Independent rounding review extension

Prepared while root proceeds through C749 onward. This document contains review questions, exact preparatory results, and suggested discriminators; it records no numbered research actions. The existing four rounded-transfer packet files remain unchanged. Use the current root journal to avoid repeating results already executed (C750–C755 already cover much of the basic defect and MT-field analysis).

The most useful remaining synthesis is: Rounded is a row transformation whose residuals propagate along the chosen chain. It is not direct rounding of already calculated dates. Cross-tradition differences are explained by the support of a few changed residuals. Their role is then visible in the Strategy's SP three-year discrepancy and its supplied Creation rectangle.

Definitions throughout: R(n)=5 floor((n+2)/5) for nonnegative integer measures; e(n)=R(n)−n; K(b,r)=R(b)+R(r)−R(b+r); D_i=Σ_{j≥i}e(L_j), with terminal displacement zero. “Source” means literal input or operation supplied by File51a/File18; “derived” means a computed comparison, not a newly adopted source chronology.

Discriminating questions and results

Component projection: does the full rule explain both MT exceptions?

Question: Is the failure of componentwise and lifespan rounding isolated, or generated by one finite rule?

Result: For a compatible row (b,r), the regular selector is b and cumulative selector is b+r. Rounding components before projection versus rounding projected values gives defect vector (0,K). The complete25-case residue table in the packet has only six nonzero entries: (1,2),(2,1),(2,2) give−5; (3,3),(3,4),(4,3) give+5. MT Methuselah and Reu occupy(2,2), so one rule explains both. This is the structural correction to File51a's globally scoped “Methuselah unique” sentence. Both literal source outputs are preserved. Root already addressed this family by C752; retain as foundation, not another step.

Reu's source landing: which rule actually produces2091?

Question: Does Reu's source-adopted theoretical death/call alignment depend on the declared component rule?

Result: Source Rounded birth2326 and component life30+205=235 give2091. Direct total-lifespan rounding239→240 would instead give2086. In the strict actual comparison, birth2327→2326 moves−1, while life239→235 moves−4, so death moves by −1−(−4)=+3:2088→2091. This links the corrected operator distinction to an already interpreted source relation. The2091 alignment is Source MOD-5 THEORETICAL;2086 is a diagnostic alternative, not a new appointed death.

LXX Lamech: can the adopted753 remain active without an MT-like death margin?

Question: What does the declared component operation imply for the main LXX182/753 row?

Result: Main182+571=753 becomes180+570=750, whereas total rounding gives755. Derived regular Rounded birth4016 produces theoretical terminal3266 from750. The corresponding derived rounded Noah3836/Flood3236 pair gives a30-year margin. The source Actual main margin is29. Using755 in the regular death calculation would give3261 and a25-year margin. Thus the two different rounding operations cannot be interchanged while keeping this derived relation. This is a derived LXX comparison; do not appoint3266 or3261 as new canonical death dates. No777 overlay or188 execution is needed.

Whole-chain reconstruction: are all boundary displacements independently chosen?

Question: Can the entire26-row MT field be reconstructed from local residuals and one fixed terminal?

Result: D_i−D_{i+1}=e(L_i) and D_terminal=0 uniquely recover the field. Both directions reproduce all26 source Actual/Rounded positions. Once rows are fixed, every displacement is forced. This is a useful minimality statement: one should not count the26 matching positions as26 independent structural successes. Already addressed by C754–C755; no new execution needed.

Unchanged intervals: can one equivalence relation explain all of them?

Question: Which whole source intervals retain their lengths under row rounding?

Result: An interval from boundary i to j has change D_i−D_j; it is unchanged iff D_i=D_j. The MT classes are:

Every preserved interval follows from these classes. For example, the nonadjacent Lamech–Shelah span remains2765 in both source cumulative columns. The classes generate38 unchanged intervals among351 source-boundary pairs; this is a dependent enumeration, not a statistical success tally. C755 apparently covers the main classification; use source-labeled illustrations only if useful.

Translation classes versus event identity: what does an unchanged interval mean?

Question: Does equal displacement identify roles or merely preserve their separation?

Result: MT Noah/Shem/Arphaxad all move+5 and preserve their950/600 cumulative spans. Terah/Abraham/Isaac/Jacob all move−4 and preserve their205/175/180 spans. Lamech and Shelah share+3 while retaining different roles. Thus equal residuals generate translated subfields, without identifying names, biographies, or cumulative boundaries with regular events. This is an explanatory consequence of the complete field, not a new free transport operator.

