File_50a: Prime Quadruplet Naturalized Master Table (Indices 1–100)

File_50b: The Prime Quadruplet Chronology — Unified Harmonic System (v2.1)

Repository: 490d Unified Chronology Knowledge Graph
Date: February 15, 2026
Scope: Raw geometric data for the Prime Quadruplet sequence (A007530), naturalized to reveal a rigid three-state digital system. This file is the foundational data layer; File_50b provides the full interpretive synthesis.
Disambiguation: This table uses the Naturalized approach (P − 11, Base-90 grid). It must not be confused with the Capsule Appendix A table (which maps primes to chronological targets via locked spans and TargetRules) or with the Cumulative Prime Table (which uses cumulative sums and interim values). These are three complementary layers of the same prime sequence, not competing versions.

1. CORE DISCOVERY

The “mysterious numbers” in ancient chronology (Biblical and Sumerian King List) are generated by a specific operation on Prime Quadruplets. When “naturalized” by removing the initial offset (P − 11) and mapped to a Base-90 grid, the chaotic distribution of primes collapses into a rigid, digital system with zero noise. Every prime quadruplet start in the sequence, without exception, falls into exactly one of three harmonic states.

2. THE MECHANICS

Source: The sequence of Prime Quadruplet starts (OEIS A007530): P = 5, 11, 101, 191, 821, 1481, …

Naturalization: Subtract 11 from the prime start value:

$$P_{nat} = P – 11$$

This anchors the system to P2 (Q = 11), the first sextuplet-embedded quad seed (see Capsule §2.6.2).

The Grid: The system operates on a base of 90. The nearest grid value is computed as:

$$V = \text{round}(P_{nat} / 90) \times 90$$

The Three States: Every naturalized prime deviates from its nearest grid point by exactly one of three values:

StateDiff ValueGrid FormulaMeaning
State 0090kPerfect grid match. Structural stability.
State +30+3090k + 30High offset. Messianic/Heptadic extensions.
State −30−3090k − 30Low offset. Historical contraction.

Critical: The Diff is not noise. It is a Gear Selector that determines whether a specific era runs on the Geometric Rail (360-based) or the Heptadic Rail (7-based). The ±30 latitude is the mechanism that allows the Prime Lattice to simultaneously satisfy Sumerian 360-degree geometry and Biblical 7-year Jubilee cycles (see File_50b §3.3 and §3.5 for the “wobble” function).

Exception: P1 (Q = 5) yields P_nat = −6, which does not conform to the three-state rule. P1 is the singular pivot preceding the first sextuplet-embedded quad seed (see Capsule §7.0.2 boundary note). Its Diff of −6 is the P1/P2 Clutch gap itself.

3. THE MOD 7 FILTER (Residue 5)

The system’s output is heavily biased toward Mod 7 Residues 0 and 2. This is a direct mathematical consequence of the rules for Prime Quadruplets.

The Constraint: For a number P to start a quadruplet (P, P+2, P+6, P+8), P cannot be congruent to 5 modulo 7 — otherwise P+2 would be divisible by 7.

The Effect: This “Missing 5” forces the primes into specific admissible residue classes (11, 101, 191 mod 210). When processed through the naturalization operation, this creates the Mod 7 Residue 0 and 2 dominance observed in Biblical chronology (e.g., the Shem ±2 anomaly of Capsule §1.5).

4. THE THREE SCALES OF TIME (Summary; see File_50b for full treatment)

The system operates as a three-gear chronometer, where each scale is a fractal magnification of the one below it.

Scale 1 — The Civil Gear (Micro):

  • Unit: 1 Year.
  • Governors: P1 (Naturalized: −6) and P2 (Naturalized: 0).
  • Function: The “Clutch.” P2 (0) provides the ideal mathematical zero (6 BC). P1 (−6) provides the civil anchor (AD 1). The gap between P1 and P2 creates the 11-year interval (6 BC to AD 6) that absorbs the “Year Zero” anomaly.

Scale 2 — The Sumerian Gear (Macro):

  • Unit: 360 Years (1 Scale-2 Day = 1 Civil Year).
  • Governors: P2 (Grid: 0), P3 (Grid: 90), P4 (Grid: 180).
  • Function: The Seasonal Quadrant. P2 = Winter Solstice (Base). P3 = Spring Equinox. P4 = Summer Solstice (Opposition). P4 (180) anchors the Exodus (1446 BC), placing it in perfect “Summer” opposition to the “Winter” of the P2 Base.

Scale 3 — The Cosmic Gear (Deep Time):

  • Unit: 25,920 Years (Precession of the Equinoxes).
  • Governors: P13 (Grid: 12,960) and P19 (Grid: 19,440).
  • Function: The Great Year. P13 (12,960) = Half a Great Year (180 × 72). Mirrors P4 on the cosmic scale. P19 (19,440) = a “Seasonal Shift” in deep time (12,960 + 6,480). The 6,480-year step matches the tier gaps of the Sumerian King List (Capsule §4.4).

5. TABLE COLUMN DEFINITIONS

ColumnDefinition
IndexOrdinal position (P_n) within the A007530 sequence
OriginalThe prime quadruplet start value (Q_n)
Naturalized (P−11)Q_n − 11. Anchored so that P2 (Q=11) = 0
(×90) to P2Nearest multiple of 90 to the Naturalized value. The harmonic grid value.
(×90) to P1 (BC)Grid value + 6. Converts from the P2 (mathematical) anchor to the P1 (civil/BC) anchor.
DiffNaturalized value minus Grid value. Always 0, +30, or −30 (except P1).

