f_Capture_Resonant — Resonance-Locked Capture

id: f_Capture_Resonant
title: "f_Capture_Resonant — Resonance-Locked Capture"
module: FFF_Gravity
version: 1.0.0
status: canonical
wave: 4
session: SES-20260813-FGRAV-033
date: 2026-08-13
authors:
  - umaywant2
invariants_enforced:
  - INV-001
  - INV-002
  - INV-003
  - INV-004
  - INV-005
  - INV-006
  - INV-008
  - INV-009
failure_modes_referenced:
  - FM-001
  - FM-002
  - FM-003
  - FM-004
  - FM-006
primitives_defined:
  - PRIM:033
  - PRIM:034
depends_on:
  - f_Capture.md
  - f_Orbit.md
  - f_Field.md
  - f_Force.md
  - OPERATORS.md

Module: FFF_Gravity Wave: 4 — Capture Variants Version: 1.0.0 Status: Canonical Session: SES-20260813-FGRAV-033 Date: 2026-08-13

§0 Session Context#

This file is the fifth of six capture-variant documents in Wave 4 of the FFF_Gravity module. It specifies the resonance-locked capture pathway: a capture that succeeds only when the entity E arrives within a narrow orbital timing window defined by the attractor A's resonance frequency ω_res.

Resonance-locked capture is the rarest canonical capture mode. Where soft capture tolerates partial binding and hard capture demands threshold excess, resonant capture demands timing precision: the approach vector must phase- align with A's field oscillation cycle. An otherwise-qualified approach (β ≥ 1.0, ρ(Φ) sufficient) is rejected as a flyby (FM-001) if it arrives outside the resonance window.

Successful resonant capture writes orbit_class = RESONANT (defined in f_Orbit.md §3) and produces the highest possible d_bind stability — resonant orbits are deeply phase-locked and resist perturbation until field coherence falls below a dedicated resonance-floor threshold.

Symbols introduced here are registered in OPERATORS.md per INV-009. No new FM IDs are created per the FM freeze protocol; FM-004 (resonance drift) serves as the recoverable warning for resonance degradation.

§1 Module Identity#

Field	Value
File path	docs/FFF_Gravity/f_Capture_Resonant.md
Parent operator	f_Capture.md (base capture contract)
Peer variants	f_Capture_Multi.md, f_Capture_Cascade.md, f_Capture_Soft.md, f_Capture_Hard.md
Successor	f_Capture_Asymmetric.md
Orbit class written	RESONANT
Condition prefix	RLC- (Resonance Lock Condition)
State flags	WINDOW_OPEN, WINDOW_CLOSED, RESONANCE_LOCKED, RESONANCE_LOST
Primitives	PRIM:033 (eval_resonance_window), PRIM:034 (lock_resonance)
Failure modes used	FM-001 (flyby), FM-002 (field null), FM-003 (saturation),
	FM-004 (resonance drift), FM-006 (phantom capture)

§2 Canonical Description#

2.1 Motivation#

Standard capture (f_Capture.md) requires β ≥ 1.0 and ρ(Φ) > 0 to bind E to A. These conditions are necessary but not sufficient for resonant capture: resonant capture additionally requires that E's arrival phase aligns with A's oscillation cycle.

A's field coherence ρ(Φ) is not static — it oscillates at angular frequency ω_res (radians per unit time). At resonance peaks, the binding force is amplified; at troughs, it is suppressed. An entity arriving at a trough may not achieve orbit even if all scalar conditions pass.

Resonance-locked capture exploits this oscillation: E must arrive within a phase window [φ_open, φ_close] relative to ω_res. If it does, d_bind is multiplied by a resonance gain factor (ρ_res_gain). If it does not, capture is rejected and E continues on its approach trajectory — recorded as FM-001 with the sub-annotation REASON: WINDOW_MISS.

2.2 Physical Analogy#

In orbital mechanics, resonance describes configurations where two bodies' orbital periods form a small integer ratio (e.g., 2:1, 3:2). Here, resonance describes the phase relationship between E's approach timing and A's field cycle. The Laplace resonances of Jupiter's moons provide the canonical physical analogue: only entities arriving in the correct phase slot achieve stable co-orbiting configurations.

2.3 Relationship to orbit_class = RESONANT#

f_Orbit.md §3 defines four orbit classes: CIRCULAR, ELLIPTICAL, ECCENTRIC, RESONANT. The RESONANT class is only reachable via this file. No other capture pathway writes orbit_class = RESONANT.

A RESONANT orbit has the following properties (inherited from f_Orbit.md):

stab_class is forced to STABLE (resonant phase-lock implies stability)
T_orb is pinned to a rational multiple of A's resonance period T_res
Perturbations are dampened by the phase-lock force until ρ(Φ) < ρ_res_floor
FM-004 (resonance drift) is the only non-fatal degradation mode

2.4 Resonance Window Geometry#

The resonance window is defined in phase space, not time space:

φ_open  = 2π × n_window_start     (n_window_start ∈ [0, 1))
φ_close = 2π × n_window_end       (n_window_end ∈ (0, 1], n_window_end > n_window_start)
window_width = φ_close - φ_open

At any clock tick t, A's current phase is:

φ_A(t) = (ω_res × t) mod 2π

E's arrival phase is φ_E = φ_A(t_arrive). Capture proceeds if and only if:

φ_open ≤ φ_E ≤ φ_close        ← RLC-1

Window width window_width is a property of A registered at f_Source.md initialization. Narrower windows produce rarer but deeper captures.

