Reject the P104 253/147 profile by endpoint demand saturation

research/p104-endpoint-saturation/README.md

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+# P104 endpoint demand saturation
+
+The 253 cheap / 147 unit-step, 400-lifetime control in [PR2896][2896] cannot
+realize the frozen X-only monotone correction. Even granting independent,
+arbitrary endpoint assignments, it can cover at most **3,972 of 4,656**
+residual demands in each orientation. The deficit is **684**.
+
+This closes PR2896's explicit endpoint-specific assignment handoff. Its
+aggregate calculation `253 + 30*147 = 4663` is correct as a necessary screen.
+The new condition accounts for demand saturation at each endpoint. Capacity
+left over at one endpoint cannot cover another endpoint's missing labels.
+
+For a budget of 400 lifetimes using only these two history types, at least
+**192 unit-step lifetimes** are necessary. This threshold is sharp for the
+separate endpoint-count relaxations. With 191 units, at least 219 cheap
+lifetimes are needed, totaling 410. With 192 units, 204 cheap lifetimes
+suffice in each relaxation, totaling 396. Four further cheap lifetimes may
+be left unproductive to give the stated 400 budget. No actual geometric,
+chronological or simultaneous source/target construction is supplied.
+
+## Contract and inherited implication
+
+Use the same PR1972/1991 104-port correction `C = I-G/2` as PR2889 and
+PR2896. Sources X are immutable, targets Y are unread, and their fixed frame
+paths increase by inclusion. New helpers start source-independent at ZERO,
+end at FULL with their logical dirt restored, and follow increasing frame
+paths. Reinitialization starts another lifetime. Original helper/private
+reads, target feedback, source writes, descending transport and changed
+original paths are outside this model. Equal-frame rational mixing, relays,
+singular mixing and cancellation remain allowed.
+
+The two types are the dimension-jump histories `(1,7,1)`, called cheap below,
+and `(1,1,1,1,1,1,1,1,1)`, called unit-step. Both visit dimension one and
+dimension eight exactly once. A nested lifetime can meet at most one
+specified source line and one specified target hyperplane.
+
+We use PR2896's written dependency-path implication. Every residual source
+label at a target endpoint must arrive on a new helper from a proper
+subspace. The dual statement applies to residual target normals after a
+source departure. A cheap lifetime carries at most one such label in each
+orientation. A unit-step lifetime carries at most the endpoint's proper
+subspace capacity. Logical dirt and cancellation cannot create a missing
+source-dependence path. This package adds demand saturation to that necessary
+implication, rather than claiming a new proof of the full network interface.
+
+The checker reconstructs the original chronological X/Y paths and the
+ordered root Gram correction directly from the two main-branch inputs. It
+reconstructs all residual sets and enumerates all 510 ambient functionals
+other than the endpoint annihilator. These give the 255 proper hyperplanes
+inside each eight-dimensional endpoint, each twice. Every proper subspace
+extends to one of these hyperplanes. The resulting 208 records exactly
+match the pinned PR2896 receipt, without importing its implementation.
+
+Both orientations have the same multiset of demand/capacity pairs:
+
+| Demand d | Unit capacity b | Endpoint count |
+|---:|---:|---:|
+| 36 | 24 | 12 |
+| 44 | 28 | 24 |
+| 45 | 30 | 8 |
+| 46 | 29 | 24 |
+| 47 | 30 | 24 |
+| 48 | 30 | 12 |
+
+## Exact allocation bound
+
+At endpoint i let `u_i` and `c_i` count assigned unit and cheap lifetimes.
+Every permitted network satisfies the relaxed necessary inequalities
+
+```text
+d_i <= b_i*u_i + c_i
+sum_i u_i <= U
+sum_i c_i <= C.
+```
+
+An endpoint can usefully receive at most `d_i` distinct demands. Its first
+unit therefore contributes at most `b_i`, its second at most `d_i-b_i`,
+and later units contribute zero. Here `b_i < d_i <= 2*b_i`. The marginal
+contributions decrease within every endpoint.
+
+Sort all 208 marginals in decreasing order, placing a first unit before
+its second in a tie. Let `F(U)` be the sum of the first `min(U,208)` entries.
+The selected entries respect every endpoint prefix, so their allocation is
+valid in this relaxation. Any U units have useful contribution at most
+F(U), since no U entries can exceed the U largest. Therefore the exact
+minimum number of cheap lifetimes in this relaxation is
+
+```text
+C_min(U) = 4656 - F(U).
