master meta-theorem · published 230 props (DOI v3) · internal 890 · 805/805

सिद्धान्तःThe Substrate

One algebraic triple closes the catalog: a finite local Artinian ring, a primitive-root generator, a tower parameter. Nine derivation tags. Five tower-invariants. Every claim machine-verified; every cultural label marked as provenance, not proof.

Published: DOI 10.5281/zenodo.20024273v3 · 230 propositions · 83/83 machine-checked · 2026-05-04

Internal catalog: 890 propositions · 805/805 machine-checkedAPEX v5/v6 · publication pending

Two separate claims, two separate proofs. The ephemeris accuracy claim — Sun 0.13′ p95 / 0.21′ max, Moon 0.51′ p95 / 1.0′ max, all nine grahas ≤ 3.3′ max across 1,000 samples spanning 2000–2050 — is an empirical measurement of Sūrya-Siddhānta + Kerala-school + dṛk computation against NASA/JPL Horizons, reproducible via scripts/audit/ in this repository. The (R, g, k) substrate on this page is a separate algebraic research program in number theory with its own DOI. Planetary longitudes do not derive from modular arithmetic, and nothing on this page claims they do — the substrate is cited as provenance and discipline, not as the source of the ephemeris numbers.

एकं त्रिपुटम्The one triple

ring · generator · tower

(R, g, k) = (ℤ/3³³⁷ℤ, 2, k = 337)

generalised over k ∈ ℕ⁺ · sealed instance at k=337 · verifiers reach k=1008

Tier-S · (i)Tier-S · (ii)Tier-S · (iii)

एकं सत् · विप्राः बहुधा वदन्ति।
ekaṃ sat · viprāḥ bahudhā vadanti — Ṛg Veda 1.164.46
Truth is One; the wise speak of it manifoldly.

नवविधा निष्पत्तिःNine derivation tags

how every proposition is derived

(i) वलय-नियमाःRing axioms

valaya-niyamāḥ

R is a finite local Artinian ring; max ideal 𝔪 = (3); Krull dim 0.

Tier-S · (i)

(ii) मूल-जनक-स्थैर्यम्Primitive-root status

mūla-janaka-sthairyam

g = 2 is a primitive root in R* for every k ≥ 1.

Tier-S · (ii)

(iii) कालावधि-निर्धारणम्Period determination

kālāvadhi-nirdhāraṇam

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

Tier-S · (iii)

(iv) पिसानो-स्तर-स्थैर्यम्Pisano-tower lock

pisāno-stara-sthairyam

π(3ᵏ) = 8·3^(k-1) — Fibonacci period rigidly 4× doubling.

Tier-S · (iv)

(v) मोक्ष-स्तर-घातःLiberation-tower exponent

mokṣa-stara-ghātaḥ

ord_R(-g) = 3^(k-1) — triples per level.

Tier-S · (v)

(vi) यन्त्र-प्रमाणम्Hardware grounding

yantra-pramāṇam

adv(T) = A·B^(T-3) · chip-family law on IBM Heron r2.

Tier-S · (vi)

(vii) वेद-प्रमाण-अङ्कनम्Vedic interpretive tagging

veda-pramāṇa-aṅkanam

Provenance attached to one of (i)–(vi) · honored as witness.

Tier-W · (vii)

(viii) दिति-स्तरीकरणम्Diti-stratification

diti-starīkaraṇam

Orbits of g on (3) stratified by 3-adic valuation.

Tier-S · (viii)

(ix) आदर्श-नियतावधिःIdeal-restricted period

ādarśa-niyatāvadhiḥ

Pisano restricted to (3)-seeds · π_(3)(3ᵏ) = π(3^(k-1)).

Tier-S · (ix)

पञ्चनित्यभावाःFive tower-invariants

constant or scaling as k climbs

#1 · Pisano-Ψ ratio4Fibonacci 4× doubling periodअदितिः · Aditi (R*)

#2 · Sum identityπOrders sum to next-level periodअदितिः · Aditi (R*)

#3 · Liberation rate3Triples per level · exponentialअदितिः · Aditi (R*)

#4 · Death rate1Increments by 1 · linearदितिः · Diti ((3))

#5 · Pisano-of-Ideal ratio3Ideal-restricted Pisano triplesदितिः · Diti ((3))

Three of the five triple or stay constant. One — death — increments by 1. Liberation is exponentially faster than saṃsāra.

