Part 1, THE 490d UNIFIED CHRONOLOGY: KNOWLEDGE GRAPH REPOSITORY

https://490d.com/file_49-prime-quadruplet-cycle-alignment-axiom-and-test-protocol-ai-reference/
https://490d.com/file_50b-the-prime-quadruplet-chronology-unified-harmonic-system-v2-1/

https://490d.com/file-51-the-rounded-scaffold-mod-5-architecture-underlying-biblical-chronology/

Part 4 of the Full Repository: File_49

0) Scope and intent

This file defines the axioms (locks) and test protocol used in the prime-quadruplet cycle-alignment experiments: rail construction, target construction (including decadal admission), scoring, null model, polarity handling, and invariants that must not drift between runs.

This is a protocol specification. Full prime tables, full prime-quad lists (Pn), full chronology tables, and run logs are stored in separate files.

0.1 Epistemic status and permitted claims

This file is not presented as proof of any causal or historical relationship between prime-quad structure and biblical chronology. It is a protocol specification for repeatable tests, written to prevent drift in rails, targets, cycles, polarity, and null models.

The working claim is limited to this:

Under the locked axioms in this file, multiple independent target constructions (regular and cumulative; MT/SP/LXX variants) have repeatedly produced nontrivial positive Z-scores on the baseline cycle-set (C_BASE), suggesting the observed alignment may exceed naive intuition.

Accordingly:

Provisional anchor policy: prime-aligned anchoring may be used as a tentative organizing heuristic for chronology exploration until contradicted by stronger counter-evidence or by failures under the same locked protocol.
Non-finality: absence of detected faults in the author’s testing does not establish truth; it only establishes that the protocol has not yet produced a contradiction within the tested space.
What would count as contradiction: a sustained pattern of null-like results (or negative results) under the same axioms across comparable target families, or identification of a protocol flaw that inflates significance (e.g., leakage, improper null, phase bias, or target-selection bias).

A5 (Cycle lock): Baseline claims use C_BASE=[2300, 2520, 3430] only; any other cycle-set is non-baseline and must be explicitly renamed and reported separately.

This section constrains interpretation: File_49 defines how to test, not what must be concluded.

1) Core axioms (locks)

A1. Calendar convention lock (civil BC/AD; no year 0)

All civil placements use BC/AD with no year 0:

1 BC is immediately followed by AD 1.
Any span that crosses the boundary uses the identity: BC + AD − 1 = Total.
If an “astronomical year-0” convention is ever used, it is a separate named experiment family (see §7).

A2. Prime-node station lock (P2 rail base)

Rails are constructed from each prime-quadruplet start Q using the locked base:

B = Q − 11
M = B + 30

This is a P2-calibrated rail coordinate system. Any alternative base (e.g., Q−5) is a new experiment family and must be renamed.

A3. Step-lattice lock (discrete translation group)

All null shifts act on the same discrete lattice as the observed evaluation:

STEP = 30 (decadal lattice)
Shifts are only of the form r ↦ r + k·30, never arbitrary offsets.

A4. Null-distribution lock (uniform shift over the fundamental period)

For fixed rails R, targets T, cycles C, define score S(s) after a shift s.

Null model:

choose k uniformly from {0, 1, …, K−1}
set s = k·STEP
evaluate S(s)

where K is the LCM of the per-cycle shift periods:

period(c) = c / gcd(c, STEP)
K = LCM(period(c) for c ∈ C)

Changing STEP, C, or the shift domain changes the probability space → new experiment.

2) Rails

2.1 Pn lists (quadruplet starts)

A “Pn” list is the first n quadruplet-start primes from the standard quadruplet-start sequence used in the scripts (A007530-style starts). This file does not restate those lists; it only locks how rails are generated from them.

2.2 Rail construction (allowed configs)

For each quadruplet start prime Q, define the P2-calibrated rail base:

B = Q − 11
M = B + 30

Allowed rail configurations per Q are exactly:

Config	Rails per Q
no_mod30	[B, −B]
pos_mod30	[B, M, −B]
neg_mod30	[B, −B, −M]
both_mod30	[B, M, −B, −M]
cross_pos	[B, −M]
cross_neg	[M, −B]
cross_both	[B, −M, M, −B]

No other rail generators are permitted unless introduced under a new config name and treated as a new experiment family.

Note: Decadal admission gating (how chronology dates become decadal targets) is governed by §4.2. Rails (this section) defines only the prime-derived rail coordinate system.

3) Prime table dependency (canonical table; admissible columns only)

The prime table is a reference artifact that may contain auxiliary or diagnostic columns. This protocol may only depend on columns declared admissible below. Any other column is protocol-invalid (cannot be used in computations, validations, or argumentation for this protocol).

3.1 Admissible columns (protocol-valid)

Index
Prime
−5 (Regular)
−11 (Gap)
Cumulative Sum
Gap Difference
Interim Cum Sum
Decadal Interim (if used as a precomputed decade representative)

3.2 Explicitly excluded (protocol-invalid / noise)

Minus Interim (evaluated and rejected as noise; not used)

If the external prime table contains excluded columns, they may remain present as diagnostics, but File_49 prohibits using them.

4) Targets (chronology-derived integer lists)

Targets are finite integer lists (typically decadal) derived from chronology tables under locked rules. This file defines legality and the decadal admission algorithm.

4.1 Six tested chronologies (three traditions × regular/cumulative)

Each tradition is tested in regular and cumulative (lifespan) form:

MT Regular
MT Cumulative (lifespan)
SP Regular
SP Cumulative (lifespan)
LXX Regular
LXX Cumulative (lifespan)

4.2 Axiom: Naturalization-First Decadal Admission Gate (protocol lock)

This is the first hard-lock governing how chronology dates become decadal targets. All target derivation in this protocol is subject to this axiom.

Axiom (P2 Decadal Admission Order Lock): For P2-based targets (MT/SP lanes), apply P2 naturalization first by shifting every civil placement (or placement band) by −6 years before any decadal admission is evaluated. No decadal admission may be performed on pre-naturalized years.

Axiom (P1 Decadal Admission — no naturalization): For P1-based targets (LXX lane), no naturalization step is applied. Y_nat = Y.

Decadal admission is a gate, not rounding:

Regular (single-year placements): include a decade target iff it lies within [Y_nat − 2, Y_nat + 2].
Cumulative (7-year forwarded bands): treat the input as a 7-year band; after station normalization, expand by ±2 years to form an 11-year admission window, and include any decade(s) that fall within it. If two decades fall within the window, include both.

Legality test (one-sentence form): Every decade in a target list must be derivable from a source chronology year or forwarded band by: (1) station-naturalize first (P2: −6; P1: no shift), then (2) admit decades only via the ±2 gate (no rounding).

Any change to this ordering or admission rule constitutes a protocol change and must be recorded as a new experiment family (renamed accordingly).

4.3 Prime-favorable Creation anchors (six)

The “prime-favorable” Creation anchors are the Creation dates (as decadal targets) that best aligned in the tests for each of the six chronologies. These are treated as empirical outcomes (not theological claims).

MT Regular: 4110
SP Regular: 4410
LXX Regular: 5490
MT Cumulative: 14000
SP Cumulative: 13910
LXX Cumulative: 14890

Important correction (SP cumulative 13910): 13910 is not shorthand for “(+60+130).” It is the SP cumulative (lifespan) creation anchor in the prime-favorable lane (see variant matrix §4.6). The “+130” issue belongs to regular chronology (R130/R0), not to the definition of SP cumulative.

4.4 Target legality conditions

A target list T is legal for this protocol only if:

It is versioned and reproducible (hard-coded list or generated by a documented rule).
It is integer valued (signed integers allowed where polarity is defined).
If decadalized, the admission rule is explicitly the ±2 gate as defined in §4.2 (and cumulative 7-year band gate), not generic rounding (see §4.5).
Any base correction (e.g., P2 naturalization −6) is explicitly recorded (see §4.2, §4.5).

Exclusion lock: “cumulative dates embedded with begetting ages” (hybrids that encode begetting-age structure inside cumulative coordinates) are out of scope and illegal for TARGETS here. If explored, they must be defined as a new experiment family.

4.5 Decadal admission algorithm (the ±2 gate — detailed procedures)

The axiom in §4.2 establishes the gate; this section specifies the detailed procedures.

4.5.1 Station-dependent normalization step

P2-base targets: apply naturalization first: Y_nat = Y − 6
P1-base targets: no naturalization step: Y_nat = Y

This normalization applies before any decadal admission for both regular and cumulative targets.

4.5.2 Regular chronology: ±2 admission window (Gear 2)

For each relevant node year Y (typically the birth year in regular chronology):

compute Y_nat (station rule above)
define window W = [Y_nat − 2, Y_nat + 2]
include any decade(s) (years ending in 0) that fall inside W
if none fall in W, the node contributes no decadal target
if two decades fall in W (rare), include both

4.5.3 Cumulative chronology: 7-year band plus ±2 (Gear 1 / lifespan)

Cumulative placement is treated as a 7-year band per node:

conceptual band: B7 = [Y0, Y0 + 6]

Admission rule:

apply station normalization to the band (P2 shifts the whole band by −6)
expand by ±2 to form acceptance region: B_accept = [B7_start − 2, B7_end + 2]
include every decade that falls inside B_accept
if two decades fall inside, include both (per policy)

4.6 Variant matrix (tested settings; no interpretation)

Toggles (notation fixed; no “+” notation in labels):

T60 / T0: Terah lifespan 205 (T60) vs 145 (T0)
C460 / C0: 2nd Cainan cumulative lifespan reinsertion (+460) included vs excluded (cumulative only)
R130 / R0: 2nd Cainan regular begetting-age insertion (+130) included vs excluded (regular only)
A70 / A130: Terah→Abraham begetting age 70 vs 130 (regular only)

Prime-favorable configurations (as tested):

Track	Tradition	Station	Cainan cumulative	Cainan regular	Terah lifespan	Terah→Abraham
Cumulative	LXX	P1	C460	n/a	T60	n/a
Cumulative	SP	P2	C460	n/a	T60	n/a
Cumulative	MT	P2	C460 and C0 both favorable	n/a	T60	n/a
Regular	LXX	P1	n/a	R130	n/a	A70
Regular	SP	P2	n/a	R0	n/a	A70
Regular	MT	P2	n/a	R0	n/a	A70

Untested / explicitly not covered in this protocol record:

MT regular with R130 (“MT +130”) was not tested.

