f_Deflect — Heading Deflection Primitive

F_force engineering primitive.
Modifies the direction of v_approach via angular redirect.
Magnitude of v_approach is invariant across deflection.
Resolves the heading_delta pending stub declared in f_Force.md §4.3.

§0 Session Context#

§0.1 Purpose of This Session#

This session produces the canonical f_Deflect.md. It is the last Wave 3 file and the last engineering primitive of the FFF_Gravity module's F_force node. Its central duty is to resolve the heading_delta pending stub referenced in f_Force.md §4.3, formalize the deflection geometry, and close Wave 3 by recording the completion milestone in §10.

§0.2 Stub Resolution Record#

f_Force.md §4.3 declared:
  heading_delta  [PENDING: defined in f_Deflect.md]

Resolution:
  This file (f_Deflect.md §4) freezes heading_delta.
  Freeze date: 2026-08-13
  Session: SES-20260813-DEFLECT-001

§1 Module Identity#

§1.1 Formal Signature#

f_Deflect(v_approach, heading_delta, r_deflect) → new_v_approach_heading

§1.2 Triadic Position#

G = F_freq · F_fluid · F_force
                         ↑
                    f_Deflect lives here.
                    It is a geometry-layer primitive
                    operating strictly within the F_force node.

§1.3 Companion Primitive Table (F_force Node)#

§2 Canonical Description#

§2.1 What f_Deflect IS#

f_Deflect is the heading-modification primitive of the F_force node. It takes an existing v_approach vector and rotates its direction by heading_delta radians while preserving the vector's scalar magnitude. The resulting new_v_approach_heading is then available to subsequent F_force evaluations, subject to Deflect Conditions (§5) and FM guards (§6).

Deflection is a geometric operation, not an energetic one in the orbital sense. However, applying a redirect to a force node consumes deflect_cost units of binding budget — this cost is computed by PRIM:024 and is audited against the system's binding state before PRIM:023 executes.

§2.2 What f_Deflect IS NOT#

§2.3 Design Motivation#

The F_force node defines v_approach as the magnitude of approach — but direction matters for capture geometry. Without heading control, the only orbital outcome is determined by approach angle at genesis. f_Deflect introduces the angular engineering layer: the ability to steer approach trajectories before capture evaluation commits. This is the F_force analogue of f_Amplify's fluid-layer engineering — both are cost-bearing interventions that expand the outcome envelope without breaking triadic invariants.

§2.4 Operating Modes#

§2.5 Relationship to Capture Pipeline#

Genesis of E approaching A
        ↓
  f_Force: compute v_approach
        ↓
  f_Deflect: redirect heading [OPTIONAL — this file]
        ↓
  f_Frame: register_capture slot check
        ↓
  f_Capture: final capture evaluation
        ↓
  Orbital states → f_Orbit, f_Decay, f_Orbit, f_Release, f_Collapse

f_Deflect is the last engineering intervention point before capture locks. Once f_Capture transitions state to CAPTURED, deflection is no longer applicable.

§3 Triadic Equation#

§3.1 Formal Operator Signature#

f_Deflect : (v_approach: vector, heading_delta: float, r_deflect: float)
          → new_v_approach_heading: vector

Constraint: |new_v_approach_heading| = |v_approach|  [magnitude invariant]

§3.2 Node Decomposition#

F_force(deflect context):

  v_approach_direction  := normalize(v_approach)
  new_direction         := rotate(v_approach_direction, heading_delta)
  new_v_approach_heading := new_direction * |v_approach|

  Where:
    rotate(d, δ)  → rotates unit vector d by δ radians
    |v_approach|  → Euclidean norm of v_approach; invariant
    heading_delta → angular deviation; domain ℝ; range (-π, π]

§3.3 G-Equation Role#

G = F_freq · F_fluid · F_force

F_force contribution in deflect context:

  F_force_deflect = redirect_force_node(
      current_heading = normalize(v_approach),
      target_heading  = rotate(normalize(v_approach), heading_delta),
      delta           = heading_delta
  ) → new_v_approach_heading

  This new heading is injected back into v_approach before
  f_Capture evaluates SC-1 (Approach Bound).
  F_freq and F_fluid are read during cost computation (§4)
  but their values are not modified by f_Deflect.

