n=14 branching case feasible with one regular uniform seam family

Full uniform-family CSP at n=14 --no-tri (4403 tiles, 193 branching) is FEASIBLE:
one family threads every tile incl. branching nodes (outer rim + both inner faces
at once). Independent candidate test threads 193/193 branching tiles. Witness is
fully regular: sigma_m = 0^m if m even (monochromatic), 0^(m-2)12 if m odd. So on
the 4CT-relevant class the chained pigeonhole is constructively resolved throughout
the tested range (n=9,12,14, incl. branching).

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

papers/medial_tire_decompositions_of_plane_triangulations/experiments/chained_seam_findings.md

@@ -169,11 +169,35 @@ Branching tile = `≥2` inner faces carrying singleton-down interfaces (a tree n
 So **n=14** is the smallest place to test the uniform family / `R_T` composition on a
 genuine *branching* no-separating-triangle tile — the conjecture's real case.
 
+## Finding 7 — the branching case (n=14) is FEASIBLE with one regular family
+
+Full uniform-family CSP at n=14 `--no-tri` (4403 tiles, 3766 bite, **193 branching**):
+**FEASIBLE** — a single uniform family threads every tile, branching nodes included
+(each branching tile shows the uniform state on its outer rim AND both inner faces at
+once). Domains `|D[m]|`: `4:2 5:1 6:2 7:2 8:3 9:3 10:5`. The witness family is fully
+regular:
+
+```
+σ_m = 0^m            (monochromatic)        if m even
+σ_m = 0^(m-2) 1 2    (one block + 1 + 2)    if m odd
+       4:0000   5:00012   6:000000   7:0000012   8:00000000   9:000000012   10:0000000000
+```
+
+A direct candidate test (just the 193 branching tiles) confirmed it independently:
+the monochromatic-even / min-cut-odd family threads **193/193**.
+
+So on the 4CT-relevant class (no separating triangles), the chained-seam pigeonhole is
+**constructively resolved throughout the tested range** (n = 9, 12, 14, including the
+first branching nodes) by one explicit regular seam family: paint every even level
+cycle one colour, and every odd level cycle as a single monochromatic block plus the
+two parity-forced off-colour vertices.
+
 ## Open threads
 
-- **Test the uniform family at n=14 (restricted).** The first branching tiles live
-  here; does `monochromatic-even / min-cut-odd` still thread them (now with 2 inner
-  faces forced simultaneously)? This is the genuine conjecture case.
-- **Does the restricted uniform family persist past n=13?** A single later failure
-  (as in the unrestricted n=12 case) would be very informative.
-- **`R_T` composition along real trees** rather than per-size uniformity.
+- **Conjecture the regular family always works.** Does `σ_m = 0^m` (even) /
+  `0^{m-2}12` (odd) thread *every* no-separating-triangle tile for all `n`? Push to
+  n=15, 16 (branching grows: 1022 branching tiles at n=15) and look for any failure.
+- **Why does the restriction make monochromatic-even universal?** Separating-triangle
+  tiles are exactly what blocked an all-one-colour rim; removing them frees it.
+- **`R_T` composition along real trees** rather than per-size uniformity — the fully
+  general conjecture.