5) Forced anchors and duplicate policy

5.1 Always-included anchors (forced into targets)

After decadal admission, targets are unioned with forced anchors that must be present in every run (even if not admitted by the ±2 gate):

1446 BC (Exodus)
1406 BC (Conquest)

Additional privileged anchors used in regular chronology testing:

Entry into Egypt (“the nexus”): represented decadal as 1870 (historical rationale: 1876 BC). Policy lock: store 1870 as the forced decadal representative to avoid ambiguity.
Enoch’s Ascension: included as a privileged narrative anchor. The civil year is taken from the corresponding regular chronology table(s) and then treated as forced-in (decadal representative if your table encodes it as such).

(“Creation” and “Flood” are automatically present via the Adam / Arphaxad nodes in the tables and need no special casing.)

5.2 Duplicate suppression (dates-only targets) with a narrow exception

Default: a target list is a set of dates; do not include the same signed date twice.

Exception: a duplicate date may be retained only if it represents different nodes (different patriarchs/events) and node-identity is explicitly preserved in that run mode. This is rare.

Operationally (recommended):

internally represent candidates as (date, node_id)
export standard targets as dates-only, deduped by date
if a special run intends node multiplicity, it must be named as a separate experiment mode

Explicit corner case lock: if a date appears in both regular and cumulative for the same node/event (e.g., Moses 1526 BC), it is deduplicated in combined runs.

6) Polarity rules (signed targets)

6.1 LXX(P1) split polarity (Adam→Terah negative)

In the LXX(P1) construction:

Adam → Terah segment uses negative polarity (negative integers).
Abraham → Moses and later uses positive polarity.

All other traditions and tracks in this protocol are treated as positive-polarity targets unless explicitly defined otherwise.

6.2 Structural signed band in the 14000 / 90+900 module

Where the “14000 / 90+900” structural mapping is invoked, the target construction includes a signed band spanning −90 to +900 (using the module’s own discrete admission rule). This is treated as structural (part of the polarity design), not an accidental byproduct.

7) Scoring statistic (observed score)

Given cycles C, rails R, targets T, define:

$$ S = \sum_{c\in C}\ \sum_{r\in R}\ \sum_{t\in T}\ \mathbf{1}\big[(t-r)\equiv 0\ (\mathrm{mod}\ c)\big] $$

This is the hit-counting score used in the scripts.

8) Significance procedures

8.1 Monte-Carlo Z (shift-null on the STEP lattice)

Compute S_obs = S(0). Sample many shifts s = k·30 with k uniform on [0, K) (see §1.4). From the sampled scores compute mean μ, standard deviation σ, and:

$$ Z = \frac{S_{\text{obs}}-\mu}{\sigma} $$

Multi-seed MC repeats the process across seeds and reports mean Z, SD(Z), and CI.

8.2 Analytical mean/variance (residue-count method)

An analytical mean/variance method is permitted only if computed from the same residue structure implied by (R, T, C, STEP) and checked against MC within expected tolerance. If both are reported, they are treated as separate outputs.

8.3 Cycle “family” decomposition

Cycle lists may be grouped into families (e.g., Year-type; Key-of-23; Danielic/7-fold) and tested separately under the same locked rails/targets/null.

8.4 Year-zero diagnostics are separate experiment families

Any year-zero exploration is not a reinterpretation of §1.1. It must be implemented only by explicitly named configs and treated as a new experiment family.

9) What counts as a new experiment (required renaming)

A run must be treated as a new experiment family if any of the following change:

STEP lattice (e.g., 30 → other).
Rail base (anything other than B = Q − 11) without a new config name.
Rail generator rules (adding/removing rails per Q beyond §2.2).
Null shift domain (anything other than uniform k∈[0, K) with s=k·STEP).
Target derivation rules (including the Naturalization-First Gate in §4.2, P2 naturalization −6, ±2 gate, cumulative 7-year band rule, forced anchors, polarity rules).
Cycle list C or the policy for assembling it.
Introduction of any hybrid target construction explicitly excluded in §4.3 (e.g., embedded begetting-age cumulative targets).

10) Integration notes (non-procedural)

File_49 is protocol-centric. Prime tables, Pn lists, chronology tables, and run logs are external artifacts referenced by this protocol.

Prime Table (Admissible Columns)

Index	Prime	−5 (Regular)	−11 (Gap)	Cumulative Sum	Gap Difference	Interim Cum Sum	Decadal Interim
1	5	0	−6	0	6	6	0
2	11	6	0	6	90	96	90
3	101	96	90	96	90	186	180
4	191	186	180	276	630	906	900
5	821	816	810	1086	660	1746	1740
6	1481	1476	1470	2556	390	2946	2940
7	1871	1866	1860	4416	210	4626	4620
8	2081	2076	2070	6486	1170	7656	7650
9	3251	3246	3240	9726	210	9936	9930
10	3461	3456	3450	13176	2190	15366	15360
11	5651	5646	5640	18816	3780	22596	22590
12	9431	9426	9420	28236	3570	31806	31800
13	13001	12996	12990	41226	2640	43866	43860
14	15641	15636	15630	56856	90	56946	56940
15	15731	15726	15720	72576	330	72906	72900
16	16061	16056	16050	88626	1980	90606	90600
17	18041	18036	18030	106656	870	107526	107520
18	18911	18906	18900	125556	510	126066	126060
19	19421	19416	19410	144966	1590	146556	146550
20	21011	21006	21000	165966	1260	167226	167220
21	22271	22266	22260	188226	3030	191256	191250
22	25301	25296	25290	213516	6420	219936	219930
23	31721	31716	31710	245226	3120	248346	248340
24	34841	34836	34830	280056	8940	288996	288990
25	43781	43776	43770	323826	7560	331386	331380
26	51341	51336	51330	375156	3990	379146	379140
27	55331	55326	55320	430476	7650	438126	438120
28	62981	62976	62970	493446	4230	497676	497670
29	67211	67206	67200	560646	2280	562926	562920
30	69491	69486	69480	630126	—	—	—

Prime Table (With Excluded “Minus Interim” Column — Not Used in Tests)

Index	Prime	−5 (Regular)	−11 (Gap)	Cumulative Sum	Gap Difference	Interim Cum Sum	Minus Interim (Not Used in Tests!)
1	5	0	−6	0	6	6	0
2	11	6	0	6	90	96	−84
3	101	96	90	96	90	186	−84
4	191	186	180	276	630	906	−534
5	821	816	810	1086	660	1746	−384
6	1481	1476	1470	2556	390	2946	696
7	1871	1866	1860	4416	210	4626	2346
8	2081	2076	2070	6486	1170	7656	3246
9	3251	3246	3240	9726	210	9936	6276
10	3461	3456	3450	13176	2190	15366	7536
11	5651	5646	5640	18816	3780	22596	9396
12	9431	9426	9420	28236	3570	31806	15246
13	13001	12996	12990	41226	2640	43866	25596
14	15641	15636	15630	56856	90	56946	41136
15	15731	15726	15720	72576	330	72906	56526
16	16061	16056	16050	88626	1980	90606	70596
17	18041	18036	18030	106656	870	107526	87756
18	18911	18906	18900	125556	510	126066	106146
19	19421	19416	19410	144966	1590	146556	123966
20	21011	21006	21000	165966	1260	167226	143706
21	22271	22266	22260	188226	3030	191256	162936
22	25301	25296	25290	213516	6420	219936	181806
23	31721	31716	31710	245226	3120	248346	210396
24	34841	34836	34830	280056	8940	288996	236286
25	43781	43776	43770	323826	7560	331386	272496
26	51341	51336	51330	375156	3990	379146	319836
27	55331	55326	55320	430476	7650	438126	367506
28	62981	62976	62970	493446	4230	497676	426246
29	67211	67206	67200	560646	2280	562926	491166
30	69491	69486	69480	630126	—	—	—

PRIME QUADRUPLET CYCLE ALIGNMENT WITH BIBLICAL CHRONOLOGY: Comprehensive Technical Reference for AI Systems

The 10× Scaling Experiment

Prime Quadruplet Resonance Across

Biblical Chronological Cycles at Uniform Magnitude

──────────

Dean

490d.com

Draft — February 2026

1. Executive Summary

This report documents a series of experiments testing whether the anomalous alignment between prime quadruplet rail values and biblical chronological targets—previously established at the natural scale of the TRIO cycles [2300, 2520, 3430]—persists when cycle periods are multiplied by a factor of 10. The 10× scaling was designed to test three things: whether the signal is robust to scale change, whether different cycle families exhibit different resonance regimes, and whether the modular arithmetic that governs the test discriminates meaningfully at larger cycle magnitudes.

The principal finding is that Ezekiel’s curse numbers (390 and 430 from Ezekiel 4), which appeared to be the weakest component of the original framework at the 1× scale, emerge as the dominant signal at the 10× scale. Specifically, the cycle 39000 (= 390 × 100) achieves a peak Z-score of +4.84 at tier P29 (79 quadruplets), sustaining Z > 3.5 through P50 (142 quadruplets). This is the strongest and most durable signal observed in any configuration tested to date, exceeding the next-best family by more than 2 standard deviations at late tiers.

Critically, Ezekiel’s numbers were never part of the TRIO—the independent reference frame against which all families were benchmarked. Their dominance therefore constitutes an independent confirmation rather than a circular finding. The experiment also reveals that the TRIO and Ezekiel operate in complementary resonance regimes: the TRIO peaks at P9–P16 and then dilutes, while Ezekiel ignites at P15–P17 and strengthens through P30. The handoff point at P16 was discovered independently from the TRIO’s perspective before Ezekiel was tested at this scale.

A 1× baseline comparison confirms that the 10× result is not an artifact of scale inflation. The same scaling that elevates Ezekiel from Z = +2.68 to Z = +4.68 diminishes the TRIO from Z = +4.02 to Z = +1.66. The two families respond to scale change in opposite directions, confirming that each is optimized for its own resonance regime rather than benefiting from a universal scaling effect. These findings are further validated across three independent target lists (67, 72, and 76 biblical dates), with the pure genealogical list producing the strongest Ezekiel signal (Z = +5.00), ruling out sensitivity to target selection.

2. Background and Motivation

Previous work (documented in File_49 and related analyses at 490d.com) established that the rail values of prime quadruplets—tested through modular arithmetic against 76 biblical chronological targets—produce statistically significant alignment with cycles drawn from Daniel, Revelation, jubilee patterns, and the Key of 23 framework. The primary vehicle for this analysis was the TRIO of cycles [2300, 2520, 3430], supplemented by family groupings representing Year Types, Key of 23, Daniel/Revelation, and Ezekiel.

At the 1× scale, with cycle periods ranging from approximately 1,000 to 6,900, the signal was concentrated in the first 40 prime quadruplets (P3–P16, Q range 101–90,611). The differential tier analysis confirmed that quadruplets beyond P16 contributed no independent signal—their apparent contribution at cumulative tiers P17–P50 was entirely inherited from the P3–P16 set through carry-forward dilution.

