490d C276 — The 598 source relation and the translation increment

24 September 2026. Continuation of the Mirror and 529 research after C275. Completed bounded step; C277 is proposed only.

Result

The retained 598-year source relation determines the difference between the Flood and Shem translations when the selected cross-index endpoints are required to align. The source duration and the 17K correspondence alone do not impose that alignment.

With the alignment imposed, the increment is uniquely 598 and the same-index 600 pair reflects about the established local junction x=1, January AD 2. One common translation remains free. This determines relative geometry without determining the familiar absolute images 5, −5 and 7.

This is a precise conditional role for 598 in the construction. It connects the source-tagged rectangle to its local reflection while distinguishing the source relation, the required alignment, and the retained absolute placement.

1. Fixed source register

The entire audit uses the three cumulative LXX OFF centers and their three reflected regular SP counterparts already retained in C269–C270. Here K=529 and T(q)=q+8993=q+17K.

Symbol and cumulative role	Center q	Field index	Full cumulative pair	T(q), reflected regular counterpart
S: Shem closing	−5883	7	5884t/5883n BC	R=3110: G3 Flood start, AD 3110t/3111n
I: Flood interior	−5285	5	5286t/5285n BC	P=3708: G2 primary Noah birth, AD 3708t/3709n
L: Flood closing	−5283	7	5284t/5283n BC	N=3710: G3 primary Noah birth, AD 3710t/3711n

Thus:

I-S=P-R=598,\qquad L-I=N-P=2,\qquad L-S=N-R=600.

The 598 span is cross-index: Shem index 7 to Flood index 5. Its regular counterpart connects G3 Flood to G2 primary Noah. The 600 span is same-index, index 7 on the cumulative side, and same-Gear G3 on the regular side. The 598 comparison is not the full matched 600 lifespan or a one-year companion shift.

The positive regular coordinates are reflected source positions. They do not claim historical AD occurrences or reverse the historical order of Noah and the Flood.

The frozen C127 graph supplies three relevant duration edges:

Stored edge	Endpoints	Width
N04_598cum	S to I	598
N04_17	I to P	8993
N04_17opposite	S to R	8993

L=−5283 is not a stored C127 node. Its retained connection to N comes from C254 selector 7, preserved and checked in C269:

-5283\to-4415\to1404\to2460\to3710, \qquad868+5819+1056+1250=8993.

C276 reuses that completed route without rerunning C254 or adding a graph edge. A graph edge typed as a duration records the source separation; it does not itself assert equality of images under two independently translated branches.

2. Let the translation increment remain free

Let f be the cumulative LXX Flood translation and h the cumulative LXX Shem translation. Define:

F(q)=q+f,\qquad H(q)=q+h,\qquad G(q)=M(H(q))=-q-h,\qquad \delta=h-f.

M(x)=−x remains the global Mirror about January AD 1. The reflected regular branch maps are:

R_N(y)=y+\alpha,\qquad R_F(y)=-y+\beta.

Require both routes through the source correspondence to agree on their declared branches:

R_N(T(q))=F(q),\qquad R_F(T(q))=G(q).

Substitution gives:

\boxed{\alpha=f-8993,\qquad \beta=8993-h.}

These are two independent equations in the four unknowns (f,h,α,β). Their rank is 2, with two free parameters. Null directions are:

(1,0,1,0),\qquad(0,1,0,-1).

Both preserve source transport, while changing δ by −1 and +1 respectively. Consequently, the retained source coordinates, their 598/600 separations, and the branch transport equations leave δ undetermined.

The calculation uses exact rational arithmetic. Integer f and h preserve the inherited annual-center and seasonal labeling convention. These are formal freedoms used to identify what the equations determine; no new historical state or fitted parameter value is selected.

3. State the extra alignment explicitly

Call the selected image-alignment condition X:

\boxed{X:\ H(S)=F(I).}

The two images have residual:

H(S)-F(I)=(-5883+h)-(-5285+f)=\delta-598.

Therefore:

\boxed{X\iff\delta=598.}

X adds exactly one independent equation. The system now has rank 3 and one free parameter, with surviving null direction:

(1,1,1,-1).

This condition can be stated without requiring the MT closing join or specifying that the common image is 5. It is nevertheless an additional alignment requirement. Choosing the source endpoints supplies the number 598; requiring their separately translated images to coincide is the step that transfers that source number into the operator increment.

The following are equivalent descriptions of this one scalar requirement:

Description	Equation
Forward cross-index alignment	H(S)=F(I)
Global reflection of that pair	G(S)=M(F(I))
Translation increment	h−f=598
Same-index local reflection	G(S)=2−F(L)
Regular cross-index image sum	R_N(P)+R_F(R)=0
Regular same-index image sum	R_N(N)+R_F(R)=2

They should not be counted as six independent determinations.

4. What the alignment determines

Without X, the image sums are:

F(I)+G(S)=598-\delta,\qquad F(L)+G(S)=600-\delta.

The two image midpoints are therefore:

\mu_{598}=\frac{598-\delta}{2},\qquad \mu_{600}=\frac{600-\delta}{2},\qquad \mu_{600}-\mu_{598}=1.

The one-year difference follows from the fixed two-year source offset for every f and h. These are diagnostic image midpoints, not newly adopted global axes.