2.5 Resonance Gain and d_bind Enhancement#

When E arrives in-window, d_bind is computed with a resonance gain multiplier:

ρ_res_gain ∈ (1.0, ∞)          (registered in f_Source.md for A)
d_bind_res = β × (ρ(Φ) × ρ_res_gain) × (1 − e)

Note: ρ(Φ) × ρ_res_gain must be capped at 1.0 for the coherence product, then d_bind_res is computed. The gain amplifies the effective field density but does not violate the ρ(Φ) ∈ [0, 1] invariant on the base field.

ρ_eff = min(1.0, ρ(Φ) × ρ_res_gain)
d_bind_res = β × ρ_eff × (1 − e)

This is the governing d_bind for RESONANT orbits. It is always ≥ standard d_bind when ρ_res_gain ≥ 1.0.

2.6 Resonance Lock vs. Resonance Drift#

Once locked, a RESONANT orbit remains locked as long as:

ρ(Φ)(t) ≥ ρ_res_floor          ← RLC-4

where ρ_res_floor is a registered threshold (default 0.30, stricter than the general field-null threshold of 0.0 in INV-003).

If ρ(Φ) drops below ρ_res_floor but remains above 0:

FM-004 (resonance drift) is raised as a recoverable warning
orbit_class degrades from RESONANT → ELLIPTICAL
stab_class degrades from STABLE → MARGINAL
Lock is not yet lost; recovery is possible if ρ(Φ) recovers above ρ_res_floor

If ρ(Φ) subsequently falls to 0: INV-003 triggers FM-002 (field null/collapse), and the orbit is terminal regardless of prior resonance.

§3 Triadic Equation Mapping#

G = F_freq · F_fluid · F_force

Node	Resonant-capture contribution
F_freq	ω_res (oscillation frequency), φ_A(t) (current phase),
	T_res (resonance period), window_width (phase gate width)
F_fluid	ρ(Φ) (field coherence), ρ_res_gain (amplification factor),
	ρ_eff (capped effective coherence), ρ_res_floor (lock floor)
F_force	β (binding coefficient), d_bind_res (resonance-enhanced depth),
	v_approach, r_capture (base capture scalars inherited from
	f_Capture.md), heading_delta (from f_Force.md §4.3)

INV-001 compliance: All three nodes participate. ω_res is a F_freq primitive; ρ_eff is a F_fluid primitive; d_bind_res is a F_force primitive. No resonant capture computation is possible with any node absent.

§4 Operator Registry#

All symbols below are registered in OPERATORS.md per INV-009.

4.1 Resonance Frequency and Phase#

Symbol	Domain	Description
`ω_res`	ℝ, > 0	Angular resonance frequency of attractor A (rad/unit time)
`T_res`	ℝ, > 0	Resonance period = 2π / ω_res
`φ_A(t)`	[0, 2π)	Current phase of A at time t = (ω_res × t) mod 2π
`φ_E`	[0, 2π)	Arrival phase of E = φ_A(t_arrive)
`φ_open`	[0, 2π)	Window open phase boundary
`φ_close`	(0, 2π]	Window close phase boundary (> φ_open)
`window_width`	(0, 2π]	φ_close − φ_open
`t_arrive`	ℝ, ≥ 0	Clock tick at which E reaches r_capture
`t_next_open`	ℝ, > t_arrive	Earliest future t where WINDOW_OPEN holds

4.2 Field Enhancement#

Symbol	Domain	Description
`ρ_res_gain`	ℝ, > 1.0	Field amplification factor during resonance window
`ρ_eff`	[0, 1]	min(1.0, ρ(Φ) × ρ_res_gain)
`ρ_res_floor`	(0, 1)	Minimum ρ(Φ) to maintain resonance lock (default 0.30)

4.3 Enhanced Binding#

Symbol	Domain	Description
`d_bind_res`	ℝ, ≥ 0	Resonance-enhanced binding depth = β × ρ_eff × (1 − e)
`orbit_class`	enum	Set to RESONANT on successful lock
`stab_class`	enum	Forced STABLE on resonance lock
`T_orb_res`	ℝ, > 0	Orbital period under resonance lock = p/q × T_res (p, q ∈ ℤ⁺)
`p_ratio`	ℤ⁺	Numerator of orbital resonance ratio p:q
`q_ratio`	ℤ⁺	Denominator of orbital resonance ratio p:q

4.4 Window State Flags#

Flag	Meaning
`WINDOW_OPEN`	φ_E is within [φ_open, φ_close] — capture eligible
`WINDOW_CLOSED`	φ_E is outside window — capture rejected (FM-001)
`RESONANCE_LOCKED`	Orbit is actively phase-locked; ρ(Φ) ≥ ρ_res_floor
`RESONANCE_LOST`	ρ(Φ) dropped below ρ_res_floor; FM-004 raised

§5 Resonance Lock Conditions (RLC-)#

All five conditions are conjunctive (INV-005). All must hold simultaneously for resonance-locked capture to complete.

