Certify the changed-history boundary of P104's lifetime screen

research/p104-three-history-threshold/README.md

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+# The P104 lifetime threshold depends on its history alphabet
+
+Adding one complete monotone history moves the first rejecting lifetime floor
+from **423 to 436** in the stated endpoint/cut/ideal-idle relaxation. This is a
+cost-screen result, not a correction circuit, lifetime lower bound or supplier.
+The selected conditional kappa remains unchanged.
+
+[Sudarsh Chaturvedi's PR2924](https://github.com/CrocSwap/integer-mult-bounds/pull/2924),
+head `d1660ed0a17b66ad0c86cd4aaea724926134e362`, correctly restricts its 423
+threshold to the histories `(1,7,1)` and `(1,1,1,1,1,1,1,1,1)`. This successor
+changes that restriction. It does not identify an error in PR2924 or claim that
+its author extended the result beyond those two histories.
+
+## Exact result
+
+Write `C`, `U` and `M` for integer multiplicities of these rank-step histories:
+
+| Name | Rank steps | Visited dimensions |
+| --- | --- | --- |
+| Cheap, C | 1,7,1 | 0,1,8,9 |
+| Unit, U | 1,1,1,1,1,1,1,1,1 | 0,1,2,3,4,5,6,7,8,9 |
+| Merged, M | 1,1,2,1,1,1,1,1 | 0,1,2,4,5,6,7,8,9 |
+
+The new history replaces two consecutive rank-one steps by one rank-two step.
+Every lifetime still pays nine rank units. Both U and M visit dimensions
+1,2,7,8, so their proper-arrival dimension and post-source departure
+codimension remain seven. Their separate endpoint hyperplane capacities and
+the credited demand-saturated allocation curve are therefore unchanged.
+This statement concerns separate endpoint constraints, not compatible joint flags.
+
+At the selected complex primitive saving
+`a = 67018507072324140917 / 40000000000000000000000`, keep 232 cheap histories
+and replace `n` of PR2924's 191 unit histories by the merged history:
+
+| C | U | M | Certified optimistic moment, decimal approximation |
+| --- | --- | --- | --- |
+| 232 | 191 | 0 | 1.000003548402828 |
+| 232 | 181 | 10 | 1.000000083777248 |
+| 232 | 180 | 11 | 0.999999737314690 |
+| 232 | 0 | 191 | 0.999937374054245 |
+
+**Eleven replacements are necessary and sufficient within this fixed-count
+one-parameter family.** The exact outward intervals are in [RESULT.json](RESULT.json).
+No floating-point value determines a sign or acceptance decision.
+
+The eleven-replacement profile has
+`3235 H1 + 69 H2 + 86 H3 + 104 H4 + 48 H5 + 24 H6 + 240 H7 + 8 H8`.
+Its local rank mass is 6,175 and its optimistic idle rank mass is 537.
+The seven cut capacities are
+`[12456,12104,12104,12456,12456,12456,12456]`, against the credited demands
+`[4656,4656,4672,4680,4672,4656,4656]`.
+Both endpoint dimension totals are 1,569, above 832. Both endpoint allocation
+curves require at least 219 cheap helpers at 191 wide endpoint histories;
+the profile has 232. All 192 members of this fixed-count family pass those
+necessary constraints.
+
+Optimizing over the entire three-history alphabet, with lifetime count at least
+L and at most 482, gives:
+
+| Floor L | Cheapest C,U,M | Optimistic moment, decimal approximation |
+| --- | --- | --- |
+| 423 | 232,0,191 | 0.999937374054245 |
+| 435 | 245,0,190 | 0.999999349571449 |
+| 436 | 246,0,190 | 1.000004768716193 |
+
+Thus 436, not 423, is the first rejecting floor **for this enlarged alphabet
+and this relaxation**. Neither number is an all-history or all-P104 barrier.
+
+## Proof and inherited premises
+
+The contract remains the frozen X-only, ZERO-to-FULL monotone relay model.
+There are no new original-helper reads, target feedback or nonmonotone paths.
+Initial segments of any ordered nine-vector basis realize these dimension
+histories as nested frames. This does not bind them to actual source/target
+frames. The merged history loses the dimension-three contact, which may matter
+to source acquisition. A dimension history is not a chronological correction
+program.
+
+The new work uses PR2675's seven coupled cut inequalities and PR2920's separate
+endpoint curve as pinned premises. It does not independently reprove their
+original P104 geometry or the network-to-inequality implications. It reconstructs
+the endpoint curve from all six endpoint types in both orientations, checks
+all history-dependent quantities and reproduces PR2924's baseline moment
+**exactly**, including both rational enclosure endpoints.
+
+For a crossing from dimensions p to q at cut k, `p <= k < q`, the credited
+capacity is zero if p=0 or q=9, otherwise `2^(7-q+p)`. The cheap history has
+capacity one at each cut. The merged history has capacities
+`[64,32,32,64,64,64,64]`. This proves the displayed fixed-family cut totals.
+The first departure from a source line is still to dimension two; the last
+proper arrival before a target hyperplane is still from dimension seven.
+Consequently all existing endpoint dimension, support and separate allocation
+rows used here survive the replacement. No simultaneous source/target
+endpoint realization follows.
+
+The whole original channel invoice is retained. At m=73, 832 families, nine
+stages and centre charge 56, the total rank mass is 6,712. A lifetime contributes
+nine local units, leaving `4344 - 9*(C+U+M)` idle units. The optimistic bill puts
+that idle mass into fractional rank-51 children, as in PR2675/2920/2924.
