The Key of 23 Capstone: Expansion Across the Mirror — Supporting Tables and Research Notes

A companion to The Key of 23 Capstone: Expansion Across the Mirror.

These tables retain the exact inputs, endpoint movements and research comparisons behind Part 5D. All plain signed values belong to its local arithmetic coordinate model. A row explicitly marked BC/AD instead uses the stated civil chronology. The main article can be read without working through every table.

Starting sets · Expansion rule · Initial eligible pairs · Worked pairs · Complete rounds · Unbounded growth · Calendar comparisons · Jared across traditions · Jared and the two-day lead · Source controls.

1. The forty values and their reflected copy

Set A is the unshifted forty-value result of Part 5B’s basic 23 branch. Set B reverses every sign. Each row pairs an A value with its own reflection; this row pairing does not assert that every such pair qualifies for expansion.

A totals 42,800 and B totals −42,800. Their combined signed sum is zero. Both contain negative and positive values; neither is simply the positive or negative half of a single list. The original forty members of A form twenty complementary pairs summing to 2140. That within-A complement is not preserved by the new cross-set expansion.

2. The rule, with both endpoints retained

For a in A and b in B, let the positive separation be d = |a − b|. Exclude zero. A pair qualifies if d = 230k for a positive integer k.

The scalar conversion is d × 25/23. Its increase is:

δ = d × (25/23 − 1) = 2d/23 = 20k.

If a is the higher endpoint, the adjusted endpoints are a + δ and b − δ. If a is lower, they are a − δ and b + δ. Each adjusted endpoint keeps its input-set membership. Old values are retained.

The resulting separation is d + 2δ = 27d/23. Thus 230 becoming 250 describes the scalar calculation that supplies the individual endpoint movement. It does not describe the final width when both ends move.

For reproducible complete rounds, freeze A and B at the start, process every eligible labeled cross-pair, and add all outputs only after the scan. Deduplicate within each set. This differs from selecting one illustrative pair or combining A and B before searching. The downloadable calculation records use complete synchronous, membership-preserving rounds.

Every initial value is 5 modulo 10, and every adjustment is 20k. The invariant survives all rounds. Differences are consequently multiples of 10. Since 10 and 23 are coprime, divisibility by 23 is equivalent to divisibility by 230 within this particular model; a relaxed 23 filter does not rescue the rejected 7290 pair.

3. All 24 initially eligible labeled cross-pairs

The live discussion supplied sixteen representative rows. The complete search below supplies all twenty-four labeled pairs, including reciprocal reflected cases. Counted pairs are input labels, not independent observations or twenty-four distinct outcomes. Processing all of them yields thirteen distinct new A members.

The greatest eligible initial separation is 5290 = 23 × 230 = 10 × 529, with 529 = 23². The greatest possible separation across the two initial sets is 5650, which is ineligible. The repeated factor 23 is a feature of these declared inputs and this filter; no sampling probability follows from it.

4. Selected chains and their stopping point

The symmetric chain moves each endpoint 460 + 540 = 1000, preserving midpoint zero. The asymmetric chain has the same widths and movements, preserving midpoint 180. Both finish with separation 7290, whose remainder on division by 230 is 160; its remainder on division by 23 is 22. Neither selected terminal pair can be expanded again under this rule.

The smaller example 575 / −575 gives d = 1150, δ = 100 and outputs 675 / −675, with width 1350. A display adding just those two outputs would contain 41 values in each set. It is a selected one-pair update; the first complete scan instead yields 53 in each.

Endpoint signs matter. |−645 − 185| = 830 is not eligible. The valid pair is −645 / −1565, separated by 920. Likewise, the geometric reflected widths 1070 from ±535 and 990 from ±495 are not expansion inputs; 1150 from ±575 is.

5. What complete rounds produce

The initial forty-value state is round 0. Each later row records the sets after that many complete frozen-input scans. In every round B = −A. The two labeled sets are disjoint in the computed rounds below; the combined distinct count is therefore twice the per-set count.

Round 1 adds these thirteen A values:

−785, −745, −605, −565, −245, −65, 675, 855, 1685, 2185, 2925, 3105, 3285.

Both 3605 and 3645 enter A at round 2. The provenance of 3605 is A: 3105 / B: −2645, with d = 5750 and δ = 500; that operation gives A: 3605 / B: −3145. The symmetric chain gives 3645. By the same round A also reaches 3825, so ±3645 are selected comparison nodes rather than the entire system’s outer limits.

6. Why this stated model grows without bound

A finite list of growing rounds alone would not establish endless growth. For the complete membership-preserving protocol, however, the round 3 values supply a short exact argument.