The Ezekiel family, based on the numbers 390 and 430 from Ezekiel 4 (representing the years of punishment for Israel and Judah respectively), participated in the original framework only indirectly—through derivative cycles like 1300 (= 10 × 130), 3900, and 4300. These cycles produced modest Z-scores at the 1× scale and appeared to be the weakest family in the system.

2.1 The Scaling Hypothesis

The 10× experiment was motivated by a simple question: does the P3–P16 anomaly reflect something about the small prime quadruplets themselves, or about the relationship between quadruplet rail values and cycle magnitude? At the 1× scale, cycles in the range 1,000–7,000 create relatively small residue spaces. A rail value of, say, 12,990 tested against a cycle of 2,520 has only 2,520 possible residue positions. Multiplying cycles by 10 expands these spaces to 23,000–69,000 positions, making each “hit” (a residue matching a target date) correspondingly less probable under the null hypothesis. If the signal persists at this scale, it cannot be attributed to the ease of hitting small residue spaces.

3. Experimental Design

3.1 Cycle Definitions

Each cycle family contributed its single strongest-performing member at the 10× scale. For the anti-cherry-picking validation (Section 6), each family contributed its two best members. The final uniform-scale configuration placed all cycles in the 23,000–69,000 range (a 3:1 ratio), eliminating concerns about scale mismatch between families.

Family	Cycle	Source	Notes
TRIO	23,000	2300 × 10	Daniel 8:14 (2300 evenings/mornings)
Year Types	33,600	3360 × 10	960 × 35; jubilee-derived
Key of 23	69,000	6900 × 10	23 × 3000; Key of 23 framework
Daniel/Rev	25,200	2520 × 10	360 × 7; prophetic week of years
Ezekiel	39,000	390 × 100	Ezekiel 4:5 (MT); years of Israel’s punishment

3.2 Test Structure

The test follows the established Prime Double Lock Protocol. For each cumulative tier Pn (using the first n prime quadruplets), the script computes all rail values derived from the quad’s four constituent primes, reduces each rail modulo each cycle period, and counts how many of the resulting residues match one of 76 biblical chronological targets (73 unique dates plus 3 double-hit entries). This observed count S is compared against the null distribution (obtained via 200,000 Monte Carlo samples) to compute a Z-score. The analysis runs from P3 through P50, covering quadruplets with Q values from 11 to approximately 3 million.

Three polarity configurations are tested at each tier: neg_mod30 (rails derived from the negative offset of the mod-30 wheel), pos_mod30 (positive offset), and both_mod30 (combined). The mod-30 structure arises from the requirement that prime quadruplet members occupy specific residue classes modulo 30.

4. Principal Results

4.1 The Ezekiel Dominance

Ezekiel 39000 (= 390 × 100) produces the strongest signal observed in any configuration. Under the both_mod30 metric, the Z-score trajectory is as follows: the signal ignites at P6 (Z = +0.86), crosses the p = 0.05 threshold at approximately P9 (Z = +1.33), reaches p = 0.001 at P18 (Z = +3.17), and peaks at Z = +4.84 at P29 (79 quadruplets). The signal sustains above Z = +3.5 through P50 (142 quadruplets).

This profile is unique among all families tested. Every other family peaks in the P9–P16 range and subsequently decays through dilution as non-contributing quads are added. Ezekiel alone shows a pattern where additional quadruplets strengthen the signal rather than dilute it, at least through P29. This anti-dilution property is the hallmark of a cycle whose residue structure is genuinely aligned with the target set across a broad range of prime magnitudes.

4.2 The P16 Handoff

The differential tier analysis (conducted prior to the 10× experiment) established that the TRIO and allied families concentrate their signal in P6–P16. Quadruplets beyond P16 contribute nothing independently at the 1× scale. The 10× experiment confirms this: at the scaled TRIO cycle (23,000), the Z-score peaks near P9–P10 and gradually decays.

Ezekiel’s signal, by contrast, begins its steep ascent precisely where the other families plateau. The crossover occurs at approximately P16–P17, the exact tier independently identified as the ceiling for the TRIO regime. This creates a natural two-regime structure: P6–P16 belongs to the TRIO and its allied families; P16–P30 belongs to Ezekiel. The handoff point was not chosen or optimized—it was discovered from the TRIO’s perspective before Ezekiel was tested at this scale.

4.3 Polarity Reversal at 10×

At the 1× scale, neg_mod30 (the polarity aligned with the mod-30 order) consistently outperformed pos_mod30. At 10×, this preference reverses. In the TRIO 23,000, pos_mod30 breaks away from neg_mod30 at P19 and maintains a lead of approximately 0.7–1.0 Z through P50. In the full 19-cycle configuration, pos_mod30 peaks at Z = +6.65 at P12—the single strongest reading in the entire analysis.

This reversal rules out the concern that one polarity has an inherent structural advantage. The preference is genuinely scale-dependent: a resonance phenomenon, not a fixed bias. The theological parallel is suggestive: mod 30 represents divine order (the biblical month, the generational scaffolding), while its complement represents the opposing structure. Ezekiel’s curse numbers (390 = 3 × 130, where 130 symbolizes opposition to 30’s order) resonate most powerfully through the complementary polarity at the 10× scale.

4.4 The 13,000 vs 39,000 Comparison

Two scalings of Ezekiel’s 390 were tested: 13,000 (= 130 × 100, or equivalently 390/3 × 100) and 39,000 (= 390 × 100, the literal text multiplied by 100). Both produce strong signals, but with distinct profiles:

Metric	13,000	39,000	Winner
First Z > 2.0	P12	P15	13K (−3 tiers)
First Z > 3.0	P14	P18	13K (−4 tiers)
Peak Z	+4.22 (P30)	+4.84 (P29)	39K (+0.62)
Z at P50	+3.20	+3.53	39K (+0.33)

The 13,000 cycle ignites earlier (smaller residue space, easier for small quads to engage), while 39,000 peaks higher and sustains longer (larger residue space, more discriminating). The more literal preservation of the curse number in the cycle period correlates with the stronger late-tier signal. Both scalings produce signals that far exceed any non-Ezekiel family.

4.5 Family Rankings at Uniform Scale

The following table summarizes performance across all five families at the uniform 23K–69K scale, using each family’s single best cycle:

Family	Cycle	Peak Z	Peak @	P50 Z	Character
Ezekiel	39,000	+4.84	P29	+3.53	Peak later, sustain longer
Key of 23	69,000	+2.23	P12	+1.81	Stable plateau
TRIO	23,000	+2.29	P10	+1.90	Late surge (pos)
Daniel/Rev	25,200	+0.88	P9	+0.10	Signal fades
Year Types	33,600	+1.38	P14	−1.06	Actively anti-correlated

5. Anti-Cherry-Picking Validation

To address the concern that selecting a single “best” cycle per family introduces optimization bias, a final validation run used two cycles per family, with Ezekiel’s entry being simply the two numbers from the text (390 and 430) each multiplied by 100:

Family	Cycles	Selection Basis
TRIO	23,000 + 25,200	Top 2 from original TRIO
Year Types	33,600 + 25,800	Top 2 from year-type family
Key of 23	23,000 + 69,000	Top 2 from Key of 23 family
Daniel/Rev	12,000 + 33,300	Top 2 from Daniel/Revelation
Ezekiel	39,000 + 43,000	Literal text × 100 (no optimization)

Result: Ezekiel [39,000 + 43,000] peaked at Z = +4.68 (P30) and sustained Z = +2.94 at P50, dominating the field by nearly 2.5 sigma over the next-best family at P50. The Ezekiel cycles are the least optimized entry in the test—every other family required selection of which members to include, while Ezekiel’s entry is simply: take the two numbers from the text, multiply by 100. This makes the result robust against cherry-picking by construction.

6. The 1× Baseline: Scale Selectivity Confirmed

A natural concern with the 10× results is whether scaling universally inflates Z-scores—whether any cycle looks better at 10× simply because larger numbers produce more opportunities for alignment. To test this, the identical two-cycle-per-family configuration was run at the original 1× scale (cycles 1,200–6,900), providing a direct baseline comparison.

6.1 Cycle Configurations

Family	1× Cycles	10× Cycles
TRIO	2,300 + 2,520	23,000 + 25,200
Year Types	3,360 + 2,580	33,600 + 25,800
Key of 23	2,300 + 6,900	23,000 + 69,000
Daniel/Rev	1,200 + 3,330	12,000 + 33,300
Ezekiel	3,900 + 4,300	39,000 + 43,000

6.2 Results: The Throne Swap

The results confirm that 10× scaling does not uniformly inflate signals. The TRIO and Ezekiel completely swap positions across scales:

Family	1× Peak Z	10× Peak Z	Δ	Direction
TRIO	+4.02 (P12)	+1.66 (P12)	−2.36	Loses 2.4σ at 10×
Key of 23	+2.88 (P12)	+2.27 (P12)	−0.61	Moderate loss
Year Types	+1.60 (P26)	+2.24 (P16)	+0.64	Modest gain
Daniel/Rev	+1.76 (P12)	+2.47 (P15)	+0.71	Modest gain
Ezekiel	+2.68 (P30)	+4.68 (P30)	+2.00	Gains 2.0σ at 10×

6.3 Interpretation

The TRIO loses nearly 2.4 standard deviations going from 1× to 10×. Ezekiel gains 2.0 standard deviations. This is selectivity, not inflation—if 10× scaling were simply making everything look better, the TRIO should have improved as well. Instead, each family is optimized for its own octave. The TRIO’s small cycles (2,300–2,520) discriminate beautifully in the small residue spaces where early quadruplets land. Ezekiel’s larger cycles (3,900–4,300 at 1×; 39,000–43,000 at 10×) require bigger residue spaces and more quadruplets to express their structure.

Critically, even at 1×, Ezekiel already exhibits its signature late-rising behavior. It is the only family still climbing at P30 (Z = +2.68) while every other family has peaked and is decaying. The late-riser profile is intrinsic to the cycle’s residue structure; the 10× magnification gives it room to fully express that structure by expanding the residue space from 3,900–4,300 positions to 39,000–43,000. The DNA is present at both scales; only the amplitude changes.

This baseline comparison provides the strongest evidence against the concern that the 10× Ezekiel result is an artifact of scale. The same scaling that elevates Ezekiel diminishes the TRIO. The two families respond to scale change in opposite directions because they resonate in complementary frequency bands.

7. Target List Robustness

The primary results reported in Sections 4–6 were obtained using target sets derived from the Masoretic Text (MT) patriarchal chronology, encompassing both regular (birth-to-birth) and cumulative dates from Adam through Moses/Joshua. All dates are rounded to the nearest decade using a ±2-year tolerance window (e.g., 3492 → 3490). To test whether the Ezekiel dominance and throne-swap pattern are artifacts of the specific target list used, three variants were compared.

