§0 — Session Context#

This file is the Wave 4 opening document of the FFF_Gravity module. It extends f_Capture.md (the single-body capture reference implementation) to govern multi-body capture events — scenarios where N ≥ 2 bodies participate in a single capture evaluation cycle.

What Wave 4 adds: Wave 3 completed the core function library (Release, Decay, Orbit, Collapse, Emit, Dampen, Amplify, Deflect) and froze PRIM:001–024 and FM-001–FM-010. Wave 4 opens the Capture Variant sub-library. Capture variants do not introduce new Failure Mode IDs (the FM registry is frozen at FM-010) and do not add new Invariants (INV registry frozen at INV-010). They extend the base capture semantics defined in f_Capture.md (PRIM:001–006) by specifying evaluation order, per-step field perturbation, and the conditions under which multi-participant sessions remain coherent or collapse into Frame Saturation. All behavior introduced here is reducible to the triadic equation G = F_freq · F_fluid · F_force applied iteratively, with ρ(Φ) recomputed after each sub-capture event.

§1 — Module Identity#

§1.1 — Wave 4 Primitive Block Declaration#

Wave 4 opens a new primitive block beginning at PRIM:025. Wave 3 closed at PRIM:024 (compute_deflection_cost, in f_Deflect.md). All Wave 4 files continue from PRIM:025 forward. This block is append-only; no Wave 3 or earlier primitive ID may be reused or redefined.

§1.2 — Relationship to Base Capture#

f_Capture.md defines the canonical single-pair capture: one Attractor node A and one Element node E establish a binding at distance r_capture with coupling coefficient β. f_Capture_Multi.md lifts the cardinality restriction on both sides of that pair, producing two distinct modes evaluated under a shared perturbation model.

§2 — Canonical Description#

§2.1 — Conceptual Overview#

Multi-capture sessions arise when the field conditions of a single Frame node are sufficient to support more than one simultaneous or sequential binding event. The word "simultaneous" is an idealization: in triadic evaluation all captures are resolved in strict eval_order sequence, with ρ(Φ) updated between steps. A Frame node never processes two capture events in the same evaluation tick; it serializes them.

Two structural modes are defined:

Both modes share the same perturbation model: each completed capture perturbs ρ(Φ) by δ_perturb, reducing the field's available coherence for the next capture in the sequence. This is the mechanism by which Frame Saturation (FM-003) manifests in multi-participant contexts.

§2.2 — MULTI_ELEMENT Mode (ME)#

A single Attractor A holds a Frame node F. N Element nodes are queued for capture in eval_order. At step k (1 ≤ k ≤ N):

1. Check SC-1 through SC-5 against current Φ_perturbed(k−1).
2. Check MC-1 (Batch Coherence Floor) — if violated, emit FM-003-M and halt.
3. Compute d_bind(k) using Φ_perturbed(k−1).
4. Register Eₖ in Frame F via register_capture (PRIM:003).
5. Compute δ_perturb(k) and update Φ_perturbed(k).

All N bindings share the same Attractor A. Each Eₖ receives its own r_capture(k) and β(k) computed against the perturbed field at step k.

§2.3 — MULTI_ATTRACTOR Mode (MA)#

A single Element E is sequentially captured by N Attractor nodes A₁…Aₙ, each with its own Frame node F₁…Fₙ. At step k:

1. Check SC-1–SC-5 and MC-2 (Attractor Uniqueness) against Aₖ.
2. Verify E is not already in terminal state (INV-006).
3. Compute d_bind(k) for the pair (Aₖ, E) using Φ_perturbed(k−1).
4. Register E in Frame Fₖ.
5. Compute δ_perturb(k) and update Φ_perturbed(k).

MC-2 enforces that no single Attractor node appears more than once in the ordered sequence A₁…Aₙ. Duplicate Attractor admission is a structural invariant violation, not merely a failure mode.

