This appendix supports Mirror Translations: How the Chronological Families Connect. It combines the selective working results of the September 25 framework and September 20 bilateral study. The current canonical files control inherited dates and permissions; the preserved research papers supply these particular applications.
Reading key: t = Tishri/autumn; n = Nisan/spring. Pb means (b+1)t/bn BC, centered at −b. Qb means AD bt/(b+1)n, centered at +b. All centers in these tables refer to January AD 1. They use the phase-resolved model, rather than the pure Rounded display grid.
Go directly to: phase rules · width family · settings · 4232 examples · regular 8464 · cumulative 12696 · one-year companions · 8395 positions · Jubilees births · central weeks · related constructions.
1. Seasonal coordinates and operations
| Operation | Exact rule | Meaning |
|---|---|---|
| BC autumn/spring | A(yt BC) = −y + 3/4; A(yn BC) = −y + 1/4 | An ordered P_b pair spans [−b−1/4, −b+1/4]. |
| AD autumn/spring | A(AD yt) = y − 1/4; A(AD yn) = y − 3/4 | An ordered Q_b pair spans [b−1/4, b+1/4]. |
| Ordinary paired Mirror | M(A) = −A; P_b ↔ Q_b | Reflect each component, interchange t/n as the phase formulas require, and reorder the pair chronologically; retain source identity and marker role. |
| Forward translation | T_W(A) = A + W; P_b + W = Q_(W−b) | The Q notation applies when the landing is AD. A landing remaining BC uses ordinary BC subtraction. |
| Landing versus reflected anchor | D_W(b,d) = W − b − d | Signed pair-center difference between the landing Q_(W−b) and Q_d. |
| Local reflection | T_(2r) M(A) = 2r − A | A composed reflection about +r; it leaves the global January AD 1 coordinate datum unchanged. |
| Civil same-season crossing | B BC → AD D: B + D − 1 | For example, (C+1)t BC → AD Rt and Cn BC → AD (R+1)n both span C+R. |
The quarter-year values encode the repository’s normalized October/April seasonal model. Their exactness concerns that model, not the unequal lengths of real lunar months. A pure Rounded coordinate has its own display and counting conventions; do not transfer these formulas to that grid without declaring the change.
File 09, Appendix A.3; File 51b, convention entry and formal mechanics. Framework §§1.2–1.4, 17; bilateral study §1.
2. Radii, doubled widths, and Key conversions
| Radius r | 529 units | Width 2r | Width ×25/23 | Width ×(25/23)² |
|---|---|---|---|---|
| 2116 | 4 | 4232 | 4600 | 5000 |
| 4232 | 8 | 8464 | 9200 | 10000 |
| 5290 | 10 | 10580 | 11500 | 12500 |
| 6348 | 12 | 12696 | 13800 | 15000 |
The three principal widths are 4232, 8464, and 12696. The 5290 radius/10580 width supplies the outer geometry used in the 8395 example; its wider domain assignment remains open in the research paper. The perfect squares include 529 = 23², 2116 = 46², and 8464 = 92². Other listed values can be multiples of squares.
September 25 framework §2; File 46 for inherited conversion protocols. Equal numbers in another source state do not make those constructions interchangeable.
3. Source settings used in this appendix
| Application | Tradition/domain | Flood selection | Sojourn | Terah +60 | Second Cainan |
|---|---|---|---|---|---|
| 4232 patriarchal/Mosaic examples | MT regular comparison | Normal baseline where applicable | Full 430 | Only named 2226 firstborn reconstruction | OFF |
| 8464 baseline | MT regular | Gear 2 | Full 430 | OFF | OFF |
| 8464 restored comparison | MT regular | Gear 2 | Full 430 | OFF | ON: +130 upstream |
| 12696 baseline/restoration | MT cumulative seven-year fields | No cumulative Gear | Source cumulative chain | No added cumulative +60 | OFF / ON: +460 through Arphaxad; Shelah onward fixed |
| 8395 first configuration | LXX cumulative OFF + regular SP or LXX OFF | Regular G1 close, then all six boundaries | Regular full 430 | Regular OFF | Cumulative OFF; regular OFF |
| 8395 second configuration | LXX cumulative ON + MT regular restored | Regular G1 close, then all six boundaries | Regular full 430 | Regular ON | Cumulative native ON; MT regular restored ON |
| Jubilees birth comparisons | BJ project-frame births | No autonomous BJ Gear | BJ source coordinates | No new shift | No new insertion |
Regular restoration adds a begetting interval of 130. Cumulative restoration adds a lifespan of 460. They are separate operations. The 8395 table compares two explicitly selected source configurations; it does not license unrestricted exchange between textual traditions. The ordinary default remains MT, Nisan, Gear 2, with no +60 or restored Cainan unless stated.
