Coordinates — Dome Cosmological Model V50.6

V12 Coordinate System (Current)

Key V50.6 discovery: H is not constant. The firmament curves downward from pole toward rim.

H(r) = 8537 × exp(−r / 8619) km

r(city) = solve numerically:  r × tan(polaris_elevation) = H(r)
θ        = Δlon × 0.9941   (EW angular scale)
distance = sqrt(r1² + r2² − 2 × r1 × r2 × cos(Δθ))
where    Δθ = Δlon × 0.9941

Performance: mean 5.2%, median 4.4% across 19 city pairs (was ~40% in V9).

V9 Reference (Superseded by V12)

V9 used flat H=4,750 km and solar noon UTC as theta proxy. V50.6 replaced both with H(r) exponential and Δlon × 0.9941.

Southern Hemisphere

Input: Solar noon altitude at June solstice + solar noon UTC Output: r, θ lat_S = 90 − noon_altitude_june + 23.44 (June dec = −23.44°) r = H / tan(lat_S) θ = (solar_noon_utc − 12.0) × 15°

Distance Formula (V9 — Law of Cosines)

d = sqrt(r1² + r2² − 2 × r1 × r2 × cos(Δθ)) Correct for polar disc geometry. Arc formula d = r_avg × Δθ used in V1–V8 is only valid for very small Δθ. Replaced at V9.

Ellipse Correction

r_ellipse(θ) = (a × b) / sqrt((b × cos(θ − orient))² + (a × sin(θ − orient))²) a ≈ 20,015 km (semi-major, pole to equatorial boundary) b ≈ 15,000 km (semi-minor, refining) b/a = 0.70–0.90 (V5–V9 range, converging) orient ≈ 60° (NE from Greenwich, Atlantic-to-Asia axis)

Ellipse-over-circle: LOCKED. Ellipse outperforms circle across all versions (WIN-030). Parameters: not yet locked — WGS84 contamination in source coordinates (OPEN-001). V10 will use sunrise/sunset times only.

Issue	V9 Error	V12 Status	Fix
Oslo-Stockholm	−89%	0.0% — RESOLVED	V12 H(r) + correct θ from longitude
Southern over-prediction	Quadratic with latitude	Reduced (V12 EW scale)	Further refinement with dome-native coords
Cross-equator	25–78% on long pairs	Under revision	V12 H(r) + ellipse

Issue

V9 Error

V12 Status

Fix

Oslo-Stockholm

−89%

0.0% — RESOLVED

V12 H(r) + correct θ from longitude

Southern over-prediction

Quadratic with latitude

Reduced (V12 EW scale)

Further refinement with dome-native coords

Cross-equator

25–78% on long pairs

Under revision

V12 H(r) + ellipse

Northern Hemisphere City Coordinates (H = 4,750 km)

City	Polaris °	Noon UTC	r (km)	θ °	x (km)	y (km)
North Pole	89.99	0.00	1	−180.0	−1	0
Reykjavik	64.10	13.93	2,306	+28.9	2,018	1,116
Anchorage	61.20	22.80	2,611	+162.0	−2,484	807
Oslo	59.90	12.23	2,753	+3.5	2,748	166
Stockholm	59.33	12.07	2,817	+1.1	2,817	54
Moscow	55.75	9.57	3,234	−36.4	2,601	−1,921
London	51.50	12.00	3,778	+0.0	3,778	0
Warsaw	52.20	10.60	3,684	−21.0	3,440	−1,320
Berlin	52.50	11.07	3,645	−13.9	3,537	−879
Paris	48.90	11.85	4,144	−2.3	4,140	−163
Chicago	41.88	17.80	5,298	+87.0	277	5,290
Istanbul	41.01	9.47	5,462	−37.9	4,307	−3,359
Madrid	40.42	12.47	5,577	+7.1	5,535	685
New York	40.71	16.87	5,520	+73.1	1,609	5,281
Denver	39.73	19.13	5,715	+106.9	−1,666	5,467
Chapel Hill NC	36.18	17.25	6,495	+78.8	1,267	6,370
Tokyo	35.70	2.73	6,610	−139.0	−4,993	−4,332
Los Angeles	34.05	19.87	7,029	+118.1	−3,305	6,203
Cairo	30.10	9.87	8,194	−32.0	6,953	−4,336
Baghdad	33.30	8.47	7,231	−52.9	4,357	−5,771
Miami	25.80	17.27	9,826	+79.0	1,866	9,647
Mexico City	19.43	18.60	13,466	+99.0	−2,107	13,300
Mumbai	19.08	6.20	13,733	−87.0	719	−13,714
Hong Kong	22.32	3.80	11,570	−123.0	−6,302	−9,704

