Triangle and short-flight trial

The default trial looks a little rounder in two starts, but it is not a clear overall improvement. Long-flight error improves slightly; short-flight and all-original-edge error worsen. Another start remains trapped in a poor local minimum. No sphere or geographical coordinates enter optimization.

The two simple policies

The triangle rule removes a repeatedly overlong edge only when at least two fully observed triangles violate the triangle inequality by more than 30 minutes. The edge with the most violations is removed first; total excess breaks ties. Triangles are checked again after each removal, and the selected minimum degree is preserved. At most 50 edges are excluded per run; that limit was not reached here. A triangle's other two edges keep the endpoints connected. One isolated contradiction cannot reliably identify a responsible edge and is left untouched.

Short-flight influence is reduced using a fixed weight min(1, (hours / 3)²). A one-hour edge receives 1/9 of full weight; a three-hour or longer edge receives full weight. This changes influence, not targets: nothing is subtracted, overhead is not learned, and all short flights remain.

Same measurements, different influence

The original 34,782 observed triangles contain 82 violations over 0.001 hours, including 26 over 30 minutes. Five exclusions remove 15 material contradictions. The retained 34,646 observed triangles still have 60 violations, including 11 over 30 minutes. Removing constraints also removes some observed triangles, so fewer violations alone cannot prove the data is better. All 201 airports remain; 3,875 edges are fitted and 1,060 short edges are downweighted.

Comparison

Every error column below is unweighted and uses the same original 3,880 edges, including the five exclusions. Long means at least three hours. Radial variation is measured around a best-fit sphere; it is diagnostic only. The main app score is weighted when weighting is enabled, so it should not be compared directly with the original unweighted score.

Policy	Random seed	All original edges	Long edges	Short edges	Radial variation
Original	71	4.57%	3.99%	14.48%	8.37%
Original	72	4.57%	3.99%	14.48%	8.37%
Original	73	4.59%	3.99%	14.57%	7.48%
Triangle only	71	4.61%	4.03%	14.49%	7.64%
Short weights only	71	4.66%	3.83%	16.85%	6.98%
Both (trial default)	71	4.69%	3.86%	17.02%	7.47%
Both (trial default)	72	4.70%	3.86%	17.08%	7.36%
Both (trial default)	73	5.75%	4.90%	19.28%	16.87%

All runs use the same source means, jets-only and minimum-four-connection filters, seeded random 3D starts, and 128-trial two-chain annealing. Each reported run settled within 100,000 steps. Baseline fits come from the previous deployed version. Restart repeats the seed; New random start changes it. For the default seed 71, the new weighted readout is 4.16%, retained unweighted error is 4.62%, and original all-edge error is 4.69%. Those are different measurements, not interchangeable improvements.

Excluded distance constraints

Their 32 original schedule observations remain available in the airport inspector and downloadable audit. These are incompatible distance targets under the trial's rule, not proven bogus flights. Route detours, wind, speed differences or schedule padding can produce such contradictions even for real nonstop schedules.

Remaining material contradictions

Each is witnessed by only one remaining triangle, so this conservative rule leaves it for review. Geometry alone cannot identify which of the three observations is responsible.

What this does not solve

It does not turn flight hours into exact straight-line distances. A network can obey every observed triangle inequality yet remain inconsistent with an embedding in three dimensions. Fewer constraints or weaker short edges can also make some point positions less certain. Seed 73's 16.87% radial variation shows that local minima remain despite the bounded annealing search. These measurements do not demonstrate reliable spherical reconstruction.

Edge	Original target	Witnesses at removal	Via airports
AMS–PEK	11.67 h	5	FRA, CPH, WAW, ARN, OSL
BKK–TLV	11.27 h	3	DXB, DEL, AUH
LHR–TPE	14.42 h	3	PEK, XMN, WUH
TLV–HKT	11.29 h	2	DXB, AUH
DEL–CPH	10.25 h	2	MUC, WAW