File 18, regular tables and §§6A–6D; framework §§3, 7, 9; bilateral §§2–4.
4. Selected 4232-year translations
| Source position | BC pair P_b | Landing Q_(4232−b) | Declared comparison |
|---|---|---|---|
| Abram birth | 2167t/2166n BC | AD 2066t/2067n | Isaac’s ordinary reflected birth |
| Isaac birth | 2067t/2066n BC | AD 2166t/2167n | Abram’s ordinary reflected birth |
| Haran / selected firstborn | 2227t/2226n BC | AD 2006t/2007n | Jacob’s ordinary reflected birth; +60 Terah reconstruction |
| Jacob birth | 2007t/2006n BC | AD 2226t/2227n | Selected firstborn’s reflected position |
| Moses birth | 1527t/1526n BC | AD 2706t/2707n | Q2706 − Q1446 = 1260 |
| Exodus | 1447t/1446n BC | AD 2786t/2787n | Q2786 − Q1406 = 1380 |
| Conquest / Moses death | 1407t/1406n BC | AD 2826t/2827n | Q2826 − Q1446 = 1380 |
The patriarchal sums are 2166+2066 = 2226+2006 = 4232. Moses’ translated biography retains 80+40 = 120. The crossed spans satisfy 4232−1526−1446 = 1260 and 4232−1446−1406 = 1380. Then 1380×25/23 = 1500. Replacing one biography endpoint and applying a Key are distinct causes of equal 120-year gains.
Framework §§3–5; File 18 for the regular chronological inputs.
5. Regular MT landings under +8464
| Node | BC Nisan base OFF | OFF landing pair | OFF to Q6 | ON landing pair | ON to Q1876 |
|---|---|---|---|---|---|
| Noah birth | 3058 | AD 5406t/5407n | 5400 | AD 5276t/5277n | 3400 |
| 120 before Flood | 2578 | AD 5886t/5887n | 5880 | AD 5756t/5757n | 3880 |
| Shem birth | 2558 | AD 5906t/5907n | 5900 | AD 5776t/5777n | 3900 |
| Flood start | 2458 | AD 6006t/6007n | 6000 | AD 5876t/5877n | 4000 |
| Noah death | 2108 | AD 6356t/6357n | 6350 | AD 6226t/6227n | 4350 |
| Shem death | 1958 | AD 6506t/6507n | 6500 | AD 6376t/6377n | 4500 |
Each source base b means P_b, not an unsuffixed instant. ON adds 130 to every upstream BC base and subtracts 130 from its AD landing. OFF uses normal Gear 2, full 430, Terah +60 OFF, Cainan OFF. ON changes only the explicitly restored regular Cainan state. Q6 = AD 6t/7n; Q1876 = AD 1876t/1877n. OFF Flood to Q1406 is 6006−1406 = 4600.
Framework §§7.1–7.3; File 18 regular MT and restoration controls.
6. Selected cumulative MT fields under +12696
| Node | OFF source field | OFF landing field | ON source field | ON landing field |
|---|---|---|---|---|
| Noah | 6387t/6386n → 6380t/6379n BC | AD 6310t/6311n → AD 6317t/6318n | 6847t/6846n → 6840t/6839n BC | AD 5850t/5851n → AD 5857t/5858n |
| Shem | 5437t/5436n → 5430t/5429n BC | AD 7260t/7261n → AD 7267t/7268n | 5897t/5896n → 5890t/5889n BC | AD 6800t/6801n → AD 6807t/6808n |
| Flood / Arphaxad boundary | 4837t/4836n → 4830t/4829n BC | AD 7860t/7861n → AD 7867t/7868n | 5297t/5296n → 5290t/5289n BC | AD 7400t/7401n → AD 7407t/7408n |
| Shelah / Cainan death | 4399t/4398n → 4392t/4391n BC | AD 8298t/8299n → AD 8305t/8306n | 4399t/4398n → 4392t/4391n BC | AD 8298t/8299n → AD 8305t/8306n |
| Second Cainan | Absent | — | 4859t/4858n → 4852t/4851n BC | AD 7838t/7839n → AD 7845t/7846n |
Every matching-season field spans seven elapsed years and contains eight annual boundary members. Its first Tishri to last Nisan outer spread is 7½ years. The cumulative Flood label follows the source convention; it does not collapse the regular Flood and Arphaxad event dates. Cainan ON shifts Arphaxad and earlier fields by 460; Shelah onward stays fixed. Source labels and marker roles remain attached.
