கணித வடிவம்The maths frame
Five sections, each reproducible. Friedmann · inflaton · Casimir · Pingala · Kāla-cakra. Real numbers. No mystical claims. Sibling to /caste/maths.
The Kāla-cakra section gives real arithmetic side-by-side with real cosmological timescales. Some numbers are close. None of this validates either tradition. The cluster's findings live in structural rhyme, not numerical match.
§01 Friedmann equation & Planck 2018
H² = (8πG / 3) ρ − k c² / a² + Λ c² / 3
The Friedmann equation describes the expansion rate H of a homogeneous, isotropic universe. For the spatially-flat case (k = 0), with the Planck 2018 fit:
- Hubble constant H₀ = 67.4 ± 0.5 km s⁻¹ Mpc⁻¹
- Matter density Ω_m = 0.315 ± 0.007
- Dark energy density Ω_Λ = 0.685 ± 0.007
- Age of the universe t₀ = 13.787 ± 0.020 Gyr
~68.5% of the universe's energy budget is in Λ — the energy of "empty" space. See Dossier 05 for the Cidambaram-as-void structural reading.
§02 Inflaton slow-roll
N ≡ ln(a_end / a) ≈ (1 / M_pl²) ∫ (V / V') dφ
The number of e-folds N during inflation, in single-field slow-roll, is the integral of the inflaton potential V over its derivative V'. The standard inflationary scenario produces N ≈ 60 — a linear scale factor expansion of e⁶⁰ ≈ 10²⁶.
The Tamil-Śaiva descent nāda → bindu → bīja → kalā (Tirumūlar, Tirumantiram Bk 3) maps structurally — not numerically — onto this sequence: vibration condenses to a point, point seeds the manifestation, manifestation differentiates into kalā. The mapping is a structural rhyme, not a derivation.
§03 Casimir effect — the cleanest Spanda handle
F / A = − (π² ℏ c) / (240 d⁴)
Two uncharged, perfectly-conducting parallel plates separated by distance d in vacuum experience an attractive force per unit area given by the formula above (Casimir 1948). For d = 1 μm, F/A ≈ 1.3 × 10⁻³ Pa — small but measurable. Lamoreaux (1997) measured the force in the 0.6–6 μm range with ~5% accuracy.
The physical content: empty space, between the plates, has fewer permitted vacuum-fluctuation modes than empty space outside. The pressure difference is the force. The "empty" space is full of pulsation; the Spanda claim — vibration prior to thing — has an empirical handle.
§04 Pingala's binary (~3rd–2nd c. BCE)
Pingala's Chandaḥ-sūtra enumerates the possible guru/laghu patterns of a Sanskrit metre of n syllables. There are 2ⁿ patterns. Pingala assigns each pattern a positional binary code and gives rules — sūtras 8.24–28 — that are functionally equivalent to modern binary arithmetic, including a halving-based decoding algorithm.
For a metre of 3 syllables: 2³ = 8 patterns.
0 → LLL · 1 → GLL · 2 → LGL · 3 → GGL · 4 → LLG · 5 → GLG · 6 → LGG · 7 → GGG
Donald Knuth (TAOCP Vol. 4A, 2011) names this as the earliest known binary enumeration system. Leibniz's Explication de l'Arithmétique Binaire postdates Pingala by approximately 1,900 years.
§05 Kāla-cakra arithmetic
| Unit | Definition | Human years |
|---|---|---|
| Divine year | 360 human years | 3.6 × 10² |
| Kali Yuga | 1,200 divine years | 4.32 × 10⁵ |
| Mahāyuga | Kṛta+Tretā+Dvāpara+Kali = 12,000 divine y | 4.32 × 10⁶ |
| Manvantara | 71 Mahāyugas | 3.07 × 10⁸ |
| Kalpa (Day of Brahmā) | 1,000 Mahāyugas | 4.32 × 10⁹ |
| Age of universe | Planck 2018 fit | 1.38 × 10¹⁰ |
One Kalpa : age of the universe ≈ 1 : 3.2 — same order of magnitude. This is a coincidence. The Purāṇic arithmetic derives from ritual-calendrical operations and theological scaling (every cycle 4:3:2:1 in length); the cosmological age comes from CMB, BAO, and Type Ia supernovae. They have no causal relation. The structural finding — that the Purāṇic tradition computed in cyclic timescales of cosmological order while Western cosmology held to linear-finite time until ~1980 — is the actual contribution. See Dossier 06.