Record that the literal P104 slot exchange misses the helper gate.

research/p104-literal-helper-gate/PROOF.md

@@ -0,0 +1,112 @@
+# Literal correction, helper entrance, and the #3954 gate
+
+This is a scoped negative certificate. It does not change κ. The selected
+value stays `0.001672660197076918`.
+
+The source contract is the two-axis scaffold in
+`research/p104-two-axis-compatibility/PROOF.md` at base
+`12fa8c5c844be21bfefb38a19503fbd867a11244` (PR #3954). The literal-slot
+identity is the one separated in `research/p104-literal-slot-defect`. This
+note does not rerun either package.
+
+## 1. Charts of one scaffold call
+
+Order the two data banks as a pair. The retained caller uses source charts
+`(T, I)` and terminal charts `(F, E)`, with `T = C_U`, `F = C_I` and
+`E = F T^{-1}`. Here `I` is the identity address, so `C_I = C_0`, and
+
+```text
+C_u = ((1+i) I + (1-i) X_u) / 2.
+```
+
+On a Fourier character `z`, `C_u` acts by `1` when `u·z` is even and by `i`
+when `u·z` is odd. Thus `C_0` is the identity operator, and `C_u = C_0` if
+and only if `u = 0`.
+
+For a call on address `u` and a next call on address `v`, the exit chart pair
+and the next entrance chart pair are
+
+```text
+exit      = (C_0, C_u^{-1}),
+entrance  = (C_v, C_0).
+```
+
+These pairs agree on both banks if and only if `C_v = C_0` and `C_u^{-1} = C_0`,
+hence if and only if `u = v = 0`. Every nondegenerate line in the scaffold is
+a nonzero address, so no adjacent pair of scaffold calls matches both data
+banks.
+
+`Π P_enc Π` is the operator which intertwines one call's literal encoding
+`A_T = Π L_T`. It is not a rewrite of the chart pair `(C_0, C_u^{-1})` into
+`(C_v, C_0)`.
+
+## 2. Helper entrance and exit
+
+Section 1 of the #3954 proof extends every compatibility map by identity on
+the 72 arbitrary dirty helper roles of a fiber. Section 5 places two disjoint
+banks of those roles in the inventory
+
+```text
+n = 104,
+helper roles = 2 n · 72 = 14976,
+W = 2 n^2 + 2 n · 72 = 36608.
+```
+
+The same section says that every persistent entrance other than the rank-one
+data source chart is the identity, and that the completed contract is full
+`F` on all `W` roles. The progress identity therefore charges `81` dimensions
+on each helper role and subtracts only the `N = n^2` data-line sources:
+
+```text
+helper progress demanded = 81 · 14976 = 1213056.
+```
+
+The identity extension acts by `1` on each helper coordinate. It sends a
+helper basis vector to itself and adds no helper component to a data vector.
+Its contribution to helper progress is `0`. It preserves a dirty helper
+state, and it does not implement the transition from the identity entrance
+to the full-`F` exit.
+
+So `Π P_enc Π`, extended in the way #3954 states, does not match a completed
+helper exit to the next call's identity helper entrance.
+
+## 3. The slot exchange is not free, and zero is already too expensive
+
+`Π` swaps the two data slots on the `96` N-ports and fixes the `8` J-ports.
+That support size is a count of coordinates, not a native tariff. The #3954
+gate records the realization of `Π` inside an unknown nonnegative physical
+charge. Under hypotheses (a)–(e) of that proof, with this charge set to an
+integer `s ≥ 0`,
+
+```text
+P_total     = 81 W − N           = 2954432,
+remaining   = P_total − 208·2312 = 2473536,
+lower mass  = 208·2576 + remaining + s = 3009344 + s,
+capacity    = 81 W               = 2965248,
+excess      = 44096 + s > 0.
+```
+
+The moment ratio at `a = 0` is already greater than `1` when `s = 0`. Every
+larger price is worse. A negative price would be a different gate; unknown
+costs in that proof are not negative.
+
+The local progress `2312` per call is the figure hypothesis (b) grants. This
+note does not move any of that grant onto the boundary map. The boundary map
+adds none.
+
+## 4. What a matching pair would have to change
+
+The conjugation identity on the two data banks remains true: on a witness
+with one J port, one N port and one helper coordinate,
+
+```text
+(Π P_enc Π ⊕ I) (A_T ⊕ I) = (A_T ⊕ I) (P ⊕ I).
+```
+
+The old correction, extended in the same way, fails that identity. This is
+the data-slot statement already separated from the census. It does not make
+the chart pairs of section 1 equal, and it does not pay section 2.
+
+A favorable adjacent pair needs a different chart contract or a helper word
+whose proved progress can be subtracted from `1213056` without a new
+nonnegative charge that puts the excess back. Neither is supplied here.

