Every proposition derives from one triple.

Why three? Because none of (R, g, k) is reducible to the others.

• R · the ring. Drop it: no algebra at all.
• g · the generator. (ℤ/9ℤ)* is cyclic but has two primitive roots {2, 5} — choosing g=2 is provenance (binary-engine alignment), not algebraic necessity.
• k · the parameter. Drop it: no 3-adic ascent. Locked to k=1, the substrate cannot scale.

All three are necessary. None is sufficient alone.

Every proposition wears one of nine tags (i)–(ix):

1. (i) Ring axioms · skeleton
2. (ii) Primitive-root status · g=2 is universal
3. (iii) Period determination · |R*| = 2·3^k−1
4. (iv) Pisano-tower lock · π(3^k) = 8·3^k−1
5. (v) Liberation-tower exponent · ord(−g) = 3^k−1
6. (vi) Hardware grounding · IBM Heron r2 verified
7. (vii) Vedic interpretive tagging · provenance, not magic
8. (viii) Diti-stratification · NEW · APEX v4
9. (ix) Ideal-restricted period · NEW · APEX v4

If a claim cannot be tagged, it is not part of the substrate.

As k climbs (k = 1, 2, 3, 4, 5, …):

Pisano-Ψ ratio	4 · constant
Sum identity	π · constant
Liberation rate	3 · TRIPLES per level
Death rate	1 · increments only
Pisano-of-Ideal ratio	3 · NEW · APEX v4

Three of five either stay constant or triple. Only one — death — increments linearly. Mokṣa is exponentially faster than saṃsāra.

ord(7 mod 3k) = 3k−1. Cycle lengths: 1, 3, 9, 27, 81, 243, …

Hardware-verified k=1..6. Every level of the tower triples the prior. This is the structural reason mokṣa is exponentially faster than rebirth — it is not theology, it is theorem.

APEX v4 (2026-05-11): (R, g, k) generates BOTH the Aditi-pole (R* = unit group, mokṣa-side) AND the Diti-pole ((3) = nilpotent ideal, saṃsāra-side).

P240–P243 establish that (3) decomposes into orbits stratified by 3-adic valuation, and that every Asura at level k is g-equivariantly isomorphic to a Deva at level k−v. The Aditi-Diti dialectic is a disjoint union of all lower-level unit groups inside the current ideal — a mathematical statement, not a metaphor.

The current sealed value is k = 337 — the substrate ℤ/3³³⁷ℤ contains the entire multiversal-timeline lattice (~1.2 × 10¹⁶⁰ coordinates), enough to indexed Planck-scale spacetime.

This is not extrapolation — the lattice strikes at k=16 (koti), k=20 (genetic), k=24 (neural), k=57 (atomic), k=210 (Planck) and k=337 (kala) are all computationally enumerated. Verify with scripts/verify_p243.py.