LXX transfer: what is the smallest support explaining its whole field?

Question: Which changes to MT residuals are necessary and sufficient for the native LXX field?

Result: The only changed common-row residuals are Lamech+4, Arphaxad−2, Shelah−2. All other common rows retain their residuals; native Cainan adds a zero-residual row. Consequently, LXX minus MT boundary displacement is−4 at Noah/Shem/Arphaxad,−2 at Shelah, and0 at every other common row. Its source lifespan-difference vector becomes:

This provides an exact row explanation of both the preserved overall differential and the altered interior. The cross-tradition rounded output remains a derived comparison from source rows and the transferred rule.

SP transfer: can the608→605 change be traced without fitted corrections?

Question: Is SP's three-year change an endpoint convention or a determined rounding effect?

Result: For the held lifespan-count vector it is a determined rounding effect. Only Methuselah(−1) and Lamech(+4) change their residuals relative to MT. The actual five-row differential becomes:

The cross-boundary displacement is+3 from Adam through Methuselah,+4 at Lamech, and0 from Noah onward. These are cumulative count calculations, preserving the SP inclusive lifespans; they are not regular death-date corrections.

Cross-tradition interval transfer: which interior structures survive together?

Question: Does a preserved Creation differential imply preserved differential at every internal boundary?

Result: No. For two traditions define H_i=D_i(second)−D_i(first). Their interval differential changes under rounding by H_i−H_j. In LXX/MT, any two endpoints outside the four nonzero common boundary positions preserve their actual differential, but intervals crossing the local−4/−2 plateau can change. In SP/MT, every common interval wholly from Noah onward preserves its actual differential; the+3 upper plateau preserves every wholly upper interval as well. Only intervals crossing the support transition change. This is a complete whole-family statement, not a list of favorable pairs.

Native and leveled states: does removing Cainan alter the residual field?

Question: Does the460 insertion generate a second rounding correction?

Result: R(130)=130, R(330)=330, R(460)=460. The explicit finite insertion/removal therefore adds no begetting, remainder, lifespan, or projection defect. Corresponding common-name residuals and all their difference fields remain unchanged. Absolute durations change by the established130/460 where the row lies on the selected path. This verifies compatibility of leveling with residual propagation without reopening a new inverse or source variant family.

Manuscript block changes: why do centenary repartitions usually commute?

Question: Is preserved rounding under100/50/130 changes accidental?

Result: R(x+5k)=R(x)+5k, hence e(x+5k)=e(x). LXX's begetting residual profile is exactly MT's at all common names, because every corresponding begetting change is divisible by5 and native Cainan has zero residual. In SP's descriptive53 count route only Lamech changes the begetting residual relative to MT, by+4; the52-completed-input route restores the common MT begetting-residual profile. This concerns source measures, not permission to move source dates between Gears.

Direct date rounding: does it reproduce the source rounded chronology?

Question: Could one round each already calculated Actual boundary to the nearest date ending1/6 instead of rounding source rows?

Result: Define the diagnostic grid projection Q_a(x)=a+R(x−a) at a=1406. Row rounding differs by D_i−e(Σ_{j≥i}L_j). It disagrees with the literal MT rounded column at11 of26 boundaries. MT Noah6381 is already on the1/6 grid, yet source row propagation sends it to6386; cumulative Abraham2435 grid-projects to2436, but source row propagation gives2431. The full derived counts of disagreements are7/27 in LXX and17/26 in SP. These counts summarize dependence on the row path; they are not independent evidence. The diagnostic Q is not a source-adopted chronology.

Rounding after summation: does endpoint agreement extend to every subpath?

Question: Does the known12600 total cancellation mean row rounding and total rounding commute throughout the source chain?

Result: For any admitted consecutive lifespan interval,

Σ R(L_j)−R(Σ L_j)=Σ e(L_j)−e(Σ L_j).

The complete MT351-interval diagnostic has defects0,−5,+5,+10. One concrete source subpath, cumulative Enoch8492 to Terah2640, has raw5852; source row rounding gives8496−2636=5860, while rounding5852 once gives5850. Their10-year difference is fully predicted by the residual field. This directly explains why endpoint agreement at the whole chain cannot license forgetting its internal subdivision. It resembles the retained-path principle already established for decimal reversal, while using a distinct operator.

