The Loopy Knot Walker, Take Two - What we predicted, what broke, and the knot that beats us on purpose

ABSTRACT
This is the second attempt. The first is in Appendix A, and it is there rather than in a separate deposit because a paper whose spine is “here is what we withdrew” wastes a reader’s afternoon. What survived from it is here as results; what did not is here as the reason the rules exist.  Five protein structures and a plan file written before any structure was fetched. It carries four falsifiable predictions and one deliberate non-prediction — the plan’s Test 2 asks whether the instrument behaves differently on hyperbolic versus torus knots and says, in its own words, “We have no prediction. That’s the point.”  Two of the four failed. We report both, unsoftened, and the plan file ships in the crate so the registration can be read rather than taken on trust.

The failure that mattered: we registered that writhe must cancel on an amphichiral knot, because a figure-eight is its own mirror image and that is a theorem. It did not cancel. It came back +42.542 and +7.039 on the two 4₁ structures — both positive, the sign that right-handed α-helices predict. The theorem is about the reduced knot
diagram; we measured the open backbone curve. Both are correct and the gap between them is the finding, because it turns the measurement into a clean readout of helix contamination on a substrate where the knot’s own contribution is provably zero.

From that failure comes the result we think transfers furthest. Above a contamination threshold bracketed between |Wr| = 7 and 42, replicate agreement hardens to 100% on an answer the knot type cannot have, while the instrument’s own internal vote gets weaker. Agreement and certainty move in opposite directions. Nearly every field treats inter-replicate agreement as validation; here it is the failure signature, and the honest internal measure was sagging the whole time. 

We also report a structure fetched under the wrong accession — 8BN1 is human ACE2, not the composite knot 8B1N — which produced the most internally consistent result in the study, at purity up to 1.000 across three runs and six filenames. The coherence was the symptom, not the safety. A single query to a reference our pipeline did not author corrected it immediately. Re-run on the right structure, the registered prediction confirms: both domains det = 3, full chain det = 9, all four seeds.

And we name the knot that beats us on purpose. The Kinoshita–Terasaka knot has the Alexander polynomial of the unknot. Our instrument reads det = 1 and calls it unknotted at high purity — exactly as the theory says it must, on a knot built in 1957 to demonstrate that limit.

This is not a better knot detector and must not be read as one. AlphaKnot would have typed the composite correctly. What is here is a reliability characterisation of a deliberately weak channel, with the failures reported at full volume.

Keywords: holonomy · parallel transport · geometric phase · Berry phase · Wilson loop · winding number · linking number · Burgers vector · topological invariant · noise floor · loop walk · competence surface · knotted proteins · amphichiral · Alexander polynomial