slot257: PATH_4 τ ≤ 8 resubmission (BEST CANDIDATE)

[kavanaghpatrick]

Resubmission of slot49 (PATH_4 case τ ≤ 8)

THIS IS THE BEST RESUBMISSION CANDIDATE

Almost everything is proven - just the final assembly remains!

Background

Aristotle previously failed to find a proof but NO counterexample was found.

Proven Lemmas (from Aristotle - extensive!)

• tau_union_le_sum
• tau_S_le_2 (200+ line proof!)
• tau_X_le_2 (100+ line proof)
• path4_A_bridges_only_to_B
• path4_triangle_partition
• tau_Te_le_4_for_endpoint

Target

tau_le_8_path4: For path A—B—C—D, τ(G) ≤ 8

Mathematical Insight

In path A—B—C—D:

• τ(S_A) ≤ 2, τ(S_B) ≤ 2, τ(S_C) ≤ 2, τ(S_D) ≤ 2
• Bridges X_{A,B}, X_{B,C}, X_{C,D} are covered by adjacent S covers
• Key: Bridges are SHARED between adjacent S covers

Proof Strategy

S_A ∪ S_B ∪ S_C ∪ S_D covers all triangles because:

• bridges_{e} share vertex v with neighbor f
• So covered by S_f structure

Remaining Sorry

• tau_le_8_path4: Final assembly showing 8 edges suffice

File

submissions/nu4_final/slot257_path4_tau_le_8_RESUB.lean

Aristotle Submission

• Submitted to Aristotle
• Result received

[kavanaghpatrick]

Resubmission File Prepared

File: slot257_path4_tau_le_8_RESUB.lean (754 lines)

Proven Scaffolding (from slot49 uuid bf83c2c2):

Lemma	Status	Lines
tau_union_le_sum	✅ FULL PROOF	~20
tau_S_le_2	✅ FULL PROOF	~200
tau_X_le_2	✅ FULL PROOF	~100
path4_A_bridges_only_to_B	✅ FULL PROOF	~45
path4_triangle_partition	✅ FULL PROOF	~55
tau_Te_le_4_for_endpoint	✅ FULL PROOF	~40

Remaining (1 sorry):

• tau_le_8_path4: Final assembly showing 8-edge cover construction

Key Mathematical Insight:

PATH_4: A—B—C—D
- τ(S_A) ≤ 2, τ(S_B) ≤ 2, τ(S_C) ≤ 2, τ(S_D) ≤ 2
- Bridges X_{A,B}, X_{B,C}, X_{C,D} absorbed by adjacent covers
- Total: 2+2+2+2 = 8 edges

Ready to Submit

./scripts/submit.sh submissions/nu4_final/slot257_path4_tau_le_8_RESUB.lean slot257 path4_tau_le_8