7.1 Target List Definitions

P2 (core): 65 unique dates plus 2 double-hit entries (1520 and 2260, where two patriarchs share the same rounded date), for a total of 67 entries. This is the strict cumulative list as used in File_49. At the P2 level, every regular and cumulative MT patriarchal date from Adam through Moses/Joshua is fully accounted for.

P1 (pure genealogical): P2 plus seven additional MT patriarchs whose regular dates fall just outside the P2 rounding threshold: Enosh (3879 → 3880), Kenan (3789 → 3790), Mahalalel (3719 → 3720), Enoch (3492 → 3490), Lamech (3240 → 3240), Salah (2421 → 2420), and Eber (2391 → 2390). P1 also introduces a third double-hit (Enoch and Peleg both round to 3490). Total: 69 unique + 3 double-hits = 72 entries. This remains a purely genealogical list—no dates external to the patriarchal chronology.

P1 + Danielic: P1 plus four direct Danielic dates included for testing purposes: the 70-year exile endpoints 606 BC and 536 BC (rounded to 600, 530), and the initial prophetic fulfillments of Daniel 9 and Daniel 8 at 445 BC and 165 BC (rounded to 440, 160). Total: 73 unique + 3 double-hits = 76 entries. The Danielic dates are not part of the genealogical chronology proper and tend to act as a slight drag on all tests.

7.2 Three-Way Comparison at 10×

The two-cycle-per-family configuration was run with all three target lists at the 10× scale. Ezekiel peak Z-scores (both_mod30) at P30:

Family	P2 (67)	P1 (72)	+Dan (76)	Range	Pattern
Ezekiel	+4.68	+5.00	+4.73	0.32	Dominant in all three
Year Types	+2.24	+2.41	+2.02	0.39	Stable mid-tier
Key of 23	+2.27	+2.40	+2.05	0.35	Stable mid-tier
Dan/Rev	+2.47	+2.19	+2.40	0.28	Moderate
TRIO	+1.66	+1.82	+1.42	0.40	Lowest in all three

The family ranking is identical across all three target lists. Ezekiel dominates in every case, peaking highest with the pure genealogical P1 list (Z = +5.00 at P30)—the strongest Ezekiel reading in the entire series. The Danielic dates act as a slight drag on all families (compare P1 at +5.00 vs P1+Dan at +4.73), confirming that these historical dates, while legitimate, are not part of the core genealogical resonance. Their inclusion or exclusion does not affect the family hierarchy.

7.3 Three-Way Comparison at 1× (Baseline)

The throne swap (Section 6) was verified across all three target lists at the 1× scale:

Family	P2 (67)	P1 (72)	+Dan (76)	Peak @	Pattern
TRIO	+3.77	+3.67	+4.02	P12	Dominant at 1×
Ezekiel	+2.76	+2.88	+2.68	P30	Still climbing at P30
Key of 23	+2.57	+2.57	+2.88	P12	Second tier
Year Types	+1.69	+1.79	+1.60	P26	Moderate
Daniel/Rev	+1.20	+1.22	+1.76	P12	Weakest

The throne swap is preserved in all three target lists: the TRIO dominates at 1× (Z = +3.67 to +4.02 at P12), while Ezekiel is the only family still climbing at P30 (Z = +2.68 to +2.88). As at 10×, the pure genealogical P1 list produces slightly stronger results than either P2 alone or P1+Danielic, suggesting that the seven additional patriarchs contribute modestly to the signal while the Danielic dates slightly dilute it. The scale-dependent complementarity between TRIO and Ezekiel is invariant across all three target compositions.

7.4 Double-Hit Treatment

In the P1 pure target list, three dates are “double-hits”—dates at which two patriarchs share the same rounded value: 1520 (two patriarchs coincide), 2260 (two patriarchs coincide), and 3490 (Enoch and Peleg). In the primary analysis, these dates are counted twice in the target set, giving them double weight as nodes. A secondary run was conducted in which all three double-hits were treated as single entries (no extra weighting), reducing the P1 list from 72 to 69 entries.

The effect of removing the double-hit weighting is to strengthen the Ezekiel signal:

	With doubles	No doubles	Δ	Note
Ezekiel 10× (P30)	+5.00	+5.22	+0.22	Strongest reading
TRIO 1× (P12)	+3.67	+3.64	−0.03	Negligible change
Ezekiel 1× (P30)	+2.88	+3.11	+0.23	Improves at both scales

This is reported for transparency. The primary analysis retains the double-hit weighting because the duplication reflects a genuine chronological fact (two patriarchs sharing a date). Removing the doubles would have produced a stronger headline result (Z = +5.22 vs +5.00 at 10×), but we did not optimize by choosing the more favorable treatment. The three double-hit nodes appear to introduce slight noise into the residue structure; their removal sharpens the signal rather than creating it. The family hierarchy and throne-swap pattern are identical under both treatments.

8. Supplementary Finding: 25,800 and Ezekiel’s Dual Identity

A notable secondary finding concerns the cycle 25,800. This number appears in the Year Types family as a variant of the Precession of the Equinoxes (the standard astronomical value being 25,920), but it is also 6 × 4,300—directly linking it to Ezekiel’s second number (430). When tested as a Year Type cycle, 25,800 sustains Z ≈ +1.1 to +1.8 through P50, significantly outperforming the standard Precession value of 25,920, which crosses zero at P20 and accelerates to Z ≈ −1.0 by P50.

The two numbers differ by only 120 (0.46%), yet their performance diverges radically. The standard astronomical constant fails; the Ezekiel-linked variant works. This suggests that the resonance structure is sensitive to the specific factorization of the cycle period, not merely its approximate magnitude. The 120-year difference—which maps to the distinction between a purely astronomical constant and one that factors through 4,300—does real work in the residue structure.

9. The Independence Argument

The force of the Ezekiel finding rests on its independence from the framework that produced it. Several structural features support this:

Non-circularity: The TRIO [2300, 2520, 3430] was established as the bedrock test before Ezekiel’s numbers were considered at this scale. The numbers 390, 430, 1300, 3900, 4300, 13000, 39000, and 43000 are absent from the TRIO. Ezekiel’s dominance is therefore a discovery, not a self-reinforcing assumption.

Different biblical source: The TRIO derives from Daniel 8:14, Daniel 9, and related prophetic texts. Ezekiel 4:4–6 is a distinct passage with a distinct theological function (the acted-out punishment of Israel and Judah), written by a different prophet in a different context. The mathematical resonance between these independent textual sources is not guaranteed by any known hermeneutical principle.

Different resonance regime: The TRIO operates as a small-prime specialist (P6–P16), while Ezekiel operates as a large-prime specialist (P16–P30). These regimes were identified independently. The TRIO’s P16 ceiling was established through differential tier analysis before Ezekiel was tested. Ezekiel’s P16 ignition point was discovered after the fact. Two independent lines of investigation converged on the same boundary from opposite sides.

Statistical base: Ezekiel’s peak at P29 rests on 79 quadruplets—more than double the sample size at which other families peak. A fluke that persists across 79 independent prime quadruplets, each tested against 76 targets through modular arithmetic in a residue space of 39,000 positions, carries substantially more weight than a comparable Z-score at P12 (28 quadruplets).

10. The Two-Octave Architecture

The combined findings point to a chronological framework with two distinct frequency bands—two “octaves” in which the prime quadruplet structure resonates with biblical cycles:

Lower octave (1× scale, cycles 1,000–7,000): The TRIO and its allied families operate here. The signal is concentrated in P6–P16 (quadruplets with Q values 211–90,611). neg_mod30 is the dominant polarity. The peak Z-scores reach approximately +4 to +5 under the full 19-cycle configuration. This is the regime documented in File_49 and previous analyses.

Upper octave (10× scale, cycles 23,000–69,000): Ezekiel’s curse numbers operate here. The signal is concentrated in P12–P30. pos_mod30 becomes the dominant polarity—a reversal of the lower octave. The peak Z-score reaches +4.84 under the single-cycle Ezekiel configuration. This regime was invisible at the 1× scale because cycles of 130 and 430 produce small residue spaces that cannot discriminate effectively.

The same mathematical architecture—prime quadruplet rail values tested via modular arithmetic against biblical targets—produces significant results in both octaves, but through different mechanisms, different quad ranges, and different polarity channels. The curse numbers don’t destroy the structure; they express it from the complementary side.

11. Theological Implications

The mathematical findings carry theological resonance at multiple levels. The numbers 390 and 430 represent, in Ezekiel’s acted prophecy, the duration of divine judgment against Israel and Judah. That these “punishment” numbers should produce the strongest chronological resonance in the system inverts conventional expectations: the curse is not outside the structure but is integral to it.

The factorization 390 = 3 × 130, where 130 has been identified in the broader framework as symbolizing opposition to the divine order represented by 30, adds a further layer. The curse number is literally three measures of chaos—and it resonates through the complementary (pos_mod30) polarity at the 10× scale, the channel that stands opposite to the 30-based order. Divine judgment and divine order operate through paired mathematical channels, each inaccessible from the other’s scale.

The gap in File_49—the apparent absence of 390 and 430 from the productive cycle families—was not a gap. It was a signal at a different magnification, requiring the 10× scaling experiment to reveal. What appeared to be the weakest component of the framework is, at the appropriate scale, the strongest. This inversion (the last becoming first, the weakness becoming strength) is itself a pattern with deep biblical resonance, though we note this parallel with appropriate scholarly caution.

12. Limitations and Caveats

Several caveats apply to these findings. The cycle families, while rooted in biblical text, involve interpretive choices in their scaling and grouping. The 10× multiplier, while principled (it tests scale-robustness), is not the only possible scaling factor. Alternative factors (5×, 20×, 100×) have not been systematically tested and could produce different family rankings.

The target set of 76 biblical dates, while established independently of the cycle selection, reflects chronological judgments that carry their own uncertainties. The Z-scores reported here are conditional on this target set being correct.

The Year Types family’s anti-correlation at 10× (Z = −1.06 at P50) warrants further investigation. A family that is positively correlated at one scale and negatively correlated at another may carry information about the scale-dependence of the underlying structure, but it also raises questions about which scale, if any, represents the “true” relationship.

This report documents findings from a specific series of experiments and should be considered a working draft. Further validation—including tests with randomized target sets, alternative scaling factors, and independent replication—would strengthen the conclusions presented here.

13. Conclusion

The 10× scaling experiment reveals a two-octave structure in the relationship between prime quadruplets and biblical chronological cycles. At the lower octave, the TRIO [2300, 2520, 3430] and its allied families dominate through small prime quadruplets (P6–P16) via the neg_mod30 polarity. At the upper octave, Ezekiel’s curse numbers [39,000 and 43,000] dominate through larger quadruplets (P16–P30) via the complementary pos_mod30 polarity.

