Record that post-call B/C helper feedback stays inside the capacity bill

research/shared-helper-feedback-bound/PROOF.md

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+# Post-call helper feedback stays inside the separated B/C bill
+
+Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.
+
+This is a scoped negative screen. It does not change κ, admit a caller, or
+replay a native word. The selected record stays
+`781451/250000000`.
+
+## What is already proved
+
+Ali Romman's capacity gate prices every whole B or C call whose helpers stay
+out of the main computation. In that separated model the complete
+target-minus-paid moment is strictly negative for every `0<t<1`. The proof
+states the boundary it does not cross: shared or productive helper feedback
+into the main computation needs a new bill. This note supplies that bill for
+ordinary post-call feedback, and the bill does not improve.
+
+The inherited data, copied here only as the objects being rechecked, are the
+audited profiles and the capacity polynomial
+
+```text
+K(t) = 28 + 140*7^t + 84*8^t.
+```
+
+| Call | Paid rank counts | Independent helper rank counts |
+| --- | --- | --- |
+| B | 1:1956, 2:450, 3:403, 4:127, 5:28, 8:8 | 3:2, 4:26, 6:7, 7:47, 8:326 |
+| C | 1:2011, 2:445, 3:427, 4:112, 5:12, 7:8 | 3:26, 4:2, 6:6, 7:48, 8:326 |
+
+The four sign-cut charges of one call's main block are `0, 112, 224, 168`.
+
+## Post-call model
+
+After a call returns, its main rows and helper rows may be mixed by any
+ordinary complex-linear operation. Those operations are invisible to the sign
+characters, as in the capacity gate. A helper row is a coordinate Gaussian of
+its audited rank, on its own field. Under a sign set `S` that row differs by
+rank exactly 1 when `|S ∩ support|` is odd, and by rank 0 otherwise.
+
+No helper is given a second life as an uncharged address permutation, a
+conjugation, or an input that rewrites the native paid profile. Those
+remain outside this screen. The paid profile stays the audited one.
+
+## Rank budget
+
+Write `D_main(S)` for the main-block rank distance and `D_H(S)` for the
+helper-block rank distance. Any linear image of the combined output, including
+an image that mixes main and helper rows, has rank distance at most
+`D_main(S) + D_H(S)`.
+
+On distinct fields, `D_H(S)` equals the number of helpers whose support meets
+`S` oddly. Averaging independent axis flips of probability `p` therefore gives
+
+```text
+E[D_H] = sum_j h_p(r_j),
+h_p(r) = (1 - (1-2p)^r) / 2.
+```
+
+The same average of `D_main` is `K`'s character form
+`140 h_p(7) + 28 h_p(1) + 84 h_p(8)` per B or C call. A coordinate-Gaussian
+target of rank `s` flips with probability `h_p(s)`, and the rank of the
+output difference is at least the number of such targets that flip. Therefore
+
+```text
+sum_i h_p(s_i) <= E[D_main] + E[D_H].
+```
+
+The integral transform in the capacity gate is linear and sends `h_p(s)` to
+`s^t`. Every family of coordinate-Gaussian jobs extracted from the combined
+output satisfies
+
+```text
+sum_i s_i^t <= (L_B + L_C) K(t) + H(t) + sum_queries r^t
+```
+
+for `0<t<1`. Here `H(t)` is the moment of the audited helper ranks, whether or
+not those rows are still labelled helpers. Consuming a helper into another
+job spends its character budget; it does not create a second copy. Queries
+cancel against their own charge exactly as before.
+
+Subtracting the audited paid profiles leaves the same ceiling as the separated
+theorem:
+
+```text
+F(t) <= -L_B Delta_B(t) - L_C Delta_C(t).
+```
+
+Both deltas are strictly positive for `0<t<1`, by the published mass split.
+For B, paid minus helpers minus `K` has positive rank mass `4575` and negative
+rank mass `4567`. Removing eight rank-1 units leaves equal masses `4567`, with
+every positive rank at most 5 and every negative rank at least 6. For C the
+two masses are already `4576`. For `g(r)=r^t`,
+
+```text
+g(5)/5 - g(6)/6 > 0
+```
+
+because `6 * 5^t > 5 * 6^t` rearranges to `(6/5)^{1-t} > 1`. Thus one or more
+B or C calls still force `F(t) < 0`.
+
+Keeping the helpers as independent jobs and also asking for one more flipping
+target exceeds the joint row budget. On the central-axis character the main
+cut is 112 and all 408 B helper rows flip, so the budget is 520. Retaining
+those 408 jobs and filling the 112 main-cut rows already uses 520.
+
+## What remains open
+
+This does not price a native rewrite that deletes a paid Gaussian by reading
+a helper inside the call. It does not price odd input gauges or address maps
+outside the coordinate algebra. Unknown compiler, precision and outer costs
+are not set to zero in order to continue; the post-call moment is already
+negative with those costs omitted.

