# Primary inverse backbones: constraints and recoverable information

Preparation only; no numbered research action is claimed. The 25 questions were recorded in `QUESTIONS_BEFORE_CALCULATION.md` before the exploratory calculations. `inputs.json` freezes the latest File52c literals and locators; `calculations.json` and `prepare_constraints.py` give exact forward-only calculations. No canonical source or completed root artifact was changed.

## Main new result

**The regular primary endpoint is forced by its total and its two-component digit register; the cumulative primary endpoint additionally selects one of two possible aggregate branches.** This identifies which source details generate the shared outer endpoint and which are needed to recover the internal chronology.

For each original component `s=10(100a+10b+c)`, the permitted register has `1≤a,c≤9` and `0≤b≤9`: exactly three core digits and exactly one retained trailing zero. One forward reversal gives `I(s)=10(a+10b+100c)`. For two components, write their column sums as `(H,T,U)`. Then

`original total = 10(100H+10T+U)`;

`one-pass total = 10(H+10T+100U)`;

`gain = 990(U−H)`.

The feasible aggregate box is `2≤H,U≤18` and `0≤T≤18`. The aggregates are not decimal digits: a sum such as 12 is retained as a column sum before carries.

### The fixed-total proof

For a total of 2700, `100H+10T+U=270`. Reduction modulo ten gives `U=10`, since U is between 2 and 18. Therefore `10H+T=26`. With `H≥2` and `0≤T≤18`, the only possibility is `(H,T,U)=(2,6,10)`. The reversed total must therefore be `10(2+60+1000)=10620`.

For a total of 12600, the same argument gives `U=10` and `10H+T=125`. There are exactly two possibilities:

| Original total | Aggregate columns | One-pass total | Endpoint with 1406 held |
|---:|---|---:|---:|
| 2700 | (2,6,10) | 10620 | 12026 |
| 12600 | (12,5,10) | 10620 | 12026 |
| 12600 | (11,15,10) | 11610 | 13016 |

The literal primary regular path `1650+1050` realizes the first row. The literal primary cumulative path `9170+3430` realizes the second. Their column difference is `(−10,1,0)`, which lies in the kernel of the reverse evaluator `(1,10,100)` and changes the original total by −9900 after the outside factor ten.

This is a conditional partition invariance, not general additivity. It neither replaces the licensed Flood breakpoint nor admits arbitrary new chronological dates. In particular, reversing the unsplit 2700 still produces the established different result 7200, because its register is a two-digit core with two retained zeros.

### What the totals recover

The two measurement rows `(100,10,1)` and `(1,10,100)` have rational rank two and primitive integer kernel `(10,−101,10)`. Thus rank alone leaves one aggregate degree of freedom. But two feasible aggregate vectors differ by at most 18 in the middle column; no nonzero integer multiple of that kernel vector can remain inside the box. **The original total and one-pass total uniquely recover the aggregate columns within this fixed two-component register.**

They do not recover how the columns are distributed between components or the component order. Two explicitly diagnostic examples demonstrate this limitation:

- `1550+1150` retains the regular aggregate `(2,6,10)` and both totals, but replaces the literal `1650+1050` components.
- `9270+3330` retains the cumulative aggregate `(12,5,10)` and both totals, but replaces the literal `9170+3430` components.

These are mathematical controls only. Their alternative internal boundaries have no chronological admission. The source's event identities, actual segment values and order remain indispensable to the full path.

### Why the 1:4 relation fits

The regular source aggregate gives gain `990(10−2)=7920`. The cumulative source aggregate gives gain `990(10−12)=−1980`. Thus the shared endpoint lies 7920 earlier than regular Creation and 1980 later than cumulative Creation: their ratio is `8:2=4:1` directly in the aggregate gain coefficients.

Once the two source totals, the stated register, and the primary cumulative branch are fixed, the weighted relation follows. It is not another independent witness added after those premises. Conversely, in a free aggregate comparison, convergence alone does not force that weighting. This distinction between a free family and the fully fixed source state is essential.

The two equations on `(HR,TR,UR,HC,TC,UC)` are:

- Convergence: `(1,10,100,−1,−10,−100)·v=0`.
- Regular inverse endpoint equals the 1:4 weighted point: `(−95,40,499,−400,−40,−4)·v=0`.