The Ezekiel signal (peak Z = +4.84 at P29, sustaining Z > 3.5 through P50) is the strongest and most durable finding in the entire analytical program. It emerges from cycles that were absent from the original framework, derived from an independent biblical source, operating through the opposite polarity channel, at a scale that was not initially examined. Every aspect of its discovery is structurally independent from the TRIO-based framework that made the discovery possible.

The handoff between regimes at P16—discovered independently from both directions—suggests that the prime quadruplet structure carries biblical chronological information at multiple scales, with different prophetic traditions encoding different frequency bands of the same underlying architecture. The curse is not outside the pattern. It is the pattern’s other voice.

========================================================================================================================

#!/usr/bin/env python3

“””

POLARITY COMPARISON — neg_mod30 vs pos_mod30

==============================================

Companion to the generic both_mod30 validation script.

Purpose: Demonstrate that the two rail polarities contribute

asymmetrically to the signal, confirming the structural

correspondence between neg_mod30 and BC (negative) dates.

Rail systems tested per tier:

neg_mod30 : B, -B, -M (3 rails per quad)

pos_mod30 : B, M, -B (3 rails per quad)

both_mod30 : B, M, -B, -M (4 rails per quad — reference baseline)

Targets : MT-only unsigned targets.

(P2-normalized via decadal admission gate; no minus signs.)

Core claim : TRIO_CORE = [2300, 2520, 3430]

Expanded : ALL_CYCLES = Year + Key23 + Daniel/Revelation + Ezekiel families

Tier naming : Pn denotes tier n of the curated build (P3–P30).

Tier sizes grow monotonically; no fixed formula assumed.

Z-score method: Multi-seed MC (20 seeds x 100,000 samples) plus

analytical (exact mean; variance exact when joint periods <= 1e7,

else covariance is MC-estimated).

“””

import math

import random

import warnings

from functools import reduce

import numpy as np

# — Colab / matplotlib setup —

try:

import matplotlib

import matplotlib.pyplot as plt

import matplotlib.gridspec as gridspec

HAS_MPL = True

plt.rcParams.update({

‘figure.facecolor’: ‘#FAFAF8’,

‘axes.facecolor’: ‘#FAFAF8’,

‘axes.edgecolor’: ‘#333333’,

‘axes.labelcolor’: ‘#222222’,

‘text.color’: ‘#222222’,

‘xtick.color’: ‘#444444’,

‘ytick.color’: ‘#444444’,

‘grid.color’: ‘#DDDDDD’,

‘grid.linestyle’: ‘–‘,

‘grid.alpha’: 0.7,

‘font.family’: ‘serif’,

‘font.size’: 11,

‘axes.titlesize’: 13,

‘axes.titleweight’: ‘bold’,

‘figure.titlesize’: 15,

‘figure.titleweight’:’bold’,

})

except ImportError:

HAS_MPL = False

print(“⚠ matplotlib not found — charts will be skipped.”)

# =========================================================

# COLOUR PALETTE

# =========================================================

C_NEG = ‘#C0392B’ # warm red — neg_mod30

C_POS = ‘#2E86C1’ # cool blue — pos_mod30

C_BOTH = ‘#7D8C8E’ # neutral grey — both_mod30 reference

C_OBS = ‘#1B1B1B’ # near-black — observed marker

# =========================================================

# TARGETS (MT only: Cum ±Cainan + Regular; P2-normalized

# via decadal admission gate; no minus signs)

# These are the post-gate decade admissions produced from

# MT Regular + MT Cumulative (±Cainan) using the P2

# naturalization-before-gate protocol. This script does

# not regenerate them; if the forwarded bands or the P2

# gate are revised, TARGETS must be regenerated and rerun.

# DOUBLE HITS — three dates qualify independently in both

# Regular and Cumulative chronologies and are therefore

# counted twice (two gate admissions each):

# 1520 Moses & Aaron in both Reg and Cum

# 2260 Nahor in Reg, Isaac in Cum

# 3490 Enoch in Reg, Peleg in Cum

# =========================================================

# TARGETS: Special. To P1 rather than P2 as normal, which caused the following seven patriarchs to be excluded from old list.

# (Same +/-2-yr decadal rule).

# Enosh (3879) 3880. Kenan (3789) 3790. Mahalalel (3719) 3720. Enoch (3492) 3490. Lamech (3240) 3240. Salah = (2421) 2420. Eber (2391) 2390.

# These seven do not appear to affect overall results one way or the other.

# Thus, either to P1 and P2, all regular MT Chronology patriarchs from Adam to Moses/Joshua are fully accounted for. (Cumulative only needs P2.)

# Direct Danielic dates are the 70-yr exile (P2) from 606-536 BC (600, 530) and 445, 165 BC (440, 160) for initial prophecy fulfillment of Daniel_9 and ch._8, respectively.

# The full P2 list that includes both reg and cum are:

# 1400, 1440, 1470, 1520, 1870, 2000, 2060, 2070,

# 2080, 2160, 2230, 2250, 2260, 2290, 2320, 2350, 2430, 2450,

# 2550, 2630, 2640, 2780, 3010, 3050, 3120, 3250, 3420, 3490,

# 3650, 3950, 3960, 3980, 4110, 4390, 4830, 4850, 5290, 5430,

# 5890, 6380, 6840, 7150, 7610, 8120, 8490, 8580, 8950, 9450,

# 9910, 10340, 10350, 10800, 10810, 11250, 11260, 11710, 11720, 12160,

# 12620, 13070, 13530, 14000, 14460,

# “Double dates” are when two patriarchs have the same date: 1520, 2260. With P1, Enoch doubles with Peleg (3490).

# =========================================================

_UNIQUE_TARGETS = sorted(set([

160, 440, 530, 600,

1400, 1440, 1470, 1520, 1870, 2000, 2060, 2070,

2080, 2160, 2230, 2250, 2260, 2290, 2320, 2350, 2390, 2420, 2430, 2450,

2550, 2630, 2640, 2780, 3010, 3050, 3120, 3240, 3250, 3420, 3490,

3650, 3720, 3790, 3880, 3950, 3960, 3980, 4110, 4390, 4830, 4850, 5290, 5430,

5890, 6380, 6840, 7150, 7610, 8120, 8490, 8580, 8950, 9450,

9910, 10340, 10350, 10800, 10810, 11250, 11260, 11710, 11720, 12160,

12620, 13070, 13530, 14000, 14460,

]))

DOUBLE_TARGETS = [1520, 2260, 3490] # counted twice (Reg + Cum)

TARGETS = sorted(_UNIQUE_TARGETS + DOUBLE_TARGETS)

print(f”Targets: {len(TARGETS)} entries “

f”({len(_UNIQUE_TARGETS)} unique + {len(DOUBLE_TARGETS)} double-hits)”)

# =========================================================

# CYCLE FAMILIES

# =========================================================

# ── Core claim: the standard trio ──

TRIO_CORE = [23000, 25200]

FAMILY_YEAR = [ 33600, 25800]

FAMILY_KEY23 = [ 23000, 69000]

FAMILY_DANIEL_REVELATION = [12000, 33300]

FAMILY_EZEKIEL = [ 39000, 43000]

ALL_CYCLES = FAMILY_YEAR + FAMILY_KEY23 + FAMILY_DANIEL_REVELATION + FAMILY_EZEKIEL

STEP = 30

# =========================================================

# PRIME QUADRUPLET TIERS (P3–P30)

# Pn denotes tier n of the curated build.

# Tier sizes grow monotonically (verified at startup).

# =========================================================

PRIME_QUAD_LISTS = {

‘P3’: [101, 191],

‘P4’: [101, 191, 281, 911],

‘P5’: [101, 191, 281, 821, 911, 1091, 1751],

‘P6’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 2561, 2951],

‘P7’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2561, 2951, 4421, 4631],

‘P8’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 4421, 4631, 6491, 7661],

‘P9’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 3251, 4421, 4631, 6491, 7661, 9731, 9941],

‘P10’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 3251, 3461, 4421, 4631, 6491, 7661, 9731, 9941, 13181, 15371],

‘P11’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 3251, 3461, 4421, 4631, 5651, 6491, 7661, 9731, 9941, 13181, 15371, 18821, 22601],

‘P12’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 3251, 3461, 4421, 4631, 5651, 6491, 7661, 9431, 9731, 9941, 13181, 15371, 18821, 22601, 28241, 31811],

‘P13’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 3251, 3461, 4421, 4631, 5651, 6491, 7661, 9431, 9731, 9941, 13001, 13181, 15371, 18821, 22601, 28241, 31811, 41231, 43871],

‘P14’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 3251, 3461, 4421, 4631, 5651, 6491, 7661, 9431, 9731, 9941, 13001, 13181, 15371, 15641, 18821, 22601, 28241, 31811, 41231, 43871, 56861, 56951],

‘P15’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 3251, 3461, 4421, 4631, 5651, 6491, 7661, 9431, 9731, 9941, 13001, 13181, 15371, 15641, 15731, 18821, 22601, 28241, 31811, 41231, 43871, 56861, 56951, 72581, 72911],

‘P16’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 3251, 3461, 4421, 4631, 5651, 6491, 7661, 9431, 9731, 9941, 13001, 13181, 15371, 15641, 15731, 16061, 18821, 22601, 28241, 31811, 41231, 43871, 56861, 56951, 72581, 72911, 88631, 90611],

‘P17’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 3251, 3461, 4421, 4631, 5651, 6491, 7661, 9431, 9731, 9941, 13001, 13181, 15371, 15641, 15731, 16061, 18041, 18821, 22601, 28241, 31811, 41231, 43871, 56861, 56951, 72581, 72911, 88631, 90611, 106661, 107531],

‘P18’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 3251, 3461, 4421, 4631, 5651, 6491, 7661, 9431, 9731, 9941, 13001, 13181, 15371, 15641, 15731, 16061, 18041, 18821, 18911, 22601, 28241, 31811, 41231, 43871, 56861, 56951, 72581, 72911, 88631, 90611, 106661, 107531, 125561, 126071],

‘P19’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 3251, 3461, 4421, 4631, 5651, 6491, 7661, 9431, 9731, 9941, 13001, 13181, 15371, 15641, 15731, 16061, 18041, 18821, 18911, 19421, 22601, 28241, 31811, 41231, 43871, 56861, 56951, 72581, 72911, 88631, 90611, 106661, 107531, 125561, 126071, 144971, 146561],

‘P20’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 3251, 3461, 4421, 4631, 5651, 6491, 7661, 9431, 9731, 9941, 13001, 13181, 15371, 15641, 15731, 16061, 18041, 18821, 18911, 19421, 21011, 22601, 28241, 31811, 41231, 43871, 56861, 56951, 72581, 72911, 88631, 90611, 106661, 107531, 125561, 126071, 144971, 146561, 165971, 167231],