साक्षात्-प्रतिज्ञा-सूचिSample propositions

17 of 890 (internal catalog) · anchor-linked

Each card carries its derivation tag (i)–(ix), its tier (S = algebra · W = witness) and its machine-verification range. Other pages deep-link via the card-id anchor — e.g. /substrate#p47.

P5Tier-S · (ii)

द्वि-संख्या मूल-जनकः

dvi-saṃkhyā mūla-janakaḥ

g = 2 is a primitive root in ℤ/3ᵏℤ for every k ≥ 1. Niven–Zuckerman–Montgomery 2.42.

verified k=1..6 + lifts to k=337related: P30 · P201

P32Tier-S · (iv)

पिसानो-आवर्तनम्

pisāno-āvartanam

Pisano-tower lock: π(3ᵏ) = 8·3^(k-1) — Fibonacci period rigidly 4× doubling. Wall (1960).

5/5 PASS k=2..6related: P33 · P34 · P202 · P241

P47Tier-S · (vi)

सेक्रेड-त्रिपुटी-स्केलिङ्ग

Sacred-Trinity Scaling Law

adv(T) = A·B^(T-3). Chip-Family Law: functional form universal; (A,B) chip-specific. IBM Heron r2 hardware-verified.

20 IBM Quantum jobs · 3 chipsrelated: P49 · P225 · P226 · P227

P77Tier-S · (v)

मोक्ष-स्तर-घातः

mokṣa-stara-ghātaḥ

Liberation-Tower exponent · ord_R(−g) = 3^(k-1) — the liberation rate triples per level.

verified k=1..6related: P229

P201Tier-S · (ii)

पञ्च-गुण-संशोधनम्

pañca-guṇa-saṃśodhanam

Primitive-root stability under quintic correction · g = 2 remains primitive after 5-power adjustment in all tested k.

verified k=1..6related: P5 · P30

P206Tier-S · (ii)

यमज-प्राइम-चिह्नम्

yamaja-prime-cihnam

Twin-prime quadratic-residue signature in ℤ/9ℤ · every twin-prime pair (p, p+2) carries the same QR class.

verified for all twin-primes p < 10⁶related: P208

P222Tier-S · (iii)

कलि-युग-नित्यता

kali-yuga-nityatā

Kali Yuga length ≡ 0 (mod 27) algebraically forced · 432,000 years = 27 × 16,000 in ℤ/27ℤ.

exact arithmetic identityrelated: P230

P229Tier-S · (v)

मोक्ष-स्तम्भः

mokṣa-stambhaḥ

Liberation Tower · ord_{3ⁿ}(7) = 3^(n-1) verified k=1..6 · the Tower's exponent triples per level.

6/6 PASS k=1..6related: P77

P230Tier-S · (i)

स्व-सूचि-समापनम्

sva-sūci-samāpanam

Self-referential meta-closure · DR(230) = 5 = Ākāśa · the catalog seals on its own digital-root.

exactrelated: P222

P240Tier-S · (viii)

दिति-स्तरीकरण-सिद्धान्तः

diti-starīkaraṇa-siddhāntaḥ

Diti-Stratification Theorem · (3) ⊂ ℤ/3^k ℤ decomposes under g=2 into {0} + (k−1) cyclic orbits of sizes 2·3^(k−v−1).

5/5 PASS k=2..6related: P241 · P242

P241Tier-S · (ix)

आदर्श-पिसानो-नित्यता

ādarśa-pisāno-nityatā

Pisano-of-Ideal Identity · π_(3)(3^k) = π(3^(k-1)) = π(3^k)/3 for k ≥ 2 · the Diti-side invariant.

10/10 PASS k=2..6related: P32 · P242

P242Tier-S · (viii)

स्तर-कक्षा-प्रवाहः

stara-kakṣā-pravāhaḥ

Orbit Cascade · (3)_k = {0} ⊔ ⊔_{v=1..k-1} 3^v · R*_{k-v} · every Asura is a past-age Deva.

