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Proposed extensions of the source-row grammar

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Proposed extensions of the source-row grammar

Immutable preparation for C1132–C1431. These are bounded questions and exact algebraic predictions, not numbered research results. The complete Strategy and the final C1131 technical companion have been read. No source or prior manuscript was edited.

Recommended direction

The strongest next link is an inverse question on the shared source object: which local biography information survives Regular, Cumulative, Rounded and capped observations? A two-generator row normal form can organize the known differences, but its coordinate change is not explanatory compression by itself. The useful tests are the source-supported rank-five LXX family, finite rounded inverse fibres, and the rank of the whole birth-plus-cap observation map. These lead to explicit unresolved source inputs rather than another endpoint catalogue.

Two promising exact predictions are worth testing first. Joint rounded begetting/remainder values plus an exact lifespan should identify MT/LXX Methuselah and Reu, and ordinary SP Reu, uniquely within their bounded integer fibres. Main LXX Lamech should remain a two-choice inverse, preserving the need for the restored source182 selection. Separately, the nine-birth/nine-cap observation map should have rank15 on its three-row clipping region: exactly the original excesses of Jared, Methuselah and Lamech remain invisible. Neither finding would establish historical source independence.

Source controls and inherited scope

The following are already proved and should be cited, not scheduled as fresh investigations:

Bounded proposals

OPX01 — A unique two-generator row normal form

Priority: foundation. Question: Can all compatible source-row changes be classified by lifespan-preserving repartition and remainder-preserving extension, while boundary conventions stay separate?

Domain and retained premises

Exact predictions

{
  "basis": [
    "R=(1,-1,0)",
    "S=(1,0,1)"
  ],
  "normal_form": "Delta(b,r,L)=alpha R+beta S; beta=Delta L; alpha=Delta b-Delta L=-Delta r",
  "regular_edge_change": "Delta b_eff=alpha+beta+Delta kappa",
  "cumulative_edge_change": "Delta L=beta",
  "bridge_generator_change": "Delta(L-b_eff)=-alpha-Delta kappa",
  "admissibility": "Each intermediate row must retain its source status and any required nonnegative components; algebra does not authorize a variant."
}

Proof route

Complete-family test: Produce one complete labelled row classification on MT19, native LXX20 and SP19 source rows, marking the three SP inclusive exceptions and every binding correction. Do not replace these rows by an unlabelled 2D vector.

What this adds: Provides a constructive operation vocabulary for the complete received rows. The coordinate change alone is not an information reduction; reduction must come from source-controlled support/equality constraints tested next.

Counterexample or limit: A remainder derived as L-b cannot be counted as a third independent datum. SP Lamech nominal53/inclusive653 and effective52 are different tags, not competing inputs to one ordinary row.

Sources: Strategy §§3.2,5B,5F; File18 §§2.1.3–2.2,3.1–3.2,4.1–4.2,4A; File51a §3.4.

Inherited baseline: C733–C748 already prove the mode bridge and boundary recovery. C779–C786 already prove the Terah triangle and its projections.

OPX02 — A five-parameter LXX change model

Priority: high. Question: Does the complete received MT→LXX shared-row comparison lie in one explicitly supported rank-five family, plus the separate native Cainan insertion?

Domain and retained premises

Exact predictions

{
  "parameters": {
    "h": 50,
    "lambda": -24,
    "a": 27,
    "e": 40,
    "n": 60
  },
  "regular_delta": "2h on Adam,Seth,Enosh,Kenan,Mahalalel,Enoch,Arphaxad,Shelah,Eber,Peleg,Reu,Serug; h on Nahor; zero otherwise",
  "life_delta": "lambda on Lamech; a on Arphaxad and Shelah; e on Eber; 2h on Peleg,Reu,Serug; n on Nahor; zero otherwise",
  "predicted_rank": 5,
  "inserted_row": [
    130,
    330,
    460
  ],
  "native_shared_heads": {
    "Delta_regular_Creation": 1380,
    "Delta_cumulative_Creation": 890
  },
  "leveled_OFF_heads": {
    "Delta_regular_Creation": 1250,
    "Delta_cumulative_Creation": 430
  }
}

Proof route

Complete-family test: Regenerate all shared regular/cumulative displacement coordinates, including every zero plateau, using the existing displacement artifact only as a comparison target. Keep native Cainan as an inserted node.

