Certify the literal-slot defect that blocks a uniform P104 adjacent c…

research/p104-literal-slot-defect/PROOF.md

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+# Literal-slot defect of the old correction
+
+Let the two banks have the same finite dimension. Write \(I\) for the identity,
+\(H\) for a linear projection (\(H^2=H\)), and \(J_0=I-H\). Then
+\(HJ_0=J_0H=0\) and \(J_0+H=I\). Let \(T\) be invertible, \(K=T^{-1}\), and let
+\(D\) be any matrix of the same size. No hypothesis makes \(D\) commute with
+\(T\).
+
+The old correction and the ordered terminal encoding are
+\[
+P=\begin{pmatrix}I&0\\K&I\end{pmatrix},
+\qquad
+L_T=\begin{pmatrix}D&HT\\0&I\end{pmatrix}.
+\]
+The literal slot swap on the image of \(H\), fixing the complementary ports, is
+\[
+\Pi=\begin{pmatrix}J_0&H\\H&J_0\end{pmatrix}.
+\]
+The literal encoding is \(A_T=\Pi L_T\).
+
+## Ordered commutator, without \(HT=TH\)
+
+Block multiplication gives
+\[
+PL_T=\begin{pmatrix}D&HT\\KD&KHT+I\end{pmatrix},
+\qquad
+L_TP=\begin{pmatrix}D+HTK&HT\\K&I\end{pmatrix}.
+\]
+Subtracting and using \(KD-K=K(D-I)\) yields the identity
+\[
+PL_T-L_TP
+=
+\begin{pmatrix}-HTK&0\\K(D-I)&KHT\end{pmatrix}.
+\]
+If in addition \(HT=TH\), then \(KT=TK=I\) implies \(HTK=H\) and \(KHT=H\), so
+\[
+PL_T-L_TP
+=
+\begin{pmatrix}-H&0\\K(D-I)&H\end{pmatrix}.
+\]
+The bottom-right block is \(H\) only under that commutation. A triangular \(T\)
+that moves mass between \(J_0\) and \(H\) keeps the unsimplified identity and
+rejects the simplified one.
+
+## Literal slots
+
+Assume now that \(H\) is a projection and \(HT=TH\), hence \(HK=KH\) and
+\(J_0K=KJ_0\). Then \(\Pi^2=I\), and
+\[
+A_T=\Pi L_T
+=
+\begin{pmatrix}J_0D&H\\HD&HT+J_0\end{pmatrix}.
+\]
+The same expansion, reduced by \(J_0H=0\), \(H^2=H\) and \(HTK=H\), gives
+\[
+PA_T-A_TP
+=
+\begin{pmatrix}-HK&0\\KJ_0(D-I)-H&KH\end{pmatrix}.
+\]
+On an input \((a,0)\) the first-bank image of this commutator is \(-HKa\).
+If \(a\) is supported on the image of \(H\), that image is \(-Ka\). If \(a\) is
+supported on the image of \(J_0\), the first bank is \(0\).
+
+Thus the ordered defect \(-Ha\) and the literal defect \(-HKa\) are different
+whenever some value of \(T\) on the image of \(H\) is not \(1\).
+
+## What a uniform scalar cannot cancel
+
+Suppose the diagonal entries of \(T\) on the image of \(H\) are not all equal.
+Then \(-Ka=\lambda a\) cannot hold for every \(H\)-supported \(a\), for any
+single scalar \(\lambda\). A uniform bank multiplier is not an intertwiner of
+\(P\) through \(A_T\).
+
+If those diagonal entries are equal to one value \(t\), then \(\lambda=-1/t\)
+does cancel the first bank for every \(H\)-supported amplitude. It still does
+not cancel a vector with a nonzero \(J_0\) component, because \(-HKa\) has no
+\(J_0\) component and \(\lambda a\) does.
+
+## Conjugation
+
+Whenever \(D\) is invertible, \(L_T\) is invertible and
+\(P_{\mathrm{enc}}=L_TPL_T^{-1}\) satisfies \(P_{\mathrm{enc}}L_T=L_TP\).
+The literal-slot operator that satisfies the same relation for \(A_T\) is
+\(\Pi P_{\mathrm{enc}}\Pi\), not \(P\) and not \(P_{\mathrm{enc}}\). This
+identity prices neither \(\Pi\) nor the rank of \(HD^{-1}\) as a native tariff.
+
+## What this does not prove
+
+It does not exhibit an adjacent completed-call pair, a helper-role match, or a
+complete bill. It does not rerun the 10,816-line census of PR #3954. Linear
+rank is not a native-call lower bound. No \(\kappa\) changes.