Framework §§9–10; File 18 §§6A–6B.
7. The one-year arrangement around Conquest/rest
| Landing field | Subtracted width | Comparison field | Relation to Q1399 → Q1406 |
|---|---|---|---|
| Flood ON: Q7400 → Q7407 | 6000 | Q1400 → Q1407 | One year later |
| Cainan: Q7838 → Q7845 | 6440 | Q1398 → Q1405 | One year earlier |
| Shelah / Cainan death: Q8298 → Q8305 | 6900 | Q1398 → Q1405 | One year earlier |
For the selected Flood member Q7406, both matching-phase comparisons to Q1406 give 6000: 7406t−1406t = 7407n−1407n. For the closing members Q7845 and Q8305, the raw 6440 and 6900 comparisons use Q1405, not Q1406. Their conversions are 6440×25/23 = 7000, 6900×70/69 = 7000, and 6900×25/23 = 7500. Crossed suffixes require their half-year information; they cannot replace matching-season equations.
Framework §§10.1–10.2; File 09 phase control.
8. The six-position 8395 correspondence
| Regular boundary | LXX cumulative OFF C0 | SP / LXX regular OFF R0 | LXX cumulative ON C1 | MT restored, +60 R1 | Both sums |
|---|---|---|---|---|---|
| G1 close | 5290 | 3105 | 5750 | 2645 | 8395 |
| G1 start | 5289 | 3106 | 5749 | 2646 | 8395 |
| G2 close | 5288 | 3107 | 5748 | 2647 | 8395 |
| G2 start | 5287 | 3108 | 5747 | 2648 | 8395 |
| G3 close | 5286 | 3109 | 5746 | 2649 | 8395 |
| G3 start | 5285 | 3110 | 5745 | 2650 | 8395 |
All numerals are BC Nisan bases of paired boundaries. Every row satisfies C0+R0 = C1+R1 = 8395. Moving one row decreases C by 1 and increases R by 1. The two last members of each full seven-year cumulative field are outside this selected six-position relation. All regular rows use full 430; R0 has Cainan and +60 OFF; R1 has restored Cainan and +60 ON. The SP values use the equalized full-430 Flood event rows; no person-only −1 adjustment is propagated into them.
Bilateral study §3; File 18 §§3.1.4, 3.2, 6D.
The exchange is C1−C0 = 460 and R0−R1 = 650−(130+60) = 460. The Mirror sum is unchanged; the same-side difference increases by 920. At the outer row, the same-side differences are 2185 and 3105. Together with M = 8395, they reconstruct the endpoints: C = (M+D)/2 and R = (M−D)/2. The sum and difference recover coordinates; source identity still needs its labels.
The first outer self-Mirror frame has width 10580 = 2185+6210+2185. The second has width 11500 = 3105+5290+3105. The common crossed span 8395 appears inside both. Uniformly multiplying the first endpoints by 25/23 instead gives 5750+3375 = 9125. Adding native Cainan to both original endpoints gives 5750+3235 = 8985. These are different constructions.
9. Three Book of Jubilees birth positions
| BJ birth | BC base | Different MT boundary with this numeral | LXX cumulative ON partner | Paired Mirror span |
|---|---|---|---|---|
| Shem | 2650 | G3 start | 5745 | 8395 |
| Ham | 2648 | G2 start | 5747 | 8395 |
| Japheth | 2645 | G1 close | 5750 | 8395 |
The MT boundary column belongs to restored regular Cainan + Terah 60, full 430. Numerical agreement preserves distinct node classes: a BJ birth is not an MT Flood boundary. Substituting the cumulative OFF partners subtracts 460 and gives 7935 = 23×345. BJ’s own Flood start/close 2549/2548 do not participate in this 8395 relation.
