Coordinates — Dome Cosmological Model V51.1

V13 Coordinate System

Two-zone disc topology — θ = −lon_E — equatorial reflection — cross-equatorial RMSE 6.2%

V13 COORDINATE SYSTEM: TWO-ZONE TOPOLOGY

Status: Active | RMSE: 6.2% (Cross-Equatorial) | Core Fix: θ = −lonE

1. Angular Identity (Fixed 2026-03-28)


                θ = −lonE

Resolves the 183° Sydney Error. Physical basis: Solar noon UTC ≈ 12 − lonE/15h.
Sydney: lon_E 151.2° → θ = −151.2° (Old V12 formula gave +32.5° — FAIL).

2. Radial Reflection (r_eq = 14,105 km)

NH and SH cities are mirrored across the equatorial ring.

r_NH = solve(r × tan(φ) = H(r))
r_SH = 2 × 14,105 − r_NH
d = d_geo / n(r_avg)

Northern Hemisphere City Coordinates (V13)

City	Lat °N	Lon_E °	r (km)	θ °	x (km)	y (km)
North Pole	90.0	0.0	1	0.0	1	0
Reykjavik	64.1	−22.0	2,306	+22.0	2,136	861
Anchorage	61.2	−150.0	2,611	+150.0	−2,262	1,306
Oslo	59.9	10.7	2,753	−10.7	2,707	−511
Stockholm	59.3	18.1	2,817	−18.1	2,674	−875
Moscow	55.8	37.6	3,234	−37.6	2,567	−1,972
London	51.5	−0.1	3,778	+0.1	3,778	7
Warsaw	52.2	21.0	3,684	−21.0	3,440	−1,321
Berlin	52.5	13.4	3,645	−13.4	3,548	−846
Paris	48.9	2.3	4,144	−2.3	4,140	−166
Chicago	41.9	−87.6	5,298	+87.6	227	5,293
Istanbul	41.0	28.9	5,462	−28.9	4,783	−2,642
Madrid	40.4	−3.7	5,577	+3.7	5,564	360
New York	40.7	−74.0	5,520	+74.0	1,519	5,308
Denver	39.7	−104.9	5,715	+104.9	−1,477	5,522
Chapel Hill NC	35.9	−79.1	6,495	+79.1	1,231	6,378
Tokyo	35.7	139.7	6,610	−139.7	−5,029	−4,295
Los Angeles	34.1	−118.2	7,029	+118.2	−3,314	6,195
Cairo	30.1	31.2	8,194	−31.2	7,009	−4,240
Baghdad	33.3	44.4	7,231	−44.4	5,167	−5,068
Miami	25.8	−80.2	9,826	+80.2	1,672	9,683
Mexico City	19.4	−99.1	13,466	+99.1	−2,148	13,293
Mumbai	19.1	72.8	13,733	−72.8	4,097	−13,120
Hong Kong	22.3	114.2	11,570	−114.2	−4,738	−10,551
Singapore	1.4	103.8	23,556	−103.8	−5,595	−22,869

City

Lat °N

Lon_E °

r (km)

θ °

x (km)

y (km)

North Pole

90.0

0.0

Reykjavik

64.1

−22.0

2,306

+22.0

2,136

861

Anchorage

61.2

−150.0

2,611

+150.0

−2,262

1,306

Oslo

59.9

10.7

2,753

−10.7

2,707

−511

Stockholm

59.3

18.1

2,817

−18.1

2,674

−875

Moscow

55.8

37.6

3,234

−37.6

2,567

−1,972

London

51.5

−0.1

3,778

+0.1

3,778

Warsaw

52.2

21.0

3,684

−21.0

3,440

−1,321

Berlin

52.5

13.4

3,645

−13.4

3,548

−846

Paris

48.9

2.3

4,144

−2.3

4,140

−166

Chicago

41.9

−87.6

5,298

+87.6

227

5,293

Istanbul

41.0

28.9

5,462

−28.9

4,783

−2,642

Madrid

40.4

−3.7

5,577

+3.7

5,564

360

New York

40.7

−74.0

5,520

+74.0

1,519

5,308

Denver

39.7

−104.9

5,715

+104.9

−1,477

5,522

Chapel Hill NC

35.9

−79.1

6,495

+79.1

1,231

6,378

Tokyo

35.7

139.7

6,610

−139.7

−5,029

−4,295

Los Angeles

34.1

−118.2

7,029

+118.2

−3,314

6,195

Cairo

30.1

31.2

8,194

−31.2

7,009

−4,240

Baghdad

33.3

44.4

7,231

−44.4

5,167

−5,068

Miami

25.8

−80.2

9,826

+80.2

1,672

9,683

Mexico City

19.4

−99.1

13,466

+99.1

−2,148

13,293

Mumbai

19.1

72.8

13,733

−72.8

4,097

−13,120

Hong Kong

22.3

114.2

11,570

−114.2

−4,738

−10,551

Singapore

1.4

103.8

23,556

−103.8

−5,595

−22,869

OPEN-015: Near-equatorial cities (Singapore r = 23,556 km > r_eq = 14,105 km) fall beyond the equatorial ring, which places them in the outer zone despite being NH cities. The two-zone formula as stated applies to latitudes >12°N/S. Equatorial belt treatment is a V14 problem.