ID	Condition	Failure if violated
RLC-1	φ_open ≤ φ_E ≤ φ_close (in-window arrival)	FM-001 (WINDOW_MISS)
RLC-2	β ≥ 1.0 (standard capture binding threshold)	FM-001 (APPROACH_REJECTION)
RLC-3	ρ(Φ) > 0 at t_arrive (field non-null)	FM-002 (FIELD_NULL)
RLC-4	ρ(Φ) ≥ ρ_res_floor post-capture (lock floor)	FM-004 (RESONANCE_DRIFT)
RLC-5	v_approach < v_escape(A) (not hyperbolic)	FM-001 (OVERSHOOT)

RLC-1 is the distinguishing condition of this variant. All other capture variants ignore φ_E entirely. A WINDOW_MISS rejection is annotated distinctly from standard FM-001 overshoot to aid diagnostics.

RLC-4 is a post-capture maintenance condition, not an entry gate. It is evaluated on every subsequent tick after capture, not at t_arrive.

§6 Failure Modes#

No new FM IDs are introduced. The following FM entries apply:

FM-001 — Flyby (two sub-cases in this variant)#

Sub-case A: WINDOW_MISS (RLC-1 violated)

FM-001 raised with annotation: REASON=WINDOW_MISS
φ_E           := computed arrival phase
t_next_open   := next tick where WINDOW_OPEN will hold
                 = t_arrive + (φ_open − φ_E + 2π) mod 2π / ω_res
orbit_class   := not written (capture did not occur)

E continues on its pre-capture trajectory. The caller may retry by holding E at a waiting state until t_next_open.

Sub-case B: APPROACH_REJECTION or OVERSHOOT (RLC-2 or RLC-5 violated) Identical to base f_Capture.md FM-001 behavior; φ_E is irrelevant if scalar conditions fail first.

Evaluation order per INV-008: RLC-3 → RLC-2 → RLC-5 → RLC-1. (Field null checked first; window checked last to avoid phase calculation on degenerate inputs.)

FM-002 — Field Null (RLC-3 violated)#

FM-002 raised
ρ(Φ) = 0 confirmed
d_bind_res is undefined (not computed)
orbit_class := not written

Behavior identical to base f_Capture.md FM-002.

FM-003 — Frame Saturation#

Evaluated against A's orbit count prior to resonant capture attempt. If A has reached its max_orbits ceiling, the capture is refused pre-phase-check. FM-003 annotation includes VARIANT=RESONANT for tracing.

FM-004 — Resonance Drift (post-capture, recoverable)#

Trigger: ρ(Φ)(t) < ρ_res_floor (RLC-4 violated post-capture)
State:   RESONANCE_LOCKED → RESONANCE_LOST
orbit_class: RESONANT → ELLIPTICAL
stab_class:  STABLE → MARGINAL
Action:  warning raised; orbit continues as ELLIPTICAL
Recovery: if ρ(Φ) recovers ≥ ρ_res_floor → RESONANCE_LOCKED re-asserted,
          orbit_class re-promoted to RESONANT

FM-004 is the only non-fatal degradation mode for a RESONANT orbit.

FM-006 — Phantom Capture#

If φ_E is in-window but ρ(Φ) is non-zero and β ≥ 1.0, yet the computed d_bind_res resolves to 0.0 (due to eccentricity e = 1.0, i.e., p_res = 0 with P_eff = 0), FM-006 is raised: the phase alignment was real but the binding force was phantom.

Phantom resonance is rare; the usual cause is a degenerate orbit where the entity has zero effective momentum. The guard condition is:

if d_bind_res == 0.0 and all RLC pass:
    raise FM-006 (PHANTOM_RESONANCE)

§7 Engineering Primitives#

PRIM:033 — eval_resonance_window#

def eval_resonance_window(
    omega_res: float,
    phi_open: float,
    phi_close: float,
    t_arrive: float,
    beta: float,
    rho_phi: float,
    v_approach: float,
    v_escape: float,
    max_orbits: int,
    current_orbit_count: int,
) -> dict:
    """
    PRIM:033 — Resonance Window Evaluator
    ======================================
    Evaluate whether entity E's arrival at attractor A satisfies all
    pre-capture Resonance Lock Conditions (RLC-1 through RLC-3, RLC-5)
    and compute the resonance arrival phase.
 
    This primitive performs the gate-check phase of resonant capture.
    It does NOT write orbit state — that is PRIM:034's responsibility.
 
    Evaluation order (INV-008):
        1. FM-003 check  — saturation guard
        2. RLC-3         — ρ(Φ) > 0
        3. RLC-2         — β ≥ 1.0
        4. RLC-5         — v_approach < v_escape
        5. RLC-1         — φ_E in [φ_open, φ_close]
 
    Parameters
    ----------
    omega_res : float
        Angular resonance frequency of attractor A (rad / unit time). > 0.
    phi_open : float
        Window open phase boundary. In [0, 2π).
    phi_close : float
        Window close phase boundary. In (0, 2π]. Must exceed phi_open.
    t_arrive : float
        Clock tick at which E reaches r_capture. ≥ 0.
    beta : float
        Binding coefficient of E with respect to A. ≥ 0.
    rho_phi : float
        Field coherence density at t_arrive. In [0, 1].
    v_approach : float
        Approach velocity of E toward A. ≥ 0.
    v_escape : float
        Escape velocity of A's capture field. ≥ 0. v_escape(A) from f_Force.md.
    max_orbits : int
        Frame saturation ceiling for A. From f_Frame.md.
    current_orbit_count : int
        Current number of bound orbits around A. ≥ 0.
 