Bilateral study §4; File 18 §3.1.1 for corrected BJ birth inputs.
10. Translated central weeks in pair-center coordinates
| Selected source | Flood week F | Shem partner H = F−2 | Reflected Shem week M(H) | Relation of F to M(H) |
|---|---|---|---|---|
| MT ON | [−6, +1] | [−8, −1] | [+1, +8] | Meet at +1; two seven-year intervals |
| SP OFF | [−1, +6] | [−3, +4] | [−4, +3] | Overlap; not the MT end-to-start completion |
| LXX OFF | [0, +7] | [−2, +5] | [−5, +2] | Overlap; not the MT end-to-start completion |
These are selected translated research configurations, not newly adopted defaults. MT ON Flood centers −5296…−5289 translate by +5290; its Shem field −5896…−5889 translates by +(5290+598). LXX OFF Flood centers −5290…−5283 translate by +5290. SP OFF Flood centers −4716…−4709 translate by +4715, with the corresponding Shem translation using an additional 598. The source separation 600 and additional translation 598 leave residual 2. For F=[a,a+7], the reflected partner is [2−a−7, 2−a]. End-to-start meeting requires a+7 = 2−a−7, hence a=−6 and junction +1. This is January AD 2, displayed AD 1t/2n; zero is still January AD 1. The two 7½-year outer seasonal envelopes share half a year, so their union is 7½+7½−½ = 14½ years.
Bilateral study §6 and its preserved geometry diagram; File 09 for inherited coordinate meaning.
11. Further connections retained in the research framework
| Construction | Explicit arithmetic | Source or selection boundary |
|---|---|---|
| Exodus-centered Creation completion | 322+138 = 460; 322×25/23 = 350; 138×70/69 = 140; 350+140 = 490 | After +12696, closing ON/OFF Creation members are 1768/1308 BC; Exodus remains 1446 BC. |
| Composite generation carriers | 12558×70/69 = 26×490; adding the separately completed 490 gives 27×490 = 13230 | A split completion combined with the 26-carrier root, not uniform scaling of the raw ON span. |
| Crossed translated Exodus/Conquest | 12696−1446 = 11250; 12696−1406 = 11290; 11250+11290 = 22540 = 46×490 = 460×49 | Crossing the reflected generated landing with the other AD landing; midpoint pair-centers −20 and +20. |
| Two continuations from 6900 | 6900×70/69 = 7000; 6900×25/23 = 7500; doubled totals 14000 and 15000 | The Year 14 completion and Year 15 renewal readings are theological interpretations of these declared totals. |
| SKL 24+25 partition | 27366−1445 = 25921 = 49×529; 25921 = 12696+13225 = (24+25)×529 | Normal central SKL Long Flood 27366 BC; Tabernacle 1445 BC; not the Exodus endpoint 1446 BC. |
These applications preserve their own anchors. They have not been shown to form one continuous path with unchanged endpoints. The 27-carrier completion uses the final seven once: 350+133+7 = 490. A different uniform 27-carrier root is 13041 = 27×483; it is not the raw ON span 13018. The numbers 7935 and 22540 also occur in File 52f under other constructions, so their appearance here is not a claim of absolute novelty.
Framework §§13–15 and Appendix E; File 09; File 22; File 21; File 34; File 52f §§3.8, 6.6 for the distinct constructions.
Preserved research and diagram
The original papers retain the broader discussion, including explicitly unresolved or quarantined proposals. They are research records rather than replacement canonical controls:
- The 4232 / 8464 / 12696 Mirror framework, September 24, addendum September 25, 2026.
- The Mirror: bilateral integration and the 8395 correspondence, September 20, 2026.
- Original companion diagram and accessible transcription, September 20, 2026.
The research papers report earlier, larger check batteries. The publication package verifies its selected arithmetic separately; it does not claim to replay every frozen research-domain enumeration. No chance probability or unrestricted uniqueness result is asserted. Source exchange, duration conversion, and translation retain their separate meanings.
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