# C07 — One Gear rule produces 48 and 52; the complete 13-fold family selects 52

21 September 2026 · Bounded continuation of C06.

**The 48- and 52-year intervals are both sourced, and one relative-Gear rule explains them. Only the 52 cases reproduce the complete `6877→7475→8125` family in the tested endpoint arrangement.** The 48 cases instead give the sourced Creation comparison `6873→7475→8125`, with gains 602 and 650. Equality of second differences therefore does not establish equality of the underlying 529 families.

The test uses the nine combinations of MT Noah-birth Gear and SP Flood-start Gear, together with the matching MT Flood starts and SP primary Noah births. Regular Gears remain confined to Noah/Shem/Flood. Creation stays Normal/G2, with members selected inside its fixed week. The frame is full-430, Terah-shift-zero and MT/SP Cainan OFF. The +30 apparent field is retained only as an explicitly identified anchor control. Primers, Rounded, localized −33 and other selector changes are inactive.

**The complete three-Gear table gives the selection rule directly.** Entries below are the interval from reflected MT Noah birth to reflected SP Flood start. Row and column coordinates are signed AD paired centers: `R` means `AD Rt/(R+1)n`.

| MT Noah birth | SP G1 Flood 3106 | SP G2 Flood 3108 | SP G3 Flood 3110 |
|---|---:|---:|---:|
| G1, 3056 | 50 | **52** | 54 |
| G2, 3058 | **48** | 50 | **52** |
| G3, 3060 | 46 | **48** | 50 |

Let `i` be Noah Gear, `j` Flood Gear, and `d=j−i`. In this report, **d is a relative Gear index**, not the comparison-layer coordinate named d in the older three-coordinate chart. For MT Flood `Fⱼ`, MT Noah `Nᵢ`, SP Flood `Hⱼ` and SP primary Noah `Sᵢ`, File_18 gives

\[
H_j-F_j=S_i-N_i=650,
\qquad N_i-F_j=S_i-H_j=600-2d.
\]

Consequently,

\[
\boxed{H_j-N_i=50+2d.}
\]

The four reflected endpoints are ordered `Fⱼ < Nᵢ < Hⱼ < Sᵢ`, with successive gaps

\[
\boxed{(600-2d),\ (50+2d),\ (600-2d).}
\]

Same-Gear comparisons give `600,50,600`. Taking Flood one Gear above Noah gives `598,52,598`; taking it one Gear below gives `602,48,602`. These are qualified cross-Gear comparisons, not substitutions for a coherent same-Gear Noah biography. Transposing the two Gear indices changes the sign of d, and the two central gaps sum to 100.

The exact seasonal witnesses are:

| Gap | Reflected MT Noah birth | Reflected SP Flood start |
|---:|---|---|
| 48 | G2: AD 3058t/3059n | G1: AD 3106t/3107n |
| 48 | G3: AD 3060t/3061n | G2: AD 3108t/3109n |
| 52 | G1: AD 3056t/3057n | G2: AD 3108t/3109n |
| 52 | G2: AD 3058t/3059n | G3: AD 3110t/3111n |

For the scalar sequence `k×529→k×575→k×625`, the gains are `46k,50k`, and the second difference is `4k`. Thus a central source gap equal to `4k` requires

\[
50+2d=4k,\qquad k=\frac{25+d}{2}.
\]

In the finite domain `d∈{−2,−1,0,1,2}`, integer k occurs only at **k=12, d=−1**, and **k=13, d=1**. This explains the paired 48/52 appearances within the Gear table. It is a finite arithmetic selection statement, not an independence or probability claim.

**Holding the two upper widths exposes what the 48 cases actually supply.** Select the ordinary SP Creation member

\[
C_i=7475-N_i.
\]

For Noah G1/G2/G3, this gives **4419/4417/4415 BC**, respectively: offsets 2/4/6 inside the same fixed ordinary SP week. With left signed center `−Cᵢ`, the three widths are

\[
\boxed{(C_i+F_j,\ C_i+N_i,\ C_i+S_i)
=(6875+2d,\ 7475,\ 8125).}
\]

The two upper stages always retain the ratio `8125/7475=25/23`. The first stage has the same ratio only when its width is **6877**, requiring **d=1**.

A particularly clear control holds Creation and both Noah endpoints fixed while changing only Flood Gear:

| Fixed ordinary SP Creation: 4418t/4417n BC; Noah G2 in MT and SP | First reflected endpoint | Three Creation widths | Gains | Second difference |
|---|---|---|---|---:|
| Flood G1 | MT AD 2456t/2457n | **6873→7475→8125** | 602, 650 | **48** |
| Flood G2 | MT AD 2458t/2459n | **6875→7475→8125** | 600, 650 | **50** |
| Flood G3 | MT AD 2460t/2461n | **6877→7475→8125** | 598, 650 | **52** |

The fixed second and third targets are MT Noah **AD 3058t/3059n** and SP primary Noah **AD 3708t/3709n**. Each cross-Mirror width obeys both civil component checks, `(C+1)+R−1=C+(R+1)−1=C+R`, without a civil year zero.