All positive members remain 5 modulo 10. Modulo 230, they can therefore belong to exactly twenty-three classes: 5, 15, …, 225. By round 3, A contains a positive witness in every class, as the following table shows.

Let M be the positive maximum of A in any round from round 3 onward. Choose a retained positive witness c with c ≡ −M modulo 230. Then M + c is a positive multiple of 230, so the cross-pair A: M / B: −c is eligible. Its adjusted A endpoint is:

M + 2(M + c)/23 > M.

Every witness remains available because the update retains old values. Thus every subsequent complete round creates a larger positive maximum. Indeed, the new maximum is greater than 25M/23 because c is positive. The system cannot settle into a finite closed set.

This proves unbounded growth for this explicit arithmetic model. It does not prove a fractal dimension, statistical rarity, ancient intentional design, or a future-event timetable. A selected pair can stop while the complete model continues.

7. Calendar comparisons and equal divisions

The source describes these comparisons as fractal-like: equal divisions and familiar small spans recur in larger arrangements. That is a descriptive analogy. The small half-span 180 and large half-span 1820 do not have an exact tenfold relationship; their ratio is 91/9. A formally self-similar fractal would require a separately specified construction and proof.

The selected four-node line −3645 → −1825 → 1825 → 3645 has widths 1820 + 3650 + 1820 = 7290. By contrast, 5 → 1825 → 3645 has widths 1820 + 1820 = 3640. The ten-unit difference comes from choosing 5 or −5 as the comparison endpoint. These are exact arithmetic distinctions, not interchangeable calendar boundaries.

The first forty-value A state also has the older geometry −685 → 575 → 1565 → 2825, with spans 1260 + 990 + 1260 = 3510. Its central gap is an initial-state feature; subsequent insertions place values inside that gap. The selected sequence −5 → 35 → 495 → 1645 → 2105 → 2145 gives 40 + 460 + 1150 + 460 + 40 = 2150. It does not sum to 2140.

Revelation 11:3, BSB supplies a biblical 1260-day reference. The source’s associations of 990, 1000 and unbounded growth with prophetic order, completeness and eternity remain symbolic readings of the construction; they are not event assignments established by those numbers alone.

The calendar states are:

• 360: a schematic twelve-by-thirty year, used in prophetic/calendar comparisons.
• 364: fifty-two complete weeks, with four quarters of thirteen weeks. The Enochic scheme does not by itself provide a complete astronomical or lunar intercalation mechanism.
• 365: Enoch’s stated biblical lifespan and a schematic solar-year count. An astronomical tropical year is approximately 365.24 days, so 365 is not its exact duration.

8. Jared’s 460 across distinct witnesses

In the MT and SP, the first five begetting intervals are 130 + 105 + 90 + 70 + 65 = 460. The shared relative interval does not make their absolute chronology identical. Current File 18 distinguishes the genealogical chain head from Creation-week and Year-6 positions.

The normal MT regular setting uses Nisan reckoning and the full 430-year Egyptian sojourn, no +60 Terah adjustment and no inserted second Cainan. The SP native frame has a different sojourn setting; its full-430 comparison adds 215 to the affected native dates. The SP preferred chain head remains 4199 BC in its native frame. The localized inclusive Noah/Shem binding does not globally move Jared or the upstream head by one year.

Jubilees 4:15 places Jared’s birth in the sixth year of the tenth jubilee: ordinal AM 461, or 460 completed years from its origin. Its Creation 3856 BC and Jared 3396 BC belong to its own chronology. Its association of his days with the Watchers remains a Jubilees witness, not a replacement for the Bible’s chronology.

1 Enoch 6:6 places the descent in Jared’s days. Chapter 12 and chapter 13 depict Enoch’s message and petition, and chapter 15 develops the judgment. These accounts do not locate the descent precisely on Jared’s birthday. Enoch acts as messenger and intercessor; judgment is God’s, rather than an independently exercised judicial office assigned to Enoch.

9. The retained Jared and two-day research comparison

The legacy study selected 4 BC as its middle comparison point:

Each 3650 span also divides into two halves of 1825. Those halves differ from the 1820 outer calendar spans in the selected arithmetic diagram.

Both 3650 spans are arithmetically correct. They use ordinary civil elapsed reckoning, with no civil year zero: a cross-era span from B BC to AD A is B + A − 1. These rows retain the selected 4 BC comparison without changing the normal 6 BC birth framework or assigning a future event to AD 3647.