What this adds: A genuine rank reduction within a declared supported linear model: common centenary magnitude, equal Arphaxad/Shelah lifespan change and the three remainder-preserving post-Flood changes are constraints, not additional outputs.

Counterexample or limit: This compresses explored data; it does not derive100,27,40,60 or the masks historically. Count the masks, Cainan row and source choices in description cost. Reject any claim of whole-repository minimality.

Sources: File18 §§2.1.3–2.2,4.1–4.2,4A,1.6; Strategy §§4.1,5B,6.

Inherited baseline: C739–C740 already give every displacement and its local support.

OPX03 — A minimal observation basis for the LXX model

Priority: high. Question: Which five source measurements determine the entire supported LXX comparison, and which printed agreements are then dependent?

Domain and retained premises

Exact predictions

{
  "observation_basis": [
    "Delta b(Adam)=100 -> h=50",
    "Delta L(Lamech)=-24 -> lambda=-24",
    "Delta L(Arphaxad)=27 -> a=27",
    "Delta L(Eber)=40 -> e=40",
    "Delta L(Nahor)=60 -> n=60"
  ],
  "expected_identification": "Five independent observations recover all five parameters; omitting any one leaves a nonzero formal direction.",
  "predicted_source_checks": [
    "Delta L(Shelah)=27",
    "Delta b(Nahor)=50",
    "Delta L(Peleg)=Delta L(Reu)=Delta L(Serug)=100",
    "All six pre-Flood centenary rows preserve L."
  ]
}

Proof route

Complete-family test: Compare the regenerated complete19-row two-measurement field and the inserted Cainan row with the frozen source table. Do not count each predicted coordinate as independent evidence.

What this adds: Separates a minimal measurement basis from its dependent whole-family consequences. This tests compression rather than redisplaying endpoint sums.

Counterexample or limit: The basis is conditionally minimal only inside OPX02. It is not an independent textual basis or evidence that five choices explain the source history.

Sources: File18 §§2,4,1.6; Strategy §§5D,6.

Inherited baseline: C746 composition consistency and C748 unitriangular recovery remain inherited.

OPX04 — Recovering local rows from partially observed two-mode fields

Priority: medium. Question: How much source information is lost when only selected Regular and Cumulative boundaries survive?

Domain and retained premises

Exact predictions

{
  "full_fields": "The joint map (b,L)->(Ub,UL) has rank2n; r=L-b is dependent.",
  "m_block_observations": "If only m disjoint block totals are retained in each mode, rank is2m and kernel dimension2(n-m).",
  "null_moves": "Within a block, opposed transfers in b or L preserve the measured totals but change internal boundaries."
}

Proof route

Complete-family test: Test the actual source segment partitions already used in the native19/20-row fields; retain omitted internal names in metadata. Use the existing Eber73 counterexample as context, not a new result.

What this adds: Quantifies the reconstruction burden of each omitted boundary on the joint two-mode object.

Counterexample or limit: Generic rank assumes unconstrained row coordinates. Source equality constraints or nonnegative boundary cases can lower dimension; state them separately.

Sources: File18 native tables; Strategy §§5B,5D,5G.

Inherited baseline: C747 already gives one swapped-row counterexample. C993 gives the universal conversion/coarsening criterion.

OPX05 — Insertion is identifiable only with its junction data

Priority: high. Question: Can a collapsed Arphaxad→Shelah comparison distinguish the native Cainan insertion from changes in the adjacent parent row?

Domain and retained premises

Exact predictions

{
  "observed_collapsed_difference": [
    230,
    487
  ],
  "source_decomposition": {
    "parent_change": [
      100,
      27
    ],
    "Cainan_measure": [
      130,
      460
    ],
    "Cainan_remaining": 330
  },
  "rank_statement": "The map (d_b,d_L,c_b,c_L)->(d_b+c_b,d_L+c_L) has rank2 and a2-dimensional kernel.",
  "kernel_generators": [
    "(1,0,-1,0)",
    "(0,1,0,-1)"
  ],
  "recovery": "Holding source Cainan130/460 recovers parent100/27; observing the named junction in both modes also separates both pairs."
}

Proof route

Complete-family test: Apply the same source-slot bookkeeping to the complete native LXX field and the explicitly admitted MT/SP restoration companions, retaining both upstream support and the inserted node.

