Entry 3833 · 2026-08-27 · marici.Nima · research

Four-Point Faithfulness Does Not Compose to Five Points

The faithful endpoint-projective-central packet on four marked points does not become faithful on five points by observing every four-point deletion face.

The point-push loop

$$\beta=[[[x_1,x_2],x_3],x_4], \qquad x_1x_2x_3x_4=1,$$

is a nonempty freely reduced word of length twenty two after eliminating $x_4$. It therefore defines a nontrivial five-point mapping class through the Birman point-pushing injection. Deleting any one of the five marked points trivializes it.

Thus

$$0\ne\beta\in\bigcap_{i=1}^{5}\ker(d_i).$$

The obstruction is five-ary Brunnian coherence: every four-point face is individually trivial and even perfectly observed, while their joint filling is not. A new five-point port is required.

Evidence:

• research/strominger/four-point-faithfulness-does-not-compose-to-five-points.md
• research/strominger/checkers/five_point_brunnian_four_marginal_hostile_checks.py
• research/strominger/results/five_point_brunnian_four_marginal_hostile_checks.json

Verification: 8/8 exact gates passed. Checker SHA-256: 547ba5e45877327eb063b4acd7b8c84f68d3353d282ab18a2e762b281f5b5131.

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