A separate suggestion adds two units to the local output 3645, yielding 3647. Another gives 180 + 2 = 182, 182 + 182 = 364, and 10 × 182 + 10 × 182 = 3640. Their arithmetic is exact. Their application to the calendar requires a stated operator: which physical datum is held fixed, which calendar supplies the coordinate, and whether the shift is translation, reflection or independent outward movement.

The existing Julian/Gregorian distinction supplies two physically distinct January 1 datums; it does not automatically authorize adding two to every derived value. Consequently, the correspondence between local 3645 and civil AD 3647 remains an interpretive research lead here. It is not derived as an automatic consequence of the current calendar appendix. Likewise, the 3650 + 3650 civil construction totals 7300, while the selected ±3645 arithmetic construction totals 7290. The ten-unit difference must remain visible.

The theological proposal reads Enoch’s lifespan, Jared’s era, the Watchers’ judgment and Christ’s restoration together. Genesis 5:23–24, BSB provides the biblical 365 and Enoch’s being taken by God. Jude 14–15, BSB associates Enoch’s prophecy with the Lord’s judgment. Malachi 4:2, BSB supplies the image of the “sun of righteousness.” The Christian identification with Christ and the proposed harmony of heavenly and earthly order are interpretive readings. The earlier interpretation proposed AD 3647 as a horizon for the second half of the 7300-year pattern, associated with future fulfillment and restored order. That event identification remains a provisional research proposal; the arithmetic does not establish it. The wider hypothesis is that recurring geometry can illuminate a common order behind biblical dates. It needs to be tested relationship by relationship, with each chronological and calendar state kept explicit. No second-coming date or timetable of final judgment is asserted from these calculations.

10. Coordinates, calendars and controlling sources

Local reflection in this study is M(c) = −c. For a separately declared astronomical-style coordinate c, positive c labels AD c, zero labels 1 BC, and −n labels (n + 1) BC. Such a coordinate must first be tied to the correct physical and symbolic datum; the Part 5B raw origin must not silently be reset.

Under the present day–year framing, each calendar’s January 1 AD 1 circumcision datum has coordinate +1. Counting back seven elapsed days gives December 25 coordinate −6, symbolic 7 BC; the eight dated positions are counted inclusively. December 24 is eight elapsed days earlier. Julian January 1 AD 1 was Saturday; the physical Gregorian January 1 datum arrived two days later. The normal Shem/Noah G2 chronology and the named G1/G3 comparisons provide the canonical analogy, but not a blanket shift of every patriarch or every set output. Detailed definitions are already provided in the calendar-datum appendix.

File 51b’s September 25 clarification defines a separate pure Rounded display grid: q(B BC) = −(B − 1) and q(AD A) = A − 1. Its cross-polarity display width is B + A − 2. Civil elapsed years instead use B + A − 1; bare label addition B + A is another operation. None of these conventions can silently replace the local signed arithmetic used to generate this study’s sets.

The current repository controls established dates and state distinctions. The local set expansion and the retained two-day hypothesis are research constructions with their operations stated here. This check covers the listed source files and their applicable displayed amendments; it does not assert that an unlocated research motif is absent from the entire repository.

11. Reproducible calculations

The following exact integer implementation produces the complete-round sets. Adjusted A endpoints remain in A; B is the exact reflected copy after each scan.

A = {-685, -645, -505, -465, -325, -285, -185, -145, -105, -5,
     35, 75, 175, 215, 315, 355, 395, 495, 535, 575,
     1565, 1605, 1645, 1745, 1785, 1825, 1925, 1965, 2065, 2105,
     2145, 2245, 2285, 2325, 2425, 2465, 2605, 2645, 2785, 2825}

def step(A):
    B = {-x for x in A}
    next_A, next_B = set(A), set(B)
    for a in A:                 # frozen inputs
        for b in B:
            d = abs(a - b)
            if d == 0 or d % 230:
                continue
            shift = 2 * d // 23
            direction = 1 if a > b else -1
            next_A.add(a + direction * shift)
            next_B.add(b - direction * shift)
    assert next_B == {-x for x in next_A}
    return next_A

for round_number in range(11):
    print(round_number, len(A), min(A), max(A))
    if round_number < 10:
        A = step(A)

Download all initial cross-pairs, round summaries, coordinate lists through round 6, or the unbounded-growth witness certificate (ZIP). The complete package includes the audit script, exact JSON states through round 10 and further pair provenance.

For related research, see the earlier Key of 23 study at 1260d.com and the original sounding-board discussion. These are further-reading records, rather than controlling replacements for the current canonical sources.

Return to the main article · Preserved original Part 5D discussion.

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