What this adds: Adds a precise inverse criterion for topology after coarsening, instead of rechecking130/460 shifts.

Counterexample or limit: Two source means of measuring a row do not by themselves establish the historical act of insertion or omission. No repeated +130 permission follows.

Sources: File18 §1.6 and §§4.2,6B–6D; Strategy §3.2.

Inherited baseline: C740,C759,C774,C782 already establish insertion arithmetic and support.

OPX06 — Typed insertion/removal as a partial inverse

Priority: medium. Question: What metadata makes the admitted OFF↔ON Cainan operation a coherent partial inverse?

Domain and retained premises

Exact predictions

{
  "law": "Remove(Insert(X,payload,slot))=X on the declared OFF domain.",
  "reverse_limit": "Insert(Remove(Y))=Y only when the removed named row, source payload and adjacency are retained.",
  "forbidden_extension": "Insert is undefined on an already-ON state under the single-switch source policy."
}

Proof route

Complete-family test: Check the four admitted downstream placements and all corresponding retained common nodes, using existing source-state packets as fixed cases.

What this adds: A composition law with retained source information, useful for a later categorical description without a universal date map.

Counterexample or limit: A coarsened chronology generally cannot recover a deleted identity. The formal inverse does not establish textual direction or an insertion history.

Sources: File18 §§1.3.1,1.6; Strategy §5E.

Inherited baseline: C772–C785 already regenerate the finite states.

OPX07 — A rounding carry law for successive coarsenings

Priority: high. Question: Can all changes caused by merging source subdivisions be reconstructed from local rounding carries independent of merge order?

Domain and retained premises

Exact predictions

{
  "carry": "kappa(x,y)=Q(x)+Q(y)-Q(x+y)",
  "associativity_identity": "kappa(x,y)+kappa(x+y,z)=kappa(y,z)+kappa(x,y+z)",
  "block_defect": "sum_i Q(x_i)-Q(sum_i x_i) is the sum of carries along any fixed binary merge tree.",
  "Cainan_prediction": "For any integer x, kappa(x,130)=kappa(x,330)=kappa(x,460)=0."
}

Proof route

Complete-family test: Test every adjacent triple of the three frozen row-measure streams b,r,L on all compatible native rows, plus the single source Cainan merge. Predict the complete coarsening defect field; do not search arbitrary partitions for targets.

What this adds: Adds the composition law needed to combine local defects coherently across whole source paths.

Counterexample or limit: Zero total carry does not identify an intermediate path. Coarsening does not authorize removing source event names or applying one-pass digit reversal to a new total.

Sources: File51a §§3.4,16–17; File18 §1.6; Strategy §5F.

Inherited baseline: C752 already proves the25-class local defect. C759 already proves Cainan residual neutrality. C765,C769 already give full-field subdivision failures.

OPX08 — Exact inverse fibres of three rounded row measurements

Priority: high. Question: How much information do Q(b),Q(r),Q(L) jointly retain on a compatible source row?

Domain and retained premises

Exact predictions

{
  "parameterization": "b=B+u, r=R+v with u,v in{-2,-1,0,1,2}; B=Q(b),R=Q(r).",
  "third_measurement": "D=Q(L)-B-R=Q(u+v), necessarily in{-5,0,5}.",
  "fibre_cardinalities": {
    "D=-5": 3,
    "D=0": 19,
    "D=5": 3
  },
  "whole_family_prediction": "The inherited six nonzero ordinary splitting-defect rows have three possible component pairs; the other49 ordinary rows have19, before further source data."
}

Proof route

Complete-family test: Output55 labelled inverse-fibre records: MT19, native LXX20, SP16 ordinary rows. Explicitly record the three excluded SP ledger rows and their counting tags.

What this adds: A constructive inverse description of what joint rounded measurements lose, rather than another rounding formula.

