LAYER 2 — GOAL DECOMPOSITION & ACTIVATION ANALYSIS
Source: Layer 0 (goal spec, chromatic assignments, control relations, los table)
        Layer 1 (domain partition, beam crossing tables, chain interference tables)

Purpose: Trace backward from each goal literal through the activation derivation
chain to determine what base-relation configurations (loc, paired) are required.
Establish gate1's state in the goal. Derive chain structure constraints.

Inputs: Layer 0 §3, §4, §6, §10; Layer 1 §1–§5, P1.1–P1.12

Outputs: Required beam chains, gate1 state determination, chain topology
constraints.


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1. GOAL LITERAL DECOMPOSITION
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The explicit goal is:
  (and (loc agent1 area4)
       (active receiver2)
       (active receiver3))

Decomposing each literal through the derivation rules:

(active receiver2)
  ← a beam of matching color reaches receiver2
  ← receiver2 has chroma red (Layer 0 §3)
  ← a red beam chain exists from transmitter1 to receiver2
  ← transmitter1 is the only red transmitter (Layer 0 §3)

(active receiver3)
  ← a beam of matching color reaches receiver3
  ← receiver3 has chroma blue (Layer 0 §3)
  ← a blue beam chain exists from transmitter2 to receiver3
  ← transmitter2 is the only blue transmitter (Layer 0 §3)

(loc agent1 area4)
  ← agent1 must be physically present in area4
  ← reaching area4 from initial area1 requires gate1 open at some point
     (Layer 1 P1.2, Layer 0 §9)

Summary: The goal requires two simultaneous beam chains of different colors
plus agent1 in area4.


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2. REQUIRED BEAM CHAINS
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Chain R: transmitter1 →(red)→ ... →(red)→ receiver2
  Minimum structure: transmitter1 → connector(s) → receiver2
  Connector(s) must have los to both upstream source and downstream target.
  Each connector in the chain acquires derived color red.

Chain B: transmitter2 →(blue)→ ... →(blue)→ receiver3
  Minimum structure: transmitter2 → connector(s) → receiver3
  Each connector in the chain acquires derived color blue.

Chains R and B must coexist without beam interference. Each beam segment
of one chain must not cross any beam segment of the other chain.

Minimum connector commitment: at least 2 connectors (one per chain).
With only 3 connectors total, at most 1 connector is uncommitted.


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3. GATE1 STATE ANALYSIS
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Gate1 is open iff receiver1 is active (Layer 0 §4). Receiver1 has chroma
red, so receiver1 is active iff a red chain exists from transmitter1 to
receiver1. The question is: can such a chain coexist with chain B?

3a. 1-hop coexistence of receiver1 and receiver3.

Layer 1 P1.12 establishes that no 1-hop chain configuration can
simultaneously serve receiver1 (red) and receiver3 (blue) without
interference. Every valid area pair has at least one crossing among
the four types (T-end, R-end, Cross-1, Cross-2).

3b. Multi-hop coexistence of receiver1 and receiver3.

Multi-hop chains (2+ connectors per chain) would require at least 4
connectors total (2 for R1 chain + 2 for R3 chain), but only 3 connectors
exist. Even dedicating all 3 connectors leaves insufficient resources for
both a multi-hop R1 chain and any R3 chain.

3c. Conclusion.

Receiver1 cannot be active when receiver3 is active.
Therefore: gate1 cannot be open in the goal state.


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4. IMPLICATIONS OF GATE1 CLOSED
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From gate1 being closed in the goal state:

4a. Transmitter access from area4.
  area4 has los to transmitter1 and transmitter2 only via los1(gate1)
  (Layer 0 §6). With gate1 closed, area4 has NO los to either transmitter.
  A connector in area4 cannot directly pair with a transmitter.

4b. Chain routing through area4.
  For a chain to reach area4, it must use a 2-hop relay: transmitter →
  connector_in_east → connector_in_area4 → receiver. The first connector
  relays the beam through window1 (Layer 1 P1.4, always transparent)
  to the second connector in area4.
  
  A 2-hop chain through area4 requires 2 connectors for that single chain.
  With only 3 connectors and 2 required chains, at most one chain can
  be 2-hop.

4c. Agent location.
  Agent1 is in area4 (goal). Gate1 is closed (§3c). Agent1 must have
  entered area4 while gate1 was previously open, and gate1 subsequently
  closed. (This temporal reasoning is deferred to Layer 4.)