    Returns
    -------
    dict with keys:
        status : str
            "WINDOW_OPEN" | "WINDOW_CLOSED" | "FM-001" | "FM-002" | "FM-003"
        phi_E : float
            Computed arrival phase = (omega_res × t_arrive) mod 2π.
        t_next_open : float | None
            If status == "WINDOW_CLOSED": earliest future tick where WINDOW_OPEN
            holds. None otherwise.
        failure_mode : str | None
            FM code if status is a failure. None on WINDOW_OPEN.
        reason : str | None
            Sub-annotation string (e.g., "WINDOW_MISS", "APPROACH_REJECTION").
 
    Invariants
    ----------
    INV-001 : F_freq (omega_res), F_fluid (rho_phi), F_force (beta) all present.
    INV-003 : rho_phi = 0 → FM-002 raised unconditionally.
    INV-004 : beta < 1.0 → FM-001 raised (flyby, approach rejection).
    INV-005 : All RLC evaluated conjunctively; first failure terminates.
    INV-008 : Evaluation order normative (saturation → null → binding → velocity → phase).
    """
    import math
 
    T_res = (2 * math.pi) / omega_res
 
    # Compute arrival phase unconditionally (used in all branches)
    phi_E = (omega_res * t_arrive) % (2 * math.pi)
 
    # Step 1: FM-003 — frame saturation
    if current_orbit_count >= max_orbits:
        return {
            "status": "FM-003",
            "phi_E": phi_E,
            "t_next_open": None,
            "failure_mode": "FM-003",
            "reason": "FRAME_SATURATION (VARIANT=RESONANT)",
        }
 
    # Step 2: RLC-3 — field non-null (INV-003)
    if rho_phi <= 0.0:
        return {
            "status": "FM-002",
            "phi_E": phi_E,
            "t_next_open": None,
            "failure_mode": "FM-002",
            "reason": "FIELD_NULL",
        }
 
    # Step 3: RLC-2 — binding threshold (INV-004)
    if beta < 1.0:
        return {
            "status": "FM-001",
            "phi_E": phi_E,
            "t_next_open": None,
            "failure_mode": "FM-001",
            "reason": "APPROACH_REJECTION",
        }
 
    # Step 4: RLC-5 — not hyperbolic
    if v_approach >= v_escape:
        return {
            "status": "FM-001",
            "phi_E": phi_E,
            "t_next_open": None,
            "failure_mode": "FM-001",
            "reason": "OVERSHOOT",
        }
 
    # Step 5: RLC-1 — phase window check
    if phi_open <= phi_E <= phi_close:
        return {
            "status": "WINDOW_OPEN",
            "phi_E": phi_E,
            "t_next_open": None,
            "failure_mode": None,
            "reason": None,
        }
    else:
        # Compute next open tick
        phase_gap = (phi_open - phi_E + 2 * math.pi) % (2 * math.pi)
        t_next_open = t_arrive + phase_gap / omega_res
        return {
            "status": "WINDOW_CLOSED",
            "phi_E": phi_E,
            "t_next_open": t_next_open,
            "failure_mode": "FM-001",
            "reason": "WINDOW_MISS",
        }

PRIM:034 — lock_resonance#

def lock_resonance(
    beta: float,
    rho_phi: float,
    rho_res_gain: float,
    rho_res_floor: float,
    eccentricity: float,
    p_ratio: int,
    q_ratio: int,
    T_res: float,
    phi_E: float,
    phi_open: float,
    phi_close: float,
) -> dict:
    """
    PRIM:034 — Resonance Lock Writer
    =================================
    Given that PRIM:033 returned WINDOW_OPEN, compute and write the full
    resonance-locked orbit state.
 
    This primitive is called only after PRIM:033 confirms WINDOW_OPEN.
    Calling it without that confirmation violates INV-008 (evaluation order).
 
    Computes:
        ρ_eff       = min(1.0, rho_phi × rho_res_gain)
        d_bind_res  = beta × ρ_eff × (1 − eccentricity)
        T_orb_res   = (p_ratio / q_ratio) × T_res
        orbit_class = RESONANT
        stab_class  = STABLE
 
    Raises FM-006 (phantom) if d_bind_res resolves to 0.0 despite all
    RLC passing — indicating degenerate eccentricity (e = 1.0).
 
    Parameters
    ----------
    beta : float
        Binding coefficient. ≥ 1.0 (already verified by PRIM:033).
    rho_phi : float
        Field coherence density. In (0, 1] (non-null verified by PRIM:033).
    rho_res_gain : float
        Field amplification factor during resonance window. > 1.0.
    rho_res_floor : float
        Minimum ρ(Φ) to maintain resonance lock post-capture. In (0, 1).
    eccentricity : float
        Orbital eccentricity e = p_res / (p_res + P_eff). In [0, 1).
        e must be < 1.0; e = 1.0 triggers FM-006.
    p_ratio : int
        Numerator of orbital period resonance ratio p:q. ≥ 1.
    q_ratio : int
        Denominator of orbital period resonance ratio p:q. ≥ 1.
    T_res : float
        Resonance period of A = 2π / omega_res. > 0.
    phi_E : float
        Arrival phase (from PRIM:033). In [phi_open, phi_close].
    phi_open : float
        Window open phase. Informational; used in output record only.
    phi_close : float
        Window close phase. Informational; used in output record only.
 