Counterexample or limit: These are formal compatible numerical candidates, not admitted biographies. Rounded r can itself be derived from b and L, so the information is not automatically independent source evidence.

Sources: File18 ordinary native rows; §§3.1.3,4A for exclusions/status; File51a §3.4; Strategy §5F.

Inherited baseline: C752–C753 classify forward defects; C770 gives Actual recovery when residuals are fully supplied.

OPX09 — When exact lifespan makes rounded parts decisive

Priority: highest. Question: Which compatible source rows are uniquely reconstructed from rounded b, rounded r and exact L, and which still require a source choice?

Domain and retained premises

Exact predictions

{
  "formula": "s=L-Q(b)-Q(r); with |s|<=4, exactly5-|s| balanced pairs satisfy u+v=s.",
  "uniqueness": "Unique iff |s|=4, within the interior integer domain.",
  "complete_unique_rows": {
    "MT": [
      "Methuselah",
      "Reu"
    ],
    "LXX": [
      "Methuselah",
      "Reu"
    ],
    "SP_ordinary": [
      "Reu"
    ]
  },
  "MT_Methuselah": "Qb185,Qr780,L969 -> s4 -> b187,r782",
  "MT_Reu": "Qb30,Qr205,L239 -> s4 -> b32,r207",
  "LXX_Lamech_counterexample": "Qb180,Qr570,L753 -> s3 -> (181,572) or(182,571). The restored source182 still selects the main calculated571 remainder.",
  "Cainan_limit": "Qb130,Qr330,L460 leaves five compatible pairs; source130 is not derived from this packet alone."
}

Proof route

Complete-family test: Freeze Qb,Qr,L first, hide actual b/r during reconstruction, then compare all55 outputs with literal source rows. Preserve the fact that this is explored-source recoverability, not an untouched holdout.

What this adds: A sharper information criterion: some exact source parts can be reconstructed conditionally, while an explicit nearby Lamech case remains ambiguous.

Counterexample or limit: Do not advertise this as historical generation of187 or32: rounded remainder is not an independently attested premise in the inherited File18 packet. No speculative candidate becomes a source variant.

Sources: File18 §§2.1.3–2.2,4.1–4.2,4A; File51a §3.4,§§16–17; Strategy §§5B,5F,6.

Inherited baseline: C750,C753 already identify Methuselah/Reu forward splitting defects.

OPX10 — From local rounded fibres to a complete genealogy inverse

Priority: high. Question: How do actual cumulative boundaries and selected actual regular boundaries constrain the remaining rounded-row fibres?

Domain and retained premises

Exact predictions

{
  "local_stage": "Use OPX09 independently on every compatible row after recovering exact L.",
  "coupling_stage": "Each retained regular block span adds one exact equation on the corresponding candidate b values, with kappa fixed.",
  "linear_rank_baseline": "Before finite residue bounds, m independent regular block constraints remove m directions from unresolved b coordinates.",
  "failure_certificate": "Any two admissible residual assignments preserving all supplied block sums prove non-identification; do not repair by another endpoint."
}

Proof route

Complete-family test: First use the fixed Adam–Noah block, then extend through Terah only after reassessment. Report each unresolved named coordinate and all source conditions used; avoid numerical tallies of unrelated endpoint matches.

What this adds: Constructively joins complete-path information to bounded local recovery and states exactly what an extra source boundary contributes.

Counterexample or limit: A complete actual regular field already contains all begetting intervals, making the inverse tautological. Use a declared proper subset of boundaries and count every retained rounded remainder premise.

Sources: File18 native blocks; File51a rounded tables; Strategy §§5B,5F,8.

Inherited baseline: C748 and C770 already supply the two reversible coordinate steps.

OPX11 — Lifting admitted row moves to rounded-plus-residual coordinates

Priority: high. Question: Can row transformations compose exactly on the recoverable rounded representation, including changes not divisible by5?

Domain and retained premises

Exact predictions

{
  "carry": "c(rho,d)=5 floor((rho+d+2)/5)",
  "lift": "T_d(y,rho)=(y+c(rho,d),rho+d-c(rho,d))",
  "composition": "c(rho,d1+d2)=c(rho,d1)+c(rho+d1-c(rho,d1),d2)",
  "rounding_only_limit": "Dropping rho prevents a well-defined induced action for arbitrary d; shifts divisible by5 descend without that extra state."
}

Proof route

Complete-family test: Apply to the complete source-local changes in OPX01, checking all three measurement streams and their fixed count bindings. Existing source multiples-of-five are regression cases; actual nonmultiple lifespan changes provide the discriminating cases.