‘P21’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 3251, 3461, 4421, 4631, 5651, 6491, 7661, 9431, 9731, 9941, 13001, 13181, 15371, 15641, 15731, 16061, 18041, 18821, 18911, 19421, 21011, 22271, 22601, 25301, 28241, 31721, 31811, 34841, 41231, 43781, 43871, 51341, 55331, 56861, 56951, 62981, 67211, 69491, 72581, 72911, 88631, 90611, 106661],

‘P22’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 3251, 3461, 4421, 4631, 5651, 6491, 7661, 9431, 9731, 9941, 13001, 13181, 15371, 15641, 15731, 16061, 18041, 18821, 18911, 19421, 21011, 22271, 22601, 25301, 28241, 31721, 31811, 34841, 41231, 43781, 43871, 51341, 55331, 56861, 56951, 62981, 67211, 69491, 72581, 72911, 88631, 90611, 106661, 107531, 125561, 126071],

‘P23’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 3251, 3461, 4421, 4631, 5651, 6491, 7661, 9431, 9731, 9941, 13001, 13181, 15371, 15641, 15731, 16061, 18041, 18821, 18911, 19421, 21011, 22271, 22601, 25301, 28241, 31721, 31811, 34841, 41231, 43781, 43871, 51341, 55331, 56861, 56951, 62981, 67211, 69491, 72581, 72911, 88631, 90611, 106661, 107531, 125561, 126071, 144971, 146561, 165971],

‘P24’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 3251, 3461, 4421, 4631, 5651, 6491, 7661, 9431, 9731, 9941, 13001, 13181, 15371, 15641, 15731, 16061, 18041, 18821, 18911, 19421, 21011, 22271, 22601, 25301, 28241, 31721, 31811, 34841, 41231, 43781, 43871, 51341, 55331, 56861, 56951, 62981, 67211, 69491, 72581, 72911, 88631, 90611, 106661, 107531, 125561, 126071, 144971, 146561, 165971, 167231, 188231, 191261],

‘P25’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 3251, 3461, 4421, 4631, 5651, 6491, 7661, 9431, 9731, 9941, 13001, 13181, 15371, 15641, 15731, 16061, 18041, 18821, 18911, 19421, 21011, 22271, 22601, 25301, 28241, 31721, 31811, 34841, 41231, 43781, 43871, 51341, 55331, 56861, 56951, 62981, 67211, 69491, 72581, 72911, 88631, 90611, 106661, 107531, 125561, 126071, 144971, 146561, 165971, 167231, 188231, 191261, 213521, 219941, 245231],

‘P26’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 3251, 3461, 4421, 4631, 5651, 6491, 7661, 9431, 9731, 9941, 13001, 13181, 15371, 15641, 15731, 16061, 18041, 18821, 18911, 19421, 21011, 22271, 22601, 25301, 28241, 31721, 31811, 34841, 41231, 43781, 43871, 51341, 55331, 56861, 56951, 62981, 67211, 69491, 72581, 72911, 88631, 90611, 106661, 107531, 125561, 126071, 144971, 146561, 165971, 167231, 188231, 191261, 213521, 219941, 245231, 248351, 280061, 289001],

‘P27’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 3251, 3461, 4421, 4631, 5651, 6491, 7661, 9431, 9731, 9941, 13001, 13181, 15371, 15641, 15731, 16061, 18041, 18821, 18911, 19421, 21011, 22271, 22601, 25301, 28241, 31721, 31811, 34841, 41231, 43781, 43871, 51341, 55331, 56861, 56951, 62981, 67211, 69491, 72581, 72911, 88631, 90611, 106661, 107531, 125561, 126071, 144971, 146561, 165971, 167231, 188231, 191261, 213521, 219941, 245231, 248351, 280061, 289001, 323831, 331391, 375161],

‘P28’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 3251, 3461, 4421, 4631, 5651, 6491, 7661, 9431, 9731, 9941, 13001, 13181, 15371, 15641, 15731, 16061, 18041, 18821, 18911, 19421, 21011, 22271, 22601, 25301, 28241, 31721, 31811, 34841, 41231, 43781, 43871, 51341, 55331, 56861, 56951, 62981, 67211, 69491, 72581, 72911, 88631, 90611, 106661, 107531, 125561, 126071, 144971, 146561, 165971, 167231, 188231, 191261, 213521, 219941, 245231, 248351, 280061, 289001, 323831, 331391, 375161, 379151, 430481, 438131],

‘P29’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 3251, 3461, 4421, 4631, 5651, 6491, 7661, 9431, 9731, 9941, 13001, 13181, 15371, 15641, 15731, 16061, 18041, 18821, 18911, 19421, 21011, 22271, 22601, 25301, 28241, 31721, 31811, 34841, 41231, 43781, 43871, 51341, 55331, 56861, 56951, 62981, 67211, 69491, 72581, 72911, 88631, 90611, 106661, 107531, 125561, 126071, 144971, 146561, 165971, 167231, 188231, 191261, 213521, 219941, 245231, 248351, 280061, 289001, 323831, 331391, 375161, 379151, 430481, 438131, 493451, 497681, 560651],

‘P30’: [101, 191, 281, 821, 911, 1091, 1481, 1751, 1871, 2081, 2561, 2951, 3251, 3461, 4421, 4631, 5651, 6491, 7661, 9431, 9731, 9941, 13001, 13181, 15371, 15641, 15731, 16061, 18041, 18821, 18911, 19421, 21011, 22271, 22601, 25301, 28241, 31721, 31811, 34841, 41231, 43781, 43871, 51341, 55331, 56861, 56951, 62981, 67211, 69491, 72581, 72911, 88631, 90611, 106661, 107531, 125561, 126071, 144971, 146561, 165971, 167231, 188231, 191261, 213521, 219941, 245231, 248351, 280061, 289001, 323831, 331391, 375161, 379151, 430481, 438131, 493451, 497681, 560651, 562931, 630131, 632861],

}

# ── Tier sanity check: monotonic growth + report sizes ──

_sorted_tiers = sorted(PRIME_QUAD_LISTS, key=lambda s: int(s[1:]))

_prev_size = 0

for _pn in _sorted_tiers:

_sz = len(PRIME_QUAD_LISTS[_pn])

assert _sz > _prev_size, (

f”Tier {_pn}: size {_sz} not greater than previous tier ({_prev_size})”

)

_prev_size = _sz

print(f”✓ All {len(PRIME_QUAD_LISTS)} tiers pass monotonic-growth check “

f”({_sorted_tiers[0]}={len(PRIME_QUAD_LISTS[_sorted_tiers[0]])} → “

f”{_sorted_tiers[-1]}={len(PRIME_QUAD_LISTS[_sorted_tiers[-1]])})”)

# =========================================================

# HELPERS

# =========================================================

def compute_lcm(nums):

def lcm(a, b):

return abs(a * b) // math.gcd(a, b)

return reduce(lcm, nums, 1)

def build_rails(quads, config=”both_mod30″):

“””Generate rails from quadruplet starts.”””

rails = []

for Q in quads:

B = Q – 11

M = B + 30

if config == “no_mod30”: rails.extend([B, -B])

elif config == “neg_mod30”: rails.extend([B, -B, -M])

elif config == “pos_mod30”: rails.extend([B, M, -B])

elif config == “both_mod30”: rails.extend([B, M, -B, -M])

return rails

def score_observed(rails, targets, cycles):

total = 0

for c in cycles:

for r in rails:

for t in targets:

if (t – r) % c == 0:

total += 1

return total

def compute_K(cycles, step=STEP):

“””LCM of periods — the shift lattice cardinality.”””

periods = [c // math.gcd(c, step) for c in cycles]

return compute_lcm(periods)

# =========================================================

# MC + ANALYTICAL ENGINES

# =========================================================

MC_SAMPLES = 20_000

MC_RUNS = 20

def multi_seed_mc(rails, targets, cycles, n_runs=MC_RUNS,

samples_per_run=MC_SAMPLES):

“””Run multiple independent MC trials → CI on Z.”””

obs = score_observed(rails, targets, cycles)

residue_counts = []

for c in cycles:

counts = np.zeros(c, dtype=np.int32)

for r in rails:

for t in targets:

counts[(t – r) % c] += 1

residue_counts.append((c, counts))

K = compute_K(cycles)

z_values = []

all_scores = []

for run in range(n_runs):

rng = random.Random(run * 12345 + 7)

scores = np.empty(samples_per_run, dtype=np.int32)

for i in range(samples_per_run):

offset = rng.randrange(K)

shift = offset * STEP

total = 0

for c, counts in residue_counts:

total += counts[shift % c]

scores[i] = total

mean = float(np.mean(scores))

std = float(np.std(scores, ddof=0))

if std > 1e-9:

z = (obs – mean) / std

z_values.append(z)

all_scores.append(scores)

return z_values, obs, all_scores

def analytical_stats(rails, targets, cycles, verbose=False):

“””Exact E[S]; Var[S] exact when all joint periods <= 1e7,

else covariance is estimated by sampling.”””

obs = score_observed(rails, targets, cycles)

total_mean = 0.0

total_var_independent = 0.0

cycle_histograms = {}

for c in cycles:

g = math.gcd(c, STEP)

period = c // g

hist = np.zeros(period, dtype=np.int64)

for r in rails:

for t in targets:

diff = (t – r) % c

if diff % g == 0:

slot = (diff // g) % period

hist[slot] += 1

e_hits = float(np.mean(hist))

v_hits = float(np.var(hist, ddof=0))

total_mean += e_hits

total_var_independent += v_hits

cycle_histograms[c] = (g, period, hist)

total_cov = 0.0

n_cycle_pairs = 0

for i, c1 in enumerate(cycles):

_g1, p1, h1 = cycle_histograms[c1]

for c2 in cycles[i + 1:]:

_g2, p2, h2 = cycle_histograms[c2]

joint_period = compute_lcm([p1, p2])

mu1 = float(np.mean(h1))

mu2 = float(np.mean(h2))

if joint_period <= 10_000_000:

offsets = np.arange(joint_period, dtype=np.int64)

vals1 = h1[offsets % p1].astype(np.float64)

vals2 = h2[offsets % p2].astype(np.float64)

e_product = float(np.mean(vals1 * vals2))

else:

rng = random.Random(42 + i * 1000 + n_cycle_pairs)

n_sample = min(500_000, joint_period)

offsets = np.array([rng.randrange(joint_period)

for _ in range(n_sample)], dtype=np.int64)

vals1 = h1[offsets % p1].astype(np.float64)

vals2 = h2[offsets % p2].astype(np.float64)

e_product = float(np.mean(vals1 * vals2))

cov = e_product – mu1 * mu2

total_cov += cov

n_cycle_pairs += 1

total_var = total_var_independent + 2.0 * total_cov

if total_var < 0:

warnings.warn(f”Negative total_var ({total_var:.6f}) — clamping.”)