City

Polaris °

Noon UTC

r (km)

θ °

x (km)

y (km)

North Pole

89.99

0.00

−180.0

−1

Reykjavik

64.10

13.93

2,306

+28.9

2,018

1,116

Anchorage

61.20

22.80

2,611

+162.0

−2,484

807

Oslo

59.90

12.23

2,753

+3.5

2,748

166

Stockholm

59.33

12.07

2,817

+1.1

2,817

Moscow

55.75

9.57

3,234

−36.4

2,601

−1,921

London

51.50

12.00

3,778

+0.0

3,778

Warsaw

52.20

10.60

3,684

−21.0

3,440

−1,320

Berlin

52.50

11.07

3,645

−13.9

3,537

−879

Paris

48.90

11.85

4,144

−2.3

4,140

−163

Chicago

41.88

17.80

5,298

+87.0

277

5,290

Istanbul

41.01

9.47

5,462

−37.9

4,307

−3,359

Madrid

40.42

12.47

5,577

+7.1

5,535

685

New York

40.71

16.87

5,520

+73.1

1,609

5,281

Denver

39.73

19.13

5,715

+106.9

−1,666

5,467

Chapel Hill NC

36.18

17.25

6,495

+78.8

1,267

6,370

Tokyo

35.70

2.73

6,610

−139.0

−4,993

−4,332

Los Angeles

34.05

19.87

7,029

+118.1

−3,305

6,203

Cairo

30.10

9.87

8,194

−32.0

6,953

−4,336

Baghdad

33.30

8.47

7,231

−52.9

4,357

−5,771

Miami

25.80

17.27

9,826

+79.0

1,866

9,647

Mexico City

19.43

18.60

13,466

+99.0

−2,107

13,300

Mumbai

19.08

6.20

13,733

−87.0

719

−13,714

Hong Kong

22.32

3.80

11,570

−123.0

−6,302

−9,704

Southern Hemisphere City Coordinates

City	Noon Alt (June)	Noon UTC	lat_S °	r (km)	θ °	x (km)	y (km)
Sydney	33.0°	14.17	33.6°S	7,160	+32.5	6,035	3,852
Cape Town	33.6°	10.00	33.0°S	7,326	−30.0	6,344	−3,663
Buenos Aires	33.5°	15.87	33.1°S	7,298	+58.0	3,862	6,192
Santiago	37.0°	16.87	29.6°S	8,375	+73.1	2,442	8,011
Melbourne	30.0°	14.40	36.6°S	6,405	+36.0	5,182	3,765
Auckland	23.5°	14.27	43.1°S	5,083	+34.0	4,212	2,846
Johannesburg	42.0°	9.73	24.6°S	10,394	−34.0	8,612	−5,820
Perth	38.0°	12.13	28.6°S	8,727	+2.0	8,722	297

City

Noon Alt (June)

Noon UTC

lat_S °

r (km)

θ °

x (km)

y (km)

Sydney

33.0°

14.17

33.6°S

7,160

+32.5

6,035

3,852

Cape Town

33.6°

10.00

33.0°S

7,326

−30.0

6,344

−3,663

Buenos Aires

33.5°

15.87

33.1°S

7,298

+58.0

3,862

6,192

Santiago

37.0°

16.87

29.6°S

8,375

+73.1

2,442

8,011

Melbourne

30.0°

14.40

36.6°S

6,405

+36.0

5,182

3,765

Auckland

23.5°

14.27

43.1°S

5,083

+34.0

4,212

2,846

Johannesburg

42.0°

9.73

24.6°S

10,394

−34.0

8,612

−5,820

Perth

38.0°

12.13

28.6°S

8,727

+2.0

8,722

297

What Changed V9 → V50.6

H(r) = 8537×exp(−r/8619) — Firmament height is not constant. Pure exponential fit.

θ = Δlon × 0.9941 — EW angular scale replaces solar noon proxy. Fixed V9 theta bug.

Numerical r solve — r × tan(polaris) = H(r) solved iteratively. No flat-H assumption.

H reconciliation — Schumann (~9,500 km), Polaris (4,750 km), model (9,086 km) all sampling same H(r) curve at different r.

Oslo-Stockholm resolved — 0.0% error. V9 bug was wrong θ + flat H.

Coordinate System