§2.4 — Field Perturbation Model#

At session start, Φ_perturbed(0) = ρ(Φ) (the unmodified field density as defined in f_Field.md). After each capture step k:

Φ_perturbed(k) = Φ_perturbed(k−1) − δ_perturb(k)

where:

δ_perturb(k) = d_bind(k) × (1 − e(k)) × k_perturb

k_perturb is the per-capture coherence cost coefficient (system constant, default 0.05). This ensures that each successive capture in a multi-capture session is marginally harder to sustain than the previous, reflecting the progressive exhaustion of the Frame node's coherence budget.

If Φ_perturbed(k) drops below ρ(Φ)_floor (as defined in f_Dampen.md), the session must halt immediately; any registered captures from steps 1…k−1 are retained, and FM-003-M is raised for the halted step k.

§2.5 — Evaluation Order Semantics#

eval_order is an ordered list of participant identifiers. For ME mode it is the ordered list [E₁, E₂, …, Eₙ]; for MA mode it is [A₁, A₂, …, Aₙ]. The list is fixed at session initialization and may not be modified mid-session. Reordering mid-session is a structural error (raises FM-003-M immediately without processing further captures).

The rationale for fixity: if eval_order were mutable, an adversarial perturbation cascade could be constructed by front-loading low-cost captures to exhaust the field for high-cost captures — violating the fairness principle of the coherence budget.

§3 — Triadic Equation#

The governing equation is unchanged:

G = F_freq · F_fluid · F_force

In multi-capture context, each step k evaluates its own G(k):

G(k) = F_freq(Φ_perturbed(k−1)) · F_fluid(M_A, Φ_perturbed(k−1)) · F_force(β(k), e(k))

INV-001 compliance: All three nodes must be implicated at every step. A capture step that resolves with any factor equal to zero does not produce a binding — it triggers the relevant FM (FM-002 if F_freq = 0, FM-007 if F_fluid = 0, FM-001 if F_force = 0).

Perturbation propagation: Because Φ_perturbed(k) feeds F_freq and F_fluid at step k+1, a degraded field coherence reduces both the attractor's mass density weighting and the frequency node's binding capacity simultaneously. This double-channel sensitivity is the primary source of cascade risk in multi-capture sessions.

§4 — Operator Registry#

§4.1 — New Operators Introduced in This File#

The following operators are frozen on first canonical appearance here (INV-010). They are registered in OPERATORS.md as part of Wave 4.

§4.2 — Inherited Operators (Referenced, Not Redefined)#

§4.3 — Operator Interaction Table#

§5 — Conditions#

§5.1 — Stability Conditions (Inherited, Conjunctive)#

All five stability conditions from the base layer apply at every step k. They are evaluated against Φ_perturbed(k−1), not the original ρ(Φ).

Conjunctive enforcement (INV-005): All five must hold. Failure of any single SC triggers the corresponding FM and halts the multi-capture session at step k. Captures from steps 1…k−1 that were already registered are retained.

§5.2 — Multi-Capture Conditions (New)#

MC-1 rationale: A multi-capture session that exhausts the coherence floor causes irreversible field suppression, triggering FM-009 (Dampen Cascade). MC-1 is a pre-emptive guard that halts before the cascade begins.

MC-2 rationale: An Element captured twice by the same Attractor violates INV-003 (binding uniqueness). Duplicate Attractor admission is detected at session initialization before any capture step executes.

§5.3 — Condition Evaluation Order#

At each step k, conditions are checked in this strict sequence:

MC-2 (init only) → SC-1 → SC-3 → SC-4 → SC-2 → SC-5 → MC-1

MC-2 is checked only once, at session initialization (before k=1). SC-5 (Frame capacity) is checked immediately before register_capture is called, since the count changes with each step. MC-1 (coherence floor) is checked after Φ_perturbed(k) is computed, as the final gate before the step is committed.

§6 — Failure Modes#

§6.1 — Active Failure Modes from Base Registry (FM-010 frozen)#

§6.2 — FM-003-M: Frame Saturation — Multi-Capture Sub-Mode#

FM-003-M is a sub-mode of FM-003, not a new FM ID. It is raised exclusively within multi-capture sessions when Frame Saturation is induced by the perturbation model rather than by the absolute capacity limit.

