Joint ROUND observations and cap information loss

Preparation for the C1132–C1431 cycle. These are bounded questions and results, not numbered root actions. The latest File52c remains the primary project control. No decimal reversal, endpoint search, target tally, historical variant admission, or root journal edit is involved.

The useful new connection is a precise account of what the Rounded representation retains when two components are observed jointly, followed by a complementary account of the information discarded by the SP cap. Both fit the source-row grammar without treating equal totals as complete biographies.

Frozen packet and dependency status

The companion files are:

• round_joint_fibres.questions.json: 18 sequential questions recorded before this packet's exploratory calculations.
• round_joint_fibres.inputs.json: exact source bindings, the separately retained observation triples, a separate source-truth register, and the three excluded SP rows.
• round_joint_fibres.results.json: all 55 ordinary row fibres, all 35 existing ordinary inter-tradition comparisons in three measurement streams, all nine cap rows, exact suffix responses, and sequential result/reassessment records.
• round_joint_fibres.py: reproducible rational/integer calculation. It does not import the inherited calculation implementations or write any root journal.

The root's C1177 packet contains 58 rows: MT19, SP19, LXX20. The ordinary domain here contains MT19, SP16, LXX20. SP Jared, Methuselah, and Lamech remain excluded from ordinary L=b+r biography interpretation because their selected lifespan values are inclusive ledger counts. The separate cap analysis below operates on those declared counts and does not remove their types. The SP Lamech 53 counted/52 completed distinction remains unchanged.

Each source row has an exact literal File18 line binding. File18 §§2.1.3–2.2, §§3.1–3.2, and §§4.1–4.2 supply the selected begetting/lifespan values. File18 §4A.1, lines 1743–1757, controls LXX Lamech: restored182, attested753, calculated571; no identified joint182/753 manuscript attestation. File51a introduction and §3.4 supply nearest-five and separate-part rounding. Its old singular Methuselah wording is already qualified by C750/C753, which also identify Reu and the main LXX Lamech case. File52c's pinned SHA256 is a5ea84562101158b60d0cf296765d6eff38e7a2abda4e74ad1b353dfd13b9530.

The observation triples are separately retained mathematical inputs, not three independently attested historical readings. The current File18 packet calculates every remainder as r=L−b. This analysis freezes L,Q(b),Q(r), hides b,r from the inverse function, and compares the returned fibre with the source truth afterward. It tests conditional recoverability on explored sources. It does not generate the source values independently.

Inherited results reused here are C748's ordered-field reconstruction; C752–C753's forward rounding-defect law; C760's multiples-of-five equivariance; C770's residual recovery; C968's exact-cap fibres; C974's strict-support derivative; and C981's cap/round support. The numerical loss totals are regression checks, not new confirmations.

Joint observations determine a bounded fibre

For integer inputs define

There are no half-way cases for integer inputs. Retain exact lifespan L and the separately specified observations B=Q(b), R=Q(r), with L=b+r and nonnegative integer parts. Then the complete fibre is

This follows by intersecting the two rounding cells with the exact lifespan line. An empty interval proves incompatibility. A one-point interval proves conditional uniqueness. The algorithm uses no actual begetting or remainder to choose a candidate.

All 55 source rows lie away from the zero boundary. Write b=B+u, r=R+v, and s=L−B−R. The residual square has −2≤u,v≤2 and the exact lifespan imposes u+v=s. For |s|≤4, this leaves exactly 5−|s| pairs. Thus interior uniqueness occurs exactly when |s|=4. The interval formula above is the general rule; the short cardinality formula must not be extended to zero-boundary cells without qualification.

The full source test gives these conditional singleton rows: MT Methuselah, MT Reu, LXX Methuselah, LXX Reu, and SP Reu. The complete unresolved fields remain in the results alongside them. These names are an inverse-fibre description, not a target-frequency result.

Lamech's source182 selects the operative571 remainder from a two-member fibre. Native Cainan's source130 selects its row from five candidates. None of the other candidates acquires chronological status. Nor do the singleton results establish historical generation of187 or32, because their rounded remainder premises were calculated from the selected source row.

The same framework states what another observation would add. Once exact L is held, Q(L)=Q(b+r) is redundant. If only the three rounded values are held, the interior fibre instead has 19 pairs when the rounded parts agree with the rounded total, or 3 pairs for either nonzero inherited defect. Exact L selects a diagonal of that earlier fibre. When 5|L, even R=L−B follows automatically from L,B; separately written observations need not be independent restrictions.

For a nonsingleton row, one independent exact residual u, or equivalently one exact part, selects a candidate. More generally, a supplied linear measurement M=αb+βr determines

when α≠β; when α=β it only repeats L. This is a sufficiency/identifiability statement within the declared observation class, not a claim about optimal global bit encoding or a new source relation.

Source shifts require a residual carry

For a source-authorized integer shift d, the lossless pair (y,ρ)=(Q(x),x−Q(x)) transforms as

The signed residual argument is an algebraic auxiliary, not a dated count. Since x=y+ρ, the rule is ordinary addition expressed in rounded-plus-residual coordinates. Its composition law is

Dropping the residual permits a shift on rounding cells for every integer input exactly when 5|d. Such shifts carry entire five-element cells onto cells. Every other residue shift splits a translated cell across a boundary, so two inputs with the same rounded value can have different rounded results. Restricted source domains can happen to avoid that split; this does not create a universal action.