research/shared-helper-feedback-bound/README.md

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+# Post-call B/C helper feedback does not raise the moment
+
+**Scoped negative screen. No new κ.** The selected conditional exponent stays
+`781451/250000000`.
+
+Ali Romman's whole-caller capacity gate shows that separated B and C calls
+have a strictly negative complete moment for every saving exponent in `(0,1)`.
+That argument leaves shared helper feedback unpriced. This package prices one
+such feedback: after the call returns, ordinary complex-linear use of its
+helper rows, including use of those rows as ingredients of later coordinate
+Gaussian targets.
+
+The character budget of those rows is exactly the separated helper moment
+`H(t)`. Mixing them into the main output can move that budget. It cannot
+duplicate it. With the audited paid profiles held fixed, the complete moment
+remains at most
+
+```text
+F(t) <= -L_B Delta_B(t) - L_C Delta_C(t) < 0
+```
+
+for every `0<t<1` as soon as one B or C call is present. The same mass split
+as the capacity gate gives the sign: B's residual rank masses are `4567`, and
+C's are `4576`, with positive ranks at most 5 and negative ranks at least 6.
+On the central-axis character the B helper block is already fully busy, 408
+flipping rows plus the main cut of 112, so one extra flipping target does not
+fit.
+
+The argument is written in [PROOF.md](PROOF.md). The machine receipt is the
+verifier below. It recomputes the 256-mask cut identity, both helper-block
+distance polynomials, and the residual masses from the profile rows pinned in
+the predecessor proof.
+
+## Scope and evidence
+
+Task card: `complex-new-helper-inventory`, scope **changed**. The card asks
+whether a changed helper inventory can earn productive credit that survives a
+complete bill. The concrete question is the open shared-producer gate in
+[the capacity-gate README](../complementary-caller-capacity-gate/README.md),
+which is the successor of handoff
+[#1166 comment 19133514](https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19133514)
+after the fixed-224 serial loss. Discussion
+[#5628](https://github.com/CrocSwap/integer-mult-bounds/discussions/5628)
+is that gate's canonical thread.
+
+This is not a decoder for the selected PR6030 word, and it does not spend
+the #5599 helper-subspace credit. The connection to the bottleneck is the
+revision-8 decoder/interface row: one changed producer contract, priced with
+its optimistic full signed bill, stops when that bill is negative.
+
+Base commit `7c49fe32c55de02c2c873655c804b16e4c3fef0a`. The live input is
+`research/complementary-caller-capacity-gate/PROOF.md` at that commit.
+
+Not run: a native word replay, a Clifford phase suite, Lean, compiler or
+outer admission, and the selected `make frontier6030-verify` cold replay.
+Internal native rewrites that delete a paid factor, and maps outside the
+coordinate algebra, are not claimed impossible.
+
+## Reproduce
+
+Python 3.12, standard library, from the repository root:
+
+```sh
+python3 -B research/shared-helper-feedback-bound/verify.py
+python3 -B scripts/research_package.py --package research/shared-helper-feedback-bound --check-only
+```
+
+The direct command prints `PASS_SHARED_HELPER_FEEDBACK_BOUND` and
+`no_new_kappa`.
+
+## Open next questions
+
+1. Can a helper read inside the native word delete a paid Gaussian and still
+   leave a positive complete moment? First step: exhibit one internal use,
+   the deleted paid factor, and the replacement endpoint, then rerun the
+   paid-minus-helper mass split. Blocked until that word exists; this screen
+   keeps the audited paid profiles fixed.
+2. Can an odd input gauge or a noncommuting address map between calls evade
+   the sign-cut charge? First step: name the gauge, its preparation cost, and
+   one character on which the rank distance falls. Blocked because those maps
+   are outside the coordinate algebra used here.
+
+## Credit
+
+The separated capacity theorem, the audited B/C profiles, and the statement
+that helper feedback needed its own bill are Ali Romman's. The fixed-224
+serial loss that this handoff continues is Chris R / Clear1nsight's, from
+Dugongue's B/C source and Cao Yifan / intqwq's audit. This package does not
+modify those sources.
+
+Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

research/shared-helper-feedback-bound/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "d59b5035e7b4cbaa9d688a1d32dd0d6eb79c466a",
+      "sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "path": "research/complementary-caller-capacity-gate/PROOF.md",
+      "usage": "input",
+      "git_blob": "41ef324bd6a15783e7a9b329514ad3c31d95feb7",
+      "sha256": "34613c90ed02402040280aeed16963a0e0787b47df704238867603a769463cb0"
+    }
+  ]
+}