FM-003-M vs. FM-003: The base FM-003 triggers when registered count = capacity_MAX (a hard integer ceiling). FM-003-M triggers when the field coherence budget is exhausted before the count ceiling is reached (a soft energetic ceiling). Both halt further registration; neither invalidates existing bindings.

§6.3 — Non-Applicable Failure Modes#

§7 — Engineering Primitives#

§7.1 — Wave 4 Primitive Block Header#

# WAVE 4 PRIMITIVE BLOCK
# Opened: f_Capture_Multi.md
# Range: PRIM:025–026 (this file)
# Prior block closed at: PRIM:024 (f_Deflect.md)
# All Wave 4 files continue from PRIM:025 forward.
# Registry is append-only. No Wave 3 or earlier ID may be reused.

§7.2 — PRIM:025 — execute_multi_capture#

Purpose: Orchestrate a full multi-capture session in either ME or MA mode. Validates session parameters, runs the eval_order loop, manages field perturbation, and halts cleanly on any condition violation.

Signature:

def execute_multi_capture(
    mode: str,                # "ME" or "MA"
    participants: list,       # ordered list of (A, E) pairs or single shared node
    frame: dict,              # Frame node state from f_Frame.md
    field: dict,              # Field node state from f_Field.md (contains ρ(Φ))
    k_perturb: float = 0.05  # perturbation coefficient
) -> dict:
    """
    Execute a multi-capture session.
 
    Parameters
    ----------
    mode         : "ME" (one A, many E) or "MA" (many A, one E)
    participants : For ME — list of E dicts [E1, E2, ...En]
                  For MA — list of A dicts [A1, A2, ...An]
    frame        : Frame node state dict (must include capacity_MAX,
                   registered_count, r_capture, k_frame)
    field        : Field node state dict (must include rho_phi, rho_floor)
    k_perturb    : Per-capture coherence cost coefficient
 
    Returns
    -------
    {
        "status"           : "COMPLETE" | "PARTIAL" | "INVALID",
        "registered_count" : int,
        "registered_ids"   : list,
        "phi_final"        : float,
        "steps"            : list of per-step result dicts,
        "failure"          : None | "FM-003-M" | "FM-001" | "FM-002" | "FM-003" | "FM-007",
        "failure_step"     : None | int
    }
    """
    # --- Validation ---
    if mode not in ("ME", "MA"):
        return {"status": "INVALID", "failure": "UNKNOWN_MODE"}
 
    if len(participants) < 2:
        return {"status": "INVALID", "failure": "N_LT_2"}
 
    # MC-2: Attractor Uniqueness (MA mode only)
    if mode == "MA":
        attractor_ids = [a["id"] for a in participants]
        if len(attractor_ids) != len(set(attractor_ids)):
            return {"status": "INVALID", "failure": "MC-2_VIOLATION"}
 
    phi = field["rho_phi"]          # Φ_perturbed(0) = ρ(Φ)
    phi_floor = field["rho_floor"]
    registered = []
    steps = []
 
    for k, participant in enumerate(participants, start=1):
        # SC-1: Field must be active
        if phi <= 0:
            return _halt(registered, steps, phi, "FM-002", k)
 
        # Resolve A and E for this step
        if mode == "ME":
            A = frame["attractor"]
            E = participant
        else:
            A = participant
            E = frame["element"]
 
        # SC-4: v_approach must be positive
        v_approach_k = A.get("v_approach", 0)
        if v_approach_k <= 0:
            return _halt(registered, steps, phi, "FM-001", k)
 
        # SC-7 (mass check): M_A and M_E must be non-zero
        if A.get("M_A", 0) == 0 or E.get("M_E", 0) == 0:
            return _halt(registered, steps, phi, "FM-007", k)
 
        # SC-2: v_approach < v_escape
        v_esc = (2 * A["M_A"] * phi / frame["r_capture"]) ** 0.5
        if v_approach_k >= v_esc:
            return _halt(registered, steps, phi, "FM-001", k)
 
