Technical Appendix: The 365-Day Sothic Companion to the Key of 23

This appendix supports The 365-Day Sothic Companion to the Key of 23. It separates calendar-day comparisons from anchored chronological operations, and preserves the complete declared endpoint groups alongside the successful paths.

How to read the tables

n means Nisan and t means Tishri. The calculations use the repository’s schematic spring/autumn positions, separated by half a year. With January 1, AD 1 at coordinate zero, a BC Nisan position has coordinate −B + ¼, and a BC Tishri position −B + ¾. AD Nisan has coordinate A − ¾, and AD Tishri A − ¼. These are phase coordinates, not reconstructed Julian dates for lunar months.

BC year labels decrease as time moves forward. Across BC and AD, there is no civil year zero: a same-season span from B BC to AD A is B + A − 1. Unless explicitly marked Rounded, the tables use actual source constructions. “Generated” identifies a calculation output whose historical identification has not thereby been established.

1. Calendar standards

The original equality is 336 × 25/23 = 360 × 70/69 = 364 × 300/299 = K. The astronomical epoch comparisons at File 12 §5.1A are model-dependent calibrations; they neither redefine these fractions nor date an Egyptian heliacal reset.

2. One-extra-year completion

Let u = 23/2 = 11½. The common form is R_m = (mu + 1)/(mu) = (23m + 2)/(23m). It organizes the following five comparisons. The form does not independently select the five multipliers.

H satisfies 365H = J − 1/11684. Its completed civil-day block minus its raw Julian block is −⅛ day. Against S_K the same subtraction is +879/80 = +10.9875 days. Thus H is close to the ideal Julian/Sothic comparison but does not preserve K. The distinct G satisfies 365G = K exactly.

The Enochian side comparison 294/293 also remains distinct: 293S_K − 294 × 364 = 19/920 day, whereas 299S_K − 300 × 364 = 299/40 = 7.475 days. For the exact side ratio use 107016/293; its finite decimal 365.2423 is approximate. In particular, 293 × 365.2423 = 107015.9939, not exactly 107016.

A calibration counter-check rejects a stronger claim for H. At 365 days against S_K, the exact one-extra-year count is 335800/223 ≈ 1505.83. Its nearest half-23 count is 1506½, giving 3015/3013 and a discrepancy of −13/80 day. This is a sensitivity check only; it is not an additional active operator selected to improve a date match.

3. Moses, Joshua, and the Tabernacle

The 80½-year component uses 1527t → 1446n BC; same-phase 1526n → 1446n BC is 80. The whole half-step is 1380 + 80½ = 1460½. Applying the Priestly ratio to the 80½ component gives 87½, a gain of seven; applying H to the whole gives 1461½. These are separate operations. Applying 25/23 to the whole would give 1587½, so their gains must not be added as though they were one execution.

The source corridor is 2936 | 2906 | 2886 | 2856 BC, with spacings 30 | 20 | 30. Holding Exodus at 1446 gives 2906 − 1446 = 4 × 365 = 1460 and 2886 − 1446 = 4 × 360 = 1440. Their difference is 20. The shared 1380 spans are 2906 − 1526 and 2856 − 1476; the source and Moses–Joshua differences are both 50.

Those results share their inputs. Given Exodus e = 1446, Q = 4 × 365, F = 4 × 360, and the 30-year rail, the corridor is (e + Q + 30, e + Q, e + F, e + F − 30). The corridor, the two 1380 spans, the 20, and the 50 therefore are correlated consequences of the same declared packet.

4. Corresponding Creation nodes

These matched Creation comparisons use the MT full-430 baseline, normal Gear 2, no +60 Terah, and expressly restored second Cainan: 130 regularly and 460 cumulatively. Thus 9890 + 460 − 130 = 10220 = 7 × 1460. Both sides include Cainan, and each row pairs the same kind of Creation node. This differs from the Rounded 9900 bridge and from mixed Cainan/Apparent comparisons.

