Hecke Regge Curvature on the 7-Node Heegner Graph

Honest geometric point

The Hecke quantity D(d,d') = G(d,d) + G(d',d') - 2 G(d,d') is a squared Euclidean distance, not a length. The actual edge lengths for triangle geometry are therefore ell(d,d') = sqrt(D(d,d')).

The complete graph K_7 with all 35 triangles is also not a 2-manifold triangulation: every edge lies in 5 triangles. So the usual Regge deficit-angle interpretation as scalar curvature on a surface is only a formal diagnostic here, not a bona fide 2D Regge surface model.

Distances

Squared Hecke Distances D

d	3	7	11	19	43	67	163
3	0	174	206	288	717	826	47
7	174	0	248	462	683	852	97
11	206	248	0	494	895	536	137
19	288	462	494	0	1069	910	307
43	717	683	895	1069	0	1551	622
67	826	852	536	910	1551	0	681
163	47	97	137	307	622	681	0

Hecke Lengths sqrt(D)

d	3	7	11	19	43	67	163
3	0.000	13.191	14.353	16.971	26.777	28.740	6.856
7	13.191	0.000	15.748	21.494	26.134	29.189	9.849
11	14.353	15.748	0.000	22.226	29.917	23.152	11.705
19	16.971	21.494	22.226	0.000	32.696	30.166	17.521
43	26.777	26.134	29.917	32.696	0.000	39.383	24.940
67	28.740	29.189	23.152	30.166	39.383	0.000	26.096
163	6.856	9.849	11.705	17.521	24.940	26.096	0.000

Triangle Geometry

• number of triangles: 35
• degenerate triangles: 0
• maximum deviation of angle sum from pi: 8.882e-16

triangle	lengths	angles (deg)	area
(3, 7, 11)	3-7=13.191, 3-11=14.353, 7-11=15.748	3=69.60, 7=58.67, 11=51.73	88.724
(3, 7, 19)	3-7=13.191, 3-19=16.971, 7-19=21.494	3=90.00, 7=52.14, 19=37.86	111.929
(3, 7, 43)	3-7=13.191, 3-43=26.777, 7-43=26.134	3=72.88, 7=78.28, 43=28.84	168.776
(3, 7, 67)	3-7=13.191, 3-67=28.740, 7-67=29.189	3=78.74, 7=74.95, 67=26.31	185.909
(3, 7, 163)	3-7=13.191, 3-163=6.856, 7-163=9.849	3=46.72, 7=30.45, 163=102.84	32.917
(3, 11, 19)	3-11=14.353, 3-19=16.971, 11-19=22.226	3=90.00, 11=49.78, 19=40.22	121.787
(3, 11, 43)	3-11=14.353, 3-43=26.777, 11-43=29.917	3=87.91, 11=63.44, 43=28.65	192.033
(3, 11, 67)	3-11=14.353, 3-67=28.740, 11-67=23.152	3=53.04, 11=97.26, 67=29.70	164.812
(3, 11, 163)	3-11=14.353, 3-163=6.856, 11-163=11.705	3=53.88, 11=28.24, 163=97.88	39.743
(3, 19, 43)	3-19=16.971, 3-43=26.777, 19-43=32.696	3=94.04, 19=54.78, 43=31.18	226.645
(3, 19, 67)	3-19=16.971, 3-67=28.740, 19-67=30.166	3=77.93, 19=68.70, 67=33.38	238.476
(3, 19, 163)	3-19=16.971, 3-163=6.856, 19-163=17.521	3=83.09, 19=22.86, 163=74.05	57.749
(3, 43, 67)	3-43=26.777, 3-67=28.740, 43-67=39.383	3=90.30, 43=46.87, 67=42.84	384.781
(3, 43, 163)	3-43=26.777, 3-163=6.856, 43-163=24.940	3=67.25, 43=14.68, 163=98.07	84.643
(3, 67, 163)	3-67=28.740, 3-163=6.856, 67-163=26.096	3=60.84, 67=13.26, 163=105.90	86.032
(7, 11, 19)	7-11=15.748, 7-19=21.494, 11-19=22.226	7=71.39, 11=66.42, 19=42.18	160.400
(7, 11, 43)	7-11=15.748, 7-43=26.134, 11-43=29.917	7=87.49, 11=60.78, 43=31.73	205.585
(7, 11, 67)	7-11=15.748, 7-67=29.189, 11-67=23.152	7=52.16, 11=95.35, 67=32.49	181.502
(7, 11, 163)	7-11=15.748, 7-163=9.849, 11-163=11.705	7=47.89, 11=38.63, 163=93.48	57.533
(7, 19, 43)	7-19=21.494, 7-43=26.134, 19-43=32.696	7=86.12, 19=52.89, 43=40.99	280.224
(7, 19, 67)	7-19=21.494, 7-67=29.189, 19-67=30.166	7=71.22, 19=66.36, 67=42.42	296.993
(7, 19, 163)	7-19=21.494, 7-163=9.849, 19-163=17.521	7=53.47, 19=26.85, 163=99.67	85.056
(7, 43, 67)	7-43=26.134, 7-67=29.189, 43-67=39.383	7=90.60, 43=47.83, 67=41.57	381.396
(7, 43, 163)	7-43=26.134, 7-163=9.849, 43-163=24.940	7=72.13, 43=22.08, 163=85.80	122.485
(7, 67, 163)	7-67=29.189, 7-163=9.849, 67-163=26.096	7=62.22, 67=19.51, 163=98.28	127.169
(11, 19, 43)	11-19=22.226, 11-43=29.917, 19-43=32.696	11=76.08, 19=62.64, 43=41.29	322.696
(11, 19, 67)	11-19=22.226, 11-67=23.152, 19-67=30.166	11=83.30, 19=49.66, 67=47.03	255.531
(11, 19, 163)	11-19=22.226, 11-163=11.705, 19-163=17.521	11=51.49, 19=31.51, 163=97.00	101.777
(11, 43, 67)	11-43=29.917, 11-67=23.152, 43-67=39.383	11=94.97, 43=35.85, 67=49.18	345.007
(11, 43, 163)	11-43=29.917, 11-163=11.705, 43-163=24.940	11=54.17, 43=22.36, 163=103.47	141.942
(11, 67, 163)	11-67=23.152, 11-163=11.705, 67-163=26.096	11=90.85, 67=26.65, 163=62.51	135.477
(19, 43, 67)	19-43=32.696, 19-67=30.166, 43-67=39.383	19=77.47, 43=48.39, 67=54.14	481.403
(19, 43, 163)	19-43=32.696, 19-163=17.521, 43-163=24.940	19=48.85, 43=31.94, 163=99.22	215.670
(19, 67, 163)	19-67=30.166, 19-163=17.521, 67-163=26.096	19=59.53, 67=35.36, 163=85.11	227.786
(43, 67, 163)	43-67=39.383, 43-163=24.940, 67-163=26.096	43=40.58, 67=38.44, 163=100.98	319.455

