# Key constraints: a smaller explanatory basis

**Preparation only for C932–C1131.** No numbered research action or canonical edit has been performed. Twenty-eight candidate questions were fixed in `questions.json` and `questions_addendum.json` before their new calculations. `sourcepacket.json` retains literal excerpts, source hashes, and selected prior results. `exactdiagnostics.json` contains exact Fraction calculations; `prepare.py` reproduces them without optional packages.

The strongest new work is not another Key completion. It is a more economical account of **which constraints determine a completion, which outputs are dependent, and when a fixed source pivot makes a whole staged grid route possible**.

## Priority and prior-index reassessment

Read against the completed-family index and C505–511, C568–586, C609, C666–667, C690, C713–715, and C872–889:

1. **High value:** affine stagewise domains with fixed pivots. This genuinely extends the earlier same-origin word theorem and earlier E² pivot example: a mixed J→P route can have an empty integer domain because of its source-appointed pivot separation.
2. **High value:** calibration/allocation constraint basis. A rank-three linear system captures four familiar relations and prevents their dependent outputs from being treated as separate evidence.
3. **High value:** retained-component totals have a two-dimensional measurement basis. Two totals recover the numerical component and retained part; the third total is a dependent check. Apply to the whole supplied flank table and macro field.
4. **Useful new transfer:** componentwise calibrated measures preserve one day-volume even when different Keys are assigned to the source299|161 parts. Its mixed-calendar interpretation must be explicitly conditional.
5. **Useful supporting theorem:** coarsening commutes with a component transform on every input exactly when its factors are constant inside each merged block.
6. **Lower priority:** the known two-part defect examples and529 ladder. Their new role is constraint compression and indexing, not rediscovery. KQ11–12 largely re-express C609/C714–715. Do not spend new numbered actions merely recalculating those old values.

## A. Calibration plus completion relations: an alternative exact basis

Sources supply the year measures336/360/364, the dual490 source partition402.5|80.5=5:1, and the Actual12558 path12075|483=25:1. The already verified partial/whole equivalences are:

- applying E to the last sixth has the same **total** as whole P;
- applying E to the last twenty-sixth has the same **total** as whole J.

Together with common calendar calibration, these give four affine equations for three unknown Key factors:

| Constraint | Equation | Source/dependency |
|---|---|---|
| A |336E−360P=0 |Strategy§3.3 / File12 calibration |
| B |336E−364J=0 |same calibration |
| C |E−6P=−5 |File63§1.3 / C568–569 partial–whole relation |
| D |E−26J=−25 |File63§§7.3–7.4,9.7 / C570 relation |

Their coefficient rank and augmented rank are both3. The unique solution is

`(E,P,J)=(25/23,70/69,300/299)`.

The exact dependency certificate is

`7A − 6B − 420C + 84D = 0`,

including the constants: `−420(−5)+84(−25)=0`.

The first calendar pair and its completion relation alone give

`E = 360(1−1/6)/(336−360/6) =25/23`.

The second pair gives

`E =364(1−1/26)/(336−364/26)=25/23`.

More generally, if a completion relation specifies `target=1+f(E−1)` and its target calendar measure isd, then

`E=d(1−f)/(336−df)`,

provided the denominator is nonzero. Calibration alone leaves one free common valueK: `(E,P,J)=K(1/336,1/360,1/364)`. One completion relation fixesK; the other can then be checked against the corresponding source partition. The first relation plus364 predicts the second allocation fraction1/26.

**Evidence ceiling:** the partitions alone do not determine the Keys. The total-completion equivalences must also be assumed. Those equivalences were previously proved using the Keys, so solving backward is a logically equivalent constraint basis, **not an independent historical derivation of25/23 and not a fresh holdout test**. The new result is explicit dependency and compression.

A predeclared negative diagnostic replaces only1/26 by1/25 while keeping336/364 and the completion premise. It yields312/287, differing from25/23 by1/6601. The source compatibility is thus not automatic for an arbitrary partition count. No25-fold chronology is adopted.

## B. Allocation composition is not sequential Key composition

Define the selection operator on a factor by

`A_f(k)=1+f(k−1)`.

The exact fractions are:

| Relation | Selected fraction |
|---|---:|
|E allocation equivalent in total toP |1/6 |
|P allocation equivalent in total toJ |3/13 |
|E allocation equivalent in total toJ |1/26 |

They satisfy

`A_g(A_f(k))=A_(gf)(k)` and `(1/6)(3/13)=1/26`.

This is nested **selection**: using a fraction of an already specified allocation. It is not the chronological operation of applying P and then J to the same duration. The latter has factor7000/6877; nested selection has factor300/299.

