Bound P104 endpoint support and reject the 406-lifetime profile

research/p104-endpoint-support-capacity/README.md

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+# P104 endpoint-support capacity
+
+The 406-lifetime necessary-screen control in [PR2889][2889] cannot realize the
+frozen X-only monotone correction. It can span the endpoint frames, but cannot
+carry enough distinct required coefficients. This package proves two further
+necessary histogram rows and checks their finite geometric premises.
+
+For each orientation there are **4,656 residual demands**. A proper subspace
+of any endpoint contains at most **30** of its residual labels. Consequently
+335 histories `(1,7,1)` and 71 unit-step histories can cover at most
+`335 + 30*71 = 2465`, strictly below 4,656. The deficit is 2,191 in each row.
+This rejects those exact histories, not every network with 406 lifetimes.
+
+A different 400-lifetime profile passes these rows and the earlier numerical
+screens, including the declared favorable moment relaxation. Thus this is a
+specific compatibility obstruction, not a new universal lifetime floor, a
+rejection of the P104 family, a correction circuit or a new exponent.
+The selected result and frozen predecessors are unchanged.
+
+## Model and predecessor boundary
+
+Use the frozen 104-port PR1972 paths retained by PR1991. Write the required
+added correction as `C = I-G/2`. Its 4,992 nonzero entries have magnitude 1/2.
+The sources X are immutable, the targets Y are write-only, and their original
+frame paths increase by inclusion. Source lines `ell_s` and final target
+hyperplanes `H_t` are distinct in the nine-dimensional binary frame space.
+
+New helpers start at ZERO with source-independent arbitrary dirty values,
+follow increasing frame paths, and end at FULL with their logical dirt
+restored. Reinitialization counts another lifetime. Added operations are
+fixed rational linear combinations within equal frames. Arbitrary relays,
+cancellation and even singular same-frame mixing are allowed. The correction
+cannot read Y or the original helper/private fields, write X, descend, change
+the original paths or use source-dependent initialization. The retained
+channel does not write X or read Y, which the replay checks.
+
+These are the same substantive boundaries used by PR2353, PR2614, PR2640,
+PR2675 and PR2889. We reuse their source-support/dependence-path reasoning.
+The new restriction counts *distinct residual labels*, rather than the binary
+dimension of their span or all source-target pairs at an interior cut.
+
+The inspected pending predecessor is Hiroki Kumazaki's PR2889 at
+`5cd56602d65c4d0767c86b518a37168d571cf3ba`. Its dimension rows correctly
+leave its 406 profile as a necessary-screen control, not a construction.
+This package answers its explicit compatible-endpoint handoff without
+changing that predecessor. Source pins for every executable input are in
+[SOURCES.json](SOURCES.json). No pending supplier is admitted here.
+
+## Endpoint residuals and proper-subspace capacities
+
+Let `q_s` be the nonzero vector of `ell_s`, and `h_t` the unique nonzero
+ordinary covector with kernel `H_t`. This does not identify the compressed
+root-label pairing with the ordinary binary dot product.
+
+For each target t, remove any demanded source whose label is in any earlier
+proper Y_t frame, or whose X_s path visits H_t. Denote the remaining labels
+by `N_t`. For each source s, remove any demanded target for which some X_s
+frame of dimension greater than one is contained in H_t. Denote the remaining
+target normals by `F_s`. These removals grant all possible bypasses without
+requiring chronological access, which favors the proposed correction.
+
+The replay reconstructs the root order and all frame paths from the two
+pinned JSON inputs, then checks:
+
+- No direct source-to-target contact at H_t, and no target visits a source line.
+- The source labels and target normals each lie in an affine odd hyperplane.
+- Each N_t spans H_t; each F_s spans `ell_s^ann`, of dimension eight.
+- Both `sum_t |N_t|` and `sum_s |F_s|` equal 4,656.
+- For every endpoint, the largest intersection of its residual set with any
+  proper subspace is 24, 28, 29 or 30. Each orientation has respectively
+  12, 24, 24 and 44 endpoints of those types.
+
+It suffices to enumerate the 255 hyperplanes inside each eight-dimensional
+endpoint. Every proper subspace extends to one of these hyperplanes.
+The primary implementation tests the 510 nontrivial ambient functionals
+other than the endpoint annihilator; they represent each proper hyperplane
+twice. RESULT.json includes a maximizing functional for every endpoint.
+The separate implementation computes all 512 character sums by a nine-stage
+integer Walsh-Hadamard transform and recovers the intersections as
+`(|N| + character_sum)/2`. It uses bitset frame paths, imports no primary or
+predecessor implementation, and agrees on every endpoint and witness.