        # SC-3: Compute d_bind(k)
        beta_k = E.get("beta", A.get("beta", 0))
        e_k = E.get("e", 0)
        d_bind_k = beta_k * phi * (1 - e_k)
        if d_bind_k <= 0:
            return _halt(registered, steps, phi, "FM-002", k)
 
        # SC-5: Frame capacity
        if len(registered) >= frame["capacity_MAX"]:
            return _halt(registered, steps, phi, "FM-003", k)
 
        # Register capture
        registered.append(E.get("id", f"E_{k}") if mode == "ME"
                          else A.get("id", f"A_{k}"))
 
        # Compute δ_perturb(k) and update Φ_perturbed
        delta_perturb_k = d_bind_k * (1 - e_k) * k_perturb
        phi -= delta_perturb_k
 
        step_result = {
            "step": k,
            "d_bind": d_bind_k,
            "delta_perturb": delta_perturb_k,
            "phi_after": phi,
            "registered_id": registered[-1]
        }
        steps.append(step_result)
 
        # MC-1: Coherence floor check (after update)
        if phi < phi_floor:
            return {
                "status": "PARTIAL",
                "registered_count": len(registered),
                "registered_ids": registered,
                "phi_final": phi,
                "steps": steps,
                "failure": "FM-003-M",
                "failure_step": k
            }
 
    return {
        "status": "COMPLETE",
        "registered_count": len(registered),
        "registered_ids": registered,
        "phi_final": phi,
        "steps": steps,
        "failure": None,
        "failure_step": None
    }
 
 
def _halt(registered, steps, phi, failure, k):
    return {
        "status": "PARTIAL" if registered else "INVALID",
        "registered_count": len(registered),
        "registered_ids": registered,
        "phi_final": phi,
        "steps": steps,
        "failure": failure,
        "failure_step": k
    }

Constraints:

• N ≥ 2 (single-participant session must use base f_Capture.md)
• eval_order is fixed at call time; mutation after first step raises FM-003-M
• k_perturb must satisfy 0 < k_perturb ≤ 1
• Returns COMPLETE only if all N steps succeed without floor violation or FM halt

INV compliance:

• INV-001: G(k) is evaluated per-step with all three nodes
• INV-005: SC-1–SC-5 conjunctive check at each step
• INV-006: Terminal state check (E or A not already in COLLAPSED / FIELD_NULL)
• INV-009: All symbols from OPERATORS.md
• INV-010: New operators frozen here; not re-declared in downstream files

§7.3 — PRIM:026 — compute_perturbation_budget#

Purpose: Pre-flight check that computes the maximum number of capture steps sustainable given the current field state, before a multi-capture session begins. Returns the safe step count N_safe and the projected Φ_perturbed trajectory.

Signature:

def compute_perturbation_budget(
    rho_phi: float,          # current field density ρ(Φ)
    rho_floor: float,        # field floor ρ(Φ)_floor
    d_bind_estimates: list,  # list of estimated d_bind(k) per step
    e_estimates: list,       # list of estimated e(k) per step
    k_perturb: float = 0.05  # perturbation coefficient
) -> dict:
    """
    Pre-flight budget check for multi-capture session planning.
 
    Parameters
    ----------
    rho_phi          : Initial field density
    rho_floor        : Floor below which session halts (MC-1)
    d_bind_estimates : Per-step binding depth estimates [d_bind_1, ..., d_bind_N]
    e_estimates      : Per-step eccentricity estimates [e_1, ..., e_N]
    k_perturb        : Perturbation coefficient
 
