Operator-family preparation for C832–C931

Preparation only. No numbered action has been executed and no canonical file has been changed. The recommended new family is the exact domains on which the three Keys preserve a declared coordinate grid, followed by a finite classification of operator order. This joins the Rounded, exact-phase, and Key families without asserting that every Key acts internally on every chronology.

Files: operator_inputs.json freezes seven source hashes and literal excerpts; operator_diagnostics.json contains preparatory exact-fraction calculations; prepare_operator_packet.py reproduces the packet. Root should separately select questions and execute numbered research actions.

Why this is new and useful

C608 already proved moving-anchor covariance. C570–C586 and C685–C693 already cover partial allocation, calendar calibration, quarter-carrier construction, and the dual4346 placement. C702 covers the460→500 pair difference. C705–C710 cover the fixed phase mirror field. C732–C831 now explain row accumulation, rounding residuals, finite variants, and BJ path subdivision. Repeating any of these as new discoveries would add little.

The new question is: which source-defined grids and intermediate states survive a specified Key word? A Key is an exact rational transformation, but whole-year, quarter-year, and Rounded five-year displays are different arithmetic domains. Membership in one domain at the start and end does not imply membership at every intermediate stage. This directly serves the Strategy's call for admissible partial actions rather than an unrestricted global transformation group.

The grammar becomes clearer: the source chooses a labelled path, an anchor, a unit/grid and an authorized operation. Arithmetic then determines both the image and the grid it occupies. No nearest-date repair is supplied. Fractions remain valid exact comparison coordinates where authorized; losing integer or quarter-grid membership is not itself an error.

1. Literal source controllers

File46's main-chain753 and the appendix777 routes remain separate. The proposed main test uses §6A.5, not its777 appendix. The exact phase seed161/4 is a comparison scalar, not a new generational unit. No primer, new Gear transport, second decimal inversion or remote extension is introduced.

2. One exact grid theorem

Let a source duration lie on the grid hZ, and let an exact scalar Key be p/q in lowest terms. Writing s=hm, its transformed value remains on the same grid precisely when q divides m:

(p/q)(hm) ∈ hZ ⇔ q | m.

Consequently:

The last row is the intersection, since lcm(23,69,299)=897. It means each separate image remains on the stated grid. It does not authorize sequential composition or claim a historically selected897-year unit. A positive source duration can be carried to an integer even if it was not itself on5Z. For example source483 gives525/490 underE/P. Thus the conditions above must always state the input domain.

The source examples do not all have the same compatibility profile:460 has E output500;690 has E/P outputs750/700;483 has E/P outputs525/490;12558 has all three integer outputs13650/12740/12600. These are illustrations of the new domain classification, not fresh discovery claims for old completions.

3. New whole-source test: the five-row quarter-carrier family

For the complete File63§1.2 family, the source durations are s_m=161m/4 with m∈{3,10,12,30,300}. Algebra gives:

E(s_m)=175m/4, P(s_m)=245m/6, J(s_m)=525m/13.

Therefore E preserves the quarter grid for every integralm; P does so exactly when3 dividesm; J does so exactly when13 dividesm. On the complete literal five-row list:

This explains why the whole483 carrier completes cleanly underP while a tenfold402.5 subbody need not stay quarter-resolved. It does not prohibit the rational1225/3 comparison, nor does it supply permission to apply each Key historically to each row. The five-row image table is a controlled diagnostic, with the source-authorized483→490/525 entries distinguished.

4. New whole-source test: File46's complete Adam head field

Use BC magnitudes only for this same-side comparison and the declared anchor14006:

D_E,14006(x)=14006+(25/23)(x−14006).

The source's920→1000 outer completion and460→500 half-width are integral; the full image has fractional interior heads. The generated fractions retain their comparison status. The existing source did not promise that the entire field becomes a canonical integer chronology.

A stronger bounded limit follows: no choice of one integer anchor can make all four literal heads map to integers underE. If two outputs are integral, their difference must be integral. But the source adjacent gaps430 and30 are not divisible23, so their E images10750/23 and750/23 are not integral. Anchor translation cannot repair this because it preserves differences.

This tests a complete independently supplied object, rather than selecting only the outer width. It adds a useful positive/negative pair: the exact field exists and preserves its affine shape, but integer closure belongs to a subset.

5. Finite two-Key words: final integrality versus stagewise integrality

For a word in execution order, write its reduced prefix coefficients c_j=p_j/q_j. The exact necessary and sufficient condition that every stage preservehZ is

input ∈ h·lcm(q_1,…,q_r)Z.

Final-only membership requires merely h·q_r Z. This distinguishes genuine intermediate domains from equality of final scalar products.

The complete two-Key matrix is:

For all-stage5Z membership, multiply the last column's generator by5. No source authorization for nine chronology routes follows from this algebraic classification.