§3.4 Magnitude Invariant Proof (Informal)#

Let v := v_approach, |v| = m (scalar magnitude).
Let d := v / m (unit direction vector).
Let d' := rotate(d, δ).
  Since rotate preserves unit length: |d'| = 1.
Let v' := d' * m.
  |v'| = |d'| * m = 1 * m = m.
Therefore |new_v_approach_heading| = |v_approach|. ∎

§4 Operator Registry#

§4.1 Operators Introduced (Frozen Here)#

heading_delta#

Formal definition:

heading_delta (δ): ℝ → (-π, π]

  The signed angular deviation, in radians, applied to
  the current approach heading vector v_approach by
  redirect_force_node (PRIM:023).

  δ = 0.0   → no rotation; NULL_DEFLECT mode
  δ > 0.0   → clockwise rotation (in approach plane frame)
  δ < 0.0   → counter-clockwise rotation
  |δ| > π   → forbidden; raises DOMAIN_VIOLATION

r_deflect#

Formal definition:

r_deflect: (0, r_capture) → ℝ+

  The distance from attractor A at which redirect_force_node
  applies the heading rotation. Must be strictly inside the
  capture radius (r_deflect < r_capture) to affect orbital
  outcome. Values ≥ r_capture are outside deflect jurisdiction
  and raise FM-001 (Overshoot boundary condition).

  Used by compute_deflection_cost (PRIM:024) to scale cost
  with proximity — closer deflections are costlier.

deflect_cost#

Formal definition:

deflect_cost: (|δ|, r_deflect, β) → ℝ+

  deflect_cost = (|δ| / π) * (r_capture / r_deflect) * β

  Components:
    |δ| / π         → normalized angular effort (0 to 1)
    r_capture / r_deflect → proximity amplifier (> 1 when close)
    β               → current binding coefficient (F_fluid)

  Property:
    deflect_cost = 0 when δ = 0 (NULL_DEFLECT; zero cost)
    deflect_cost → ∞ as r_deflect → 0 (singularity guard applied)

§4.2 Operators Inherited (Read-Only in This File)#

§4.3 State Flags#

§5 Deflect Conditions#

All four Deflect Conditions are conjunctive (AND).
A single failure blocks deflection and triggers the appropriate FM guard.

DC-1: Approach Live#

Condition:  capture_state(E, A) ∉ {CAPTURED, RELEASED, COLLAPSED}
Rationale:  Deflection is pre-capture geometry. Once capture commits,
            v_approach is no longer an active quantity.
Guard:      If violated → DEFLECT_BLOCKED; operation not applied.

DC-2: Field Present#

Condition:  ρ(Φ) > 0
Rationale:  Deflection requires an active frequency field to define
            the approach plane geometry. Null field (ρ = 0) raises FM-002
            upstream; deflect inherits that guard.
Guard:      If violated → DEFLECT_BLOCKED; FM-002 already active upstream.

DC-3: Binding Floor#

Condition:  β ≥ 1.0
Rationale:  SC-4 must hold. A deflection attempted under β < 1.0 constitutes
            a phantom redirect into a non-binding medium. Raises FM-006.
Guard:      If violated → FM-006 (Phantom Capture); DEFLECT_BLOCKED.

DC-4: Post-Deflect Approach Bound#

Condition:  |new_v_approach_heading| < v_escape(A)
            i.e., v_approach_magnitude < v_escape(A)  [magnitude invariant applies]
Rationale:  Deflection does not change magnitude, so this condition reduces
            to confirming SC-1 still holds on the un-deflected magnitude.
            If SC-1 already fails before deflect, FM-001 is raised.
Guard:      If violated → FM-001 (Overshoot); DEFLECT_BLOCKED.

§5.1 Deflect Condition Summary#

§6 Failure Modes#

f_Deflect introduces no new failure modes.
It activates guards for FM-001 and FM-006, both frozen in f_Force.md.

FM-001 — Overshoot (active in Deflect context)#

Symbol:     FM-001
Name:       Overshoot
Frozen in:  f_Force.md
Trigger:    v_approach ≥ v_escape(A) — magnitude already exceeds escape;
            deflection cannot recover orbital binding.
Effect:     redirect_force_node not invoked; DEFLECT_BLOCKED set.
            E continues on escape trajectory.
Recovery:   None within this approach. New approach required.

