अङ्क-शास्त्रम् · aṅka-śāstra — digit-root arithmetic
Numerology in ℤ/9ℤ
Mūlāṅka, bhāgyāṅka, and nāmāṅka computed by digit-root — the same reduction the engine's Ψ-cycle Ψ(n) = 2ⁿ mod 9 lives in. The arithmetic is exact; the planet correspondences are classical tradition, cited from Bṛhat-Saṃhitā Ch. 52 and Cheiro (1926).
Aṅka-śāstra is a numerical tradition, not a science-claim. The digit-root algebra is exact; the graha tables are cultural correspondence — sources cited — no remedies, no guarantees.
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Input
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Result
digit-roots in ℤ/9ℤ
mūlāṅka · मूलाङ्क6Śukra · Venus
bhāgyāṅka · भाग्याङ्क6Śukra · Venus
nāmāṅka · नामाङ्क—enter a name above
inner relation · mūlāṅka ↔ bhāgyāṅka6mitra · friend6aṅka-maitrī projected from BPHS Ch. 3 graha-maitrī
| Aṅka | Value | Graha | Archetype | Day | Colour | Gem |
|---|---|---|---|---|---|---|
| Mūlāṅka | 6 | शुक्र Śukra | Beauty · harmony · refinement | Friday (Śukravāra) | White / pastel-rose | Diamond (vajra) — substitute: white sapphire |
| Bhāgyāṅka | 6 | शुक्र Śukra | Beauty · harmony · refinement | Friday (Śukravāra) | White / pastel-rose | Diamond (vajra) — substitute: white sapphire |
Day, colour, and gem columns are classical tradition (Cheiro 1926 · Bṛhat-Saṃhitā lineage) — listed as cultural record, not advice.
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The ring underneath
Ψ(n) = 2ⁿ mod 9 · period 6
The mūlāṅka reduction (digit-root) is the same algebraic operation as the engine's Ψ-cycle: both compute in ℤ/9ℤ. The powers of 2 walk the six-step unit orbit; the trinity 3·6·9 never appears on it — those are the nilpotent residues of the ring, not members of the cycle.
124875
Related algebra: every twin-prime pair (p, p+2) carries the same quadratic-residue class in ℤ/9ℤ, verified for all twin primes below 10⁶ — see P206 on the substrate page. The aṅka-cakra inherits its residue arithmetic from the same ring.
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9 × 9 compatibility matrix
सङ्गति-यन्त्रम् · aṅka × aṅka
| a \ b | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| 1 | M | M | M | M | N | Ś | N | Ś | M |
| 2 | M | M | N | M | N | N | M | Ś | N |
| 3 | M | N | M | N | M | Ś | M | N | M |
| 4 | M | M | N | M | N | N | Ś | M | N |
| 5 | N | N | M | N | M | M | N | M | Ś |
| 6 | Ś | N | Ś | N | M | M | N | M | N |
| 7 | N | M | M | Ś | N | N | M | N | M |
| 8 | Ś | Ś | N | M | M | M | N | M | Ś |
| 9 | M | N | M | N | Ś | N | M | Ś | M |
M mitra — friendN sama — neutralŚ śatru — adversary
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The nine aṅka–graha records
नव-ग्रह-अङ्क-संयोगः · classical correspondences
| Aṅka | Graha | Archetype | Day | Colour | Gem | Direction | Ψ-cycle position |
|---|---|---|---|---|---|---|---|
| 1 | सूर्य Sūrya · Sun | Leadership · fame · sovereign-will | Sunday (Ravivāra) | Gold / saffron | Ruby (māṇikya) | East | unit orbit · index 0 |
| 2 | चन्द्र Candra · Moon | Emotion · intuition · receptivity | Monday (Somavāra) | Silver / pearl-white | Pearl (muktā) | North-West | unit orbit · index 1 |
| 3 | गुरु Guru · Jupiter | Wisdom · expansion · dharma | Thursday (Guruvāra) | Yellow / golden-amber | Yellow sapphire (puṣparāga) | North-East | Diti pole (3·6·9) |
| 4 | राहु Rāhu · Rāhu | Innovation · restlessness · paradox | Saturday (Śanivāra) — secondary | Blue-grey / smoky | Gomedha (hessonite) | South-West | unit orbit · index 2 |
| 5 | बुध Budha · Mercury | Communication · agility · intellect | Wednesday (Budhavāra) | Green / pale-jade | Emerald (marakata) | North | unit orbit · index 5 |
| 6 | शुक्र Śukra · Venus | Beauty · harmony · refinement | Friday (Śukravāra) | White / pastel-rose | Diamond (vajra) — substitute: white sapphire | South-East | Diti pole (3·6·9) |
| 7 | केतु Ketu · Ketu | Mysticism · isolation · jñāna | Tuesday (Maṅgalavāra) — secondary | Multicolour / iridescent | Cat's eye (vaiḍūrya) | Centre / Ūrdhva | unit orbit · index 4 |
| 8 | शनि Śani · Saturn | Discipline · structure · longevity | Saturday (Śanivāra) | Dark blue / black | Blue sapphire (nīlamaṇi) | West | unit orbit · index 3 |
| 9 | मङ्गल Maṅgala · Mars | Energy · action · valour | Tuesday (Maṅgalavāra) | Red / coral-orange | Red coral (pravāla) | South | Diti pole (3·6·9) |
SRC
Source śāstras
स्रोत-शास्त्राणि · cited lineage
- Bṛhat-Saṃhitā, Varāhamihira — Ch. 52 vāstu-prakaraṇam, including aṅka-vinyāsa references (the 81-pada count is itself a 9-fold aṅka identity).
- Sūrya-Siddhānta — graha-aṅka mapping (Sun = 1, Moon = 2, …).
- Cheiro (Count Louis Hamon), Cheiro's Book of Numbers (1926) — Western continuation; identical graha-aṅka correspondences.
- Faith Javane & Dusty Bunker, Numerology and the Divine Triangle (1979) — compatibility matrices.
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aṅka-śāstra · Bṛhat-Saṃhitā 52 + Cheiro 1926 · mūlāṅka ≡ Ψ(n) ≡ digit-root in ℤ/9ℤ · ॐ अङ्क-देवतायै नमः