Attempts to prove item 1 (non-emptiness of state at L_n in closed
SR+PDS chains ending at outer triangle). Results:
PROVEN:
- S_3-closure preserved by chain propagation.
- State at L_n is either empty OR equals all 6 permutations of {1,2,3}
(the only non-empty S_3-closed subset of permutations).
- Non-emptiness propagates through intermediate tires under outward
PDS via step-1 saturation.
REMAINING GAP (conjecture, empirically true): state at L_{n-1}
intersects the "perm-paired" subset of T_n's σ_D-projection (the
σ_D values that pair with permutation σ_U). At the final step T_n
has m_n=3 < k_n, so saturation fails — chain state at L_{n-1} could
in principle lie entirely in the (non-perm-paired) parity-matching
σ_D's, but empirically doesn't.
KEY STRUCTURAL FINDING: for T=(3, k), the σ_D's paired with a
permutation σ_U equal exactly the (parity-matching σ_D's) ∩ (T's
σ_D-projection). Verified for k=3..10.
HONEST OBSERVATION: a structural proof of the remaining conjecture
(without invoking 4CT) would constitute a new proof of 4CT under
the SR+PDS modelling assumption. The chain-pigeonhole framework
reduces to this single reachability question.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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