Deflect-specific note: Since deflection preserves magnitude, if v_approach ≥ v_escape(A) before deflect, it will be ≥ v_escape(A) after. FM-001 fires at DC-4 evaluation before PRIM:023 is invoked.

FM-006 — Phantom Capture (active in Deflect context)#

Symbol:     FM-006
Name:       Phantom Capture
Frozen in:  f_Force.md
Trigger:    β < 1.0 at deflect invocation time.
Effect:     Deflection into a sub-binding medium produces a phantom
            heading change with no orbital anchoring. redirect_force_node
            not invoked; DEFLECT_BLOCKED set.
Recovery:   β must be restored to ≥ 1.0 (via f_Amplify or f_Emit
            upstream intervention) before deflect is re-attempted.

§6.1 FM Coverage Matrix#

§7 Engineering Primitives#

PRIM:023 — redirect_force_node (Impure)#

import math
from typing import NamedTuple
 
 
class Vector2D(NamedTuple):
    """Minimal 2D vector for approach-plane deflection geometry."""
    x: float
    y: float
 
    def norm(self) -> float:
        """Euclidean magnitude."""
        return math.sqrt(self.x**2 + self.y**2)
 
    def normalize(self) -> "Vector2D":
        """Return unit vector; raises if zero vector."""
        m = self.norm()
        if m == 0.0:
            raise ValueError("Cannot normalize zero vector — undefined heading.")
        return Vector2D(self.x / m, self.y / m)
 
    def rotate(self, radians: float) -> "Vector2D":
        """Rotate this vector by `radians` (positive = clockwise in approach plane)."""
        cos_r = math.cos(radians)
        sin_r = math.sin(radians)
        return Vector2D(
            self.x * cos_r - self.y * sin_r,
            self.x * sin_r + self.y * cos_r,
        )
 
    def scale(self, factor: float) -> "Vector2D":
        """Scale by scalar factor."""
        return Vector2D(self.x * factor, self.y * factor)
 
 
class DeflectState(NamedTuple):
    """Mutable approach-heading state managed by the F_force node."""
    v_approach: Vector2D          # Full approach velocity vector
    capture_state: str            # 'APPROACHING' | 'CAPTURED' | 'RELEASED' | 'COLLAPSED'
    rho_phi: float                # ρ(Φ) — field density at Φ
    beta: float                   # β — binding coefficient
    v_escape: float               # v_escape(A) — scalar escape velocity
    r_capture: float              # capture radius from f_Frame.md
 
 
class DeflectResult(NamedTuple):
    """Output of redirect_force_node."""
    success: bool
    new_v_approach: Vector2D      # Redirected heading (same magnitude)
    heading_delta_applied: float  # Actual delta applied (0.0 if blocked)
    deflect_cost: float           # Cost deducted from binding budget
    mode: str                     # 'NOMINAL_DEFLECT' | 'NULL_DEFLECT' | 'DEFLECT_BLOCKED'
    failure_mode: str | None      # 'FM-001' | 'FM-006' | None
    message: str
 
 
def redirect_force_node(
    state: DeflectState,
    heading_delta: float,
    r_deflect: float,
) -> DeflectResult:
    """
    PRIM:023 — redirect_force_node (Impure)
    ========================================
    Modifies the direction of state.v_approach by heading_delta radians.
    Magnitude of v_approach is strictly preserved.
 
    Resolves the heading_delta pending stub from f_Force.md §4.3.
 
    Parameters
    ----------
    state : DeflectState
        Current F_force node state. Contains v_approach, capture_state,
        rho_phi, beta, v_escape, r_capture.
    heading_delta : float
        Angular deviation in radians. Domain: (-π, π].
        Positive = clockwise rotation in approach plane.
        0.0 → NULL_DEFLECT (no-op, zero cost).
    r_deflect : float
        Radial distance at which deflection is applied.
        Domain: (0, r_capture). Used in deflect_cost computation.
 
    Returns
    -------
    DeflectResult
        Contains redirected heading vector, cost, mode, and any FM raised.
 