What this adds: Supplies a coherent composition law on the richer retained object, rather than declaring rounding an affine group action.

Counterexample or limit: The lossless augmented representation does not reduce source information by itself. It does not open shifts or variant compositions absent from source controls.

Sources: File51a §§3.4,16–17; File18 main lifespan changes; Strategy §§5C,5F,5G.

Inherited baseline: C760 proves equivariance for multiples of5; C770 proves recovery with the residual.

OPX12 — Rank of the full birth-plus-cap observation map

Priority: highest. Question: Exactly which baseline information is absent after the whole birth field and all nine capped lives are observed?

Domain and retained premises

Exact predictions

{
  "Jacobian": "[[U,0],[A U,I-A]], A=diag(1_S)",
  "rank": 15,
  "domain_dimension": 18,
  "kernel_dimension": 3,
  "kernel": "Birth observations force db=0; only dL on the three clipped rows remains invisible.",
  "inverse_fibre": "Each uncapped baseline is fixed by y; clipped baselines are independently bounded below by their capacities."
}

Proof route

Complete-family test: Bind all nine printed births/capacities/outputs in the existing cap packet, demonstrate all15 independent observable directions, and keep the three named losses explicit.

What this adds: One whole-family information theorem identifies precisely the three missing source quantities after both observed representations are retained.

Counterexample or limit: The result does not explain the original962/969/777 or establish MT priority. At a threshold the differential formula changes; a closed inverse fibre remains the safer global statement.

Sources: File18 §§3.1,3.1.3; Strategy §§5B,5D,5G.

Inherited baseline: C968 already gives rowwise inverse conditions; C971 gives a different late-variable constraint packet; C974 gives the begetting response at fixed baseline.

OPX13 — What additional source measurements recover the lost excesses?

Priority: high. Question: Which additional independent source measurements would close the three-dimensional cap information gap?

Domain and retained premises

Exact predictions

{
  "one_total": "Holding the inherited total reduction488 imposes z_J+z_M+z_L=488 and raises local rank15 to16, leaving two directions.",
  "two_more_rows": "Two independently supplied named excesses or baseline lives plus that total identify the third, provided the measurements are independent.",
  "inherited_solution": [
    115,
    249,
    124
  ]
}

Proof route

Complete-family test: Assess source support for the entire three-row excess vector together; coordinate with the source-choice investigation. Do not enumerate arbitrary triples as chronological alternatives.

What this adds: Makes the next source research burden exact: at most three independent named measurements are missing, and a known sum supplies only one.

Counterexample or limit: The total488 is already calculated from the retained baseline packet. It cannot become independent evidence when that same packet is what the inverse is meant to reconstruct.

Sources: File18 §3.1.3 and baseline tables; Strategy §§5D,6,8.

Inherited baseline: C957 gives488; C969 shows that even an ancestral multiset does not restore every row identity.

OPX14 — A finite cap response across support changes

Priority: medium. Question: Can the local derivative be extended to a single exact finite-response law without authorizing new source states?

Domain and retained premises

Exact predictions

{
  "formula": "Delta y=(L-c)_+-(L-c-v)_+, componentwise; c=Ub+k1.",
  "piecewise_reduction": "Inside one strict support region this reduces to A v, recovering C974; threshold crossings are handled by the same positive-part formula.",
  "source_limit": "Source-admitted frame translations have v=0 when Noah/Flood move together and preserve all capacities."
}

Proof route

Complete-family test: Evaluate the full nine-row response for every existing source-qualified counting path; reserve unsupported threshold changes as algebraic negative controls only.

What this adds: A global piecewise formula makes the earlier local model composable across mathematical regions and exposes exactly when its simpler derivative ceases to apply.

Counterexample or limit: This is a mathematical extension of the comparison model, not a wider chronology domain. It should be lower priority than identifying source-supported excess constraints.