research/p104-literal-helper-gate/README.md

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+# Literal slot exchange against the #3954 helper gate
+
+**Negative interface certificate. No new κ.** Selected κ stays
+`0.001672660197076918`.
+
+Open question 1 of `research/p104-literal-slot-defect` asks whether one
+adjacent pair that uses `Π P_enc Π`, rather than `P`, matches every helper
+role as well as both data banks, and whether the bill is favorable. On the
+PR #3954 scaffold the answer is no.
+
+## Result
+
+The retained caller has source charts `(T, I)` and terminal charts `(F, E)`,
+with `T = C_U`, `F = C_0` and `E = F T^{-1}`. For every nonzero address `u`
+and every address `v`, the exit pair `(C_0, C_u^{-1})` is not the next
+entrance pair `(C_v, C_0)`. Both pairs agree only for the degenerate address
+`u = v = 0`, which the scaffold excludes. `Π P_enc Π` intertwines one call's
+literal encoding. It does not rewrite those chart pairs.
+
+#3954 extends every compatibility map by identity on the 72 dirty helper
+roles of a fiber. The inventory has `14976` such roles. The completed
+contract charges `81` dimensions on each of them, `1213056` in total, while
+the identity extension contributes `0`. A helper basis vector is fixed, so
+the boundary map does not turn the identity helper entrance into the full-`F`
+exit.
+
+`Π` itself swaps the `96` N-ports. That count is the support of the
+permutation, not a native tariff. Under the published hypotheses (a)–(e),
+every nonnegative added rank-mass `s` leaves the excess equal to
+`44096 + s`. The zero-price gate is already over capacity `2965248` by
+`44096`. The bill is not favorable.
+
+The block arithmetic and the integer identities are in [PROOF.md](PROOF.md).
+`verify.py` recomputes the charts on `(F_2)^4`, the helper direct sum on a
+two-port witness, and the gate.
+
+## Scope
+
+Task `p104-side-correction`, scope **refined**. The card asks for a changed
+producer/consumer with a complete bill. This package answers the first open
+question of the literal-slot certificate instead of building that producer.
+The scalar obstruction in that certificate does not mention helpers or the
+gate; the commutation-gap certificate has no helper roles. Those two results
+stay as they are.
+
+Base `12fa8c5c844be21bfefb38a19503fbd867a11244`, priority revision 7.
+The #3954 head retained by the family record is the source of the entrance,
+exit and gate integers. This package does not import that code.
+
+## Reproduce
+
+Python 3.12, standard library, from the repository root:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/p104-literal-helper-gate
+```
+
+Direct check:
+
+```sh
+python3 -B research/p104-literal-helper-gate/verify.py
+```
+
+Expected marker:
+
+```text
+PASS_P104_LITERAL_HELPER_GATE charts=F2^4 nonzero_mismatch=true helper_extension=identity helper_progress=1213056 excess0=44096 ports_n=96 nonnegative_price=unfavorable no_new_kappa
+```
+
+The same command passes under `-O`. Not run: Lean, Docker, a selected cold
+replay, the #3954 native census, and hosted CI.
+
+## Open next questions
+
+1. Can one named helper word advance one of the 72 roles per fiber and be
+   subtracted from the `1213056` helper-progress charge without a new
+   nonnegative cost that restores the excess? The identity extension
+   contributes none of that progress. First step: freeze one fiber, write
+   the identity entrance and the full-`F` exit for one role, and give the
+   rank-mass of the word that connects them. Blocker: no such word is in
+   the #3954 extension, and unknown physical costs are not negative
+   rank-mass.
+2. Is there an adjacent pair whose charts are not `(T, I)` and `(F, E)`,
+   with an operator other than `Π P_enc Π`, that matches every data and
+   helper role under a freshly derived gate? First step: name the two chart
+   pairs and the helper schedule, then recompute capacity. Blocker: a
+   different inventory or coverage law needs its own gate. The excess
+   `44096` is not a theorem about that law.
+
+## Credit
+
+The entrance, exit, identity helper extension, N-port slot swap and the
+`44096` gate are Chris R / Clear1nsight's PR #3954. The request for an
+adjacent pair on every data and helper role is discussion #1166 comment
+19034066. The question of whether `Π P_enc Π` answers that request is open
+question 1 of `research/p104-literal-slot-defect`. The chart formula is the
+one used in `research/p104-uniform-commutation-gap`. Predecessor lineage
+named in those packages remains theirs, including Rohan Arun, Chafik
+Boukhalfa and Adam Gall.
+
+Prepared by unpaidmoon with substantial Cursor Grok assistance. No
+independent human or formal-kernel review is claimed. Apache-2.0.

