Construct a 22-channel monotone delivery counterexample (no new κ)

research/monotone-noisy-delivery-counterexample/PROOF.md

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+# A finite monotone-tariff counterexample in the immutable-channel quotient
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+## Statement
+
+Fix the original backward word retained in the archive with SHA256
+`595f988e748b1f3b25e7525a226da173966e4278a70f511b2919245cec1b45f7`.
+Its compressed word has SHA256
+`934fea11e51f674ce2031b7d3127b994a921a9d83317b00359292752563a1775`.
+Use the immutable-channel common-fixed-vector scalar response model of
+Ali Romman's [PR5745](https://github.com/CrocSwap/integer-mult-bounds/pull/5745),
+as retained by Nicholas Stone's
+[PR5845](https://github.com/CrocSwap/integer-mult-bounds/pull/5845).
+
+In this model a fixed-frame channel has an immutable rational linear
+combination of the eight original modes as its payload. It may add rational
+multiples of that payload at any recorded cut to a field in its frame.
+Its canonical delivery charge is H_r for the grade r of that frame.
+Repeated contacts in the same frame use one channel. Ground delivery costs
+H_0=0. Payload preparation, all scatter, channel lifetime and compensation
+of originally ground-frame helpers are granted without charge, exactly as
+in the response-space relaxation. Main outputs, all Y outputs and all
+originally non-ground helpers remain retained constraints.
+
+The finite witness in WITNESS.json has 22 channels with delivery profile
+
+    H_0 + 14 H_7 + 7 H_8.
+
+It realizes all eight desired modes on the retained outputs. Therefore an
+8 H_8 delivery floor for arbitrary finite channel counts is false for
+general positive nondecreasing tariffs in this quotient. For example,
+
+    H_0=0, H_1=...=H_7=1, H_8=H_9=15
+
+gives charge 119, strictly less than 120. More generally this witness beats
+8 H_8 exactly when H_8>14 H_7. No claim of channel-count or tariff optimality
+is made.
+
+## Response identity
+
+Index modes by 0 through 7. Let R be the archived 8 by 112 rational mode
+matrix, satisfying RR^T=56 I. Write E_j for the output vector whose first
+112 coordinates are row j of R and whose other retained coordinates vanish.
+
+The new joint response uses frames
+
+    0, 661, 662, 663, 664, 678, 679, 691, 692,
+    728, 730, 989, 993, 1003, 1007.
+
+Frame 0 has grade zero. Each other listed frame has grade seven. There
+are 61 contacts, each with an exact rational coefficient and a literal
+cut/field pair in WITNESS.json. Let v_f be the retained output response to
+unit payload at all contacts of frame f with those coefficients. Exact
+forward evaluation of the frozen word gives
+
+    sum_f v_f = E_4 + E_5 + E_6 + E_7.                  (1)
+
+This is the new finite identity. It concerns the sum of noisy channels,
+with cancellation on every retained nuisance coordinate. No census of
+single-frame or pair obstructions is used to prove it.
+
+PR5745 supplies clean grade-eight responses E_j. The certificate uses its
+seven directions with j in {0,1,2,3,5,6,7}, with their cited frame/contact
+witnesses. Their frames are respectively 254, 274, 619, 279, 280, 624 and
+281. The last direction has two contacts; each other direction has one.
+Checking those contacts is part of the new complete witness replay, not
+an independent audit or a new claim about predecessor construction.
+
+For an arbitrary original mode vector u in Q^8, assign payload u_4 to
+each of the fifteen channels in (1). Give the clean E_j channel payload
+u_j for j<4, and u_j-u_4 for j>4. All these payloads are immutable original
+mode combinations. The resulting retained response is
+
+    u_4 (E_4+E_5+E_6+E_7)
+      + sum_(j<4) u_j E_j + sum_(j>4) (u_j-u_4) E_j
+      = sum_j u_j E_j.                                (2)
+
+There are 15+7=22 channels and 61+8=69 contacts. Equations (1) and (2)