The second 48 construction uses Creation **4416t/4415n BC**, MT G2 Flood **AD 2458t/2459n**, MT G3 Noah **AD 3060t/3061n** and SP G3 primary Noah **AD 3710t/3711n**. It again gives `6873→7475→8125`.

The 52 constructions are exactly C01's two completed chains: Noah G1/Flood G2 with Creation 4419, and Noah G2/Flood G3 with Creation 4417. C06's two named 52 witnesses are thereby recovered directly from File_18.

**A second difference is insufficient to identify a 529 family.** The relevant comparison is

| Sequence | Widths | Gains | Second difference |
|---|---|---|---:|
| 12-fold scalar family | 6348, 6900, 7500 | 552, 600 | 48 |
| Sourced 48-case Creation sequence | 6873, 7475, 8125 | 602, 650 | 48 |
| 13-fold sourced family | 6877, 7475, 8125 | 598, 650 | 52 |

The first two sequences differ by **525,575,625**. That difference is an arithmetic progression, so its second difference is zero. This is the precise ambiguity: a second difference leaves an affine sequence undetermined. In the sourced 48 case, the first 25/23 step fails even though the second step remains exact.

The obstruction also follows directly from the endpoint roles. The last gain, MT Noah to same-Gear SP primary Noah, is always **650**. Matching `50k` therefore forces **k=13**. Its first gain must then be `46×13=598`, forcing **d=1**. No choice of Creation anchor can turn that fixed 650 interval into the 12-fold gain 600. This proves the absence of a complete 12-fold chain **in this endpoint arrangement**, without a wider source search.

The dilation anchors give an independent formulation of the same condition. For `Pₐ(x)=a+(25/23)(x−a)`, the unique anchors mapping `Fⱼ→Nᵢ` and `Nᵢ→Sᵢ` are

\[
a_1=F_j-\frac{23}{2}(N_i-F_j),
\qquad
a_2=N_i-\frac{23}{2}(S_i-N_i).
\]

Their difference is

\[
\boxed{a_2-a_1=25(1-d).}
\]

They coincide precisely at d=1. For the two 48 cases, the required first/second BC anchors are **4467/4417** and **4465/4415**. Each pair differs by 50; the first member lies outside both retained SP Creation fields. The sourced `6873→7475→8125` comparisons remain valid at the ordinary second anchor, but they do not form two iterations of one dilation. For the same-Gear G1 control, the anchors are 4444 and 4419, recovering C06's 25-year apparent/ordinary separation.

**The existing 552 contacts belong to the 52 cases.** MT Shem birth is 500 below the corresponding MT Noah birth in the reflected display. Thus

\[
H_j-\text{MT Shem}_i=550+2d=500+(H_j-N_i).
\]

| Relative Gear d | Noah/Flood selected span | Noah-to-SP-Flood gap | MT-Shem-to-SP-Flood contact |
|---:|---:|---:|---:|
| −1 | 602 | **48** | 548 |
| 0 | **600** | 50 | 550 |
| +1 | **598** | **52** | **552** |

C05's retained contacts **2556→3108** and **2558→3110**, each 552, occupy d=1. Accordingly, the three scalar numbers in `552+48=600` have witnesses at different relative Gear choices and different endpoint roles. The compared 552 and 48 intervals do not concatenate head to tail. One pair shares a terminal boundary: `2556→3108=552` and `3060→3108=48`. That overlap gives the sourced subtraction **3060−2556=504**, not a 600-year concatenation.

C05's separate result remains intact: the apparent SP closing at 4444 supplies **6900 to MT G1 Flood** and **7500 to MT G1 Noah**. The aligned 6348 comparison still requires the generated target 1904, which C05 found absent from its retained Noah/Shem/Flood registry. The new 48 intervals do not supply that missing endpoint.

**Evidence and checkpoint.** `evidence/check_c07.py` passes **133 exact checks**, with a byte-identical isolated replay. The six frozen dependencies are unchanged File_18 and File_09, C01/C05/C06 result records, and the C06 report. The checker independently extracts the source tables, verifies all nine Gear cases and seasonal widths, identifies the common-anchor condition, recovers the two C01 chains, and records the 48/50 controls and 552 endpoint limitation. The accompanying JSON retains every tested case and source label. No historical campaign is rerun, no canonical file changes, and no completion of interrupted Reports 11–25 is inferred.

**Best next bounded step:** test whether the same **598 selection jointly singles out the 13-, 17- and 2-fold families**—6877, 8993 and 1058—using the neighboring Gear cases as controls and distinguishing complete source joins from formal width equations. **High is sufficient.**