Under X, put p=f−5285. Then:

\boxed{F(I)=H(S)=p,\qquad G(S)=-p,\qquad F(L)=p+2.}

The cross-index pair (p,−p) is globally mirrored about x=0. The same-index pair (p+2,−p) is locally mirrored about x=1. The latter has radius |p+1|. The radius is not forced to be six, and this selected pair need not remain the outer pair under every formal common translation.

The retained baseline supplies the absolute placement:

Quantity	Baseline reference
f	5290
h	5888
δ	598
α	−3703
β	3105
p	5
F(I), H(S), G(S), F(L)	5, 5, −5, 7

This reference is read from the existing C270 maps. It is not imposed in either derivation to recover a desired landing.

In particular, the Noah subtraction is 8993−f. It becomes 3703=7K at f=5290, but X alone does not fix it at 7K. Likewise, X alone does not fix the previously obtained 5750/6250 phase bracket: that numerical bracket also depends on the retained absolute placement and chart.

5. Physical phases commute with the transport

Each source pair consists of its signed center plus offsets ε=±1/4. For all three retained routes:

T(q+\varepsilon)=T(q)+\varepsilon.

The Flood-to-Noah route has slope +1, so it preserves component order. The Shem-to-reflected-Flood route has slope −1, so it reverses order. Both preserve the physical half-year separation.

For the same inherited source offset:

H(S+\varepsilon)-F(I+\varepsilon)=\delta-598,

F(L+\varepsilon)+G(S+\varepsilon)=600-\delta.

At X, the second sum is 2 for each physical component, as required by the local reflection J₁(x)=2−x. There is no phase reattachment. Original regular BC ancestry, its reflected AD position, and the cumulative source ancestry retain their distinct labels even where their images coincide.

The phase ledger records all six component routes: three source correspondences, each with two seasonal components.

6. Dependencies that must remain visible

C270's fixed regular intercepts already contain the original placements. The maps y−3703 and 3105−y were induced from f=5290 and h=5888. Freezing those intercepts as independent source evidence would import δ=598 into the premise set. C276 instead allows the induced intercepts to vary with f and h.

The reduced MT closing equation already contains 598. If the increment is free, the corresponding MT closing join is:

m-5289=5889-(m+\delta), \qquad\boxed{2m+\delta=11178}.

The earlier equation 2m=10580 substitutes δ=598. It cannot then independently determine that increment. Neither version of the MT join is imposed in the C276 source-transport cases.

A fixed local reflection depends on the increment. The general regular identity is:

R_F(y-600)=600-\delta-R_N(y).

It becomes J₁ only at δ=598. C271's completed fixed-J₁ matching results remain valid at their retained baseline; C276 does not extend their counts to free δ or rerun the 36-pair campaign.

This is one LXX rectangle. It does not establish that every chronology profile must share the same increment. Co-location of transformed points does not identify their source events or calendars. The historical or structural reason for requiring the tagged alignment X remains a separate question from the exact consequences once it is required.

7. Supplemental LXX observation retained

Dean's corrected comparison remains an awareness supplement, as verified and recorded in C275:

LXX source and +8464 landing base	Reflected AD anchor	Gap	Expanded by 25/23
Creation 5494 BC → AD 2970	Conquest 1406	1564=68×23	1700
Jared 4534 BC → AD 3930	Exodus 1446	2484=108×23	2700

The full landing pairs remain AD 2970t/2971n and AD 3930t/3931n. LXX Creation→Jared is 960 versus MT's 460; the offsets are 1380 at Creation and 880 at Jared. The gap difference is 960−40=920, expanding to 1000. AD 3746/2300 is superseded for this observation.

The expansion applies to these comparison gaps, not to the 8464 bridge. The supplement supplies no equation to C276 and introduces no upstream Gear state.

8. Evidence and completion

Sixteen existing inputs are frozen with SHA-256 hashes. The exact verifier passes 70 targeted checks. Its six generated outputs replay byte identically: source register, transport constraints, alignment identities, phase ledger, assumption ledger and results.

The checks cover source bindings, selected graph edges, retained route endpoints, exact affine expressions, matrix ranks and null directions, physical phases, and baseline consistency. The author review separates these arithmetic checks from the interpretation of what counts as an additional requirement. No new independent-agent review is claimed.

The original dependencies remain unchanged. File_18, File_09, the C265 certificate and C127 graph pass their standing integrity checks. C127 remains 208 nodes and 382 duration edges.

The evidence packet contains the report, compact checkpoint, executable verifier, six readouts, review notes, replay record, source and artifact manifests, control integrity record, predecessor certificate, and a new certificate binding this step. Verification requires only the Python standard library and the packet's frozen inputs.

Current scope remains Mirror and 529. Regular Gears remain Noah/Shem/Flood only. No canonical amendment, graph edit, new event search, parameter fit, primer reopening, new global axis or z-law is made. Capture now, canonize later.

Best next bounded step

C277: extend this same tagged 598 alignment across the existing six C271 bindings. Check which cross-index pairs have both regular counterparts, and identify where the existing source-index or register boundaries stop the extension. Use the retained register; add no events and fit no landing.

This would test the alignment's coverage beyond the selected closing rectangle. High is sufficient. C277 has not been executed.

The598 source relation and translation increment