+give the advertised finite quotient completion and its invoice.
+
+## What is checked
+
+The verifier imports no predecessor executable. It loads the authenticated
+source word and R from the existing archive, then propagates 22 independent
+channel symbols forward through all 22,585 events. Each contact occurs
+immediately before its zero-based event index. The current frame must match
+the channel frame. The source has 641 scalar slots: 632 original fields,
+eight extra owned C slots and the unused C0 slot.
+
+The quotient treats native transport as identity on the common fixed vector,
+while checking entry and exit frames. Adapters and phase markers are erased
+as in the cited scalar quotient. Scalar ADD, SCALE and CLEAR act literally.
+All contacts follow the original mode-collection prefix. The verifier checks
+that every relaxed centre-tail physical ADD has zero channel response on
+its C input; those tail events consequently add zero to this impulse replay.
+This is not a native phase or common-fixed-vector existence proof.
+
+There are 274 retained original fields: 112 main fields, 112 Y fields and
+50 non-ground helpers. The verifier checks (1) on each of these fields and
+checks every one of the 274*8=2,192 coefficients in (2). All nine C-slot
+responses vanish at return. Of 358 exempt original ground helpers, 189 have
+a response to some independent channel symbol and 113 have a nonzero
+residual after substituting the payloads in (2).
+
+Eight controls change a contact coefficient, move a contact to cut zero,
+change a frame, understate a grade, reverse a payload coefficient, delete
+the ground channel, replace the strict tariff gain by equality, or violate
+tariff monotonicity. Each must fail. The generated RESULT.json binds the
+complete result to hashes of the verifier, witness and source archive.
+
+## Consequences and limits
+
+The result answers the arbitrary-monotone continuation in PR5845 with a
+larger positive witness. It does not settle the initially proposed
+three-frame question, or the minimum number of noisy channels. PR5845's
+at-most-nine-channel theorem remains outside this contribution's audit
+scope and is compatible with 22 channels.
+
+Ali Romman's [PR5823](https://github.com/CrocSwap/integer-mult-bounds/pull/5823)
+already supplies an arbitrary-finite-channel floor for concave power tariffs
+H_r=r^q H_1, 0<q<=1. Our witness has H_8/H_7=15, whereas those powers have
+H_8/H_7<2. This finite example therefore does not challenge that floor.
+The 14 H_7+7 H_8 invoice is also plainly greater than 8 H_8 for those powers.
+
+A response-model completion need not lift to an operator identity on the
+full native field. No such lift, preparation circuit, channel storage or
+return word is supplied. The 113 ground-helper residuals are not physically
+repaired here. Scalar operations, native phases, borrowed-channel restoration,
+ownership, precision, stock, tape movement and a complete recursive caller
+bill remain unproved or unpriced. This is a finite quotient certificate,
+not a physical supplier, a new kappa or a multiplication theorem.
+
+## Credit
+
+Nicholas Stone's PR5845 posed the monotone-tariff continuation and supplies
+the two-frame/nine-channel context. Ali Romman's PR5745 supplies the word's
+canonical delivery model and clean directions. Its temporal response code,
+retained byte-for-byte in PR5845, was used for discovery. It is not imported
+by this verifier. Ali Romman's PR5808 and PR5823 supply the distinct
+mixed-rank and concave-power obstructions, which are cited without replay.
+
+Retain flyingviper's PR5720/PR5755, Cao Yifan, Cinna Davis, Dugongue and all
+mode, native, controller and caller contributors credited in the unchanged
+source archive. The new joint response, complete quotient payload assembly
+and displayed general-monotone tariff counterexample are this contribution.
+Independent human and formal proof-kernel review are unclaimed.