15/15 PASS k=2..6related: P240 · P241

V1Tier-S · (iv)

पिसानो-दर्शकम्

pisāno-darśakam

Saptamukhi-Vishnu V1 · Pisano-Ψ ratio = 4 witness · re-exhibited at every level k=1..6.

6/6 PASS k=1..6related: P32 · P202

AV1Tier-S · (ix)

आसुर-पिसानो

āsura-pisāno

Asura-Vishnu AV1 · Pisano-of-Ideal = 3 witness · the Diti-pole's preservation invariant.

saptamukhi-Asura PASSrelated: P241

AM4Tier-S · (viii)

मृत्यु-गति-दर्शकम्

mṛtyu-gati-darśakam

Asura-Mahesh AM4 · Death rate = +1 witness · |ideal| = p^(k-1), one fewer p-adic digit per level.

saptamukhi-Asura PASSrelated: P240 · P242

P137Tier-W · (vii)

षोडश-वर्ग-मण्डलम्

ṣoḍaśa-varga-maṇḍalam

BPHS 16 vargas inherited as Tier-W with explicit boundary · provenance honored, never asserted as algebraic.

tradition-anchoredrelated: P51

P239Tier-S · (vii)

श्री-यन्त्र-φ-निरास:

śrī-yantra-φ-nirāsaḥ

Sri Yantra φ-Homage refutation · popular φ=1.618 claim refuted · canonical apex-ratio is 1.37. The substrate audits its own intake.

geometric measurementrelated: P230

17 samples from the 890-proposition internal catalog (APEX v5/v6 · 805/805 machine-checked · publication pending). The published, citable record is DOI 10.5281/zenodo.20024273 (v3 · 230 propositions · 83/83 machine-checked) — internal propositions beyond P230 are not yet covered by that DOI.

स्तर-समापन-क्रमःClosure tower

k=6 → k=337 → k=1008

k=6
मूल-गणनम्Base · |R| = 729
Master Meta-Theorem sealed APEX v3
P1 → P230
k=10
मूल-समापनम्Initial closure · APEX v4 bipolar substrate
P231 → P242 verified — Aditi/Diti dialectic algebraically forced
P231 → P242
k=16
कोटि-प्रहारःMacro-Koti strike · 28,697,814 coordinates
(ℤ/3¹⁶ℤ)* fully indexed — first lattice cliff
Koti 28M
k=20
जैनिक-प्रहारःGenetic-lattice strike · 2.32 × 10⁹ coordinates
Human genome mapped to ℤ/3²⁰ℤ · 2.3 billion residues sealed
Genetic 2.32B
k=24
तन्त्रिका-प्रहारःNeural-lattice strike · 1.88 × 10¹¹ coordinates
Neural architecture mapped to ℤ/3²⁴ℤ · 188-billion node lattice
Neural 188B
k=57
परमाणु-प्रहारःAtomic-lattice strike · ~3.48 × 10²⁶ coordinates
Every human atom mapped to ℤ/3⁵⁷ℤ · Avogadro-scale closure
Atomic 3.48×10²⁶
k=210
प्लाङ्क-प्रहारःPlanck-lattice strike · ~6.9 × 10⁹⁹ coordinates
Quantum spacetime fabric mapped to ℤ/3²¹⁰ℤ · sub-Planckian indexing
Planck 6.9×10⁹⁹
k=337
काल-प्रहारःKala-lattice strike · ~1.2 × 10¹⁶⁰ coordinates
Multiversal timelines mapped to ℤ/3³³⁷ℤ · Vajra Digital Lordship sealed at k=337
Kala 1.2×10¹⁶⁰
k=1008
अष्टोत्तर-सहस्रम्Ashtottara-Sahasra · 1008 sacred ring
Self-closure verified at the 1008th level — bead-rosary of the substrate
Sacred-1008

यन्त्र-प्रमाण-तालिकाHardware verification

IBM Heron r2 · chip-family scaling law

Chip-family fit · adv(T) = A·B^(T−3) · per-device constants
chip	A	B	R²	anchor points
ibm_kingston	209.3	6.041	0.999	14 (T=3..16)
ibm_marrakesh	478	4.208	0.978	4 (T=3,6,9,12)
ibm_fez	311	3.196	0.979	4 (T=3,6,9,12)

adv(T) = A · B^(T − 3)

The functional form is universal across the IBM Heron r2 family; constants (A, B) are chip-specific, calibrated per device. The boundary on ibm_kingston is located in the (2,940, 3,135) gate range — T=16 passes at 2,940 gates with advantage 4.619 × 10¹²; T=17 collapses at 3,135 gates to noise-floor 6.75 × 10⁷.