    Returns
    -------
    dict with keys:
        status : str
            "RESONANCE_LOCKED" | "FM-006"
        rho_eff : float
            Effective coherence used in binding computation.
        d_bind_res : float
            Resonance-enhanced binding depth.
        T_orb_res : float
            Pinned orbital period for this resonant orbit.
        orbit_class : str
            "RESONANT" on success; None on FM-006.
        stab_class : str
            "STABLE" on success; None on FM-006.
        failure_mode : str | None
            "FM-006" on phantom; None on success.
        lock_record : dict
            Structured record for appending to f_Source.md orbit registry.
 
    Invariants
    ----------
    INV-001 : All three nodes contribute to d_bind_res.
    INV-002 : Ω is frozen upon RESONANCE_LOCKED; orbit_class = RESONANT.
    INV-006 : RESONANCE_LOCKED is a terminal capture state (reversible only
              via FM-004 drift degradation, not arbitrary release).
    INV-008 : Must be called after PRIM:033 confirms WINDOW_OPEN.
    """
    # Compute effective coherence (ρ_eff capped at 1.0)
    rho_eff = min(1.0, rho_phi * rho_res_gain)
 
    # Compute resonance-enhanced binding depth
    d_bind_res = beta * rho_eff * (1.0 - eccentricity)
 
    # FM-006: phantom resonance guard
    if d_bind_res == 0.0:
        return {
            "status": "FM-006",
            "rho_eff": rho_eff,
            "d_bind_res": 0.0,
            "T_orb_res": None,
            "orbit_class": None,
            "stab_class": None,
            "failure_mode": "FM-006",
            "lock_record": None,
        }
 
    # Compute pinned orbital period (rational multiple of T_res)
    T_orb_res = (p_ratio / q_ratio) * T_res
 
    # Build lock record for f_Source.md orbit registry
    lock_record = {
        "orbit_class": "RESONANT",
        "stab_class": "STABLE",
        "d_bind_res": d_bind_res,
        "rho_eff": rho_eff,
        "T_orb_res": T_orb_res,
        "p_ratio": p_ratio,
        "q_ratio": q_ratio,
        "rho_res_floor": rho_res_floor,
        "phi_lock": phi_E,
        "phi_open": phi_open,
        "phi_close": phi_close,
        "state_flag": "RESONANCE_LOCKED",
    }
 
    return {
        "status": "RESONANCE_LOCKED",
        "rho_eff": rho_eff,
        "d_bind_res": d_bind_res,
        "T_orb_res": T_orb_res,
        "orbit_class": "RESONANT",
        "stab_class": "STABLE",
        "failure_mode": None,
        "lock_record": lock_record,
    }

§8 Canonical Examples#

Example 1 — Clean In-Window Resonant Capture (2:1 Resonance)#

Scenario: Entity E approaches attractor A at exactly the resonance peak. A has a 2:1 orbital resonance configuration; E arrives perfectly centered in the window.

Given:

omega_res       = π / 5          (T_res = 10 time units)
phi_open        = π / 3          (≈ 1.047 rad)
phi_close       = π              (≈ 3.142 rad)
window_width    = 2π / 3         (≈ 2.094 rad; wide window)

t_arrive        = 7.5
phi_A(7.5)      = (π/5 × 7.5) mod 2π
                = (1.5π) mod 2π
                = 3π/2           (≈ 4.712 rad)

Wait — φ_E = 4.712 > φ_close = 3.142. RLC-1 fails. Let us recalibrate:

t_arrive        = 3.5
phi_A(3.5)      = (π/5 × 3.5) mod 2π
                = 0.7π           (≈ 2.199 rad)

RLC-1: 1.047 ≤ 2.199 ≤ 3.142 ✓ — WINDOW_OPEN

Scalar checks:

rho_phi         = 0.75
beta            = 1.8
v_approach      = 3.2, v_escape = 5.0   ← RLC-5: 3.2 < 5.0  ✓
current_orbits  = 3, max_orbits = 10    ← FM-003: clear       ✓

PRIM:033 result:

status      = WINDOW_OPEN
phi_E       = 2.199 rad
failure_mode = None

PRIM:034 inputs:

rho_res_gain    = 1.4
rho_eff         = min(1.0, 0.75 × 1.4) = min(1.0, 1.05) = 1.0
eccentricity    = 0.15  (p_res=0.3, P_eff=1.7)
d_bind_res      = 1.8 × 1.0 × (1 − 0.15) = 1.8 × 0.85 = 1.530
p_ratio=2, q_ratio=1
T_orb_res       = (2/1) × 10 = 20.0 time units

PRIM:034 result:

status      = RESONANCE_LOCKED
orbit_class = RESONANT
stab_class  = STABLE
d_bind_res  = 1.530
rho_eff     = 1.000 (gain saturated — field fully coherent)
T_orb_res   = 20.0

Observation: ρ_res_gain of 1.4 on ρ(Φ) = 0.75 saturates the ρ_eff cap, demonstrating that resonance gain does not produce super-unity coherence. d_bind_res (1.530) exceeds standard d_bind (1.8 × 0.75 × 0.85 = 1.148) by 33% — the practical benefit of resonance capture.

