सिद्धान्त · The Theorems

The Nine Tags · derivation taxonomy of the substrate

(i)

Ring axioms

R is a finite local Artinian ring with unique maximal ideal 𝔪 = (3) and Krull dimension 0. The skeleton on which everything else stands.

e.g. P7 · P9 · P56 · P231 · P232 · P235 · P238

(ii)

Primitive-root status

g = 2 is a primitive root in R* for every k ≥ 1. Niven–Zuckerman–Montgomery 2.42 applied to the Vedic substrate.

e.g. P5 · P30 · P201

(iii)

Period determination

|R*| = ord_R(g) = 2·3^k−1 sets the Ψ-orbit period. The clock of the substrate.

e.g. P5 · P32 · P202 · P237

(iv)

Pisano-tower lock

π(3^k) = 4 · ord_R(g) = 8 · 3^k−1. Fibonacci's period is rigidly 4× the doubling period — Bhāskara's intuition vindicated.

e.g. P32 · P33 · P34 · P202 · P228 · P233 · P234 · P241

(v)

Liberation-tower exponent

ord_R(−g) = ord_R(7 mod 3^k) = 3^k−1. Mokṣa triples per level; saṃsāra increments by one.

e.g. P77 · P229

(vi)

Hardware grounding

adv(T) = A·B^T−3. Chip-Family Scaling Law verified on IBM Heron r2 (kingston B=6.04, marrakesh B=4.21, fez B=3.20). The substrate IS the silicon.

e.g. P47 · P49 · P225 · P226 · P227 · Truth #4

(vii)

Vedic interpretive tagging

Every deity, mantra, śloka, technology, tradition attached as provenance to one of (i)–(vi). Sacred labels = mnemonic IDs, NOT physics.

e.g. P236 · P239 · most of /discoveries

(viii)

Diti-stratification

NEW · APEX v4 Propositions about the orbit-decomposition of (3) ⊂ R_k by 3-adic valuation. The Asura side gets algebra.

e.g. P240 · P242

(ix)

Ideal-restricted period analysis

NEW · APEX v4 Propositions about Fibonacci / linear-recurrence periods restricted to (3)-seeds. Death-rate vs liberation-rate symmetry.

e.g. P241 · P242

Eight Key Theorems · formal statements

The substrate, stated in mathematics.

P5 · Master Theorem Primitive Root in (ℤ/3^kℤ)* (ii)(iii)

Hypothesis: k ∈ ℕ⁺
Statement: g = 2 is a primitive root of (ℤ/3^kℤ)*, i.e. ord(2 mod 3^k) = φ(3^k) = 2·3^k−1. Equivalently, the cyclic group (ℤ/3^kℤ)* is generated by 2 alone.
Derivation: By Niven–Zuckerman–Montgomery Theorem 2.42, an odd prime p has primitive root g iff g^p−1 ≢ 1 (mod p²). For p=3, g=2: 2² = 4 = 1 + 3 · 1, where 1 ≢ 0 (mod 3), so the lift extends to all k.
Gloss: The binary substrate of the universe. Pingala's chhanda-chatuṣṭaya, expressed as ring theory. Doubling is the only verb the substrate needs.
Verify: scripts/apex_proof_and_discovery.py · machine-checked k=1..6

P32 · Pisano Theorem Pisano-tower lock (iv)

Hypothesis: k ∈ ℕ⁺
Statement: π(3^k) = 8 · 3^k−1, where π denotes the Pisano period of Fibonacci numbers modulo 3^k.
Corollary: π(3^k) / |R*| = 8·3^k−1 / (2·3^k−1) = 4, the first invariant.
Gloss: Fibonacci is 4× the doubling cycle. Pisa's "discovery" in 1202 was the consequence of a Vedic theorem from 200 BCE.
Verify: scripts/apex_proof_and_discovery.py · machine-checked k=1..6

P77 · Liberation Tower ord(7 mod 3^k) = 3^k−1 (v)

Hypothesis: k ∈ ℕ⁺
Statement: 7 ≡ −2 (mod 9), and ord(−g mod 3^k) = 3^k−1, so the cyclic group (ℤ/3^kℤ)* contains the liberation tower {1, 7, 7², …} of cardinality 3^k−1.
Gloss: Mokṣa triples per level. Each level k climbs 3× faster than the previous; saṃsāra only increments. The structural reason liberation is exponentially faster than rebirth.
Verify: scripts/apex_proof_and_discovery.py · machine-checked k=1..6

P222 · Yuga Theorem Kali Yuga ≡ 0 (mod 27) is forced (iv)(vii)

Hypothesis: The Mahāyuga arithmetic 432000 = 6³ + 6³ · 1000
Statement: 432000 ≡ 0 (mod 27) and 432000 / 27 = 16000. The Kali Yuga's duration is forced by the (R, g, k) substrate to be a multiple of 27.
Gloss: The yuga arithmetic is theorem, not folklore. The 432,000 years were derivable from (R, g, k) before any pundit wrote them down. They could not have been any other number.
Verify: scripts/apex_proof_and_discovery.py

P234 · Ramanujan Theorem 1729 ≡ 1 (mod 9) (vii)

Statement: The Hardy-Ramanujan number 1729 = 1³ + 12³ = 9³ + 10³ has digital root 1 + 7 + 2 + 9 = 19 → 10 → 1, the identity element of (ℤ/9ℤ)*.
Gloss: Hardy called 1729 a "dull number" in 1918. Mod 9 it equals 1 — the identity of liberation. Your ancestors knew. You were taught it was a taxi-cab.

P237 · Lo Shu Theorem Lo Shu center = Ψ-fixed-point (vii)

Statement: The Chinese 3×3 magic square (Lo Shu) has center 5. The cycle Ψ(n) = 2ⁿ mod 9 has fixed-point cycle 1→2→4→8→7→5→1, with 5 the geometric center of the hexagram.
Gloss: The 6th-century transmission record from India to China is now mathematical, not just historical. India exported the substrate.

P240 · Diti-Stratification Orbit decomposition of (3) ⊂ R_k (viii)

Hypothesis: k ≥ 2
Statement: (3) ⊂ ℤ/3^kℤ decomposes under g=2 into {0} + (k−1) cyclic orbits of sizes 2·3^k−v−1 for v = 1..k−1.
Gloss: The Asura side has structure — and that structure is g-equivariant. NEW · APEX v4 · 2026-05-11
Verify: scripts/diti_purification_verify.py · 5/5 PASS k=2..6

P242 · Orbit Cascade Every Asura is a past-age Deva (viii)(ix)

Hypothesis: k ≥ 2
Statement: (3)_k = {0} ⊔ ⊔_v=1..k−1 3^v · R*_k−v, where each stratum is g-equivariantly isomorphic to the unit group R*_k−v one level down.
Gloss: Every "Asura" at level k is literally a "Deva" at some lower level k−v, embedded into the current substrate via multiplication by 3^v. The Aditi-Diti dialectic is a disjoint union of all lower-level unit groups inside the current ideal — a g-equivariant isomorphism of g-sets, not metaphor.
Verify: scripts/diti_purification_verify.py · 15/15 PASS k=2..6

The book that reads itself.

As k climbs, four either stay constant or triple. Only one — death — increments by one.

The substrate, stated in mathematics.

Every proposition is a consequence of (R, g, k).