Global date map: can one affine transformation replace the row rules?

Question: Are the MT cumulative Actual/Rounded dates related by one affine date map?

Result: The held terminal1406 and Moses boundary1526 both remain fixed. Any affine map fixing these two distinct coordinates is identity, but the source Amram1663→1661 contradicts identity. Thus no global affine date map can reproduce this source pair of complete columns. Local equal-residual fields are translations; the whole relation needs source-row residual data. This conclusion uses Moses-line columns, not the separately selected14004 completion endpoint.

SP count interpretation: can a one-year input distinction be corrected after rounding?

Question: Does rounding preserve the distinction between53 counted and52 completed as exactly one year?

Result: R(53)=55, R(52)=50; the difference becomes5. An “interpret first” versus “round first then subtract1” comparison differs by4. Holding the rest of the row diagnostic and Jacob2006 gives count-route4411 versus completed-route4406. These match the source's adopted lower SP block member and alternate local comparison respectively, but File51a does not explicitly identify this exact derivation as its construction. Therefore this is a conditional reconstruction, not a source-ownership claim and not upstream companion transport.

Inclusive cap rows: does algebraic decomposition supply an ordinary biography?

Question: May SP inclusive L values and ordinary b intervals be treated as the same unrestricted (b,r) row type?

Result: No. File18 explicitly distinguishes653 inclusive from52 completed to primary Noah, and inclusive cap counting at Jared/Methuselah/Lamech. L−b is an algebraic count difference in the packet; it is not automatically an ordinary elapsed remainder. SP cumulative rounding can use the source count vector, while ordinary regular death reconstruction requires the source inclusive endpoint term. SP Reu132+107=239 remains an ordinary compatible row and gives a valid−5 defect. This is a source tension that limits the domain of a universal biography map without blocking the full cumulative residual analysis.

Strategy SP rectangle: can the supplied+10 be decomposed into known operations?

Question: What source components can explain13396→13406 on the cumulative side of the Strategy rectangle?

Result: The exact conditional decomposition is

13396 +2 =13398 (completion member to source Moses-line member),

13398 +3 =13401 (the new SP row-rounding residual),

13401 +5 =13406 (File51a rounded lifespan-branch replacement125−120).

Thus+10 can be reconstructed as+2+3+5, using declared selections and the newly explained residual. The final+5 is the rounded lifespan branch, not the exact Actual Aaron/Tishri phase shift+3.5. File18 §6C.1 separately labels13401 an Aaron/Adam Day4 phase-anchor display member; its equality to the derived Rounded Moses13401 does not identify the constructions. The Strategy gives13406 directly, but assigning it specifically to this composed rounded route remains a proposed explanation until the controlling ReportIV route is available. This is the strongest prospective connection to the Strategy's larger goal.

Lossless relationship: what data must accompany Rounded to recover Actual?

Question: Can regular/cumulative/Rounded be representations of a shared object without pretending rounding is invertible?

Result: Retaining (R(x),e(x)) recovers x=R(x)−e(x) exactly. On a chain, retaining rounded boundaries and the field D recovers Actual boundaries; local differences of D recover individual row residuals. Rounding alone loses this information. Thus a shared source object with selected measurement, rounded component, and residual supports both families. This is an exact reconstruction rule for supplied integers, not a second decimal inverse and not a claim that the residual has an independently established historical origin.

Source and review controls

• Latest File52c remains primary for its downstream inverse study; this extension performs no decimal inversion.
• MT theoretical deaths remain theoretical; transferred LXX death diagnostics are derived comparisons.
• Cross-tradition direct lifespan-rounding columns remain derived from source counts and the transferred rule until matched to controlling rounded tables.
• Native LXX includes Cainan; MT/SP native omit it. Leveled comparison changes one explicit row only.
• LXX182/753 remains main; SP653 inclusive and53rd/52 completed remain distinct.
• No Gear, chronological phase, event identity, or primer is created by a residual-field equality.
• The25-case defect classification, complete interval enumeration, and date-grid comparison are dependent consequences of frozen inputs. No statistical claim or new evidential tally is proposed.
• Existing packet files were read without modification. Only this extension document was written.