File52c: a complete paired-anchor displacement field
Preparation for C832–C931; no numbered research step claimed here.
Recommendation
The strongest new bounded File52c test is the entire frozen 32-source, two-anchor map, read as a displacement field. It explains why its two endpoint columns are related yet cannot be substituted for one another. It does not add sources, targets, probability calculations, or an inverse-of-an-inverse operation.
The original source path and its selected anchor remain the input object. A digit-position vector, its length, and its retained zero-placeholders supply a small additional coordinate register. Reversal is linear in that fixed digit register; the register changes when subtraction changes the core length or requires a borrow. This supplies a concrete meaning for a state-aware decimal module alongside the row/path modules developed through C831.
Frozen inputs and verification
inverse_inputs.json: the 32 source rows from latest File52c §2.1, paired with the 64 existing endpoint literals from Appendices A.2–A.3. It includes original line locators, source hash, permitted primary path tables, and scope guards. Target columns were not extracted.inverse_paired_diagnostics.json: all 64 direct single-pass reconstructions and the predicted/literal differences for all 32 source pairs.prepare_inverse_inputs.py: extraction and preparation script. It never executes the source's reproduction block (which contains deferred and statistical operations). Every decimal call receives an original source-to-anchor duration.- Source SHA256:
a5ea84562101158b60d0cf296765d6eff38e7a2abda4e74ad1b353dfd13b9530. - Frozen packet SHA256:
dbcb35bf6cfe88b318b4c8c327e97aa0be06583d5e945c43c9aac8b7f7b80911.
All 64 existing endpoints reconstruct; all 32 pair formulas match. These are verification quantities, not a new target-frequency tally.
The common digit register
Write an original duration as
s = 10^k Σ[j=0..m−1] d_j 10^(m−1−j),
with the first and last core digits nonzero. Its permitted one-pass output is
I(s) = 10^k Σ[j=0..m−1] d_j 10^j.
For each fixed (k,m), this is the ordinary reversal permutation of the digit vector followed by numeric evaluation. It is not one affine function of the integer date. This is a forward encoding and evaluation only; it does not apply reversal to an output.
Define the forward gain g(s)=I(s)−s. Rebuilding from anchor A gives the particularly useful expression
A+I(D−A) = D+g(D−A).
Thus the source coordinate stays visible, and all anchor dependence is in which original duration enters the gain function. C498 already derived the elementary two-/three-digit gain formulas; the new work is their complete paired-field realization and its explicit change-of-register mechanism.
The complete source-specific paired law
Let a0 be 6 or 1 according to the source ending. Write D=a0+10n; the second anchor is a0+1400. Both constructions therefore use original durations 10n and 10(n−140). The Conquest-minus-Nativity endpoint difference is
Δ(D)=g(10(n−140))−g(10n).
Write n=100a+10b+c. The following cases cover the whole existing manifest, with no fitted constants:
| Source-digit condition | Mechanism | Exact Δ |
|---|---|---|
c≠0, n≥240, b≥4 | Three-digit cores; subtract 140 without a tens borrow | 990 |
c≠0, n≥240, b<4 | Three-digit cores; tens borrow changes the leading digit by two | 1980 |
c≠0, 150≤n≤239 | Conquest core shortens to two digits | 1260+90a−90b−900c |
c≠0, 144≤n≤149 | Conquest core shortens to one digit | 990−990c |
n=200 | Jacob: 2000 and 600 retain all zero-placeholders | 1400+600−2000=0 |
n=410 | Adam: 4100 and 2700 use two-digit cores with two placeholders | 1400+7200−1400=7200 |
For the first two cases, the source core abc becomes (a−1)(b−4)c or (a−2)(b+6)c. Digit reversal moves the common last digit into the same leading place, so it cancels between routes. The remaining positions force 990 or 1980. This is why long runs in the paired field have one constant gap. Shorter cores explain the later nonconstant, often negative gaps.
The complete ordered difference vector is:
[7200,990,990,990,990,990,990,990,1980,1980,990,990,990,−6930,−3330,−630,−6840,−4140,−1440,−4050,−3960,0,−360,−1800,−900,−5940,990,−6660,−5670,−2970,−2970,1980].
Its order is exactly the §2.1 manifest, not chronological sorting of generated dates. The first 23 source rows are the regular genealogical sequence; the remaining rows retain their named birth/event roles.
Why this advances the synthesis
- The same source gives both endpoint columns; the difference field is generated by one declared 1400 anchor change plus the source's digit register. It is not 32 independently chosen shifts.
- The established 990 decimal effect now explains a complete source family, including the 1980 borrow branch and all shortening/placeholder cases, rather than a few favorable segments.
- The digit register is distinct from the chronological row register. A chronological interval can remain unchanged under one operation while its decimal representation changes under another. The shared grammar needs both registers.
- The complete paired field is not a uniform translation. For example, Seth/Enosh have the same +990 while Lamech has +1980. Those two distinct same-gap points already fix any candidate affine map to slope 1, which then fails at Lamech. No new outside control case is needed.
- Existing self-inverse source rows and Jacob's paired equality are explained by zero gains in their declared source-duration registers. Their source status does not become a new source or target admission.
- The field can be attached to C831's ordered-path grammar as a forward date displacement. It should not be confused with componentwise path evaluation, which retains its admitted segmentation and has already been established in C495–499, C610 and C813.
Suggested bounded sequence (root chooses/reassesses)
- Freeze the existing paired endpoint field and explain the research question in terms of source dependence.
- Reconstruct every endpoint using the digit-vector register; do not execute Appendix B.
- Derive
endpoint=D+g(original span)and the paired difference law. - Establish the no-borrow and borrow families on all applicable source rows.
- Establish all shortened-core rows from their positional formula.
- Resolve both retained-zero rows and integrate them into the same register model.
- Test the complete field and its non-affine compatibility result.
- Connect this nonlinear finite source map to the inherited path-gain / rounding-residual descriptions, while preserving their distinct domains.
This is a useful eight-step-sized family, not a recommendation to stretch it into a large block by counting each row as a separate discovery.
Novelty audit and limits
Checked the available C482–C831 journals for prior inverse, decimal, digit, carry, borrow, anchor-change and paired-route results, with particular attention to C495–499 (path additivity, refinements, decimal gains), C512–520 (the nine-member pillar field and its affine operations), C610, C813 and the C831 synthesis. Those already establish scalar decimal gains, segmented backbone differences, affine expansion families and the need to retain component boundaries. The retrieved prior findings do not give the complete 32-row paired inverse-anchor displacement law above. File52c itself supplies the endpoint tables and acknowledges dependence, but does not give this digit-borrow/length classification.
The mathematical explanation does not determine why the received chronological source values have their particular digits, establish historical scribal use of reversal, or calibrate rarity. No higher-order decimal operation, new Gear, primer, statistical scan, or new chronological path subdivision is opened.