4d. Receiver-end line of sight.
  area4 retains los0 to receiver2 and receiver3 (Layer 0 §6) regardless
  of gate1 state. Connectors in area4 can still serve receiver2 or
  receiver3 on the downstream side, provided they receive beam power
  from an upstream relay.


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5. CHAIN TOPOLOGY CONSTRAINTS
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Given gate1 is closed and 3 connectors are available:

5a. 1-hop chain structure.
  transmitter → connector_in_area → receiver
  Requires: connector's area has los to transmitter (los0 only, since
  gate1 is closed) AND los to receiver.
  
  Viable areas for 1-hop red chain to receiver2:
    area2: los0 to transmitter1, los0 to receiver2. VIABLE.
    area3: los0 to transmitter1, los0 to receiver2. VIABLE.
    area4: NO los to transmitter1 (gate1 closed). NOT VIABLE for 1-hop.
    area1: NO los to receiver2. NOT VIABLE.

  Viable areas for 1-hop blue chain to receiver3:
    area2: los0 to transmitter2, los0 to receiver3. VIABLE.
    area3: los0 to transmitter2, los0 to receiver3. VIABLE.
    area4: NO los to transmitter2 (gate1 closed). NOT VIABLE for 1-hop.
    area1: NO los to receiver3. NOT VIABLE.

5b. 2-hop chain structure.
  transmitter → relay_connector → terminal_connector → receiver
  The relay connector must have los to the transmitter and visibility to
  the terminal connector's area. The terminal connector must have los to
  the receiver.

  If either chain uses a 2-hop structure through area4:
    transmitter → connector_in_east → connector_in_area4 → receiver
  This consumes 2 of 3 connectors, leaving only 1 for the other chain
  (which must therefore be 1-hop).

5c. Both chains 1-hop (2 connectors used).
  If both chains are 1-hop, only 2 connectors are committed to chains.
  The third connector is uncommitted.

  From Layer 1 P1.11, the non-interfering 1-hop configurations for
  serving receiver2 (red) and receiver3 (blue) are:
    (a) Red in area2, Blue in area4
    (b) Red in area3, Blue in area2
    (c) Red in area3, Blue in area4

  Applying the gate1-closed constraint (§4a eliminates area4 for 1-hop):
    (a) ELIMINATED — area4 cannot support 1-hop (no los to transmitter2).
    (b) SURVIVES — both area3 and area2 have los0 to their transmitters.
    (c) ELIMINATED — same reason as (a).

  Remaining: only configuration (b) survives — red connector in area3
  serving receiver2, blue connector in area2 serving receiver3.

5d. One chain 2-hop, one chain 1-hop (3 connectors used).
  If one chain routes through area4 (2-hop), it uses 2 connectors.
  The other chain uses the remaining connector (1-hop).

  Possible 2-hop structures:
    Red 2-hop to R2: transmitter1 → relay_in_east → connector_in_area4 → R2
    Blue 2-hop to R3: transmitter2 → relay_in_east → connector_in_area4 → R3

  Each of these must be checked for interference with the other chain's
  1-hop structure. This analysis is deferred to Layer 3 (resource and
  exclusion constraints), which will determine whether the 2-hop option
  is viable or whether both chains must be 1-hop.


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6. OUTPUT PROPOSITIONS
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P2.1  The goal requires a red beam chain (Chain R) from transmitter1 to
      receiver2, and a blue beam chain (Chain B) from transmitter2 to
      receiver3, coexisting without interference.

P2.2  At least 2 of the 3 connectors must be committed to chains R and B.

P2.3  Receiver1 cannot be active when receiver3 is active. No 1-hop
      configuration exists (Layer 1 P1.12), and multi-hop requires more
      connectors than available.

P2.4  Gate1 is closed in the goal state.

P2.5  With gate1 closed, area4 has no line of sight to either transmitter.
      A connector in area4 cannot directly pair with a transmitter.

P2.6  A chain routed through area4 requires a 2-hop relay structure
      (transmitter → east_connector → area4_connector → receiver),
      consuming 2 connectors for that single chain.

P2.7  With gate1 closed and both chains 1-hop, the only non-interfering
      configuration is: red connector in area3 serving receiver2, blue
      connector in area2 serving receiver3. (From Layer 1 P1.11, filtered
      by gate1-closed constraint.)

P2.8  Agent1 is in area4 with gate1 closed. Agent1 must have entered area4
      while gate1 was previously open. (Temporal details deferred to
      Layer 4.)