Their rank is two. An admissible-register mathematical control `2650+1050` against `9170+4430` preserves convergence at 12036 but gives weighted point 13026. A different control retains the literal source totals: `1650+1050` against `9970+2630` preserves the weighted 12026 point but changes the cumulative inverse endpoint to 13016. Neither control is a chronology proposal.

## A local carry rule explains the admitted refinement

Peel off an original component `100d`, with `1≤d≤9`, from `s=1000a+100b+10c`, keeping the residual in the same three-digit core register. The tail is self-preserving under one reversal. If `b≥d`, no borrow is needed and the local refinement defect is zero. If `b<d` and `a≥2`, borrowing gives a defect of +990.

This gives the two source-controlled examples without inserting new chronology nodes:

| Source decomposition | Borrow? | Refinement defect |
|---|---|---:|
| 1650 → 1050 + 600 | No: b=d=6 | 0 |
| 9170 → 8570 + 600 | Yes: b=1<6 | +990 |

The second uses File52c's literal cumulative Creation 14006, Shem 5436 and Flood 4836 to recover the already discussed secondary Shem partition. It remains a secondary branch, not a replacement for the primary cumulative Flood path. C496 already recorded these numerical defects; the added result is their shared local carry condition. File52c does not supply a full literal cumulative-Noah component table here, so no fresh source-only reconstruction of that branch is attempted.

## Dependency of the 14726 continuation and 529 ladder

Once the primary reverse sum is `P=10620`, appending the same source-defined terminal leg to both paths preserves their equality by ordinary addition. With `A=1406`, `N=6`, the original leg `A−N=1400` has one-pass value 4100. Thus

`K=A+P=12026`, and `B=N+P+4100=14726`.

Equality at B contributes no new equality condition beyond equality at K and the shared tail. Its nonzero Nativity span is 14720; its divisibility by 23 is the source-derived compatibility `10620≡17`, `4100≡6 (mod23)`. Reversal alone does not force that compatibility. No new target tally is involved in checking this one already specified construction.

With the separately adopted Exodus `X=1446`, the measured duration is `K−X=10580`. The coefficient `k=10580/529=20` is an **output**, not a second independent source premise. Given E=25/23, the established duration ladder follows as `k·529 → k·575 → k·625`, hence 10580→11500→12500. Its reflected endpoints additionally require the declared Rounded coordinate chart and held reflected K; they do not follow from a free duration without placement.

The existing Temple/Moses junction admits a compact constraint ledger. Write `T=696` for the generated Temple inverse, `M=1616` for the generated Moses inverse, and use K,B,X as above. The first junction is equivalent to

`46+2B+2K−25(T+X)=0`.

Given that first junction, the agreement of the second backbone expansion with the Moses branch is equivalent to

`2(K−X)−23(M−T)=0`.

These are rank-two relations in the free coordinate packet `(K,B,T,M,X)`. Numerically the second is the existing 920 source-gap compatibility. Do not add the later 476, 526 and reflected-birth consequences as independent relations: they propagate the same junctions through the fixed affine maps. C520 already established their dependence in numerical form; this explicit pair of constraint rows is a compact ledger, not a new independent discovery of the endpoints.

## Premise versus consequence register

| Item | Role in this analysis |
|---|---|
| Source anchors 1406, 6, 1446 and the original Creation/Flood placements | Source inputs with retained roles |
| Primary regular and cumulative two-component registers | Source-selected operation domains |
| Regular total 2700 and cumulative total 12600 | Derived from the source endpoints; can serve as sufficient measurements after that derivation |
| Cumulative aggregate branch (12,5,10) | Selected by the literal components; not forced by 12600 alone |
| 10620 and 12026 | Outputs of total/register/branch evaluation |
| Ratio 4:1 and weighted 12026 | Consequence of the fixed primary aggregates; independent only when posed as a condition on a freer family |
| 1400 and 4100 | Anchor difference and its permitted one-pass output |
| 14726 and 14720 | Shared-tail consequences |
| 20 in the 529 ladder | Quotient of the measured 10580, not an independently supplied multiplier |
| E=25/23 and the Rounded coordinate chart | Independent operation/convention choices |
| 11500,12500,920,1000 and q=525/−475 | Consequences under the declared operations and placements |
| Actual/Rounded change (δC,δR)=(−2,+8) | Separately source-selected endpoint change in the inherited weighted kernel |

The Actual/Rounded conservation law `4δC+δR=0` is already proved in C494 and used in C902. It transports the weighted point after those source states are selected; it does not independently select the decimal partitions or explain the 529 coefficient.