Example 2 — Window Miss: Arrival Out of Phase (FM-001, WINDOW_MISS)#

Scenario: E has fully qualifying scalar properties but arrives between resonance windows. FM-001 is raised; t_next_open is computed for retry.

Given:

omega_res       = π / 4          (T_res = 8 time units)
phi_open        = π/6            (≈ 0.524 rad)
phi_close       = π/2            (≈ 1.571 rad)
window_width    = π/3            (≈ 1.047 rad; narrow window)

t_arrive        = 6.0
phi_A(6.0)      = (π/4 × 6.0) mod 2π
                = 1.5π           (≈ 4.712 rad)

RLC-1: 0.524 ≤ 4.712 ≤ 1.571 ✗ — WINDOW_CLOSED

Scalar checks (all pass):

rho_phi         = 0.80   ← RLC-3 ✓
beta            = 1.5    ← RLC-2 ✓
v_approach      = 2.0, v_escape = 4.5  ← RLC-5 ✓
current_orbits  = 0, max_orbits = 5    ← FM-003 clear ✓

PRIM:033 result:

status          = WINDOW_CLOSED
phi_E           = 4.712 rad
failure_mode    = FM-001
reason          = WINDOW_MISS

phase_gap       = (0.524 − 4.712 + 2π) mod 2π
                = (0.524 − 4.712 + 6.283) mod 2π
                = 2.095 mod 2π
                = 2.095 rad

t_next_open     = 6.0 + 2.095 / (π/4)
                = 6.0 + 2.095 / 0.785
                = 6.0 + 2.668
                = 8.668 time units

Retry guidance:

Hold E on approach trajectory.
Re-attempt PRIM:033 at t_arrive = 8.668.
phi_A(8.668) = (π/4 × 8.668) mod 2π ≈ 0.524 rad = φ_open  ← window just opens

Observation: The WINDOW_MISS path provides a concrete retry timestamp — this is the key operational difference between resonant and standard capture. A caller that treats FM-001 as terminal (rather than retry-able) would incorrectly abandon a qualifying entity. The t_next_open return value exists specifically to support retry scheduling.

Example 3 — Resonance Drift Mid-Orbit (FM-004 Triggered, Recovery Succeeds)#

Scenario: E is already in a RESONANCE_LOCKED orbit. A field perturbation drops ρ(Φ) below ρ_res_floor. FM-004 triggers. Field recovers; lock is re-asserted.

Initial locked state:

orbit_class     = RESONANT
stab_class      = STABLE
d_bind_res      = 1.200
rho_res_floor   = 0.30

Tick-by-tick ρ(Φ) sequence:

t=10: ρ(Φ) = 0.72  ← RESONANCE_LOCKED  ✓ (0.72 ≥ 0.30)
t=11: ρ(Φ) = 0.48  ← RESONANCE_LOCKED  ✓ (0.48 ≥ 0.30)
t=12: ρ(Φ) = 0.26  ← RLC-4 violated!   ✗ (0.26 < 0.30)

FM-004 trigger at t=12:

state:        RESONANCE_LOCKED → RESONANCE_LOST
orbit_class:  RESONANT         → ELLIPTICAL
stab_class:   STABLE           → MARGINAL
FM-004 raised (recoverable warning)
d_bind_res    retained (binding depth does not vanish — orbit continues as ELLIPTICAL)

Recovery sequence:

t=13: ρ(Φ) = 0.28  ← still below floor  (RESONANCE_LOST, MARGINAL)
t=14: ρ(Φ) = 0.35  ← above floor        ← RLC-4 re-satisfied

Re-lock at t=14:

state:        RESONANCE_LOST → RESONANCE_LOCKED
orbit_class:  ELLIPTICAL     → RESONANT
stab_class:   MARGINAL       → STABLE
FM-004 cleared

Observation: FM-004 (resonance drift) is the sole recoverable degradation for RESONANT orbits. Unlike FM-005 (decay spiral, fatal) or FM-007 (mutual dissolution, fatal), FM-004 preserves the orbit as ELLIPTICAL during drift and allows full restoration. The orbit never enters a terminal state unless ρ(Φ) drops to 0, which escalates to FM-002.

Example 4 — Phantom Resonance (FM-006) on Degenerate Eccentricity#

Scenario: E arrives in-window with β = 1.2 and ρ(Φ) = 0.6, but has reached maximum eccentricity (e = 1.0) due to a prior failed capture that left it in a radial fall trajectory. PRIM:033 passes; PRIM:034 raises FM-006.

PRIM:033 evaluation:

omega_res       = π/3
phi_open        = π/4    (≈ 0.785 rad)
phi_close       = 3π/4   (≈ 2.356 rad)
t_arrive        = 4.5
phi_E           = (π/3 × 4.5) mod 2π = 1.5π mod 2π = (4.712) — MISS?