research/p104-literal-helper-gate/SOURCES.json

@@ -0,0 +1,18 @@
+{
+  "schema_version": 1,
+  "base_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "64bfc5af6709ed9811171e0a2108f05e1219cb65",
+      "sha256": "352a6184713b8843d96fd2c2071eb7aeaa655906b7c51d45ad444912666c8a96"
+    }
+  ]
+}

research/p104-literal-helper-gate/TASK.json

@@ -0,0 +1,139 @@
+{
+  "schema_version": 1,
+  "task_id": "p104-side-correction",
+  "priority_revision": 7,
+  "base_commit": "12fa8c5c844be21bfefb38a19503fbd867a11244",
+  "original_card": {
+    "id": "p104-side-correction",
+    "question": "Can a changed P104 producer/consumer contract escape the scoped slot, chart and completion obstructions with a favorable full bill?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "research/p104-literal-slot-defect/README.md",
+      "research/p104-uniform-commutation-gap/README.md",
+      "research/p104-uniform-encoding-gap/README.md",
+      "research/p104-source-chart-shortcut/README.md",
+      "research/p104-fiber-tag-seam/README.md"
+    ],
+    "methods": [
+      "Construct one side-correction word at an explicit chronological cut with the needed frames still available.",
+      "Derive a lower bound on the side correction and hosting costs that decides whether this channel can beat the selected budget."
+    ],
+    "acceptance_gate": "An exact arbitrary-input identity, source/target frame availability and restoration at the chosen cut, plus the complete channel+correction+idle/hosting bill or an explicit favorable bound. The native channel alone is not a supplier. A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read the named successor obstructions before extending #2981 static flags. Distinguish literal-slot action, commutation, uniform encoding, source-chart shortcut and fiber-tag seam scopes; none excludes all P104 mechanisms.",
+      "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately."
+    },
+    "first_experiment": "Name one hypothesis changed from these exact interfaces. Write a literal producer-to-consumer timeline and an arbitrary-dirty-input positive/control pair, then charge acquisition, coefficients, completion, hosting and restoration. Stop if the same slot/fiber mismatch persists or the favorable full budget already fails. A scoped independent audit remains useful; another static count is not a new completion.",
+    "bottleneck_link": "Completing an already realized changed channel tests a concrete construction opportunity, instead of refining the incumbent packing. A positive local result still needs global and all-size admission.",
+    "allowed_variations": [
+      "One correction factorization, one chronological cut, or one explicit hosting obstruction.",
+      "A changed correction schedule with an updated full bill and native frame contract."
+    ],
+    "scope_change_triggers": [
+      "Recompiling the already certified low-rank channel without closing a new obligation.",
+      "Treating a pending ancestor, local histogram or unpriced side correction as an admitted global supplier.",
+      "Repeating a correction or physical conversion already supplied by a pending follow-up without adding independent coverage or a different unresolved obligation.",
+      "Repeating the 208-flag static matching or calling a static nested cover a chronological correction program. Independent audit of that certificate is distinct and remains useful.",
+      "Repeating a static cover, source-chart shortcut or fiber-tag seam without changing the hypothesis responsible for its obstruction."
+    ],
+    "overlap_terms": [
+      "P104",
+      "I-G/2",
+      "side correction",
+      "native channel",
+      "1972",
+      "1991",
+      "2586",
+      "endpoint transport",
+      "physical dual",
+      "relay composition",
+      "2981",
+      "joint unit flags",
+      "acquisition delivery",
+      "static nesting",
+      "4133",
+      "4136",
+      "4140",
+      "4153",
+      "4154",
+      "p104-literal-slot-defect",
+      "p104-uniform-commutation-gap",
+      "p104-uniform-encoding-gap",
+      "p104-source-chart-shortcut",
+      "p104-fiber-tag-seam"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4133: literal-slot defect",
+        "evidence": "research/p104-literal-slot-defect/README.md",
+        "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.",
+        "status": "author-checked",
+        "source_commit": "88666f1dee93fbd85b31fd5d6f91a219e111b5f4",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4136: uniform commutation gap",
+        "evidence": "research/p104-uniform-commutation-gap/README.md",
+        "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.",
+        "status": "author-checked",
