Formalizing Ancient Knowledge
with Modern Mathematics

We formalize the mathematical relationship between tropical, sidereal, and Krishnamurti Paddhati (KP) sub-lord coordinate systems as isomorphic transforms under a time-dependent ayanamsa function ψ(t). We derive closed-form expressions for the Lahiri, Raman, and KP ayanamsa variants, prove that all three systems share identical topological structure under the transform group, and show that Vedika Ephemeris planetary state vectors provide a sufficient base representation. Benchmarks on 10,000 charts demonstrate simultaneous three-system computation in 47ms mean latency with <0.001" positional divergence from reference ephemerides.

Standard domain-grounded AI systems attribute at document or passage level, which is insufficient for domains where individual verses carry distinct legal or doctrinal authority. We propose a multi-field provenance model P = (Book, Chapter, Verse, Translator, Edition) and formalize citation verification as a verification problem over a verified knowledge base. A multi-stage verification system achieves 91.25% verse-level citation accuracy on a 200-entry evaluation corpus, compared to 23% for document-level attribution. We prove that accuracy is monotonically non-decreasing with verification depth via ablation.

The Ashtakavarga system, codified in Brihat Parashara Hora Shastra chapters 66-73, assigns binary benefic/malefic contributions across 8 planets, 12 zodiacal signs, and 8 contributing bodies. We formalize this as a third-order binary tensor A ∈ {0,1}^8×12×8 and show that the Sarvashtakavarga (aggregate strength) emerges as a tensor contraction S = Σ_k A[:,:,k]. We prove an O(n²) bound on full bindhu computation via sparse tensor factorization, achieving 23x speedup over the naive O(n³) implementation. All classical lookup tables are encoded as verifiable sparse binary tensors with provenance to specific BPHS verses.

The Vimshottari Dasha system partitions a 120-year human lifespan into hierarchically nested planetary periods. We model this as a 3-level hierarchical finite state machine (HFSM) with 9 states per level, deterministic cyclic transitions, and duration weights w = [7, 20, 6, 10, 7, 18, 16, 19, 17] satisfying Σw = 120. We prove completeness (the system covers exactly 120 years), determinism (every state has exactly one successor), and structural self-similarity (sub-periods recurse with identical proportional structure). The starting state is derived from a continuous function of the Moon's nakshatra-longitude. Formal verification via explicit state enumeration confirms zero ambiguity across all 729 Antardasha and 6,561 Pratyantardasha states.

Domain-expert AI systems face a unique challenge: citation accuracy must remain high across extended conversations, yet retrieval context dilutes with each turn. We formalize the multi-turn accuracy maintenance problem by defining a accuracy state G_t = f(G_t-1, q_t, R_t) and prove that without active restoration, citation accuracy decays exponentially: acc(t) = acc(0) · e^-λt. We derive bounds on the decay rate λ as a function of domain specificity and propose a accuracy restoration strategy that maintains acc(t) ≥ τ for all turns t. Empirical results on 10-turn conversations show the accuracy restoration strategy sustains 89% citation accuracy at turn 10, versus 34% without intervention.