research/shared-helper-feedback-bound/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "complex-new-helper-inventory",
+  "priority_revision": 8,
+  "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+  "original_card": {
+    "id": "complex-new-helper-inventory",
+    "question": "Can a changed helper inventory earn productive credit that survives completion, compilation and the actual caller?",
+    "starting_points": [
+      "docs/research/current-priorities.md",
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md"
+    ],
+    "methods": [
+      "Construct and independently check one source-bound instance of the stated first experiment.",
+      "Seek an exact counterexample or target-specific complete-budget obstruction to that same proposed change."
+    ],
+    "acceptance_gate": "A source-bound arbitrary-input identity and legal producer/consumer boundary with the complete paid and productive target invoice, including restoration, ownership, native alignment and extra charges. Price under the actual current model; historical ceiling.py assumptions must be established before use. A favorable optimistic relaxation keeps a route open but is not a completed supplier. A scoped local result remains useful; expansion requires the family-level budget gate.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_selected_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read revision 8, the selected review and live exact-source handoffs. Then follow this card's bounded first_experiment.",
+      "scope": "Source-bound orientation only, not a supplier replay or theorem proof."
+    },
+    "first_experiment": "Specify all arbitrary inputs, owned fields, restored outputs and a physically legal consumer. Compare complete P and T, including target debits and precision/stock costs. #5599 is a positive control for target refinement, not permission to split every helper for free. #5614 is a hypothetical compiler budget, not an emitted 145-split word. Stop at the first unpaid boundary or a failing optimistic full bill.",
+    "bottleneck_link": "Judge the actual current supplier contract and complete signed target/paid bill; frozen-profile ceilings require an applicability proof.",
+    "allowed_variations": [
+      "One exact source head, helper/cut region or quantified proposition and changed hypothesis; check live handoffs first. Different seeds are not different scopes."
+    ],
+    "scope_change_triggers": [
+      "Changing source head, helper/cut domain, operation alphabet or caller boundary without recording the new scope.",
+      "Treating an archived proposal or local numerical pass as complete supplier admission."
+    ],
+    "overlap_terms": [
+      "productive helper",
+      "target refinement",
+      "5599",
+      "5614"
+    ],
+    "existing_checks": [
+      {
+        "scope": "Selected PR5599: complete helper-subspace construction and independent caller replay; five framework assumptions remain.",
+        "evidence": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+        "remaining": "The next experiment must identify uncovered source-specific work. Successor archive intake does not transfer this admission.",
+        "status": "reviewed",
+        "source_commit": "09dd0965711ca7eeb169057a420abab1c9052137",
+        "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Can ordinary post-call linear feedback of the audited B/C helper rows raise the complete moment above the separated capacity-gate bill?",
+  "scope_change_reason": "The catalogue card asks for a helper-inventory credit that survives a complete bill. Handoff 19133514, after the fixed-224 serial loss, asks for one changed shared-producer premise. This prices post-call helper feedback, which Ali Romman left explicitly open, and stops when that optimistic bill stays negative. It is not a #5599 target split.",
+  "bottleneck_link": "Revision 8 asks for one changed producer contract with its optimistic full signed bill. This bill is strictly negative for every 0<t<1, so the separated B/C route does not gain from post-call helper mixing.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop this feedback model if the joint character budget cannot exceed the separated helper moment plus K(t). Do not continue into a native rewrite unless a deleted paid factor is exhibited.",
+  "outcome": "negative-result",
+  "evidence": "PROOF.md; verify.py recomputes the 256-mask cut identity, both helper distance polynomials, residual masses 4567 and 4576, and the central-axis 520-versus-521 row check.",
+  "author": "unpaidmoon, with substantial Cursor Grok assistance",
+  "ai_assistance": "Substantial Cursor Grok assistance for the sharing argument, the verifier and the package text. No independent human review.",
+  "open_questions": [
+    {
+      "question": "Can a helper read inside the native B or C word delete a paid Gaussian and leave a positive complete moment?",
+      "why_it_matters": "Post-call mixing cannot beat the separated bill while the audited paid profile stays fixed. An internal rewrite is the remaining shared-producer opening.",
+      "first_step": "Exhibit one internal helper use, the deleted paid factor and the replacement endpoint, then rerun the paid-minus-helper mass split.",
+      "blocked_by": "No such word is supplied here. The screen keeps the audited paid profiles fixed."
+    },
+    {
+      "question": "Can an odd input gauge or a noncommuting address map between calls evade the sign-cut charge?",
+      "why_it_matters": "The character argument uses the coordinate algebra. A map outside it can change the rank distance.",
+      "first_step": "Name the gauge, its preparation cost, and one character on which the rank distance falls below the tabulated cut.",
+      "blocked_by": "Those maps are outside the algebra used by this screen and by the capacity gate."
+    }
+  ]
+}

research/shared-helper-feedback-bound/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/complementary-caller-capacity-gate/PROOF.md"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "256-mask cut identity, B and C helper-distance polynomials, residual rank masses 4567 and 4576, ratio gap, and the central-axis extra-target rejection"
+    }
+  ]
+}