Reading example: Index 6 (P6). Original = 1481. Naturalized = 1470. Grid = 1440. P1(BC) = 1446. Diff = +30. Interpretation: P6 sits 30 above the grid at 1440, which with the +6 civil offset yields 1446 BC = the Exodus. The +30 Diff means P6 operates in the Messianic/Heptadic state.

6. THE NATURALIZED MASTER TABLE (Indices 1–100)

IndexOriginalNaturalized (P−11)(×90) to P2(×90) to P1 (BC)Diff
15−606−6
2110060
31019090960
41911801801860
58218108108160
61,4811,4701,4401,44630
71,8711,8601,8901,896−30
82,0812,0702,0702,0760
93,2513,2403,2403,2460
103,4613,4503,4203,42630
115,6515,6405,6705,676−30
129,4319,4209,4509,456−30
1313,00112,99012,96012,96630
1415,64115,63015,66015,666−30
1515,73115,72015,75015,756−30
1616,06116,05016,02016,02630
1718,04118,03018,00018,00630
1818,91118,90018,90018,9060
1919,42119,41019,44019,446−30
2021,01121,00020,97020,97630
2122,27122,26022,23022,23630
2225,30125,29025,29025,2960
2331,72131,71031,68031,68630
2434,84134,83034,83034,8360
2543,78143,77043,74043,74630
2651,34151,33051,30051,30630
2755,33155,32055,35055,356−30
2862,98162,97063,00063,006−30
2967,21167,20067,23067,236−30
3069,49169,48069,48069,4860
3172,22172,21072,18072,18630
3277,26177,25077,22077,22630
3379,69179,68079,65079,65630
3481,04181,03081,00081,00630
3582,72182,71082,71082,7160
3688,81188,80088,83088,836−30
3797,84197,83097,83097,8360
3899,13199,12099,09099,09630
39101,111101,100101,070101,07630
40109,841109,830109,800109,80630
41116,531116,520116,550116,556−30
42119,291119,280119,250119,25630
43122,201122,190122,220122,226−30
44135,461135,450135,450135,4560
45144,161144,150144,180144,186−30
46157,271157,260157,230157,23630
47165,701165,690165,690165,6960
48166,841166,830166,860166,866−30
49171,161171,150171,180171,186−30
50187,631187,620187,650187,656−30
51194,861194,850194,850194,8560
52195,731195,720195,750195,756−30
53201,491201,480201,510201,516−30
54201,821201,810201,780201,78630
55217,361217,350217,350217,3560
56225,341225,330225,360225,366−30
57240,041240,030240,030240,0360
58243,701243,690243,720243,726−30
59247,601247,590247,590247,5960
60247,991247,980247,950247,95630
61257,861257,850257,850257,8560
62260,411260,400260,370260,37630
63266,681266,670266,670266,6760
64268,811268,800268,830268,836−30
65276,041276,030276,030276,0360
66284,741284,730284,760284,766−30
67285,281285,270285,300285,306−30
68294,311294,300294,300294,3060
69295,871295,860295,830295,83630
70299,471299,460299,430299,43630
71300,491300,480300,510300,516−30
72301,991301,980301,950301,95630
73326,141326,130326,160326,166−30
74334,421334,410334,440334,446−30
75340,931340,920340,920340,9260
76346,391346,380346,410346,416−30
77347,981347,970347,940347,94630
78354,251354,240354,240354,2460
79358,901358,890358,920358,926−30
80361,211361,200361,170361,17630
81375,251375,240375,210375,21630
82388,691388,680388,710388,716−30
83389,561389,550389,520389,52630
84392,261392,250392,220392,22630
85394,811394,800394,830394,836−30
86397,541397,530397,530397,5360
87397,751397,740397,710397,71630
88402,131402,120402,120402,1260
89402,761402,750402,750402,7560
90412,031412,020412,020412,0260
91419,051419,040419,040419,0460
92420,851420,840420,840420,8460
93427,241427,230427,230427,2360
94442,571442,560442,530442,53630
95444,341444,330444,330444,3360
96452,531452,520452,520452,5260
97463,451463,440463,410463,41630
98465,161465,150465,120465,12630
99467,471467,460467,460467,4660
100470,081470,070470,070470,0760

7. DISTRIBUTION SUMMARY (Indices 2–100)

Excluding the P1 exception:

StateCountPercentage
0 (Perfect Grid)3737.4%
+303131.3%
−303131.3%

The ±30 states are perfectly balanced. The slight excess of State 0 reflects the structural stability of the geometric rail.

8. CROSS-REFERENCE TO COMPANION FILES

FileLayerFunction
File_50a (this file)GeometricRaw naturalized grid data. The “source code.”
File_50bInterpretiveFull synthesis of P1–P23 against the Three-Scale system, Cumulative Chronology, and SKL/Berossus lattice.
Capsule Appendix ATargetingMaps primes to civil chronological anchors via locked spans (S) and TargetRules (§7.0.4).
Cumulative Prime TableCumulativeUses cumulative sums and interim sums of prime gaps. Contains the Interim Cum Sum column used in File_50b §2.4 (P5 chiasm). Referenced at https://490d.com/file_49-prime-quadruplet-cycle-alignment-axiom-and-test-protocol-ai-reference/

Relationship between layers: Capsule Appendix A provides the what (which civil date each prime targets). File_50a provides the where (which geometric state each prime occupies on the Base-90 grid). File_50b provides the why (how the three states, three scales, and harmonic keys generate ancient chronology as a unified system).

End of File_50a