Each Cumulative source is held fixed in its own corresponding span map. The target week moves seven years without changing its seven-year width. This is not one dilation of the complete multi-source envelope.

The corresponding ten-cycle relations are 14466 BC → AD 135 = 14600 = 10 × 1460, partitioned at 4246 BC into seven and three cycles. The Creation head 14471 BC → AD 135 gives 14605 = 10 × 1460½. Their difference is 10 × ½ = 5 years. AD 135 is the repository’s schematic era-boundary comparison; completing either ten-cycle span generates AD 145, to which this study assigns no historical identification.

5. The complete twelve-position Short field

The six seasonal pairs are listed in Table S6. In January-centered coordinates their twelve individual positions are X = {−2921 + 15r + 2g + p/4}, where r and p are −1 or +1, and g is −1, 0, or +1. The extrema are −2938¼ and −2903¾, with width 34½ and midpoint −2921.

Let M(x) = −x. The uniform translation T(x) = x + 5842 satisfies T(X) = M(X), since 5842 = 4 × 1460½. Applying the inverse Mirror after translation gives the reflection R(x) = −5842 − x about −2921. This maps X to itself and doing it twice returns each original position.

The center is the opening of January 1, 2921 BC. The existing 2922t/2921n pair brackets that instant; it is not the instant itself. For the fixed template, a translation of 5842 reaches its epoch Mirror only when its center is −2921. The period fixes this placement conditional on the stated template; it does not generate the template’s shape independently.

The finite comparison can be extended by opposed ±720 shifts: T₅₈₄₂(X + k) = M(X − k). Adding three macro positions produces 36 distinct nodes with the same reflection. This proves the declared finite extension; it does not prove propagation through every distant Creation/Flood node or the complete 360:1-scaled King List.

The Mosaic placement supplies the center independently of the field’s internal symmetry: 1460 + (1476 + 1446)/2 = 2921. These are already controlled repository targets, not independent archaeological dating. Joshua to the paired Mirror of Exodus has same-phase span 2922; its outer cross-phase span is 2922½ and its inner span 2921½. The outer midpoint is January 1, 15 BC, rather than the opening of AD 1.

6. The Long variant: retain the missing member

The Long variant has valid 2920-year routes: 2888n BC → AD 33n; 2886n BC → AD 35n; and 2856n BC → AD 65n. Its twelve-position center is −2871 and its own self-Mirror translation is 5742. It therefore does not separately satisfy the Short field’s 5842 theorem.

The matrix contains 3 × 1460 = 4380, 3 × 1460½ = 4381½, and 3 × 1461½ = 4384½. It does not contain 3 × 1461 = 4383. Adding an unspecified adjacent endpoint would change the test. The missing member remains absent from this fixed field.

7. The Pillar: two operators, different outcomes

The fixed Pillar endpoint is AD 29465n. The source comparison 2936 BC → AD 29465 gives 32400 = 180². Actual Year-6 Adam at 4116n BC gives 33580 = 23 × 1460. This seed has two distinct 365-day comparisons.

For H, 33614½ − 33580 = 34½ = 30 + 4½. The valid decomposition does not remove the unresolved 4½-year difference. For G, the exact day volumes agree: 36500 × 336 = 33600 × 365 = 12264000. The actual 4146 and Rounded 4136 coordinates remain distinct.

8. Results and limits

Sources and research history

• File 12, Calendrical Physics, §§5.1A and 5A: standards, completion ratios, calibration and scope.
• File 22, Cumulative MT Harmonics, §§5A and 9B: Creation and Pillar applications.
• File 70 Supplement A, §13: the complete Short field, Mosaic selection and Long endpoint audit.
• Sothic Report V, September 6, 2026: the consolidated original research discussion.
• Discussion and Amendment Ledger v0.3, September 6, 2026: dated preparation and verification history.
• Censorinus, De die natali, chapter 18: ancient calendar analogue.

The current canonical catalog governs source definitions.