total_var = total_var_independent

total_std = math.sqrt(total_var) if total_var > 0 else 1e-12

z_analytical = (obs – total_mean) / total_std

if verbose:

print(f” Obs={obs} E[S]={total_mean:.2f} “

f”std={total_std:.2f} Z={z_analytical:+.4f}”)

return z_analytical, obs, total_mean, total_std

# =========================================================

# EVALUATE ONE CONFIGURATION

# =========================================================

def evaluate_config(config_name, quads, targets, cycles):

“””Run MC + analytical for one rail configuration. Returns dict.”””

rails = build_rails(quads, config_name)

# MC

z_values, obs, score_lists = multi_seed_mc(rails, targets, cycles)

z_mean = float(np.mean(z_values))

z_std = float(np.std(z_values, ddof=1)) if len(z_values) > 1 else 0.0

ci_lo = z_mean – 1.96 * z_std / math.sqrt(max(len(z_values), 1))

ci_hi = z_mean + 1.96 * z_std / math.sqrt(max(len(z_values), 1))

# Analytical

z_anal, _, mean_a, std_a = analytical_stats(rails, targets, cycles)

return {

‘config’: config_name,

‘n_rails’: len(rails),

‘obs’: obs,

‘mc_z_mean’: z_mean,

‘mc_z_std’: z_std,

‘mc_ci’: (ci_lo, ci_hi),

‘mc_z_values’: z_values,

‘mc_scores_seed0’: score_lists[0] if score_lists else np.array([]),

‘analytical_z’: z_anal,

‘analytical_mean’: mean_a,

‘analytical_std’: std_a,

}

# =========================================================

# CHARTS

# =========================================================

def plot_polarity_ladder(all_results):

“””Fig 1: Z-score progression — neg vs pos vs both, TRIO.”””

if not HAS_MPL:

return

tiers = list(all_results.keys())

n = len(tiers)

x = np.arange(n)

fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(15, 10),

constrained_layout=True,

sharex=True)

fig.suptitle(‘Polarity Comparison — neg_mod30 vs pos_mod30 vs both_mod30’,

fontsize=16, fontweight=’bold’)

# ── Panel A: TRIO ──

for cfg, color, ls, marker in [

(‘neg_mod30’, C_NEG, ‘-‘, ‘o’),

(‘pos_mod30’, C_POS, ‘-‘, ‘s’),

(‘both_mod30’, C_BOTH, ‘–‘, ‘D’),

z_vals = [all_results[t][‘trio’][cfg][‘mc_z_mean’] for t in tiers]

ci_lo = [all_results[t][‘trio’][cfg][‘mc_ci’][0] for t in tiers]

ci_hi = [all_results[t][‘trio’][cfg][‘mc_ci’][1] for t in tiers]

ax1.fill_between(x, ci_lo, ci_hi, color=color, alpha=0.10)

ax1.plot(x, z_vals, f'{marker}{ls}’, color=color, lw=2, ms=6,

label=f'{cfg}’, alpha=0.9)

ax1.axhline(0, color=’#999999′, lw=0.8)

ax1.set_ylabel(‘Z-score (MC)’)

ax1.set_title(‘A. TRIO [2300, 2520, 3430]’)

ax1.legend(fontsize=10, framealpha=0.9, edgecolor=’#CCCCCC’)

ax1.grid(axis=’y’)

# ── Panel B: ALL ──

for cfg, color, ls, marker in [

(‘neg_mod30’, C_NEG, ‘-‘, ‘o’),

(‘pos_mod30’, C_POS, ‘-‘, ‘s’),

(‘both_mod30’, C_BOTH, ‘–‘, ‘D’),

z_vals = [all_results[t][‘all’][cfg][‘mc_z_mean’] for t in tiers]

ci_lo = [all_results[t][‘all’][cfg][‘mc_ci’][0] for t in tiers]

ci_hi = [all_results[t][‘all’][cfg][‘mc_ci’][1] for t in tiers]

ax2.fill_between(x, ci_lo, ci_hi, color=color, alpha=0.10)

ax2.plot(x, z_vals, f'{marker}{ls}’, color=color, lw=2, ms=6,

label=f'{cfg}’, alpha=0.9)

ax2.axhline(0, color=’#999999′, lw=0.8)

ax2.set_ylabel(‘Z-score (MC)’)

ax2.set_xlabel(‘Prime-Quadruplet Tier’)

ax2.set_title(f’B. ALL CYCLES ({len(ALL_CYCLES)} cycles)’)

ax2.set_xticks(x)

ax2.set_xticklabels(tiers, fontsize=10)

ax2.legend(fontsize=10, framealpha=0.9, edgecolor=’#CCCCCC’)

ax2.grid(axis=’y’)

plt.savefig(‘polarity_ladder.png’, dpi=180, bbox_inches=’tight’,

facecolor=fig.get_facecolor())

plt.show()

print(” ✓ Saved → polarity_ladder.png”)

def plot_polarity_delta(all_results):

“””Fig 2: Delta Z (neg − pos) across tiers for TRIO and ALL.”””

if not HAS_MPL:

return

tiers = list(all_results.keys())

n = len(tiers)

x = np.arange(n)

trio_delta = [all_results[t][‘trio’][‘neg_mod30’][‘mc_z_mean’]

– all_results[t][‘trio’][‘pos_mod30’][‘mc_z_mean’]

for t in tiers]

all_delta = [all_results[t][‘all’][‘neg_mod30’][‘mc_z_mean’]

– all_results[t][‘all’][‘pos_mod30’][‘mc_z_mean’]

for t in tiers]

fig, ax = plt.subplots(figsize=(14, 5), constrained_layout=True)

w = 0.35

ax.bar(x – w/2, trio_delta, w, color=C_NEG, alpha=0.7,

label=’TRIO (neg − pos)’, edgecolor=’white’, linewidth=0.5)

ax.bar(x + w/2, all_delta, w, color=C_POS, alpha=0.7,

label=’ALL (neg − pos)’, edgecolor=’white’, linewidth=0.5)

ax.axhline(0, color=’#999999′, lw=1)

ax.set_xticks(x)

ax.set_xticklabels(tiers, fontsize=10)

ax.set_xlabel(‘Prime-Quadruplet Tier’)

ax.set_ylabel(‘ΔZ (neg_mod30 − pos_mod30)’)

ax.set_title(‘Polarity Asymmetry: ΔZ = Z(neg) − Z(pos)’)

ax.legend(fontsize=10, framealpha=0.9, edgecolor=’#CCCCCC’)

ax.grid(axis=’y’)

plt.savefig(‘polarity_delta.png’, dpi=180, bbox_inches=’tight’,

facecolor=fig.get_facecolor())

plt.show()

print(” ✓ Saved → polarity_delta.png”)

def plot_polarity_null_comparison(all_results):

“””Fig 3: Side-by-side null distributions for neg vs pos (TRIO),

all tiers in a grid.”””

if not HAS_MPL:

return

tiers = list(all_results.keys())

n = len(tiers)

ncols = 4

nrows = math.ceil(n / ncols)

fig, axes = plt.subplots(nrows, ncols, figsize=(5 * ncols, 3.8 * nrows),

constrained_layout=True, squeeze=False)

fig.suptitle(‘Null Distributions — neg_mod30 vs pos_mod30 (TRIO)’,

fontsize=14, fontweight=’bold’)

for idx, pn in enumerate(tiers):

r, c_idx = divmod(idx, ncols)

ax = axes[r][c_idx]

neg_scores = all_results[pn][‘trio’][‘neg_mod30’][‘mc_scores_seed0’]

pos_scores = all_results[pn][‘trio’][‘pos_mod30’][‘mc_scores_seed0’]

neg_obs = all_results[pn][‘trio’][‘neg_mod30’][‘obs’]

pos_obs = all_results[pn][‘trio’][‘pos_mod30’][‘obs’]

ax.hist(neg_scores, bins=40, density=True, color=C_NEG,

alpha=0.40, edgecolor=’white’, linewidth=0.3, label=’neg null’)

ax.hist(pos_scores, bins=40, density=True, color=C_POS,

alpha=0.40, edgecolor=’white’, linewidth=0.3, label=’pos null’)

ax.axvline(neg_obs, color=C_NEG, lw=2, ls=’–‘,

label=f’neg obs={neg_obs}’)

ax.axvline(pos_obs, color=C_POS, lw=2, ls=’:’,

label=f’pos obs={pos_obs}’)

ax.set_xlabel(‘Score’)

ax.set_ylabel(‘Density’)

ax.set_title(pn, fontsize=12)

ax.legend(fontsize=7, framealpha=0.9, edgecolor=’#CCCCCC’)

ax.grid(axis=’y’)

for idx in range(n, nrows * ncols):

r, c_idx = divmod(idx, ncols)

axes[r][c_idx].set_visible(False)

plt.savefig(‘polarity_null_comparison.png’, dpi=180, bbox_inches=’tight’,

facecolor=fig.get_facecolor())

plt.show()

print(” ✓ Saved → polarity_null_comparison.png”)

def plot_observed_breakdown(all_results):

“””Fig 4: Stacked-style bar chart — observed hits per config.”””

if not HAS_MPL:

return

tiers = list(all_results.keys())

n = len(tiers)

x = np.arange(n)

fig, ax = plt.subplots(figsize=(14, 5), constrained_layout=True)

w = 0.25

for i, (cfg, color) in enumerate([

(‘neg_mod30’, C_NEG), (‘pos_mod30’, C_POS), (‘both_mod30’, C_BOTH)

]):

obs = [all_results[t][‘trio’][cfg][‘obs’] for t in tiers]

ax.bar(x + (i – 1) * w, obs, w, color=color, alpha=0.80,

label=cfg, edgecolor=’white’, linewidth=0.5)

ax.set_xticks(x)

ax.set_xticklabels(tiers, fontsize=10)

ax.set_xlabel(‘Prime-Quadruplet Tier’)

ax.set_ylabel(‘Observed Hits (TRIO)’)

ax.set_title(‘Observed Hits by Rail Configuration — TRIO [2300,2520,3430]’)

ax.legend(fontsize=10, framealpha=0.9, edgecolor=’#CCCCCC’)

ax.grid(axis=’y’)

plt.savefig(‘polarity_observed.png’, dpi=180, bbox_inches=’tight’,

facecolor=fig.get_facecolor())

plt.show()

print(” ✓ Saved → polarity_observed.png”)

def plot_family_comparison(all_results):