The1/6 and1/26 chronological comparisons already have their source-appointed subdivisions. The computed3/13 fraction alone supplies no new named cut. At12558 it would select2898, but locating that portion against a chosen terminal would be an additional construction. The existence of2898 elsewhere in a source does not itself authorize that placement. No such new event or path is entered in this packet.

## C. What a total forgets, and when coarsening is legitimate

For a two-part source vector `v=(fS,(1−f)S)`, scale only the first part byk. Its effective whole-span factor is `r=1+f(k−1)`. The difference between the selective and uniform images is

`diag(k,1)v − r v = (k−1)f(1−f)S · (1,−1)`.

The difference belongs to the kernel of total evaluation. Reversing the component order reverses the sign of the vector. The four old source cases reduce to this one form:

| Source partition and selected conversion | Difference: selective minus uniform |
|---|---|
|402.5 \| 80.5, finalE |(−35/6,+35/6) |
|12075 \| 483, finalE |(−525/13,+525/13) |
|9660 \| 2940, upperP |(+98/3,−98/3) |
|9660 \| 2940, upperE |(+196,−196) |

These numbers are inherited context. The new general statement concerns a complete operator square. LetC sum fine components into declared coarse blocks, letD be component factorsdiag(k_i), and letDbar apply one factor per coarse block. Then

`C D = Dbar C` **for every source vector**

if and only if allk_i inside each coarse block are equal to that block's factor. Proof: compare the coefficient of each independent component. One particular weighted vector can satisfy total equality even when the operator square fails. Its computed effective factor then depends on that vector; it is not a universal replacement Key.

This criterion explains why subdivision can be removed under genuinely uniform conversion, but must remain when source-appointed legs are converted differently.

## D. Two independent measurements recover the retained-part family

For a converted componentu and retained componentv, write

`T0=u+v`, `TP=P u+v`, `TE=E u+v`.

The measurement matrix is

`[[1,1],[70/69,1],[25/23,1]]`.

It has rank2. Its unique left-kernel direction is `(5,−6,1)`, so

`5T0−6TP+TE=0` or `TE−T0=6(TP−T0)`.

Two measurements recover the numerical components exactly:

`u=69(TP−T0)`, `v=T0−u`.

All four predeclared rows satisfy the same criterion:

| Source object |T0 |TP |TE |Recoveredu,v |
|---|---:|---:|---:|---|
|SupplementA core |690 |700 |750 |690,0 |
|One-flank bracket |720 |730 |780 |690,30 |
|Two-flank bracket |750 |760 |810 |690,60 |
|Cumulative retained-lower family |12600 |12740 |13440 |9660,2940 |

This compresses the complete source table and macro family into one two-dimensional construction. The Priestly total is dependent after the native and Prophetic totals are fixed. Inverting the measurements recovers numericalparts, **not their roles, order, endpoint placement or source authorization**. The source9660|2940 split remains an independently readable source premise; deriving it backward from the totals checks consistency rather than adding historical evidence.

## E. Affine stagewise domains with source-appointed pivots

This is the strongest new modular result.

Apply a reduced factorp1/q1 about integer pivota, thenp2/q2 about integer pivotb. The first output is integral exactly when

`x=a+q1 t`, for integert.

That output is `a+p1 t`. The second is integral exactly when

`p1 t ≡ b−a (mod q2)`.

Let `g=gcd(p1,q2)`. Therefore:

- an all-stage integer input exists **iffg dividesb−a**;
- when it exists, the input set is one affine residue class;
- its period is `q1 q2/g`.

This generalizes the previous same-origin prefix-denominator result. Settinga=b=0 recovers C881's domains. It also generalizes the older E² source-pivot condition without searching new pivots.

For the complete fixed nine-pair diagnostic, retain File46's already supplied pivot labels14006 and4836. Their order is fixed here: first map about14006, second about4836. Neither pivot is fitted to obtain an integral completion.

| Order |Residue class for sourcex |Period |Status |
|---|---:|---:|---|
|E→E |68 |529 |nonempty |
|E→P |1126 |1587 |nonempty |
|E→J |68 |6877 |nonempty |
|P→E |206 |1587 |nonempty |
|P→P |206 |4761 |nonempty |
|P→J |4967 |20631 |nonempty |
|J→E |2345 |6877 |nonempty |
|J→P |— |— |**empty** |
|J→J |43607 |89401 |nonempty |

The residue representatives are congruence labels, not event dates. These mixed compositions are diagnostics; File46 does not thereby authorize nine chronological routes.