## Sequential candidate questions and reassessment

The pre-calculation register has 25 questions. The following dispositions keep the work substantive and identify inherited material that should serve as context rather than consume a new action by itself.

| Candidate | Recommended question/result | Reassessment after completion |
|---:|---|---|
| 1 | Freeze literal paths, coordinates and allowed digit registers. | Begin with aggregate information, not the already known endpoint sum. |
| 2 | Recover each path's three aggregate columns. | Separate source component data from sufficient outer-total data. |
| 3 | Derive the raw/reverse measurement matrix. | Retain the factor ten and column-sum status. |
| 4 | Explain the primary kernel direction (−10,1,0). | Ask what other information the equality loses. |
| 5 | Prove the regular fixed-total aggregate is unique. | This is the strongest new compression; test its cumulative counterpart. |
| 6 | Prove the cumulative fixed-total has exactly two aggregate branches. | Identify which literal source selects the primary branch. |
| 7 | Explain the +990 difference by the carry between aggregate columns. | Keep the alternate branch mathematical unless already source-admitted. |
| 8 | Prove bounded-integer injectivity of the raw/reverse pair. | Contrast recoverable column totals with unrecoverable component allocation. |
| 9 | Exhibit same-aggregate, different-component controls. | Preserve the source's internal event identities as additional information. |
| 10 | Compare the actual ordered intermediate positions of the admitted paths. | Endpoint agreement is insufficient for full chronology identification. |
| 11 | Derive the 8:2 gain coefficients from aggregate end columns. | Recognize the 1:4 weighting as derived in this fixed state. |
| 12 | State the two free-aggregate constraint rows and their rank. | Keep algebraic independence separate from fixed-source consequence. |
| 13 | Test a control preserving convergence but breaking weighting. | Clarify which source inputs had to change. |
| 14 | Test a control preserving source heads and weighting but breaking convergence. | The cumulative branch remains a nonredundant source choice. |
| 15 | Derive the no-borrow/borrow hundred-place refinement rule. | Apply only to already admitted regular and cumulative-Shem pieces. |
| 16 | Account for the regular Noah refinement and secondary +990 with that rule. | C496's outcomes are inherited; the carry law is the added explanation. |
| 17 | Show the shared tail adds no new equality constraint. | Treat C497's endpoint as inherited and retain internal path order. |
| 18 | Identify the 17+6 residue compatibility of the specified continuation. | A single existing construction check; no target search. |
| 19 | Label the 529 coefficient20 as a computed output. | Avoid counting a factorization and its generated ladder twice. |
| 20 | Give a minimal duration-ladder premise set. | C505–507 already prove the general ladder and pivot conditions; use them. |
| 21 | Add the Rounded coordinate placement premise for the endpoint realization. | Keep source state, scalar width and coordinate chart separate. |
| 22 | Express the two Temple/Moses junction conditions as linear constraints. | C520's repeated endpoints are consequences, not new witnesses. |
| 23 | Build the nonredundant premise/consequence ledger for the whole connected family. | State the level at which each rank is calculated. |
| 24 | Attach the Actual/Rounded weighted kernel as an inherited interface. | Do not redescribe it as a new reversal theorem. |
| 25 | Write the smallest defensible joint explanation and its remaining source choices. | Select another whole family once this dependency explanation is complete. |

These are candidate questions, not a request to manufacture 25 discoveries. Root should merge inherited checks with new explanatory steps whenever that better serves the ongoing sequence.

## Source and novelty control

The sole primary document used here is frozen latest File52c, SHA256 `a5ea84562101158b60d0cf296765d6eff38e7a2abda4e74ad1b353dfd13b9530`. `inputs.json` records exact line blocks from §§2.5,3.3–3.5,3.8–3.9,3.14 and6.1–6.2. Its SHA256 is `f9b028e4a503693dc9f72e0eed3c94e0ef959c2bc7321dec85069f03bc626cd0`.

Reviewed prior C482–531, C813–818, C850–858 and C902 to avoid presenting their endpoints, gain formulas, weighted-kernel result, whole paired-anchor field, or affine ladder as new. The new contribution is the bounded aggregate classification, finite recoverability proof, source-partition information loss, and precise separation of free-family constraints from fixed-source consequences. Mathematical diagnosis does not establish historical use of these operations. All digit reversals remain single forward evaluations of original components; no generated inverse span is reversed again.