    Failure Modes
    -------------
    FM-001  Raised when v_approach_magnitude ≥ v_escape (DC-4 violation).
    FM-006  Raised when β < 1.0 (DC-3 violation).
 
    Invariant
    ---------
    |new_v_approach| = |state.v_approach| — magnitude is never altered.
    """
    # ── DC-1: Approach Live ──────────────────────────────────────────────
    terminal_states = {"CAPTURED", "RELEASED", "COLLAPSED"}
    if state.capture_state in terminal_states:
        return DeflectResult(
            success=False,
            new_v_approach=state.v_approach,
            heading_delta_applied=0.0,
            deflect_cost=0.0,
            mode="DEFLECT_BLOCKED",
            failure_mode=None,
            message=(
                f"DC-1 FAILED: capture_state='{state.capture_state}' is terminal. "
                "Deflection not applicable post-capture."
            ),
        )
 
    # ── DC-2: Field Present ──────────────────────────────────────────────
    if state.rho_phi <= 0.0:
        return DeflectResult(
            success=False,
            new_v_approach=state.v_approach,
            heading_delta_applied=0.0,
            deflect_cost=0.0,
            mode="DEFLECT_BLOCKED",
            failure_mode=None,
            message=(
                f"DC-2 FAILED: ρ(Φ)={state.rho_phi:.4f} ≤ 0. "
                "FM-002 should be active upstream. Deflect blocked."
            ),
        )
 
    # ── DC-3: Binding Floor (FM-006 guard) ───────────────────────────────
    if state.beta < 1.0:
        return DeflectResult(
            success=False,
            new_v_approach=state.v_approach,
            heading_delta_applied=0.0,
            deflect_cost=0.0,
            mode="DEFLECT_BLOCKED",
            failure_mode="FM-006",
            message=(
                f"DC-3 FAILED: β={state.beta:.4f} < 1.0. "
                "FM-006 (Phantom Capture) — deflect into sub-binding medium blocked."
            ),
        )
 
    # ── domain validation: heading_delta ─────────────────────────────────
    if not (-math.pi < heading_delta <= math.pi):
        raise ValueError(
            f"heading_delta={heading_delta:.6f} rad out of domain (-π, π]. "
            "Caller must normalize angular input before invoking PRIM:023."
        )
 
    # ── domain validation: r_deflect ─────────────────────────────────────
    if r_deflect <= 0.0:
        raise ValueError(
            f"r_deflect={r_deflect} must be > 0. "
            "Singularity at r_deflect=0 — deflect_cost undefined."
        )
    if r_deflect >= state.r_capture:
        return DeflectResult(
            success=False,
            new_v_approach=state.v_approach,
            heading_delta_applied=0.0,
            deflect_cost=0.0,
            mode="DEFLECT_BLOCKED",
            failure_mode="FM-001",
            message=(
                f"r_deflect={r_deflect:.4f} ≥ r_capture={state.r_capture:.4f}. "
                "Deflection outside capture zone — FM-001 (Overshoot boundary)."
            ),
        )
 
    # ── NULL_DEFLECT: zero-angle no-op ───────────────────────────────────
    if heading_delta == 0.0:
        return DeflectResult(
            success=True,
            new_v_approach=state.v_approach,
            heading_delta_applied=0.0,
            deflect_cost=0.0,
            mode="NULL_DEFLECT",
            failure_mode=None,
            message="heading_delta=0.0 — NULL_DEFLECT mode; no heading change applied.",
        )
 
    # ── DC-4: Post-Deflect Approach Bound (FM-001 guard) ─────────────────
    v_magnitude = state.v_approach.norm()
    if v_magnitude >= state.v_escape:
        return DeflectResult(
            success=False,
            new_v_approach=state.v_approach,
            heading_delta_applied=0.0,
            deflect_cost=0.0,
            mode="DEFLECT_BLOCKED",
            failure_mode="FM-001",
            message=(
                f"DC-4 FAILED: v_approach_magnitude={v_magnitude:.4f} ≥ "
                f"v_escape={state.v_escape:.4f}. FM-001 (Overshoot) — "
                "deflection cannot restore orbital binding."
            ),
        )
 