The unique obstruction is `gcd(300,69)=3` for J→P. It requires `a≡b mod3`. Here4836−14006 is not divisible3, so no integer input can keep both stages integral under those fixed pivots. The same-origin J→P route was integral on6877Z; changing the two pivots independently is not an innocuous substitution.

For the two inherited E² examples:

| Existing object |Fixed pivots |Required affine coset |Existing source points |
|---|---|---|---|
|C480 |3620 then2756 |32+529Z |2148 and3206 both qualify |
|fixed12026 diagnostic |12026 twice |388+529Z |4114 and3056 both fail |

In each case the source width is a529 multiple. The new normal form explicitly separates **difference-lattice compatibility** from **one placement residue**, while retaining the old success/failure instead of recounting it as new evidence.

For an entire finite fieldX, a nonempty admissible coset containsX iff one source point has the required residue and every pair difference is divisible by its period. Translating all points and both pivots by the same integert shifts the residue byt, preserving period and obstruction. This is domain-level covariance, extending the already known affine covariance identity.

The same result works onhZ after normalizing points and pivots byh, provided those pivots belong to that grid. It does not silently recode civil, exact-phase, and Rounded charts as the same object.

## F. An exact rational image is a coherent comparison field

File46's frozen five-head set has anchor14006 and the representation

`X=14006+10·{92,89,46,43,0}`.

Its anchoredE image is exactly

`E_14006(X)=14006+(250/23)·{92,89,46,43,0}`.

The finite index pattern is unchanged. The greatest common divisor of the source differences is10; the image differences therefore generate the affine mesh250/23. This gives a coherent exact field without rounding any head to an integer or moving its anchor.

The integer subset corresponds to indices divisible23:92,46,0. This is the earlier integral subset in a more informative normal form. The mesh is an algebraic comparison object, not an adopted calendar or a new historical date grid.

## G. Conditional mixed-calendar realization of the source299|161 partition

The source Covenant path299|161 and the inherited mixed comparison300|175 already exist. The new test fixes **all nine** assignments ofE/P/J to the two source components before evaluation.

If each transformed component is measured with its Key's matching calendar lengthd, then

`d_i k_i s_i = K s_i`, hence `Σ d_i k_i s_i=KΣs_i`.

Every one of the nine assignments therefore has component volumes109200 and58800, total168000. This is a complete componentwise calibration identity, not nine independent witnesses.

The inherited mixedJ/E numbers give the clearest example:

`300×364 +175×336 =168000 =500×336`.

Under this **conditional mixed-calendar reading**, the naked sum475 does not identify a common-calendar count. Remeasuring the same volume gives500 in336-day units,1400/3 in360-day units, or6000/13 in364-day units. These agree with the uniformE/P/J images of the460 source scalar.

Retaining the same numerical outputs but swapping only their unit labels gives

`300×336+175×364=164500`,

a difference of−3500 while475 is unchanged. The unit tags therefore carry real information.

**Scope:** the source's chronological475 comparison is not being rewritten as a literal mixed-calendar elapsed timeline. This packet supplies a possible calibrated scalar realization under explicitly declared measures. The nine assignments other than existing source comparisons are formal diagnostics, not new chronological routes. Calendar equivalence does not prove historical use of those units by an ancient author.

## H. The529 ladder's limited role in this packet

The inherited primitive ladder is23²→23×25→25², or529→575→625. The first and second gains46/50, total gain96, geometric-mean identity575²=529×625, and second difference4 all follow from the same two integers. They are dependent arithmetic displays. The intermediate575 lies2 below the arithmetic midpoint577.

This is a low-priority dependency certificate, not a new529 discovery. Source-appointed pivots still control endpoint realizations; duration constraints alone do not choose them. Do not expand this into another target search.

## Suggested work distribution across the28 frozen questions

- KQ02–09: alternate constraint basis, rank certificate, controlled perturbation, selection triangle.
- KQ13–17: complete coarsening criterion and retained-part measurement basis.
- KQ19–24: affine congruence theorem, complete nine-pair fixed-pivot field, whole-field placement and exact comparison mesh.
- KQ25–28: conditional componentwise calibration and unit-preserving remeasurement.
- KQ01,10,11,12,18: source/dependency preparation or compression context; reduce or omit numbered treatment if it only repeats completed results.

The shared explanation is now sharper: a Key family's relations can be represented with a small constraint basis, but the source still supplies the partition, the calendar interpretation, the pivot placement and the licensed route. Removing any one of those tags can preserve a total while changing either the object or the domain on which the construction works.