research/monotone-noisy-delivery-counterexample/README.md

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+# A 119-versus-120 monotone delivery counterexample
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+**Claim type: finite certificate in the fixed immutable-channel scalar
+quotient.** An explicit 22-channel witness has delivery profile
+
+    H0 + 14 H7 + 7 H8.
+
+It realizes the complete eight-mode response in the retained PR5745 quotient.
+The nondecreasing tariff H0=0, H1=...=H7=1, H8=H9=15 gives cost **119**
+against the eight-call comparator **120**. Thus the quotient's 8H8 floor
+cannot extend to arbitrary finite channel counts under every positive
+nondecreasing tariff. No optimal channel count or new kappa is claimed.
+
+The new identity combines fourteen grade-seven frames and one ground frame
+into mode (0,0,0,0,1,1,1,1). Seven cited clean grade-eight directions complete
+the other modes with immutable payloads. [WITNESS.json](WITNESS.json) gives
+all 69 contacts and coefficients; [PROOF.md](PROOF.md) gives the contract,
+linear identity and exact assembly. [RESULT.json](RESULT.json) is generated
+by the verifier and bound to the source, witness and verifier hashes.
+
+## Scope and predecessor credit
+
+Nicholas Stone's [PR5845](https://github.com/CrocSwap/integer-mult-bounds/pull/5845)
+at `7b8ed4b168890d90e100e75a4062174a145be246` poses the continuation to
+three-frame sums and ten or more channels under arbitrary monotone tariffs.
+The initially proposed three-frame question remains open here. This first
+positive witness uses 22 channels; its channel count is not optimized.
+
+Ali Romman's [PR5745](https://github.com/CrocSwap/integer-mult-bounds/pull/5745)
+at `e82b64e91e57c838ff2028bcf2219203966bf9ed` supplies the canonical
+delivery model and clean grade-eight directions. Its retained temporal
+response code was used for discovery. Verification uses a separate forward
+impulse implementation and imports no predecessor executable. The predecessor
+obstruction censuses are not replayed or submitted as this contribution.
+
+The certificate is compatible with PR5845's at-most-nine-channel theorem and
+Ali Romman's [PR5823](https://github.com/CrocSwap/integer-mult-bounds/pull/5823)
+arbitrary-channel concave-power floor. Its tariff is outside that power class.
+Also retain [PR5808](https://github.com/CrocSwap/integer-mult-bounds/pull/5808),
+flyingviper's PR5720/PR5755, Cao Yifan, Cinna Davis, Dugongue and every source
+contributor credited in the unchanged archive. Full attribution and the
+distinction between these results are in PROOF.md.
+
+## Reproduction
+
+From the repository root, using CPython 3.12 and the standard library:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/monotone-noisy-delivery-counterexample/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/monotone-noisy-delivery-counterexample/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/monotone-noisy-delivery-counterexample
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/monotone-noisy-delivery-counterexample --check-source-history
+```
+
+An equivalent CPython 3.12 path works on another machine. In a sparse checkout,
+first materialize `research/filtered-native-kappa004/inputs`, the sole live
+input, with `git sparse-checkout add research/filtered-native-kappa004/inputs`.
+The archive already belongs to main; no pending predecessor commit is needed
+for replay. SOURCES.json pins its exact bytes and the revision-8 baseline.
+
+Expected output starts `PASS_MONOTONE_NOISY_DELIVERY` and reports 22 channels,
+69 contacts, all 22,585 source events, 2,192 retained mode equalities, eight
+rejection controls, charge 119 and comparator 120. Both registered commands
+run the same full scientific check, normally and with assertions disabled.
+Their local timings and interpreter environment accompany the PR evidence;
+the generated receipt records the actual environment used to create it.
+
+## Limitations
+
+This is a positive witness in a necessary scalar quotient. A full native
+operator identity and a physical restoration circuit are not supplied.
+Of 358 ground helpers exempted by the cited model, 113 have a nonzero residual
+after the displayed payload substitution. The certificate restores all 112 Y
+responses and all 50 non-ground-helper responses, and all nine C-slot impulse
+responses vanish, including the eight owned slots. It does not restore the
+113 exempt residuals physically. Delivery prices alone exclude payload
+preparation, scatter arithmetic, channel ownership/return, precision, stock,
+tape movement and the complete supplier/caller invoice. No selected frontier
+or inherited framework assumption changes.
+
+The verifier uses direct forward evaluation of the word and has no recursive
+or unbounded search. Its memory policy has a managed-runtime deviation:
+Python/Fraction allocates during replay. Live scalar state is bounded by
+641 rows times 22 channel coefficients, with eight payload coordinates and
+69 contacts; controls run sequentially. Source bytes are hash-fixed. The
+compensating check is the complete normal/optimized replay. A future change
+to dimensions or streaming behavior must revisit these bounds. Warning-strict
+Python and the repository metadata/source checks provide the available
+diagnostics; no repository-wide configuration changes are made.
+
+Independent human and formal proof-kernel review are unclaimed.
+
+## Open next questions
+
+1. Do three, or any fewer than fifteen, low-grade frames have a nonzero clean
+   response in this exact quotient? Start with a triple-span search that retains
+   literal contacts. No exhaustive triple computation or minimum-channel proof
+   is supplied here.
+2. Can this joint response lift to exact native operators and physically
+   compensate the 113 exempt helper residuals? Start with the full native action
+   for the fourteen named grade-seven charts and price the resulting cleanup.
+   Scalar fixed-vector responses do not imply operator equality or a favorable
+   complete bill.
+
+The relevance, first steps and blockers are also in [TASK.json](TASK.json).

research/monotone-noisy-delivery-counterexample/RESULT.json

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+{
+  "schema": "immutable-channel-evidence/v1",
+  "hashes": {
+    "verify.py": "a9faa4d82aff0976690111682e015277017f0e47b4fcf4883044f051ed9257eb",
+    "WITNESS.json": "43d4e3ceb7354b10138b854ac95abf505155d08ea6c10dfedca95445a42789c8",
+    "source_archive": "595f988e748b1f3b25e7525a226da173966e4278a70f511b2919245cec1b45f7"
+  },
+  "result": {
+    "events": 22585,
+    "channels": 22,
+    "contacts": 69,
+    "profile": {
+      "0": 1,
+      "7": 14,
+      "8": 7
+    },
+    "retained_fields": 274,
+    "retained_mode_equalities": 2192,
+    "owned_fields_zero": 8,
+    "c_slots_zero": 9,
+    "exempt_ground_helpers": 358,
+    "exempt_helpers_with_channel_response": 189,
+    "exempt_helpers_with_payload_residual": 113,
+    "charge": "119",
+    "baseline": "120",
+    "gain": "1",
+    "rejection_controls": 8
+  },
+  "environment": {
+    "python": "3.12.15",
+    "platform": "macOS-27.0.1-arm64-arm-64bit"
+  }
+}

research/monotone-noisy-delivery-counterexample/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "de5ec2e7a6074068452f45b1a191b5c72832df4a",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "581f8d87008354eb5b869f7ed9de1317ad837ce8",
+      "sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "path": "research/filtered-native-kappa004/inputs/original-packet.tar.gz",
+      "usage": "input",
+      "git_blob": "7f904307bd99f2d778bc7b64e98e353715cbc041",
+      "sha256": "595f988e748b1f3b25e7525a226da173966e4278a70f511b2919245cec1b45f7"
+    }
+  ]
+}