Let us use t_arrive = 1.5:

phi_E           = (π/3 × 1.5) mod 2π = π/2 ≈ 1.571 rad
RLC-1: 0.785 ≤ 1.571 ≤ 2.356  ✓  — WINDOW_OPEN
rho_phi = 0.60   ✓  (RLC-3)
beta    = 1.20   ✓  (RLC-2)
v_approach = 2.0 < v_escape = 3.5  ✓  (RLC-5)
PRIM:033 → status = WINDOW_OPEN

PRIM:034 evaluation:

eccentricity    = 1.0    ← degenerate radial trajectory
rho_res_gain    = 1.3
rho_eff         = min(1.0, 0.60 × 1.3) = 0.78
d_bind_res      = 1.2 × 0.78 × (1 − 1.0)
                = 1.2 × 0.78 × 0.0
                = 0.000

FM-006 raised:

status          = FM-006
reason          = PHANTOM_RESONANCE
d_bind_res      = 0.000
orbit_class     = not written
stab_class      = not written

Diagnosis and remediation:

Root cause: eccentricity = 1.0 (p_res = 0, P_eff = 0 — radial infall)
            Phase alignment was genuine but force was absent.
Remediation: f_Force.md heading_delta adjustment to give E non-zero
             transverse momentum before next approach attempt.
             Until e < 1.0, resonant capture is structurally impossible.

Observation: FM-006 is the rarest failure mode in this file. It requires the improbable combination of correct phase alignment AND degenerate orbital geometry. It is not a retry-able condition without structural correction — the caller must address e before re-attempting.

§9 Cross-Module References#

Reference	Symbol used	Direction
f_Capture.md	β, ρ(Φ), v_approach, v_escape, r_capture, d_bind base formula	Parent
f_Orbit.md	orbit_class = RESONANT, T_orb, stab_class, classify_orbit (PRIM:007)	Peer
f_Field.md	ρ(Φ) oscillation model, field source	Parent
f_Force.md	v_escape(A), heading_delta	Parent
f_Source.md	ω_res, T_res, φ_open, φ_close, ρ_res_gain, ρ_res_floor, max_orbits	Read-only (INV-007)
f_Decay.md	d_warn, d_collapse monitoring for RESONANT orbits post-lock	Downstream
f_Dampen.md	ρ(Φ) floor enforcement; cascade guard relevant if resonance chain	Downstream
f_Capture_Cascade.md	Ω_cascade — resonant capture can be a cascade step	Peer
OPERATORS.md	Symbol authority for all operators in §4	Authority (INV-009)

§10 Operator Integration Notes#

10.1 OPERATORS.md Registration Block#

The following symbols are added to OPERATORS.md upon this file's ratification:

| ω_res          | Angular resonance frequency       | ℝ, > 0       | f_Capture_Resonant.md §4.1 |
| T_res          | Resonance period (2π / ω_res)    | ℝ, > 0       | f_Capture_Resonant.md §4.1 |
| φ_A(t)         | Current phase of A at time t     | [0, 2π)      | f_Capture_Resonant.md §4.1 |
| φ_E            | Arrival phase of E               | [0, 2π)      | f_Capture_Resonant.md §4.1 |
| φ_open         | Window open phase boundary       | [0, 2π)      | f_Capture_Resonant.md §4.1 |
| φ_close        | Window close phase boundary      | (0, 2π]      | f_Capture_Resonant.md §4.1 |
| window_width   | φ_close − φ_open                 | (0, 2π]      | f_Capture_Resonant.md §4.1 |
| t_arrive       | Clock tick at E reaching r_cap   | ℝ, ≥ 0       | f_Capture_Resonant.md §4.1 |
| t_next_open    | Earliest future WINDOW_OPEN tick | ℝ, > t_arrive| f_Capture_Resonant.md §4.1 |
| ρ_res_gain     | Field amplification in window    | ℝ, > 1.0     | f_Capture_Resonant.md §4.2 |
| ρ_eff          | min(1.0, ρ(Φ) × ρ_res_gain)     | [0, 1]       | f_Capture_Resonant.md §4.2 |
| ρ_res_floor    | Minimum ρ(Φ) for lock maintenance| (0, 1)       | f_Capture_Resonant.md §4.2 |
| d_bind_res     | Resonance-enhanced binding depth | ℝ, ≥ 0       | f_Capture_Resonant.md §4.3 |
| T_orb_res      | Pinned orbital period (p/q×T_res)| ℝ, > 0       | f_Capture_Resonant.md §4.3 |
| p_ratio        | Numerator of resonance ratio p:q | ℤ⁺           | f_Capture_Resonant.md §4.3 |
| q_ratio        | Denominator of resonance ratio   | ℤ⁺           | f_Capture_Resonant.md §4.3 |

10.2 f_Source.md Fields Required#

Per INV-007 (f_Source.md read-only), the following fields must be set at source initialization and never modified by this file:

omega_res       : float    — registered by source author at A creation
phi_open        : float    — window geometry, source-specific
phi_close       : float    — window geometry, source-specific
rho_res_gain    : float    — amplification factor, source-specific
rho_res_floor   : float    — lock maintenance floor, default 0.30
p_ratio         : int      — resonance ratio numerator
q_ratio         : int      — resonance ratio denominator

10.3 Condition Prefix Uniqueness#

RLC- (Resonance Lock Condition) is unique to this file. No other Wave 4 file uses this prefix. Full prefix registry across Wave 4:

MC-    f_Capture_Multi.md
CAS-   f_Capture_Cascade.md
SCS-   f_Capture_Soft.md
HLC-   f_Capture_Hard.md
RLC-   f_Capture_Resonant.md    ← this file
       f_Capture_Asymmetric.md  (prefix: AC-, to be assigned)