Sources: File18 §§3.1,3.1.3; Strategy §§5B,5E.

Inherited baseline: C962 gives the fixed support interval; C974–C977 give local response and frame/count invariance.

OPX15 — Joint cap-loss and rounding-residual bookkeeping

Priority: high. Question: Can the information lost by capping and rounding be separated in one complete reconstruction identity?

Domain and retained premises

Exact predictions

{
  "identity": "Q(L)-Q(y)=z+epsilon(L)-epsilon(y)",
  "exact_reconstruction": "L=Q(y)-epsilon(y)+z",
  "source_changed_rows": {
    "actual_losses": [
      115,
      249,
      124
    ],
    "rounded_losses": [
      115,
      250,
      120
    ],
    "rounded_minus_actual": [
      0,
      1,
      -4
    ]
  },
  "whole_field": "Apply U to each retained vector; the source total488 becomes485 after separately rounding the baseline and capped lives, with no new endpoint parameter."
}

Proof route

Complete-family test: Produce all nine zero/nonzero rows, not just the three changed examples, and their complete cumulative suffix responses. Treat numerical values as inherited regression targets.

What this adds: A single loss-register interface connects two irreversible operations while identifying the additional information needed to invert their composite.

Counterexample or limit: Recording every lost quantity makes a map reversible by bookkeeping but does not compress or explain those quantities. Do not count the rounded loss identity as independent corroboration.

Sources: File18 §3.1.3; File51a rounding policy; Strategy §§5F–5G.

Inherited baseline: C981 already explains the0,-1,+4 comparison-residual support and -608→-605 consequence.

OPX16 — A source-costed reconstruction ledger for the combined row grammar

Priority: synthesis. Question: Which proposed reductions explain new structure, and which merely re-encode source inputs?

Domain and retained premises

Exact predictions

{
  "ledger_columns": [
    "source input and provenance",
    "formal operation and exact domain",
    "full generated field",
    "independent equation count conditional on held values",
    "unrecovered source information",
    "counterexample",
    "description cost including masks and exceptions"
  ],
  "expected_distinctions": [
    "OPX01 and lossless residual encoding are coordinate changes, not reductions.",
    "OPX02/03 reduce a declared supported family to five parameters, conditional on its masks.",
    "OPX09 can give singleton bounded inverse fibres without an independent historical derivation.",
    "OPX12/13 locate missing baseline data without generating it."
  ]
}

Proof route

Complete-family test: Publish one integrated row-operation diagram and complete reconstruction ledger, with native/leveled states and inclusive/ordinary rows still separate.

What this adds: Keeps the cycle aimed at explaining how families connect and what remains source-supplied, rather than accumulating endpoint observations.

Counterexample or limit: No unique smallest grammar, statistical rarity, or historical sequence follows merely from exact algebra or a successful finite reconstruction.

Sources: Strategy entirety, especially §§1,5,6,8–9; C1131 final technical companion.

Inherited baseline: The C1131 operation inventory and source-cost cautions remain the baseline.

Adoption and reassessment

Before executing a selected question, root should bind its exact source rows, tags, masks and withheld measurements. Use the existing full fields as regression targets, with native Cainan and SP inclusive exceptions preserved. Do not recast these explored sources as untouched holdouts. For inverse questions, hide the requested values from the computation but acknowledge when retained rounded remainders were themselves calculated from them.

At each50-action review, ask whether a result reconstructs an entire source family, identifies an independent constraint, gives a counterexample, or merely re-encodes known inputs. Count masks, partitions, source choices, anchors and exceptions in description cost. A proposal that only repeats an existing equality should be dropped or used as a short regression check.

No new family computations are certified by this preparation. The numerical predictions and rank claims are intended for exact, independently bound root verification. The JSON retains the complete literal row packet and inherited comparison targets, together with source hashes. This prevents an apparent success from depending on an unstated row or source-state change.

Bound sources

Linked sources and evidence

Edition and provenance

operator_extension_proposals.md

SHA-256 183824eb955e6de2bf54ca1a175961a337764ef836a691a9edaead764e65849a

C480–C1634/Research_Cycles/C1132_C1431_Recovered/prep/operator_extension_proposals.md