    # ── PRIM:024: compute deflect_cost ───────────────────────────────────
    cost = compute_deflection_cost(
        heading_delta=heading_delta,
        r_deflect=r_deflect,
        r_capture=state.r_capture,
        beta=state.beta,
    )
 
    # ── NOMINAL_DEFLECT: apply heading rotation ───────────────────────────
    unit_dir = state.v_approach.normalize()
    rotated_dir = unit_dir.rotate(heading_delta)
    new_v_approach = rotated_dir.scale(v_magnitude)
 
    # Magnitude invariant assertion (defensive)
    new_magnitude = new_v_approach.norm()
    magnitude_error = abs(new_magnitude - v_magnitude)
    if magnitude_error > 1e-9:
        raise RuntimeError(
            f"MAGNITUDE INVARIANT VIOLATED: original={v_magnitude:.10f}, "
            f"new={new_magnitude:.10f}, error={magnitude_error:.2e}. "
            "This is a bug in redirect_force_node — report immediately."
        )
 
    return DeflectResult(
        success=True,
        new_v_approach=new_v_approach,
        heading_delta_applied=heading_delta,
        deflect_cost=cost,
        mode="NOMINAL_DEFLECT",
        failure_mode=None,
        message=(
            f"NOMINAL_DEFLECT: heading rotated by δ={heading_delta:.6f} rad "
            f"at r_deflect={r_deflect:.4f}. "
            f"deflect_cost={cost:.6f}. "
            f"|v_approach| preserved at {v_magnitude:.6f}."
        ),
    )

PRIM:024 — compute_deflection_cost (Pure)#

import math
 
 
def compute_deflection_cost(
    heading_delta: float,
    r_deflect: float,
    r_capture: float,
    beta: float,
) -> float:
    """
    PRIM:024 — compute_deflection_cost (Pure)
    ==========================================
    Computes the binding budget cost of one deflection operation.
 
    Formula
    -------
    deflect_cost = (|δ| / π) × (r_capture / r_deflect) × β
 
    Components:
      |δ| / π          — normalized angular effort in [0, 1]
      r_capture / r_deflect — proximity amplifier; increases as
                            deflection occurs closer to the attractor
      β                — binding coefficient scaling; heavier binding
                         makes heading changes more expensive
 
    Properties
    ----------
    - deflect_cost = 0.0 when heading_delta = 0.0 (null deflect)
    - deflect_cost is monotonically increasing in |heading_delta|
    - deflect_cost → ∞ as r_deflect → 0 (singularity; caller must guard)
    - deflect_cost is always ≥ 0.0
 
    Parameters
    ----------
    heading_delta : float
        Angular deviation in radians. Domain (-π, π].
    r_deflect : float
        Deflection radius. Must be > 0 and < r_capture.
    r_capture : float
        Capture radius from f_Frame.md. Must be > 0.
    beta : float
        Current binding coefficient β from f_Force.md. Must be ≥ 1.0
        (caller enforces DC-3 before invoking this function).
 
    Returns
    -------
    float
        deflect_cost ≥ 0.0
 
    Raises
    ------
    ValueError
        If r_deflect ≤ 0, r_capture ≤ 0, or r_deflect ≥ r_capture.
    """
    # Guard: singularity prevention
    if r_deflect <= 0.0:
        raise ValueError(
            f"r_deflect={r_deflect} must be > 0 — singularity undefined."
        )
    if r_capture <= 0.0:
        raise ValueError(
            f"r_capture={r_capture} must be > 0."
        )
    if r_deflect >= r_capture:
        raise ValueError(
            f"r_deflect={r_deflect} must be < r_capture={r_capture}. "
            "Deflection outside capture zone is meaningless."
        )
 
    # Normalized angular effort: |δ| / π ∈ [0, 1)
    angular_effort = abs(heading_delta) / math.pi
 
    # Proximity amplifier: r_capture / r_deflect > 1 (always, given r_deflect < r_capture)
    proximity_amp = r_capture / r_deflect
 
    # Binding scale: β ≥ 1.0 (DC-3 enforced by caller)
    cost = angular_effort * proximity_amp * beta
 
    return cost

§8 Canonical Examples#

Four worked examples cover: nominal deflect, null deflect, FM-001 guard, FM-006 guard.