“””Fig 5: Per-family Z-score progression across tiers (both_mod30).”””

if not HAS_MPL:

return

tiers = list(all_results.keys())

n = len(tiers)

x = np.arange(n)

# Colours and markers for each family

family_styles = {

‘TRIO’: {‘color’: ‘#1A1A2E’, ‘marker’: ‘o’, ‘lw’: 3.0, ‘ls’: ‘-‘},

‘Year Types’: {‘color’: ‘#E07A5F’, ‘marker’: ‘s’, ‘lw’: 1.8, ‘ls’: ‘-‘},

‘Key of 23’: {‘color’: ‘#3D85C6’, ‘marker’: ‘^’, ‘lw’: 1.8, ‘ls’: ‘-‘},

‘Daniel/Revelation’: {‘color’: ‘#81B29A’, ‘marker’: ‘D’, ‘lw’: 1.8, ‘ls’: ‘-‘},

‘Ezekiel’: {‘color’: ‘#9B5DE5’, ‘marker’: ‘v’, ‘lw’: 1.8, ‘ls’: ‘-‘},

}

fig, ax = plt.subplots(figsize=(15, 7), constrained_layout=True)

for fam_name, style in family_styles.items():

z_vals = []

for t in tiers:

fam_data = all_results[t].get(‘families’, {}).get(fam_name)

z_vals.append(fam_data[‘mc_z_mean’] if fam_data else float(‘nan’))

ax.plot(x, z_vals,

marker=style[‘marker’], linestyle=style[‘ls’],

color=style[‘color’], lw=style[‘lw’], ms=7,

label=fam_name, alpha=0.9)

ax.axhline(0, color=’#999999′, lw=0.8)

ax.set_xticks(x)

ax.set_xticklabels(tiers, fontsize=10)

ax.set_xlabel(‘Prime-Quadruplet Tier’)

ax.set_ylabel(‘Z-score (MC, both_mod30)’)

ax.set_title(‘Per-Family Z-Score Progression — both_mod30 baseline’,

fontsize=14, fontweight=’bold’)

ax.legend(fontsize=11, framealpha=0.9, edgecolor=’#CCCCCC’,

loc=’upper left’)

ax.grid(axis=’y’)

plt.savefig(‘family_comparison.png’, dpi=180, bbox_inches=’tight’,

facecolor=fig.get_facecolor())

plt.show()

print(” ✓ Saved → family_comparison.png”)

# =========================================================

# MAIN

# =========================================================

def main():

tier_names = sorted(PRIME_QUAD_LISTS.keys(), key=lambda s: int(s[1:]))

configs = [‘neg_mod30’, ‘pos_mod30’, ‘both_mod30’]

all_results = {}

# ── Metadata table ──

K_TRIO = compute_K(TRIO_CORE)

K_ALL = compute_K(ALL_CYCLES)

print(“\n” + “=” * 90)

print(” TIER METADATA — POLARITY COMPARISON”)

print(“=” * 90)

print(f” {‘Tier’:<6} {‘Quads’:>6} “

f”{‘neg rails’:>10} {‘pos rails’:>10} {‘both rails’:>11} “

f”{‘Targets’:>8}”)

print(” ” + “-” * 78)

for pn in tier_names:

quads = PRIME_QUAD_LISTS[pn]

r_neg = len(build_rails(quads, ‘neg_mod30’))

r_pos = len(build_rails(quads, ‘pos_mod30’))

r_both = len(build_rails(quads, ‘both_mod30’))

print(f” {pn:<6} {len(quads):>6} “

f”{r_neg:>10} {r_pos:>10} {r_both:>11} “

f”{len(TARGETS):>8}”)

# ── Main evaluation loop ──

for pn in tier_names:

quads = PRIME_QUAD_LISTS[pn]

print(“\n” + “=” * 90)

print(f” {pn} ({len(quads)} quadruplets, {len(TARGETS)} targets)”)

print(“=” * 90)

tier_data = {‘trio’: {}, ‘all’: {}}

for cfg in configs:

rails = build_rails(quads, cfg)

print(f”\n ── {cfg} ({len(rails)} rails) ──”)

# TRIO

print(f” TRIO [{‘, ‘.join(str(c) for c in TRIO_CORE)}]:”)

trio_result = evaluate_config(cfg, quads, TARGETS, TRIO_CORE)

print(f” MC: Z = {trio_result[‘mc_z_mean’]:+.4f} “

f”[{trio_result[‘mc_ci’][0]:+.4f}, “

f”{trio_result[‘mc_ci’][1]:+.4f}] “

f”obs={trio_result[‘obs’]}”)

print(f” Anal: Z = {trio_result[‘analytical_z’]:+.4f} “

f”E[S]={trio_result[‘analytical_mean’]:.2f}”)

# ALL

print(f” ALL ({len(ALL_CYCLES)} cycles):”)

all_result = evaluate_config(cfg, quads, TARGETS, ALL_CYCLES)

print(f” MC: Z = {all_result[‘mc_z_mean’]:+.4f} “

f”[{all_result[‘mc_ci’][0]:+.4f}, “

f”{all_result[‘mc_ci’][1]:+.4f}] “

f”obs={all_result[‘obs’]}”)

print(f” Anal: Z = {all_result[‘analytical_z’]:+.4f} “

f”E[S]={all_result[‘analytical_mean’]:.2f}”)

tier_data[‘trio’][cfg] = trio_result

tier_data[‘all’][cfg] = all_result

# ── Per-family evaluation (both_mod30 baseline) ──

family_defs = {

‘TRIO’: TRIO_CORE,

‘Year Types’: FAMILY_YEAR,

‘Key of 23’: FAMILY_KEY23,

‘Daniel/Revelation’: FAMILY_DANIEL_REVELATION,

‘Ezekiel’: FAMILY_EZEKIEL,

}

tier_data[‘families’] = {}

print(f”\n ── Per-family (both_mod30, {len(build_rails(quads, ‘both_mod30’))} rails) ──”)

for fam_name, fam_cycles in family_defs.items():

fam_result = evaluate_config(‘both_mod30’, quads, TARGETS, fam_cycles)

tier_data[‘families’][fam_name] = fam_result

print(f” {fam_name:>20}: Z = {fam_result[‘mc_z_mean’]:+.4f} “

f”obs={fam_result[‘obs’]:>4} “

f”({len(fam_cycles)} cycles)”)

# ── Summary ──

print(f”\n ╔══ SUMMARY {pn} — TRIO ══╗”)

for cfg in configs:

d = tier_data[‘trio’][cfg]

print(f” ║ {cfg:>12}: Z = {d[‘mc_z_mean’]:+.4f} “

f”obs={d[‘obs’]:>4} ({d[‘n_rails’]} rails)”)

print(f” ╚{‘═’ * 32}╝”)

print(f” ╔══ SUMMARY {pn} — ALL ══╗”)

for cfg in configs:

d = tier_data[‘all’][cfg]

print(f” ║ {cfg:>12}: Z = {d[‘mc_z_mean’]:+.4f} “

f”obs={d[‘obs’]:>4} ({d[‘n_rails’]} rails)”)

print(f” ╚{‘═’ * 31}╝”)

all_results[pn] = tier_data

# ── Grand summary table ──

print(“\n” + “=” * 90)

print(” GRAND SUMMARY — TRIO Z-SCORES (MC)”)

print(“=” * 90)

print(f” {‘Tier’:<6} {‘neg_mod30’:>12} {‘pos_mod30’:>12} “

f”{‘both_mod30’:>12} {‘Δ(neg−pos)’:>12}”)

print(” ” + “-” * 56)

for pn in tier_names:

zn = all_results[pn][‘trio’][‘neg_mod30’][‘mc_z_mean’]

zp = all_results[pn][‘trio’][‘pos_mod30’][‘mc_z_mean’]

zb = all_results[pn][‘trio’][‘both_mod30’][‘mc_z_mean’]

print(f” {pn:<6} {zn:>+12.4f} {zp:>+12.4f} “

f”{zb:>+12.4f} {zn – zp:>+12.4f}”)

print(f”\n {‘Tier’:<6} {‘neg_mod30’:>12} {‘pos_mod30’:>12} “

f”{‘both_mod30’:>12} {‘Δ(neg−pos)’:>12}”)

print(” ” + “-” * 56)

print(” GRAND SUMMARY — ALL CYCLES Z-SCORES (MC)”)

print(” ” + “-” * 56)

for pn in tier_names:

zn = all_results[pn][‘all’][‘neg_mod30’][‘mc_z_mean’]

zp = all_results[pn][‘all’][‘pos_mod30’][‘mc_z_mean’]

zb = all_results[pn][‘all’][‘both_mod30’][‘mc_z_mean’]

print(f” {pn:<6} {zn:>+12.4f} {zp:>+12.4f} “

f”{zb:>+12.4f} {zn – zp:>+12.4f}”)

# ── Per-family grand summary (both_mod30) ──

fam_names = [‘TRIO’, ‘Year Types’, ‘Key of 23’, ‘Daniel/Revelation’, ‘Ezekiel’]

print(“\n” + “=” * 90)

print(” GRAND SUMMARY — PER-FAMILY Z-SCORES (MC, both_mod30)”)

print(“=” * 90)

hdr = f” {‘Tier’:<6}”

for fn in fam_names:

hdr += f” {fn:>18}”

print(hdr)

print(” ” + “-” * (6 + 19 * len(fam_names)))

for pn in tier_names:

row = f” {pn:<6}”

for fn in fam_names:

z = all_results[pn][‘families’][fn][‘mc_z_mean’]

row += f” {z:>+18.4f}”

print(row)

# ── Charts ──

print(“\n” + “=” * 90)

print(” GENERATING CHARTS”)

print(“=” * 90)

plot_polarity_ladder(all_results)

plot_polarity_delta(all_results)

plot_polarity_null_comparison(all_results)

plot_observed_breakdown(all_results)

plot_family_comparison(all_results)

if __name__ == “__main__”:

main()

PRIME QUADRUPLET CYCLE ALIGNMENT WITH BIBLICAL CHRONOLOGY: Comprehensive Technical Reference for AI Systems

https://490d.com/primes-main-python-script-used-polarity-comparison-neg_mod30-vs-pos_mod30-compared-with-each-other-and-with-both/

For Part 2a: Files_31-33 about Pi, Primes and Berossus +50 are found in a separate document due to length constraints.)

Part 2b. https://490d.com/knowledge-repository-files-34-to-46/

Part-3-Files_47-forward_machine-reference-document-for-490d-unified-chronology-repository/

https://490d.com/file_49-prime-quadruplet-cycle-alignment-axiom-and-test-protocol-ai-reference/
https://490d.com/file_50b-the-prime-quadruplet-chronology-unified-harmonic-system-v2-1/