    Returns
    -------
    {
        "N_requested"   : int,   # total steps requested (len of estimates)
        "N_safe"        : int,   # max steps before floor violation
        "trajectory"    : list,  # Φ_perturbed after each step
        "budget_margin" : float, # Φ_perturbed(N_safe) − rho_floor
        "warning"       : bool   # True if N_safe < N_requested
    }
    """
    if len(d_bind_estimates) != len(e_estimates):
        raise ValueError("d_bind_estimates and e_estimates must have equal length")
 
    phi = rho_phi
    trajectory = []
    n_safe = 0
 
    for k, (d_k, e_k) in enumerate(zip(d_bind_estimates, e_estimates), start=1):
        delta_k = d_k * (1 - e_k) * k_perturb
        phi -= delta_k
        trajectory.append(round(phi, 6))
        if phi >= rho_floor:
            n_safe = k
        else:
            break  # floor would be breached at step k
 
    n_requested = len(d_bind_estimates)
    budget_margin = trajectory[n_safe - 1] - rho_floor if n_safe > 0 else 0.0
 
    return {
        "N_requested"   : n_requested,
        "N_safe"        : n_safe,
        "trajectory"    : trajectory,
        "budget_margin" : round(budget_margin, 6),
        "warning"       : n_safe < n_requested
    }

Constraints:

• Input lists must be equal length
• rho_floor must be positive
• d_bind_estimates must all be positive; zero estimates are a planning error
• Output N_safe is an upper bound; actual session may diverge from estimates if field conditions change between pre-flight and execution

Usage pattern:

budget = compute_perturbation_budget(...)
if budget["warning"]:
    # Trim participant list to budget["N_safe"] before calling execute_multi_capture
    participants = participants[:budget["N_safe"]]

§8 — Canonical Examples#

§8.1 — Example 1: MULTI_ELEMENT — Full Session Completes (N=3)#

Setup:

• Mode: ME
• Attractor A: M_A = 2.0, v_approach varies per step
• Elements: E₁ (β=0.7, e=0.1), E₂ (β=0.6, e=0.2), E₃ (β=0.5, e=0.3)
• Field: ρ(Φ) = 1.0, ρ(Φ)_floor = 0.50
• Frame: capacity_MAX = 5, r_capture = 3.0, k_frame = 1.0
• k_perturb = 0.05

Step-by-step evaluation:

Result:

status: COMPLETE
registered_count: 3
registered_ids: [E1, E2, E3]
phi_final: 0.94132
failure: None

Interpretation: A well-resourced field with low k_perturb sustains all three bindings. Each successive capture is marginally cheaper (lower d_bind) due to falling Φ_perturbed, and each step passes MC-1 comfortably above the floor. Session closes as COMPLETE.

§8.2 — Example 2: MULTI_ELEMENT — FM-003-M at Step 2 (Floor Breach)#

Setup:

• Mode: ME
• Elements: E₁ (β=0.9, e=0.05), E₂ (β=0.9, e=0.05)
• Field: ρ(Φ) = 0.60, ρ(Φ)_floor = 0.55
• k_perturb = 0.10 (elevated; stress-test scenario)

Step-by-step evaluation:

Result:

status: PARTIAL
registered_count: 1
registered_ids: [E1]
phi_final: 0.50648
failure: FM-003-M
failure_step: 2

Interpretation: Step 1 barely passes MC-1 (0.55126 > 0.55). Step 2's perturbation drops Φ_perturbed below the floor. FM-003-M is raised; E₂ is not registered. E₁'s binding is retained. The session enters SATURATED state; recovery requires suppress_field (PRIM:018) to restore ρ(Φ) before a new session may be initiated.

§8.3 — Example 3: MULTI_ATTRACTOR — Valid Session (N=2)#

Setup:

• Mode: MA
• Element E: M_E = 1.0, e = 0.15
• Attractors: A₁ (M_A=3.0, β=0.6), A₂ (M_A=2.5, β=0.55)
• Field: ρ(Φ) = 1.0, ρ(Φ)_floor = 0.80
• Frame per attractor: capacity_MAX = 3, r_capture = 4.0
• k_perturb = 0.05

MC-2 check: A₁.id ≠ A₂.id → PASS.