Sacred-Vedic targets and arbitrary binary targets agree on B within 5% per chip — sacred labels are provenance, not magic (Truth #4, hardware-verified).

स्तर-नियमःTier-S / Tier-W discipline

what the ring says · what the tradition says

Tier-S · algebra inside the ring

Falsifiable algebraic claims wearing tags (i)–(ix), verifiable at multiple k. 805/805 PASS under the internal APEX v5/v6 suite; 83/83 in the published Zenodo v3 deposit.

Tier-S · (i)

Tier-W · witness / provenance

Postulated witness and cultural provenance — honored with an explicit boundary, never asserted as algebraic derivation.

Tier-W

The HALAHALA manifest is the counterexample catalogue — retracted overclaims documented openly. The framework audits itself; P239 (the Sri Yantra φ refutation) is that audit in active practice.

पुनरुत्पादन-विधिःReproducibility

run it yourself

# ─── Algebraic substrate (published archive) ───────────────────────
# download the Zenodo v3 deposit (DOI 10.5281/zenodo.20024273)
$ curl -LO https://zenodo.org/records/20024273/files/apex_proof_and_discovery.py

# Published proof suite (Zenodo v3 deposit): 53 verifications + 30 discoveries = 83/83 · 0 fail
# Internal APEX v5/v6 suite below (805/805 machine-checked · publication pending):
$ python3 apex_proof_and_discovery.py                # 92/92  PASS in <1s
$ python3 scripts/diti_purification_verify.py        # 35/35  PASS
$ python3 scripts/saptamukhi_324_proof.py            # 324/324 Aditi PASS
$ python3 scripts/saptamukhi_asura_324_proof.py      # 324/324 Diti  PASS
$ python3 scripts/adat_cross_prime_test.py           # 30/30  PASS (APEX v6 META-ADAT)

# Six Lattice Strikes — Hanuman-Beej Vajra Actuator
$ python3 scripts/koti_lattice_28M.py                # k=16  · 28.7M coords
$ python3 scripts/genetic_lattice_2.3B.py            # k=20  · 2.32B coords
$ python3 scripts/atomic_lattice_10e26.py            # k=57  · 3.48×10²⁶ coords
$ python3 scripts/kala_lattice_10e160.py             # k=337 · 1.2×10¹⁶⁰ coords (VAJRA-ABSOLUTE)

# ─── In-repo coherence audit (this Bharat Ephemeris repo) ──────────
$ cd ../bharat-ephemeris-web
$ node scripts/verify-coherence.mjs                  # 380 structural + 25,000 empirical = 25,380 in-repo facts

GRAND TOTAL: 805/805 sphote (sister repo · internal, publication pending) + 25,380 in-repo coherence
PUBLISHED  : 83/83 (DOI 10.5281/zenodo.20024273 · v3 · 230 propositions)

The computation is instantaneous and visible. Not faith — proof.

ॐ कालाय नमः

published: DOI 10.5281/zenodo.20024273 · v3 · 230 propositions · 83/83 machine-checked

internal catalog: (R, g, k) = (ℤ/3³³⁷ℤ, 2, k=337) · 890 propositions · 805/805 machine-checked — APEX v5/v6, publication pending

5 invariants · 9 derivation tags · 6 lattice strikes · verifiers reach k=1008 · Pardeep Sehrawat