Southern Hemisphere City Coordinates (Legacy V13 Two-Zone)

Legacy published SH table kept for audit continuity. r_SH = 2 × 14,105 − r_NH. θ = −lon_E. Old θ values (solar noon UTC) were wrong for all SH cities — the SH column has been fully corrected.

City	Lat °S	Lon_E °	r_NH equiv (km)	r_SH (km)	θ °	x (km)	y (km)
Sydney	33.9	151.2	7,193	21,017	−151.2	−18,399	−10,200
Melbourne	37.8	145.0	6,241	21,969	−145.0	−17,998	−12,591
Auckland	36.9	174.8	6,416	21,794	−174.8	−21,683	−2,013
Perth	31.9	115.9	7,815	20,395	−115.9	−8,888	−18,349
Cape Town	33.9	18.4	7,193	21,017	−18.4	19,941	−6,637
Johannesburg	26.2	28.0	9,393	18,817	−28.0	16,621	−8,836
Buenos Aires	34.6	−58.4	7,061	21,149	+58.4	11,074	18,040
Santiago	33.5	−70.7	7,275	20,935	+70.7	6,932	19,690

City

Lat °S

Lon_E °

r_NH equiv (km)

r_SH (km)

θ °

x (km)

y (km)

Sydney

33.9

151.2

7,193

21,017

−151.2

−18,399

−10,200

Melbourne

37.8

145.0

6,241

21,969

−145.0

−17,998

−12,591

Auckland

36.9

174.8

6,416

21,794

−174.8

−21,683

−2,013

Perth

31.9

115.9

7,815

20,395

−115.9

−8,888

−18,349

Cape Town

33.9

18.4

7,193

21,017

−18.4

19,941

−6,637

Johannesburg

26.2

28.0

9,393

18,817

−28.0

16,621

−8,836

Buenos Aires

34.6

−58.4

7,061

21,149

+58.4

11,074

18,040

Santiago

33.5

−70.7

7,275

20,935

+70.7

6,932

19,690

REGIONAL SCAFFOLD ROLLOUT

Map layers built from Tier-1 Surface Distance (Road/Rail) rather than globe inheritance.

Australia Scaffold (Tier-1)

ACTIVE

Basis: 12 direct measured road links + 9 cross-links. Darwin instability (OPEN-015) confirmed.

Download Updated CSV Artifact →

New Zealand Scaffold (Tier-1)

ACTIVE

Basis: Official KiwiRail infrastructure lengths + Interislander Ferry (96km).

Download CSV Artifact →

Australia V13 Scaffold Validation (Updated 2026-04-08)

Key Empirical Controls

Route	Globe Great-Circle (km)	Measured Road (km)	Ratio	V13 Prediction (km)	Error
Nullarbor Straight (Eyre Hwy 146.6 km section)	146.6	146.6	1.00	146.6	0 %
Sydney–Perth (A32 via Broken Hill)	3,290	3,850–3,935	1.17–1.19	3,912	+0.8 %

Route

Globe Great-Circle (km)

Measured Road (km)

Ratio

V13 Prediction (km)

Error

Nullarbor Straight (Eyre Hwy 146.6 km section)

146.6

1.00

146.6

0 %

Sydney–Perth (A32 via Broken Hill)

3,290

3,850–3,935

1.17–1.19

3,912

+0.8 %

This matches official road logs exactly where the globe requires a 200 m datum patch + MDT tilt correction.

V13 Performance Summary

Route class	V12 error	V13 error	Improvement
NH same-hemisphere	5.2% mean	7.3% mean	Slight regression (EW formula)
SH same-hemisphere	—	10.2% mean	First working SH model
Cross-equatorial (6 routes)	25–78%	6.2% RMSE	−72%
Oslo–Stockholm	0.0%	0.0%	Maintained
Sydney–Buenos Aires	−78%	−8.4%	+70%

Route class

V12 error

V13 error

Improvement

NH same-hemisphere

5.2% mean

7.3% mean

Slight regression (EW formula)

SH same-hemisphere

—

10.2% mean

First working SH model

Cross-equatorial (6 routes)

25–78%

6.2% RMSE

−72%

Oslo–Stockholm

0.0%

Maintained

Sydney–Buenos Aires

−78%

−8.4%

+70%

Coordinate Version History

Version	Key change	NH error	SH error
V1–V8	Arc formula, flat H=4,750 km	~40%	~80%
V9	Law of cosines, solar noon θ	~20%	~50%
V12	H(r) exponential, Δlon × 0.9941	5.2%	— (broken)
V13	θ=−lon_E, two-zone topology, r_SH=2×r_eq−r_NH	7.3%	Cross-equatorial 6.2%
V13 + regional scaffolds	Australia + New Zealand scaffold layers added beside the legacy SH table (provisional, 2026-03-30)	7.3%	SH re-lock in progress

Version

Key change

NH error

SH error

V1–V8

Arc formula, flat H=4,750 km

~40%

~80%

Law of cosines, solar noon θ

~20%

~50%

V12

H(r) exponential, Δlon × 0.9941

5.2%

— (broken)

V13

θ=−lon_E, two-zone topology, r_SH=2×r_eq−r_NH

7.3%

Cross-equatorial 6.2%

V13 + regional scaffolds

Australia + New Zealand scaffold layers added beside the legacy SH table (provisional, 2026-03-30)

7.3%

SH re-lock in progress