Add Section 3.5: why the forward branch wins (directional bias)

Three mechanisms explain sword spark straightness: axial field
concentration at the leader tip, directional thermal pre-conditioning
of gas ahead, and cold-air confinement restricting lateral breakout.
Connects to the "too long ramp" regime where forward bias disappears
and branching returns. Prompted by external reviewer question.

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

context/branching-physics.md

@@ -173,6 +173,35 @@ At higher RF frequencies, the channel receives more heating cycles per unit time
 
 This is the physical basis for the 300-600 kHz frequency threshold for QCW sword sparks documented in [[qcw-operation]] [T3 — mechanism inferred; the frequency threshold itself is T2]. The frequency threshold is not about breakdown physics — it is about whether the thermal competition can resolve fast enough to suppress branching during the QCW ramp. See [[thermal-physics]] Section 7 for community observations.
 
+### 3.5 Why the Forward Branch Wins: Directional Bias
+
+The current-hogging instability (Section 3.1) explains why **one** branch wins, but not why the winner is consistently the **forward-pointing** branch — the one continuing along the parent leader's axis. Three mechanisms converge to bias the competition in favor of the forward direction, explaining why QCW sword sparks grow straight rather than random-walking.
+
+**Mechanism 1: Axial field concentration** [T0]
+
+The leader is a long, thin conductor protruding from the topload into a background electric field. The field at its tip is concentrated axially — field lines converge at the tip and point outward along the channel axis. The tip enhancement factor (kappa ~ 2-5, see [[field-thresholds]] Section 2) is highest in the forward direction. A branch propagating forward enters the region of strongest driving field; a branch going sideways sees a much weaker field because it propagates perpendicular to the field lines.
+
+Since streamer velocity is proportional to tip potential [T1, Bazelyan & Raizer 2000, Ch 2, Eq. 2.6], the forward branch propagates faster, draws more current from the outset, and enters the current-hogging positive feedback loop with an initial advantage. Even a small initial current asymmetry is sufficient — the b = 1.84 exponent amplifies it exponentially (Section 3.2).
+
+**Mechanism 2: Directional thermal pre-conditioning** [T3]
+
+The gas immediately ahead of the leader tip is pre-conditioned by four mechanisms documented in [[field-thresholds]] Section 4.7:
+
+- **UV photoionization** from the leader corona creates seed electrons ahead of the tip, densest along the forward axis where the corona is most intense
+- **Thermal pre-conditioning**: heat conducts and convects forward from the 5,000-20,000 K leader trunk, warming gas ahead to 600-1,000 K and reducing its density
+- **Residual ionization** from previous streamer bursts is concentrated along the prior propagation axis
+- **Gas expansion** from rapid channel heating pushes lowest-density gas forward via shock/pressure waves
+
+All four mechanisms are strongest directly ahead of the tip and decay rapidly off-axis. A forward branch propagates into warm, pre-ionized, rarefied gas with a reduced effective E_propagation. A lateral branch must propagate into cold, un-ionized, full-density air requiring the full cold-air E_propagation (~0.5 MV/m). The forward branch requires less field to sustain propagation — so it grows faster — so it wins.
+
+**Mechanism 3: Cold-air confinement** [T1]
+
+Bazelyan describes the confinement explicitly for lightning leaders: the dense cold air surrounding the hot leader channel restricts radial expansion because it acts as a high-breakdown-threshold wall [Bazelyan & Raizer 2000, Ch 5, p. 271]. The E/N ratio (which controls ionization rate) drops abruptly at the channel boundary where the temperature transitions from thousands of Kelvin to ambient. Lateral streamer propagation must overcome this barrier; forward propagation does not.
+
+**Combined effect:** The forward direction offers the path of least resistance in three independent senses — strongest driving field (electrostatics), lowest propagation threshold (thermal pre-conditioning), and least opposition from surrounding gas (cold-air confinement). The current-hogging instability then amplifies this directional bias into winner-take-all within ~120-200 us. Each successive generation of streamer competition at the advancing leader tip is similarly biased, producing a straight sword spark.
+
+**Connection to "too long" ramp regime:** This directional bias explains why lateral breakouts occur when the QCW ramp exceeds ~25 ms (Section 4.3). Once the leader stalls because E_tip < E_propagation in the forward direction, the thermal pre-conditioning advantage disappears — the tip is no longer advancing, so no fresh pre-conditioned zone forms ahead. The superheated trunk, now at peak temperature, makes lateral breakout competitive with (stalled) forward growth. The forward bias switches off, branching returns, and the spark becomes "hot, fat, and bushy."
+
 ## 4. Branching Regimes in Tesla Coil Operation
 
 ### 4.1 Burst Mode: Branching Dominates