§11 Document Metadata#

11.1 INV Compliance Table#

Invariant	Description (abbreviated)	Status in this file
INV-001	G = F_freq · F_fluid · F_force	✅ All three nodes in §3
INV-002	f_Capture → Ω frozen	✅ RESONANCE_LOCKED freezes Ω
INV-003	ρ(Φ) = 0 → FM-002	✅ RLC-3 + PRIM:033 step 2
INV-004	β < 1.0 → flyby	✅ RLC-2 + PRIM:033 step 3
INV-005	Conditions conjunctive	✅ RLC-1–5 all required
INV-006	Terminal states irreversible	✅ FM-006 terminal; LOCKED→LOST reversible only via FM-004
INV-007	f_Source.md read-only	✅ §10.2 lists read-only fields
INV-008	Evaluation order normative	✅ PRIM:033 docstring + §5 table
INV-009	OPERATORS.md is symbol authority	✅ §10.1 registration block
INV-010	Frozen symbols unrenameable	✅ No renames; new symbols only

11.2 Primitive Registry (this file)#

PRIM	Name	Type	Pure?	Description
033	eval_resonance_window	Guard	Yes	Phase-gate check; returns WINDOW_OPEN/CLOSED or FM code
034	lock_resonance	Writer	No	Computes ρ_eff, d_bind_res, T_orb_res; writes RESONANCE_LOCKED state

Running total after this file: PRIM:034

11.3 Failure Mode Summary (this file)#

FM	Trigger in this file	Fatal?	Sub-annotation
FM-001	RLC-1, RLC-2, or RLC-5 violated	No	WINDOW_MISS / APPROACH_REJECTION / OVERSHOOT
FM-002	RLC-3 violated (ρ(Φ) = 0)	Yes	FIELD_NULL
FM-003	max_orbits ceiling reached	Yes	FRAME_SATURATION (VARIANT=RESONANT)
FM-004	RLC-4 violated post-capture	No	RESONANCE_DRIFT (recoverable)
FM-006	d_bind_res = 0 despite RLC pass	Yes	PHANTOM_RESONANCE

11.4 State Flag Registry#

Flag	Set by	Cleared by	Meaning
WINDOW_OPEN	PRIM:033	next tick	E's arrival phase is in window
WINDOW_CLOSED	PRIM:033	next tick	E's arrival phase is out of window
RESONANCE_LOCKED	PRIM:034	FM-004 trigger	Orbit is actively phase-locked
RESONANCE_LOST	FM-004	ρ(Φ) recovery	Lock broken; orbit downgraded

11.5 Wave 4 Status Tracker#

File	Status	PRIM range
f_Capture_Multi.md	✅ Complete	025–026
f_Capture_Cascade.md	✅ Complete	027–028
f_Capture_Soft.md	✅ Complete	029–030
f_Capture_Hard.md	✅ Complete	031–032
f_Capture_Resonant.md	✅ Complete	033–034
f_Capture_Asymmetric.md	⏳ Pending	035–036

Wave 4 is 5/6 complete.

11.6 Changelog#

## [1.0.0] — 2026-08-13
### Added
- Initial canonical release of f_Capture_Resonant.md
- PRIM:033 (eval_resonance_window): phase-gate guard with FM-001/002/003 routing
- PRIM:034 (lock_resonance): resonance-enhanced d_bind_res writer
- RLC-1 through RLC-5 conditions (conjunctive, INV-005 compliant)
- ω_res, T_res, φ_open, φ_close, φ_E, t_next_open operator family
- ρ_res_gain, ρ_eff, ρ_res_floor field enhancement family
- d_bind_res, T_orb_res, p_ratio, q_ratio binding/period family
- WINDOW_OPEN/WINDOW_CLOSED/RESONANCE_LOCKED/RESONANCE_LOST state flags
- FM-004 drift/recovery cycle for post-capture maintenance (RLC-4)
- FM-006 phantom resonance guard on degenerate eccentricity
- Four canonical examples: clean lock (2:1), window miss with retry,
  FM-004 drift + recovery, FM-006 phantom resonance
- Full INV-001–010 compliance table
- OPERATORS.md registration block for all 14 new symbols

11.7 Suggested Commit Message#

feat(FFF_Gravity): add f_Capture_Resonant.md — PRIM:033–034, Wave 4 file 5/6

Introduces resonance-locked capture variant for orbit_class=RESONANT.
Defines ω_res-gated capture windows (RLC-1), resonance gain (ρ_res_gain,
ρ_eff), enhanced binding depth (d_bind_res), and orbital period pinning
(T_orb_res = p/q × T_res). PRIM:033 evaluates phase gate with FM-001/002/003
routing and t_next_open retry computation. PRIM:034 writes RESONANCE_LOCKED
state with stab_class=STABLE forcing. FM-004 drift/recovery cycle defined
for post-capture ρ(Φ) maintenance (RLC-4). FM-006 phantom resonance guard
added for degenerate eccentricity case. Wave 4 now 5/6 complete; running
primitive total: PRIM:034.

f_Capture_Resonant.md — FFF_Gravity v1.0.0 — Session SES-20260813-FGRAV-033 Wave 4 file 5/6 — PRIM:033–034 — 2026-08-13

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