Example 1 — Nominal Deflect: Shallow Correction#

Scenario: Entity E is approaching attractor A on a trajectory that will result in a fly-by. A small clockwise heading correction at mid-range brings E into capture geometry.

Parameters#

Deflect Condition Trace#

Computation Trace#

Step 1: angular_effort  = |0.2618| / π     = 0.08333
Step 2: proximity_amp   = 10.0 / 6.0       = 1.6667
Step 3: deflect_cost    = 0.08333 × 1.6667 × 1.3 = 0.1806

Step 4: unit_dir        = (3.5, 0.0) / 3.5 = (1.0, 0.0)
Step 5: rotated_dir     = rotate((1.0, 0.0), +0.2618 rad)
                        = (cos(0.2618), sin(0.2618))
                        = (0.9659, 0.2588)
Step 6: new_v_approach  = (0.9659, 0.2588) × 3.5
                        = (3.3807, 0.9058)

Step 7: magnitude check = √(3.3807² + 0.9058²)
                        = √(11.4291 + 0.8205)
                        = √12.2496 ≈ 3.5000  ✅ invariant holds

Post-State Analysis#

Example 2 — Null Deflect: Zero-Angle Pass-Through#

Scenario: f_Deflect is called with δ = 0.0 as a no-op pipeline pass-through. No heading change; no cost.

Parameters#

Computation Trace#

heading_delta = 0.0 → NULL_DEFLECT mode triggered immediately.
deflect_cost  = (0.0 / π) × (r_capture / r_deflect) × β = 0.0
new_v_approach = v_approach (unchanged) = (2.1, 1.4)

Post-State Analysis#

Example 3 — FM-001 Guard: Overshoot Blocked#

Scenario: Entity E is already on an escape trajectory (v_approach ≥ v_escape). Deflection is requested but cannot restore binding — FM-001 fires.

Parameters#

Deflect Condition Trace#

Post-State Analysis#

Key insight: Deflection preserves magnitude. If E is already escaping, no angular redirect can change that — the magnitude invariant makes heading correction powerless against a speed violation.

Example 4 — FM-006 Guard: Phantom Binding Blocked#

Scenario: The F_fluid node has degraded below β = 1.0 (binding floor violated). Deflection is attempted but enters a phantom binding state — FM-006 fires.

Parameters#

Deflect Condition Trace#

Post-State Analysis#

Recovery path: Invoke f_Amplify to restore β ≥ 1.0, then re-attempt deflect. Alternatively, f_Emit may restore ρ(Φ) gradient effects that indirectly support binding recovery.

§9 Cross-Module References#

§9.1 Dependency Table#

§9.2 OPERATORS.md Updates Required#

The following additions must be made to OPERATORS.md when this file is published:

§F_force — New Operators (f_Deflect.md canonical, 2026-08-13):

  heading_delta   δ   float   (-π, π]   Angular deviation (radians); +CW; frozen
  r_deflect           float   (0, r_capture)  Deflection radius; frozen
  deflect_cost        float   [0, ∞)    Binding budget consumed by deflect; frozen

§Primitives — New Entries:
  PRIM:023  redirect_force_node    Impure  F_force  f_Deflect.md
  PRIM:024  compute_deflection_cost Pure   F_force  f_Deflect.md

§Stub Resolution:
  heading_delta — previously marked [PENDING: f_Deflect.md] in f_Force.md §4.3.
  Now frozen. Remove pending annotation from OPERATORS.md §F_force stub entry.

§9.3 f_Force.md Annotation Update Required#

The following note must be appended to f_Force.md §4.3 when this file is published:

heading_delta — RESOLVED: see f_Deflect.md §4.1
Freeze date: 2026-08-13 | Session: SES-20260813-DEFLECT-001