+At a>0, `cost(2) < 2*cost(1)`, so each U-to-M replacement strictly lowers the
+moment without changing idle mass. The certified signs at n=10 and n=11
+therefore prove the minimal replacement count.
+
+For the full three-history minimum, replace all U by M. For each wide-helper
+count K from zero through 482, the least feasible cheap count is the inherited
+endpoint curve, or `L-K` if larger. Every feasible such all-merged profile
+passes the seven cuts and endpoint dimension/count rows. The minimum cut
+capacity over the endpoint-feasible minimal profiles is 6,248, above every
+forced cut demand. Additional cheap helpers only improve those constraints.
+Thus replacing U by M is feasible throughout this optimization domain.
+The extra-helper prices are strictly positive, so only the least cheap count
+at each K needs checking. Exact interval comparisons certify unique minima
+at L=423,435,436. A passing profile at 435 and exclusion at 436, together with
+nested floor constraints, establish the first rejecting floor.
+
+## Independent arithmetic and reproduction
+
+Standard-library Python 3.12 or later, no network or solver:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/p104-three-history-threshold
+python3.12 -I -O -B research/p104-three-history-threshold/verify.py
+python3.12 -B research/p104-three-history-threshold/test_verify.py
+```
+
+The primary intervals reuse Douglas Colkitt's authenticated logarithm/exponential
+engine. [independent.py](independent.py) instead encloses
+`(r/73) * (1-z)^(-a)`, where `z=1-r/73`, by the positive binomial series.
+Successive term ratios are `z*(a+k)/(k+1) <= z` for 0<a<1. Integer floor/ceiling
+recurrences at denominator 2^128 enclose every term; the first omitted upper
+term divided by `1-z` bounds the tail. A 4,096-term limit fails closed.
+Both implementations certify every reported sign and overlap before decimal
+outward rounding. The independent implementation does not import the primary
+engine, use logarithms or use floating point.
+
+Controls check exact rational-power oracles, an independent endpoint-allocation
+DP, independent optimization including nonminimal cheap counts, invalid
+histories/counts, insufficient endpoint and cut capacity, a changed endpoint
+signature, duplicate keys, isolated optimized Git-free execution, source
+corruption before engine import and complete receipt corruption. Source bytes
+and the retained PR2924 result are authenticated before use. The full selected
+cold replay is not required or claimed for this scoped successor.
+
+## Family decision and limits
+
+Finish this audit and stop. The result identifies a concrete hypothesis change
+that a lifetime-floor argument must account for. It does not justify a new
+construction campaign or infer a favorable *complete* bill.
+
+Proved here: history arithmetic, the stated finite screen optima, independent
+moment sign certificates and the eleven-replacement minimum. Inherited: original
+channel correctness, binary endpoint geometry, cut implications and the
+separate endpoint relaxation. Unknown: compatible joint flags, chronological
+source acquisition, scalar coefficients, arbitrary dirty restoration, native
+lowering, actual idle children, owners/routing, precision/tape, consumer and
+all-size admission. Unknown costs are not zero. Passing an optimistic lower
+cost screen is not a constructive upper bound on a supplier's cost.
+
+Power of Ten inspection covers the new Python files only. All loops have
+finite input-derived bounds: 482 lifetimes, nine dimensions, seven cuts, 104
+endpoints, 4,096 binomial terms, 128-bit rational saving arguments and bounded
+source JSON. The interpreter and
+JSON decoder allocate dynamically, a managed-runtime deviation bounded by these
+sizes. The independent cache holds at most 16 child intervals. The fixed primary
+engine retains its existing recursive logarithm and convergence loops; it is
+unchanged, its rank arguments here are at most 73, and the independent bounded
+binomial checker guards all reported decisions. Dynamic imports are limited
+to two explicit local files. Warning-as-error compilation, lint and optimized
+execution are the local diagnostics. There is no physical-memory admission.
+
+## Open next questions
+
+Can the third history participate in an actual joint flag/dependency assignment
+within the available cost margin? First bind its dimension-two departure and
+dimension-seven arrival to actual source/target endpoint pairs. Reject an
+incompatible assignment before scalar synthesis. Compatible acquisition,
+restoration and the complete consumer/physical cost ledger remain blockers.
+A further numeric threshold search alone does not close them.
+
+## Credit and provenance
+
+Sudarsh Chaturvedi/PR2924 supplies the audited two-history threshold. Credit Nick
+Petalas/PR2920/2909/2896 for endpoint geometry and saturation; Arham Tahir/PR2675/2682
+for the coupled profile budget; Hiroki Kumazaki/PR2889/2353 for endpoint spans and
+the relay model; ywz65/PR2614; Sagar Thakrar's mixer handoff; Philippe Van Velzen's
+relay and cost contract; and Douglas Colkitt's interval engine. Retain the Rohan
+Arun/PR1972, Chafik Boukhalfa/PR1991, Adam Gall/PR1940, Josie Elliston/PR1071,
+Ali Romman/PR1125 and native/compiler/geometry lineage credited by those sources.
+The PR2924 result is copied unchanged with its exact head, blob and SHA256.
+Other predecessors remain live pinned inputs, not copied source packages.
+
+Task p104-side-correction, refined to a changed-history cost-screen audit.
+Scientific base `914cd73f563acaaa682d733a9fbfcc07272336a3`, priority revision 5.
+[Task, freshness and verification limits](TASK.json), [source pins](SOURCES.json),
+[canonical claim](https://github.com/CrocSwap/integer-mult-bounds/discussions/2245#discussioncomment-18973831).
+Evidence is locally checked; no independent human or proof-assistant review is
+claimed. Apache-2.0.