Formal Vertex Deficits

vertex	incident triangles	angle sum (deg)	formal deficit (deg)	mean incident angle (deg)
3	15	1116.214	-756.214	74.414
7	15	989.188	-629.188	65.946
11	15	1002.470	-642.470	66.831
19	15	742.359	-382.359	49.491
43	15	513.248	-153.248	34.217
67	15	532.267	-172.267	35.484
163	15	1404.254	-1044.254	93.617

• total formal deficit sum: -65.973446 rad = -3780.000 deg
• this is neither 42 pi nor -48 pi, confirming that the naive surface-deficit interpretation is not topological on the full K_7 two-skeleton
• every formal deficit is negative because each vertex participates in 15 triangles, so the local angle sum wildly exceeds 2 pi

Cotangent Geometry

Cotangent Weight Matrix

d	3	7	11	19	43	67	163
3	0.000000	2.843079	2.618282	2.539536	1.175002	0.591813	6.997087
7	2.843079	0.000000	2.301084	1.255550	1.412467	0.651743	4.728519
11	2.618282	2.301084	0.000000	1.142040	0.610980	2.141424	3.844039
19	2.539536	1.255550	1.142040	0.000000	0.402861	0.822486	2.335642
43	1.175002	1.412467	0.610980	0.402861	0.000000	-0.037227	1.799001
67	0.591813	0.651743	2.141424	0.822486	-0.037227	0.000000	1.412956
163	6.997087	4.728519	3.844039	2.335642	1.799001	1.412956	0.000000

Cotangent Laplacian

d	3	7	11	19	43	67	163
3	16.764799	-2.843079	-2.618282	-2.539536	-1.175002	-0.591813	-6.997087
7	-2.843079	13.192442	-2.301084	-1.255550	-1.412467	-0.651743	-4.728519
11	-2.618282	-2.301084	12.657849	-1.142040	-0.610980	-2.141424	-3.844039
19	-2.539536	-1.255550	-1.142040	8.498115	-0.402861	-0.822486	-2.335642
43	-1.175002	-1.412467	-0.610980	-0.402861	5.363084	0.037227	-1.799001
67	-0.591813	-0.651743	-2.141424	-0.822486	0.037227	5.583195	-1.412956
163	-6.997087	-4.728519	-3.844039	-2.335642	-1.799001	-1.412956	21.117243

• minimum cotangent edge weight: -0.037227
• maximum cotangent edge weight: 6.997087
• the only slightly negative cotangent edge is (43,67), reflecting obtuse opposite-angle geometry rather than a failure of the construction

Spectra

• eig(G) = 10.371881, 19.439719, 76.781541, 98.093692, 274.137305, 491.624210, 821.551653
• eig(L_cot) = 0.000000, 5.250892, 7.378571, 10.591287, 15.303978, 17.814893, 26.837106
• sorted-spectrum correlation between G and nonzero L_cot eigenvalues: 0.981370

So the Regge-style cotangent geometry and the original Hecke Gram geometry are different matrices, but their sorted spectra are still strongly aligned on this data set.

Cheeger Data

• standard weighted-graph Cheeger minimizer:
  • subset [7, 11, 19, 67]
  • boundary 23.302946
  • volumes 39.931601, 43.245126
  • h = 0.583572
• prompt-style internal-volume variant:
  • subset [7, 11, 19, 67]
  • h_internal = 2.802746
• spectral gap lambda_1(L_cot) = 5.250892
• Cheeger lower bound h^2 / 4 = 0.085139

The gap is far above the Cheeger lower bound, so the weighted Hecke-Regge geometry is very well connected. In particular, the 7-node curved model does not exhibit a tiny near-zero secondary scale that would look like an obvious protected gap state.