§10 Document Metadata#

§10.1 INV Compliance Table#

§10.2 Primitive Registry (This File)#

§10.3 Operator Registry (This File)#

§10.4 Failure Mode Registry (This File)#

§10.5 ██ WAVE 3 COMPLETION MILESTONE ██#

╔══════════════════════════════════════════════════════════════════════╗
║              FFF_Gravity Module — WAVE 3 COMPLETE                   ║
║              All 8 Wave 3 files are now canonical.                  ║
╠══════════════════════════════════════════════════════════════════════╣
║                                                                      ║
║  Date:     2026-08-13                                                ║
║  Session:  SES-20260813-DEFLECT-001                                  ║
║  Author:   umaywant2                                                 ║
║                                                                      ║
╠═══════════════════════════╦══════════════════╦══════════════════════╣
║  File                     ║  Node            ║  Status              ║
╠═══════════════════════════╬══════════════════╬══════════════════════╣
║  f_Release.md             ║  F_force         ║  ✅ CANONICAL         ║
║  f_Decay.md               ║  F_fluid/F_freq  ║  ✅ CANONICAL         ║
║  f_Orbit.md               ║  F_freq          ║  ✅ CANONICAL         ║
║  f_Collapse.md            ║  F_force/F_fluid ║  ✅ CANONICAL         ║
║  f_Emit.md                ║  F_freq          ║  ✅ CANONICAL         ║
║  f_Dampen.md              ║  F_freq          ║  ✅ CANONICAL         ║
║  f_Amplify.md             ║  F_fluid         ║  ✅ CANONICAL         ║
║  f_Deflect.md             ║  F_force         ║  ✅ CANONICAL         ║
╠═══════════════════════════╩══════════════════╩══════════════════════╣
║                                                                      ║
║  PRIMITIVE REGISTRY COMPLETE: PRIM:001 — PRIM:024  (all frozen)     ║
║  FAILURE MODE REGISTRY COMPLETE: FM-001 — FM-010   (all frozen)     ║
║  OPERATOR REGISTRY: all symbols frozen; stub resolved               ║
║                                                                      ║
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║              WAVE 4 — FULLY UNLOCKED                                 ║
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§10.6 Wave 4 Unlock Manifest#

Wave 4 contains the capture variant files — specialized f_Capture extensions for non-standard attractor/entity configurations. All 6 files are now unlocked.

§10.7 Full Module Wave Completion Status#

§10.8 Changelog Entry#

- version: 1.0.0
  date: 2026-08-13
  session: SES-20260813-DEFLECT-001
  author: umaywant2
  changes:
    - Initial canonical publication of f_Deflect.md
    - Freezes heading_delta operator (resolves f_Force.md §4.3 pending stub)
    - Freezes r_deflect operator
    - Freezes deflect_cost operator and formula
    - Introduces PRIM:023 redirect_force_node (Impure, F_force)
    - Introduces PRIM:024 compute_deflection_cost (Pure, F_force)
    - Defines 4 Deflect Conditions (DC-1 through DC-4, conjunctive)
    - Activates FM-001 and FM-006 guards in deflect context
    - Establishes NULL_DEFLECT, NOMINAL_DEFLECT, DEFLECT_BLOCKED modes
    - Provides 4 canonical worked examples
    - Records Wave 3 Completion Milestone — all 8 files canonical
    - Unlocks Wave 4 manifest (6 capture variant files)

§10.9 Suggested Commit Message#

docs(FFF_Gravity): publish canonical f_Deflect.md — Wave 3 complete

- Introduces heading_delta, r_deflect, deflect_cost operators (frozen)
- Resolves heading_delta pending stub from f_Force.md §4.3
- PRIM:023 redirect_force_node (Impure) — modifies v_approach direction
- PRIM:024 compute_deflection_cost (Pure) — cost = (|δ|/π)(r_c/r_d)(β)
- Magnitude invariant enforced: |new_v_approach| = |v_approach|
- FM-001 and FM-006 guards active via DC-3 and DC-4
- 4 Deflect Conditions (conjunctive); 4 canonical examples
- Wave 3 Completion Milestone: all 8 files canonical
- Wave 4 (capture variants) fully unlocked — 6 files pending
- Primitive registry now complete: PRIM:001–PRIM:024 all frozen
- Failure mode registry now complete: FM-001–FM-010 all frozen

Refs: SES-20260813-DEFLECT-001
Closes: Wave 3

End of f_Deflect.md — FFF_Gravity Module, Wave 3, File 8 of 8.
All Wave 3 primitives frozen. Wave 4 unlocked.