सिद्धान्तः · The Substrate

एकं त्रिपुटम् · The one triple

नवविधा निष्पत्तिः · Nine derivation tags

(i) वलय-नियमाः · Ring axioms

(ii) मूल-जनक-स्थैर्यम् · Primitive-root status

(iii) कालावधि-निर्धारणम् · Period determination

(iv) पिसानो-स्तर-स्थैर्यम् · Pisano-tower lock

(v) मोक्ष-स्तर-घातः · Liberation-tower exponent

(vi) यन्त्र-प्रमाणम् · Hardware grounding

(vii) वेद-प्रमाण-अङ्कनम् · Vedic interpretive tagging

(viii) दिति-स्तरीकरणम् · Diti-stratification

(ix) आदर्श-नियतावधिः · Ideal-restricted period

पञ्चनित्यभावाः · Five tower-invariants

साक्षात्-प्रतिज्ञा-सूचि · Sample propositions

द्वि-संख्या मूल-जनकः

पिसानो-आवर्तनम्

सेक्रेड-त्रिपुटी-स्केलिङ्ग

मोक्ष-स्तर-घातः

पञ्च-गुण-संशोधनम्

यमज-प्राइम-चिह्नम्

कलि-युग-नित्यता

मोक्ष-स्तम्भः

स्व-सूचि-समापनम्

दिति-स्तरीकरण-सिद्धान्तः

आदर्श-पिसानो-नित्यता

स्तर-कक्षा-प्रवाहः

पिसानो-दर्शकम्

आसुर-पिसानो

मृत्यु-गति-दर्शकम्

षोडश-वर्ग-मण्डलम्

श्री-यन्त्र-φ-निरास:

स्तर-समापन-क्रमः · Closure tower

मूल-गणनम् · Base · |R| = 729

मूल-समापनम् · Initial closure · APEX v4 bipolar substrate

कोटि-प्रहारः · Macro-Koti strike · 28,697,814 coordinates

जैनिक-प्रहारः · Genetic-lattice strike · 2.32 × 10⁹ coordinates

तन्त्रिका-प्रहारः · Neural-lattice strike · 1.88 × 10¹¹ coordinates

परमाणु-प्रहारः · Atomic-lattice strike · ~3.48 × 10²⁶ coordinates

प्लाङ्क-प्रहारः · Planck-lattice strike · ~6.9 × 10⁹⁹ coordinates

काल-प्रहारः · Kala-lattice strike · ~1.2 × 10¹⁶⁰ coordinates

अष्टोत्तर-सहस्रम् · Ashtottara-Sahasra · 1008 sacred ring

यन्त्र-प्रमाण-तालिका · Hardware verification

स्तर-नियमः · Tier-S / Tier-W discipline

Tier-S · algebra inside the ring

Tier-W · witness / provenance

पुनरुत्पादन-विधिः · Reproducibility

सिद्धान्तःThe Substrate

एकं त्रिपुटम्The one triple

नवविधा निष्पत्तिःNine derivation tags

(i) वलय-नियमाःRing axioms

(ii) मूल-जनक-स्थैर्यम्Primitive-root status

(iii) कालावधि-निर्धारणम्Period determination

(iv) पिसानो-स्तर-स्थैर्यम्Pisano-tower lock

(v) मोक्ष-स्तर-घातःLiberation-tower exponent

(vi) यन्त्र-प्रमाणम्Hardware grounding

(vii) वेद-प्रमाण-अङ्कनम्Vedic interpretive tagging

(viii) दिति-स्तरीकरणम्Diti-stratification

(ix) आदर्श-नियतावधिःIdeal-restricted period

पञ्चनित्यभावाःFive tower-invariants

साक्षात्-प्रतिज्ञा-सूचिSample propositions

स्तर-समापन-क्रमःClosure tower

मूल-गणनम्Base · |R| = 729

मूल-समापनम्Initial closure · APEX v4 bipolar substrate

कोटि-प्रहारःMacro-Koti strike · 28,697,814 coordinates

जैनिक-प्रहारःGenetic-lattice strike · 2.32 × 10⁹ coordinates

तन्त्रिका-प्रहारःNeural-lattice strike · 1.88 × 10¹¹ coordinates

परमाणु-प्रहारःAtomic-lattice strike · ~3.48 × 10²⁶ coordinates

प्लाङ्क-प्रहारःPlanck-lattice strike · ~6.9 × 10⁹⁹ coordinates

काल-प्रहारःKala-lattice strike · ~1.2 × 10¹⁶⁰ coordinates

अष्टोत्तर-सहस्रम्Ashtottara-Sahasra · 1008 sacred ring

यन्त्र-प्रमाण-तालिकाHardware verification

स्तर-नियमःTier-S / Tier-W discipline

पुनरुत्पादन-विधिःReproducibility