+
+## Written program-to-histogram implication
+
+Project each helper's logical row onto the independent X inputs. Its source
+support lies in its current frame. Initialization has empty support;
+equal-frame operations preserve the invariant; increasing transport cannot
+violate it. Restored lifetime boundaries have zero X row, so reuse does not
+introduce an uncounted source-dependency edge.
+
+Fix target t. Any nonzero coefficient indexed by N_t must reach H_t through
+a new helper entering from a proper frame A. It cannot be delivered earlier,
+and cannot arrive by direct X read, by the definition of N_t. Expand a finite
+linear program into its chronological dependency graph. A nonzero final
+coefficient has a nonzero path even with cancellation. That path supplies
+an entrance whose A contains its source label. Mixing at H_t cannot create
+a source coefficient absent from every entrance.
+
+For an entrance from dimension d, its number of distinct N_t labels is at
+most 30 by the checked proper-subspace bound. It is also at most `2^(d-1)`
+for d>0, since all source labels are odd under a common nonzero functional.
+A ZERO entrance carries none. Define
+
+```text
+f(0) = 0
+f(d) = min(2^(d-1), 30), for 1 <= d <= 7.
+```
+
+The union bound gives `|N_t| <= sum_entrances f(dim A)`. It permits duplicate
+labels, unproductive entrances, arbitrary relays and dirty-state compensation;
+none makes the bound stronger than justified.
+
+Dually, for source s, each coefficient indexed by F_s must be read at ell_s
+and leave on a helper's first departure to B. A later source read reaching
+the target would have been removed from F_s. Target reads and source writes
+are forbidden, so the dependency cannot bypass that departure. Its target
+normal belongs to `B^ann`, a proper subspace of `ell_s^ann`. The same argument
+bounds its distinct F_s normals by `f(9-dim B)`.
+
+A nested lifetime visits at most one designated source line and at most one
+designated target hyperplane. Thus summing the endpoint inequalities counts
+each lifetime at most once in each orientation. For a positive composition
+h of nine, define:
+
+```text
+a(h) = f(dimension immediately before visiting dimension 8), or 0 if absent
+d(h) = f(9 - dimension immediately after leaving dimension 1), or 0 if absent
+```
+
+For the complete history histogram x, every permitted correction satisfies
+
+```text
+sum_h a(h) x_h >= 4656
+sum_h d(h) x_h >= 4656.
+```
+
+Visiting dimension eight directly from ZERO, or leaving a line directly for
+FULL, contributes zero. Reversing a history exchanges its two weights.
+The replay checks all 256 histories, including asymmetric and empty-contact
+controls. The universal implication above is a written proof, not finite
+network enumeration or a proof-assistant theorem.
+
+## Rejected and surviving profiles
+
+Let c count histories `(1,7,1)` and u count nine unit jumps. Their support
+weights are respectively 1 and 30 in both orientations. The earlier
+span-dimension weights are 1 and 7.
+
+| c | u | Lifetimes | Span total | Support total | New rows |
+|---:|---:|---:|---:|---:|---|
+| 329 | 71 | 400 | 826 | 2459 | fail |
+| 335 | 71 | 406 | 832 | 2465 | fail |
+| 253 | 147 | 400 | 1282 | 4663 | pass |
+
+The last profile also has 9,661 capacity at every old interior cut, 400 proper
+arrivals and departures, rank mass 3,600, and 744 ideal idle rank units.
+It exceeds the reported mixer lifetime floors, stays in the quoted 387 to
+482 investigation window, and passes PR2889's dimension rows.
+
+Reusing Douglas Colkitt's exact rational ceiling engine, the primary replay
+certifies its moment upper bound **0.999729278053145** at the selected complex
+primitive saving `67018507072324140917/40000000000000000000000`. This is only
+the inherited `m=73`, 832-family, nine-stage, centre-56, ideal rank-51-idle
+relaxation, including the original channel profile. The original transport
+profile is recounted from the literal word. The engine is source-authenticated
+but not independently reimplemented. No actual geometry, chronology,
+coefficients, dirty restoration word, owners or complete consumer is supplied
+for the surviving profile. Unknown additional costs are not zero.
+
+## Reproduce and validation limits
+
+Python 3.12 or later, standard library only:
+
+```sh
+python3.12 scripts/research_package.py --package research/p104-endpoint-support-capacity
+```
+
+Direct checks, including optimized execution:
+
+```sh
+python3 -I -O -Werror -B research/p104-endpoint-support-capacity/verify.py
+python3 -I -O -Werror -B research/p104-endpoint-support-capacity/independent.py
+python3 -I -O -Werror -B research/p104-endpoint-support-capacity/test_verify.py
+```
+
+The first command ends with
+`PASS_P104_ENDPOINT_SUPPORT required=4656 capacity=30 rejected=406 control=400`.