+        "source_commit": "281ce601a8be9db470de630759179c23e1dfaa88",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4140: uniform encoding gap",
+        "evidence": "research/p104-uniform-encoding-gap/README.md",
+        "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.",
+        "status": "author-checked",
+        "source_commit": "de25c6525aa866962309df5d107b32d55e6e5628",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4153: source-chart shortcut exhausts its optimistic allowance",
+        "evidence": "research/p104-source-chart-shortcut/README.md",
+        "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.",
+        "status": "author-checked",
+        "source_commit": "9123241fea53df77e51ffe30d9aca1625d4bf3a8",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4154: fiber-tag seam mismatch",
+        "evidence": "research/p104-fiber-tag-seam/README.md",
+        "remaining": "Independently validate the stated scope or change its producer/consumer premise and pay the complete boundary.",
+        "status": "author-checked",
+        "source_commit": "4ffcdaee5d6255a9cea1fe53a5424435068089d0",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "On the PR #3954 two-axis scaffold, the literal correction Pi P_enc Pi extended by identity on the 72 helper roles does not match an adjacent call's data charts (T, I) to (F, E), does not advance a helper from the identity entrance to the full-F exit, and leaves the published gate over capacity for every nonnegative slot-exchange price.",
+  "scope_change_reason": "Open question 1 of research/p104-literal-slot-defect asks whether Pi P_enc Pi matches helper roles as well as both data banks and whether the bill is favorable. This package answers that question on the pinned #3954 entrance, exit and gate. It does not construct a new chronological correction word.",
+  "bottleneck_link": "The literal-slot identity already supplies an intertwiner on the two data banks. A favorable adjacent pair would still have to match helpers and beat the scaffold capacity. This certificate shows that pair is not the #3954 contract, so the obstruction stays in force and kappa does not move.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop after the chart mismatch on every nonzero address of (F_2)^4, the identity helper extension on one witness, and the nonnegative gate. Do not search another gauge, price a native word, or treat the 96-port support of Pi as a tariff.",
+  "outcome": "negative-result",
+  "evidence": "PROOF.md derives the chart inequality, the zero helper-boundary progress, and the excess 44096+s. verify.py recomputes those integers and the Gaussian-integer charts, and checks that Pi P_enc Pi plus an identity helper coordinate intertwines the literal encoding while fixing the helper.",
+  "author": "unpaidmoon",
+  "ai_assistance": "Substantial Cursor Grok assistance in the proof, checker, and write-up. Not independent human or formal-kernel review.",
+  "open_questions": [
+    {
+      "question": "Can one named helper word advance one of the 72 roles per fiber and be subtracted from the 1213056 helper-progress charge without a new nonnegative cost that restores the excess?",
+      "why_it_matters": "The identity extension contributes none of that progress. A real helper schedule is the only way this boundary could move the gate.",
+      "first_step": "Freeze one fiber, write the helper entrance and the full-F exit for one of its 72 roles, and give the rank-mass of the word that connects them.",
+      "blocked_by": "No such word is in the #3954 identity extension. Unknown physical costs cannot be entered as negative rank-mass."
+    },
+    {
+      "question": "Is there an adjacent pair whose entrance and exit charts are not (T, I) and (F, E), for which some operator other than Pi P_enc Pi matches every data and helper role under a freshly derived gate?",
+      "why_it_matters": "This certificate rejects only the retained #3954 charts, the identity helper extension, and hypotheses (a)-(e).",
+      "first_step": "Name the two chart pairs and the helper schedule, then recompute capacity before any search.",
+      "blocked_by": "A different inventory or coverage law needs its own gate. The old excess is not a theorem about that law."
+    }
+  ]
+}

research/p104-literal-helper-gate/VERIFY.json

@@ -0,0 +1,19 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": ["{python}", "-B", "{package}/verify.py"],
+      "timeout_seconds": 60,
+      "checks": "Chart mismatch on every nonzero address of (F_2)^4, identity helper extension, and nonnegative gate excess 44096+s"
+    },
+    {
+      "argv": ["{python}", "-O", "-B", "{package}/verify.py"],
+      "timeout_seconds": 60,
+      "checks": "Same certificate with assertions removed, so the explicit require() failures still fire"
+    }
+  ]
+}