decay risk due to asymmetric mass distribution PRIM:036

f_Capture_Temporal Operators (Wave 4 · PRIM:037–038)#

Symbol Type Domain Description Frozen in
t_open float ≥ 0 Timestamp at which temporal capture window opens PRIM:037
t_close float > t_open Timestamp at which temporal capture window closes PRIM:037
t_span float > 0 Duration of window; t_span = t_close − t_open PRIM:037
t_elapsed float ≥ 0 Time elapsed since t_open at evaluation moment PRIM:037
t_remaining float ≥ 0 Time remaining until t_close at evaluation moment PRIM:037
window_id string unique Identifier for this temporal capture window instance PRIM:037
proximity_ratio float [0, 1] t_elapsed / t_span; urgency metric for temporal capture PRIM:038
temporal_decay_factor float (0, 1] Decay multiplier applied to d_bind as window approaches close PRIM:038
d_bind_temporal float > 0 Temporally-adjusted binding distance PRIM:038
temporal_margin float ≥ 0 Buffer interval before t_close that triggers margin alert PRIM:038

f_Capture_Networked Operators (Wave 4 · PRIM:039–040)#

Symbol Type Domain Description Frozen in
N_net int ≥ 2 Number of nodes participating in the capture network PRIM:039
G_net graph connected Network graph structure (nodes + weighted directed edges) PRIM:039
w_i float ≥ 0 Edge weight for node i within G_net PRIM:039
d_bind_net float > 0 Network-consensus binding distance (weighted aggregate) PRIM:040
ρ(Φ)_net float ≥ 0 Aggregate network field density across all participating nodes PRIM:040
resilience_threshold float (0, 1) Minimum connectivity fraction required to sustain network capture PRIM:040

Wave 5 — Dismissal Operators (PRIM:041–042)#

Session: SES-20260814-DISMISS-001 | Sealed: 2026-08-14

f_Dismiss Operators (Wave 5 · PRIM:041–042)#

Symbol Type Domain Description Frozen in
F_dismiss float ≥ 0 Scalar dismissal force magnitude (|G_D|) PRIM:041
ρ_D(Φ) float (−1, 0] Dismissal field density (negative-polarity extension) PRIM:042
d_dismiss float > 0 Initial Dismissal Well depth at t = 0 PRIM:042
T_dismiss float > 0 Dismissal persistence time; governs exponential well decay PRIM:042
r_dismiss float (0, r_capture] Spatial extent of dismissal repulsion zone around A PRIM:041
ψ_dismiss enum {INTENTIONAL, STRUCTURAL, DRIFT} Dismissal mode flag PRIM:041
t_dismiss float ≥ 0 Absolute clock time of dismissal event PRIM:042
v_depart float ≥ 0 Entity departure velocity post-dismissal PRIM:042
β_D float (−∞, 0] Repulsive coupling coefficient active during expulsion PRIM:042

Well decay formula (conceptual authority: GravityOfDismissal.md §3.2):

ρ_D(Φ, t) = −d_dismiss × exp(−t / T_dismiss)

Re-capture gate (operational form of GravityOfDismissal.md §3.3):

d_bind_approach(t) > |ρ_D(Φ, t)|   →   re-capture structurally eligible

Updated

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