All 35 ordinary existing comparisons—MT→LXX for 19 common rows and MT→SP for the 16 ordinary common rows—satisfy the same rule in the begetting, remainder, and total-lifespan streams. The native LXX Cainan insertion has no fictitious MT row in this comparison; it is already retained in the 55-row fibre field. The two scalar contributions to Δr=ΔL−Δb give the same lifted result in either order. This does not claim additional intermediate chronology routes.

The source LXX Lamech change provides a discriminating case. The lifespan changes by−24, from777 to753; its remainder also changes by−24, from595 to571. But the first lifespan has residual2 while the first remainder has residual0. Therefore rounded lifespan changes from775 to755, a shift of−20, while rounded remainder changes from595 to570, a shift of−25. One source displacement with different retained residues explains both. The residual encoding itself is lossless bookkeeping, not reduced source information.

Capping and rounding have a complete composite fibre

Keep a fixed nonnegative integer capacity c, set y=min(L,c), and observe T=Q(y). Put C=Q(c). Because nearest-five rounding is monotone,

Thus capping commutes with rounding when both arguments are represented on the same rounded grid. Keeping the capacity exact creates a different comparison. For Jared, Q(min(962,847))=845 but min(Q(962),847)=847; for Lamech, the corresponding values are655 and653. Their differences−2 and+2 arise from the mixed representation.

For T a nonnegative multiple of5, the complete original-lifespan fibre is

An off-grid observation is also impossible. The first case is an ordinary rounding cell strictly below capacity. In the second, every input from the start of the capacity's rounding cell through and beyond the cap has the same observation. The plateau therefore can begin below the true capacity: a rounded cap observation does not alone prove that clipping occurred.

This is the whole nine-row source cap field, with exact capacities inherited from File18 §3.1.3 and the already reconstructed birth field. All alternatives are inverse-fibre integers, not proposed ancestral lifespans. In particular the Jared fibre includes values843–846 that were not clipped at all, even though they share the source's rounded cap observation.

Loss registers are complementary on each branch

Define the discarded cap excess z=L−y, the capped residual ρ_y=y−Q(y), and the baseline residual ρ_L=L−Q(L). Then

These give two named loss registers, but they are not two independent missing numbers on every row. Valid metadata satisfy

The conditions are necessary and sufficient for the reconstructed L=y+z to cap back to y.

• On known strict active support L>c, y=c and ρ_y=c−Q(c) are fixed by the capacity. Only the positive excess z is missing. Indeed c and z>0 already give L=c+z without an additional rounded-residual input.
• On known strict inactive support L<c, z=0. Only the finite rounding residual is missing.
• At a separately declared tie L=c, the original lifespan is already fixed.

Branch support must be retained or inferred from additional observations; it cannot be read automatically from T=Q(c). Supplying ρ_y alone recovers the exact capped value but still leaves the excess unknown if y=c. Supplying z=0 alone generally leaves finite rounding ambiguity. This is the sharper minimum-missing-information statement supplied by the composite analysis.

For the complete source field the nonzero exact excesses are115,249,124 and the separately rounded reductions are115,250,120. Their totals488 and485, and their suffix responses, are inherited numerical consequences. The complete exact reduction suffixes are

488,488,488,488,488,488,373,373,124;

the rounded suffixes are

485,485,485,485,485,485,370,370,120.

Their difference has the already established local support0,+1,−4 on Jared, Methuselah, Lamech. The sign is rounded-loss minus exact-loss; it is the negative of C981's comparison-residual convention.

With exact births and exact capped lives retained, the strict-support observation Jacobian is

where U accumulates the nine completed begettings, and A selects Jared, Methuselah, Lamech. Invertibility of U fixes all begetting inputs; the six uncapped lives are observed directly; only three ancestral excesses remain. The rank is15 on18 variables. The global capped fibres, rather than this local differential statement, govern thresholds.

Replacing exact capped lives by their rounded observations additionally discards the finite residual choice on each of the six inactive rows. It does not create a second independent loss on each clipped row, because that capped residual is fixed by capacity. A separately supplied total excess would give one equation on the three excesses, leaving two independent directions; the existing488 was calculated from those same baseline lives and cannot be promoted to an independent input that regenerates them. Further source measurements must be named and costed. A source-supported exact life can be retained, but retaining it is different from explaining its origin.

Sequential adoption recommendation

Use RJ01–RJ07 to establish the full joint-observation fibre and its dependency limits. Follow with RJ08–RJ11 to show how the same row grammar acts coherently on Rounded plus residuals. Then RJ12–RJ16 connect the SP cap through complete composite fibres and complementary losses. RJ17–RJ18 give the minimum source burden and distinguish new structural explanations from inherited regression values.

The principal new results are the conditional inverse fibres, their complete source bindings, the carry action on the full ordinary comparison family, the global cap-plus-round fibres, and the branchwise loss constraint. They explain relationships among complete representations while leaving the original source choices visible. No claim of historical generation, statistical significance, or automatic source identification follows from these mathematical reconstruction results.

Complete ordinary-row audit

For each row the interval lists all compatible begetting values; the remainder is L−b. These are formal candidates only. The actual source pair is used exclusively for the final containment check.