+```
+
+The lower bound follows by summing the remaining demands. It is attained
+in the relaxed count problem by placing one cheap lifetime on each remaining
+demand. This attainment does not assert that actual subspaces realize the
+capacities or that the assignments in opposite orientations can be joined.
+
+For U=147, the first 104 marginals sum to 2,976. The next 43 are
+`12*18 + 31*17 = 743`, so `F(147)=3719` and `C_min(147)=937`.
+With C=253, the total useful contribution is at most 3,972. This is already
+an obstruction in either orientation separately.
+
+A second implementation uses integer dynamic programming. After each
+endpoint it minimizes the cheap count for each exact unit count, considering
+zero, one or two units there. It does not use the sorted marginals. The two
+methods agree at all 209 budgets in both orientations. RESULT.json records
+the complete curve and explicit relaxed allocations at the 192-unit threshold.
+
+## Reproduce and evidence
+
+Python 3.12, standard library only, from the repository root:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/p104-endpoint-saturation
+```
+
+The main command prints
+`PASS_P104_ENDPOINT_SATURATION deficit=684 minimum_units_at_400=192`.
+Use `verify.py --emit` to print the recomputed receipt without changing files.
+The registered normal and optimized checks compare the complete typed receipt.
+Five behavioral test groups cover 780 exhaustive small allocation problems,
+the literal rejection/threshold, invalid domain bounds, a semantic mutant
+that removes demand saturation, a Git-free export, source corruption and an
+integer-to-float receipt corruption. Subprocesses have 30-second timeouts.
+
+Evidence is author-checked. The allocation algorithms are separate
+implementations, while both use the same reconstructed geometry and inherited
+program-to-endpoint implication. No independent human or formal-kernel review
+is claimed. Source hashes, the exact PR2896 snapshot and main provenance are
+in SOURCES.json; scope and freshness are in TASK.json.
+
+No moment or new exponent is claimed for any replacement profile. The
+selected frontier and frozen predecessors are unchanged. Native execution,
+rounded restoration, physical hosting/movement, all-size lifting and a complete
+consumer/supplier invoice remain open. The selected cold replay is not run
+because this package does not change the selected construction.
+
+Engineering choices follow Build the Lever, through the reproducible integer
+certificate, and Prove It Works, through original-input reconstruction and
+rejecting controls. All functions are below 60 lines. Loop bounds come from
+104 endpoints, 4,058 events, 512 frame vectors, 209 DP states and the explicitly
+enumerated 780 toy cases. The input boundary limits each JSON file to 2 MB.
+Python's managed allocation is a Power of Ten rule 3 deviation; the fixed
+finite domains bound live collections, and temporary test exports are cleaned
+on success and failure. A fixed-array port would be appropriate if this checker
+became a resident or streaming service. The inherited source contract and
+all-size theorem are not strengthened by these implementation checks.
+
+## Open next questions
+
+Can a named two-type profile with at least 192 unit-step lifetimes satisfy
+actual subspace-union coverage, joint source/target incidence and chronology,
+while retaining a favorable complete cost? First price a concrete candidate
+against the inherited optimistic budget, then test its actual endpoint sets.
+The count relaxation supplies neither that budget nor a construction.
+
+Family decision: stop after the endpoint-assignment audit. Do not start another
+construction round merely because the 192-unit count threshold passes. Any
+continuation needs a stated candidate, complete boundary/cost ledger and an
+identified geometric or chronological constraint that can change the decision.
+
+## Credit and provenance
+
+Nick Petalas. New work is Apache-2.0. The source-bound allocation implication
+and checker are the contribution here. Credit Nick Petalas/PR2896 for the
+residual endpoint capacities and handoff, Hiroki Kumazaki/PR2889/2353 for the
+endpoint-span and monotone-history model, ywz65/PR2614 for source-departure
+handling, Arham Tahir/PR2675/2682 for coupled screens, Sagar Thakrar for the
+mixer handoff, and Philippe Van Velzen for the relay construction/window bill.
+Retain Rohan Arun/PR1972, Chafik Boukhalfa/PR1991, Adam Gall/PR1940,
+Josie Elliston/PR1071, Ali Romman/PR1125 and the geometry/native lineage.
+The byte-exact predecessor RESULT.json is data from PR2896, not a new result
+of this package. Substantial AI assistance was used; details are in TASK.json.
+
+[2896]: https://github.com/CrocSwap/integer-mult-bounds/pull/2896