The distinctive counterexample uses6877=13×23²:

• J→P: 6877→6900→7000, integer at both stages.
• P→J: 6877→20930/3→7000, same final scalar image, nonintegral first stage.

Thus scalar commutation and commutation as partial integer-grid actions are different statements. This does not ban rational intermediate values. It identifies what extra condition an integer-completion claim requires. The appearance of6877 is forced by the reduced denominators, not independently discovered chronological evidence for a13×529 family.

The existing File51a civil-count examples provide source-authorized positive routes:10051 and55016 are529 multiples; both E² chains stay integral. The55016→59800→60000 E→J route is also integral, since55016=8×6877. Its source role remains a civil inputC; neighboring Rounded width55015 is not substituted. This can relate the new theorem directly to primary Rounded-source protocols without any new digit inversion.

6. Optional three-distinct-Key closure

All six scalar orders have the same final coefficient 175000/158171, with158171=13×23³. Their stagewise integer domains split into two classes:

• J occurs beforeP:158171Z.
• P occurs beforeJ:474513Z=3×158171Z.

The fixed six-word family gives a concise confirmation of the prefix-denominator rule. It must not be presented as six source-authorized chronological operations or an invitation to unlimited word searches. Stop at the predetermined finite inventory if this test adds nothing beyond the two-Key distinction.

7. A separate kind of order effect: different held anchors

The two sources of order dependence should not be conflated. For ordinary linear coordinates:

D_k,a(x)=kx+(1−k)a.

Then

D_k,a D_l,b(x) − D_l,b D_k,a(x) = (k−1)(l−1)(a−b).

The difference is constant over every point of a labelled path. Both outputs have the same scaled interval pattern, but their absolute placements differ. Ifa=b, the affine maps commute; the stagewise-grid issue in§5 can still remain.

Use source anchorsa=14006,b=4836 and source KeysE,P only as a bounded formal diagnostic. Their two composed paths differ by18340/1587 everywhere. File46 independently licenses the originalE andP constructions at these anchors, but does not thereby license the mixed compositions as chronology states. The diagnostic explains why multiplying the scalar Keys and ignoring their held anchors loses information. It is not a new historical route and no fractional anchor is fitted to remove the discrepancy.

The already established C608 moving-anchor covariance should be cited as inherited context, not re-executed as novelty. Ordinary same-side reflection transports an anchor too: R_c D_k,a = D_k,2c−a R_c. This is a derived companion identity, not permission to replace civil/phase/Rounded Mirror charts by one generic reflection.

8. Optional whole-block family if root needs another independent source test

File68 AppendixK freezes five complete three-boundary translation instances: JoshuaN21 at carriers70/72; EnochN49 at70/72; LukeN76 at70 only. In ordinary civil coordinatesz(BCB)=1−B,z(ADA)=A, the shared terminal triple is T_g=(65−g,65,65+g) and each source triple is T_g−Ng. This preserves the two equalg intervals while retaining distinct depthN and carrierg. Five full source triples are reconstructed by one formula; it does not invent the deferred72 Luke rail.

This family is source-explicit, but C593–C607 already contain related four-rail grid work; root must decide whether the new depth hierarchy is sufficiently distinct. It is secondary to the domain family above.

Source tension: AppendixK.7's table and following guards say76, but one sentence still says “21,49,and77 counts.” Treat the77 as inherited prose tension; do not promote a76→77 correction to a new result or change the canonical file. The literal table and current correction header control the packet.

Suggested bounded root sequence

1. Freeze the grid-domain question and literal packet.
2. Prove the one-Key lattice theorem.
3. Classify all three integer and Rounded domains.
4. Execute the full five-row quarter-phase image table.
5. Separate quarter-grid preservation from permitted exact rational comparisons.
6. Execute the complete File46 Adam head image.
7. Prove its all-integer-anchor obstruction from the adjacent differences.
8. Connect the positive460/920 completions to the integral subset.
9. Derive the prefix-denominator criterion.
10. Execute the complete nine-word matrix.
11. Isolate theP/J intermediate-domain asymmetry.
12. Validate source-authorized File51a E² andE→J paths against the criterion.
13. Preserve civilC versus RoundedW_R inputs in that transfer.
14. If useful, run the fixed six-word triple test and stop.
15. Separate same-anchor scalar commutation from different-anchor affine order.
16. Verify the whole-path constant-displacement formula on a frozen set of source nodes.
17. Integrate this as grid and anchor tags in the common path grammar, not a new universal date map.

Reassess after each. These are candidates; root may compress or discard them if they do not advance the explanation. The strongest publishable addition is that a shared operator may preserve a whole shape while leaving its original display grid, and operator order can matter through either anchoring or intermediate admissibility.