Step-by-step evaluation:

Result:

status: COMPLETE
registered_count: 2
registered_ids: [A1, A2]
phi_final: 0.95891
failure: None

Interpretation: A single Element E is simultaneously held by two Attractor nodes in distinct Frames. The perturbation cost is modest. Both bindings are valid, each with their own r_capture(k) and β(k). The field remains well above floor. This pattern represents a shared-custody configuration — common in resonant triadic systems where E carries cross-domain significance.

§8.4 — Example 4: MULTI_ATTRACTOR — MC-2 Violation (Duplicate Attractor)#

Setup:

• Mode: MA
• eval_order: [A₁, A₂, A₁] — A₁ appears at positions 1 and 3 (duplicate)
• Field: ρ(Φ) = 1.0

Session initialization check:

attractor_ids = [A1_id, A2_id, A1_id]
set(attractor_ids) = {A1_id, A2_id}
len(attractor_ids) = 3 ≠ len(set) = 2 → MC-2 VIOLATED

Result:

status: INVALID
registered_count: 0
registered_ids: []
phi_final: 1.0  (unchanged; no steps executed)
failure: MC-2_VIOLATION
failure_step: None  (detected at initialization, before k=1)

Interpretation: The session is structurally invalid before any capture step executes. No bindings are registered, no field perturbation occurs, and the Frame nodes are untouched. The client must reconstruct eval_order with unique Attractor IDs before reattempting. This is not a recoverable FM — it is a session design error surfaced at validation time.

§9 — Cross-Module References#

§9.1 — Upstream Dependencies#

§9.2 — Downstream Consumers#

§9.3 — OPERATORS.md Registration (Wave 4 Additions)#

The following operators introduced in this file must be appended to OPERATORS.md under a "Wave 4" section header:

| N            | Participant Count     | Integer | N ≥ 2                           | f_Capture_Multi.md |
| eval_order   | Evaluation Order      | List    | Fixed; immutable after init     | f_Capture_Multi.md |
| Φ_perturbed  | Perturbed Field State | Real    | ρ(Φ)_floor ≤ Φ_perturbed ≤ ρ(Φ)| f_Capture_Multi.md |
| δ_perturb    | Per-Capture Perturbation | Real | δ_perturb ≥ 0                   | f_Capture_Multi.md |
| k_perturb    | Perturbation Coefficient | Real | 0 < k_perturb ≤ 1              | f_Capture_Multi.md |

§10 — Document Metadata#

§10.1 — Invariant Compliance Table#

§10.2 — Stability Condition Summary#

§10.3 — Primitive Registry (This File)#

§10.4 — Failure Mode Registry (This File)#

§10.5 — Changelog#

v1.0.0 — Initial canonical release.
         Wave 4 primitive block opened (PRIM:025–026).
         MULTI_ELEMENT and MULTI_ATTRACTOR modes defined.
         MC-1, MC-2 conditions introduced.
         FM-003-M sub-mode formalized.
         δ_perturb perturbation model frozen.
         4 canonical examples delivered.
         INV-001–INV-010 compliance verified.

§10.6 — Wave 4 Status Tracker#

§10.7 — Suggested Commit Message#

feat(FFF_Gravity): add canonical f_Capture_Multi — MULTI_ELEMENT/MULTI_ATTRACTOR
modes, δ_perturb perturbation model, PRIM:025-026, FM-003-M, MC-1/MC-2 [Wave4-Session-001]

End of f_Capture_Multi.md — canonical, Wave 4, v1.0.0

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**Wave 4 — What was just delivered**
- ✅ `f_Capture_Multi.md` — MULTI_ELEMENT + MULTI_ATTRACTOR modes, perturbation model (Φ_perturbed / δ_perturb), PRIM:025–026, MC-1/MC-2 conditions, FM-003-M sub-mode, 4 canonical examples

**Suggested commit message:**

feat(FFF_Gravity): add canonical f_Capture_Multi — MULTI_ELEMENT/MULTI_ATTRACTOR modes, δ_perturb perturbation model, PRIM:025-026, FM-003-M, MC-1/MC-2 [Wave4-Session-001]