+`--emit` prints a recomputed receipt; normal replay checks RESULT.json and
+writes nothing. The tests cover normal/optimized CLIs, all boundary weight
+cases, a read-only Git-free export, source corruption before code execution,
+a typed receipt fault, the weaker affine-only bound and a phantom endpoint
+visit. Deliberately replacing the geometric cap by the old affine bound fails.
+
+Local validation passed on Linux x86_64 / CPython 3.12.15: the registered
+three-command replay, five behavioral test groups, Ruff 0.17.1, warning-as-error
+source compilation and the selected-result binding check. Direct CPython
+3.13.5 optimized primary and independent replays also passed. These checks
+were rerun after fast-forwarding the publication branch to canonical main
+`a1bfbc677fdcbc2a27a9b3bf6c78cac8fd6da9e2`. The scientific pins, priority
+revision, selected record and PR2889 head were unchanged. Final overlap at
+08:29 to 08:30 UTC covered 488 open PRs, 2,103 closed PR bodies and 1,558
+Discussion records with complete pagination. It found the credited PR2889
+handoff; arbitrary code diffs and unpublished work remain outside coverage.
+
+Native phases, full corrected-word execution, rounding, physical hosting,
+movement, supplier admission, all-size lifting, the selected cold replay and
+the outer multiplication theorem are not checked. No independent human,
+formal-kernel or hosted-CI review is claimed.
+
+Engineering limits: Python allocates managed objects during these bounded
+finite computations. Frames have at most 512 members, paths at most ten
+frames, residual sets at most 48 labels, input JSON files at most 2 MB, and
+there are exactly 104 endpoints per orientation and 256 histories. The one
+`exec` is limited to the exact SHA256-authenticated, self-contained ceiling
+source bytes to avoid source-adjacent bytecode caches. Its narrow static
+warning suppression does not apply to untrusted input or any other module.
+A static import can replace it if the inherited cost module gains a stable
+cache-independent import contract. All changed functions are under 60 lines.
+
+## Open next questions
+
+Does a profile surviving the new rows admit simultaneous endpoint assignments
+and chronological reads? Pin the 253/147 profile above as a first finite test,
+using the endpoint-specific capacities in RESULT.json rather than the relaxed
+uniform maximum. This matters because the seven units of support slack leave
+little room for assigning the 24/28/29-capacity endpoints. Interior sharing,
+coefficient compatibility and a complete paid consumer remain blockers.
+
+Family decision: stop after this necessary-condition audit. A favorable
+surviving screen does not justify a construction campaign or supplier
+admission. Reopen only for a named geometric/chronological constraint or a
+fully stated changed producer/consumer boundary with its complete cost ledger.
+
+## Credit and task scope
+
+Nick Petalas. Apache-2.0. Task `p104-side-correction`, refined to the explicit
+PR2889 endpoint-assignment handoff. The single chooser nonce is
+`e5cc9f5989ecea2fa6060e7b440c4314`; scientific base is canonical main
+`2b4c46fbb19b2df3685b751965877fa7c7fa34af`, priority revision 5.
+[The canonical claim][claim] records coordination and freshness.
+
+Credit Hiroki Kumazaki for PR2889's endpoint-span audit and PR2353's monotone
+model; ywz65 for PR2614's later-read-aware source departures; Arham Tahir for
+PR2675/2682's coupled screens; Sagar Thakrar for the newer mixer handoff at
+`743640530229e1821ae11dc104798871272d6f00`; Philippe Van Velzen for the relay
+construction and window bill; Nick Petalas/PR2640/2604 for crossing capacities
+and the legal cheap carrier; and Douglas Colkitt for the inherited ceiling
+engine. None of those earlier facts or methods is claimed as new here.
+
+Retain the Rohan Arun/PR1972, Chafik Boukhalfa/PR1991, Adam Gall/PR1940,
+Josie Elliston/PR1071, Ali Romman/PR1125 and geometry/native/compiler lineage.
+No predecessor source is copied or modified. Exact inspected predecessor
+heads and the remaining obligations are recorded in TASK.json.
+
+[2889]: https://github.com/CrocSwap/integer-mult-bounds/pull/2889
+[claim]: https://github.com/CrocSwap/integer-mult-bounds/discussions/2245#discussioncomment-18967463