# C62 — Ten-step synthesis: the 529 Mirror has source-qualified members and exact phase

22 September 2026 · Step 10 of the authorized ten-step Mirror continuation (C53–C62).

**C53–C62 complete the authorized ten-step continuation.** The central advance is an explanation of the endpoint selections behind the existing 20/22/24 Mirror and its relation to the 13×529 Creation family. The result keeps the exact Aaron phase, distinguishes cumulative week members from regular Gears, and explains why matching widths can have different source decompositions.

| Step | Question | Completed result |
| --- | --- | --- |
| C53 | Named source member | 6348=12K is the derived whole-year Aaron-side ON-Shem member; its exact Aaron phase is 6348.5. |
| C54 | Cumulative member selection | 1058 requires Flood offset minus Shem offset =2; six admitted pairs, j=0…5. |
| C55 | Week boundaries | Both full 6877 branches have cut support; valid envelope members need not be cut boundaries. |
| C56 | Canonical-axis Mirror | OFF-Flood opening 5290 and ON-Shem 6348 form the cumulative-only 20/22/24 rectangle. |
| C57 | Source displacement | MT ON minus LXX OFF is 6=460−454, using named lifespan differences. |
| C58 | Equal source-qualified margins | 1058=600+460−2=600+454+4; the adjustments differ by six. |
| C59 | Joined N01 head | Only j=2 combines the Noah join with all four cumulative members in the 2+2+3 cut set. |
| C60 | Inverse limit | 24-fold admission allows formal Δ=2…9; even common-cut support does not uniquely recover six. |
| C61 | Exact seasonal phase | 3.5-year Aaron phase and paired 24K coexist; correct centres alone cannot certify suffix order. |
| C62 | Integrated checkpoint | Retain these source/phase explanations together with the existing 6877 chains and family permissions. |

The source-derived whole-year Shem cell is

\[
6350\xrightarrow{2}6348\xrightarrow{3}6345\xrightarrow{2}6343.
\]

Its named 6348 member is 12K, where K=529=23². Paired with the derived-OFF Flood opening 5290=10K, it gives the canonical-axis width matrix

\[
K\begin{pmatrix}20&22\\22&24\end{pmatrix}.
\]

The two same-side margins are 1058=2K. In the translated mixed-source rectangle, those equal margins instead have distinct source accounts:

\[
600+460-2=1058=600+454+4,
\qquad 460-454=6.
\]

Here 454 is the summed LXX-minus-MT lifespan surplus in the retained post-Flood rows. The six-year difference is the MT restored-ON versus LXX derived-OFF cumulative Flood displacement. It explains the different member adjustments; it is not a regular Gear displacement or an independently predicted value from 24-fold closure.

For F₋=5290−j (LXX OFF), F₊=5290+j (MT ON), and S₋/S₊=6348∓j (native-ON Shem), all four source fields admit j=0,1,2. N01 at j=2 alone adds the regular Noah join. Its four Moses-relative offsets belong to the 2+2+3 cut template. At j=0 the geometry coincides with the single-tradition rectangle, while the source roles remain distinct. These numeric week templates do not assign regular G1/G2/G3.

The exact phase gives a stronger endpoint check. In File_09's simplified Julian model,

\[
6349t/6348n\ \mathrm{BC}\longrightarrow\mathrm{AD}\ 6348t/6349n
\]

has half-year internal spacing on each side and same-season component spans 12696=24K. The Aaron/Tishri component stands exactly 3.5 years before the 6345n Moses line. The 6348.5 Aaron-column label therefore coexists with the pair's centre 6348. Treating that phase label as a shifted centre would compute a different state. An incorrect AD suffix order can preserve the numerical centre while making the pair 1.5 years wide; component checks are required.

The 6877 family remains the Creation-facing core of the joined construction. The retained complete branches are:

| Branch / j | SP Creation BC | Reflected ordinary MT Flood | Reflected SP companion death | Reflected MT Noah | Reflected SP Noah |
| --- | --- | --- | --- | --- | --- |
| N02 / 3 | 4419 | 2458 | 2757 | 3056 | 3706 |
| N04 / 5 | 4417 | 2460 | 2759 | 3058 | 3708 |

The table uses the existing paired-coordinate numerals, not a new choice of standalone dates. Each branch has Creation-to-event spans 6877, 7176, 7475, 8125: the increments after 6877 are 299, 299, 650. The 13K interval lies inside 17K with flanks 868/1248 and inside 19K with flanks 1926/1248. Both full branches have cumulative cut-boundary support in C55. Their indices j=3/5 exceed the upper-Shem limit j≤2 for the four-field 24K head, so that head remains a distinct N01 attachment. N01's existing simple 13K interval and its blocked full Noah expansion are unchanged.

This batch adds source interpretation, a cumulative-only rectangle, and exact phase certificates. The retained joined graph stays at 81 typed nodes, 161 duration records and 19 reflection relations. Its selected family labels remain 2/10/11/12/13/15/17/19/20/22/24; partition legs 3/5/7 remain separately present. These shared-endpoint constructions are dependent relations, not additional independent calibrations of 529.

All ten reports have individual evidence checkpoints. C62 freezes and reads C53–C61 certificates and checks their integration with the older graph; it does not rerun those steps or any historical campaign. This is a delta packet for the existing compact restart kit. The interrupted historical Reports 11–25 remain unestablished as completed. Stop here pending the next instruction.

**Validation.** 17 exact arithmetic/structural checks passed. The 22 frozen dependencies passed hash verification; all three generated outputs replayed byte for byte in isolation, and the original dependencies remained unchanged. This step reads completed certificates and does not execute their campaigns.

**Scope and provenance.** Controlling File_18 and File_09 remain unchanged. Regular Gears apply only to Noah/Shem/Flood; SP person companions only to Noah/Shem. Creation and Conquest fields remain fixed. Native LXX ON, explicitly derived OFF, independent Flood start/close, and paired suffix inversion at January AD 1 remain in force. No civil year zero, upstream Gear transport, primers, Rounded campaign, or localized −33 is introduced. These are new continuation reports, not recovered historical Reports 11–25. Evidence is conditional on the supplied project model, not a proof of literary intention or independent historical dating.

**Source locators.** File_09 §0.4 rules 21–22 and the Creation Cell / Aaron Phase guard (week anatomy), §3 and Appendix A (paired Mirror and exact phase); File_18 §§4.2 and 6A–6D (lifespans and cumulative fields); frozen C15/C16/C22 and C43/C46/C48/C50/C52 certificates (retained graph, unit, and domains).

**Best next bounded step — High sufficient.** Carry the component-level phase certificate into the two full N02/N04 6877→299→299→650 Creation/Noah chains, checking source-bound seasonal pairs at each handoff without reopening completed family searches. Await user instruction.