research/p104-literal-slot-defect/README.md

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+# Literal-slot defect of the old P104 correction
+
+**Negative interface certificate. No new κ.** Selected κ stays
+`0.001672660197076918`.
+
+Discussion #1166 comment 19034066 asks for one adjacent completed-call pair
+whose encoded exit matches the next entrance on every role. This package does
+not supply that pair. It certifies the obstruction that pair would have to
+cancel if it reused the old correction, and it shows a uniform scalar bank
+multiplier cannot cancel it when the N-port scales differ.
+
+## Result
+
+Let \(H\) be a projection, \(J_0=I-H\), \(T\) invertible, \(K=T^{-1}\), and
+\(D\) any matrix of the same size. With
+\[
+P=\begin{pmatrix}I&0\\K&I\end{pmatrix},
+\qquad
+L_T=\begin{pmatrix}D&HT\\0&I\end{pmatrix},
+\qquad
+\Pi=\begin{pmatrix}J_0&H\\H&J_0\end{pmatrix},
+\qquad
+A_T=\Pi L_T,
+\]
+the ordered commutator is
+\[
+PL_T-L_TP=\begin{pmatrix}-HTK&0\\K(D-I)&KHT\end{pmatrix}
+\]
+with no assumption that \(H\) commutes with \(T\). When \(HT=TH\) and \(H\) is a
+projection, \(\Pi^2=I\) and the literal-slot commutator is
+\[
+PA_T-A_TP=\begin{pmatrix}-HK&0\\KJ_0(D-I)-H&KH\end{pmatrix}.
+\]
+On an N-port amplitude the first bank of that defect is \(-Ka\). If two N-port
+diagonal entries of \(T\) differ, no scalar \(\lambda\) satisfies \(-Ka=\lambda a\)
+for every such amplitude. If they agree at \(t\), then \(\lambda=-1/t\) works
+on the image of \(H\) and still fails as soon as a complementary port is
+present. The ordered conjugated correction \(L_TPL_T^{-1}\) does intertwine
+\(L_T\); the literal slots require the extra conjugation by \(\Pi\).
+
+The block multiplication is in [PROOF.md](PROOF.md). `verify.py` recomputes it
+over the rationals on one \(4\times 4\) witness with unequal N-port scales and
+a noncommuting \(D\), on the constant-scale sharpness case, and on the two
+dropped hypotheses (commutation, idempotence).
+
+## Scope
+
+Task `p104-side-correction`, scope **changed**. The card asks for a
+chronological correction program with a complete bill. Comment 19034066 is
+item 4 of this session's eligible handoff list, and the list has fewer than
+21 entries, so the session refines that item instead of constructing the pair.
+The refinement is the commutator, not a caller.
+
+Base `bb4bccbf42844d2801487450727c2f63e10d1d0f`, priority revision 6.
+PR #3954 already records these block formulas for the P104 shapes
+\(H=\mathrm{diag}(0_8,I_{96})\) and checks them inside its full root census.
+This package separates the dimension-free identity and the scalar obstruction
+from that census. It does not import PR #3954.
+
+## Reproduce
+
+Python 3.12, standard library, from the repository root:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/p104-literal-slot-defect
+```
+
+Direct check:
+
+```sh
+python3 -B research/p104-literal-slot-defect/verify.py
+```
+
+Expected marker:
+
+```text
+PASS_P104_LITERAL_SLOT_DEFECT ordered=general literal=projection_and_commutation distinct_scales=no_scalar constant_scale=-1/2 pi_required=1 no_new_kappa
+```
+
+The same command passes under `-O`. Not run: Lean, Docker, a selected cold
+replay, the 10,816-line census, and hosted CI.
+
+## Open next questions
+
+1. Does one adjacent pair that uses \(\Pi P_{\mathrm{enc}}\Pi\), rather than
+   \(P\), match every helper role as well as both data banks, and is the
+   resulting bill favorable? The scalar obstruction does not answer that.
+   First step: pin one helper entrance and exit from the PR #3954 endpoint
+   and price \(\Pi\) as a real slot exchange. Blocker: this identity has no
+   helper schedule and does not convert linear rank into a native tariff.
+2. Which portwise diagonal corrections, other than \(K\) on the image of
+   \(H\), can cancel the first bank for every N-port amplitude? The identity
+   says the first bank is exactly \(-HKa\), so any such correction is fixed
+   on that image. First step: state the allowed diagonal alphabet and test
+   it on two distinct N-port scales. Blocker: a new alphabet is a different
+   contract from the old correction \(P\).
+
+## Credit
+
+The block shapes, the literal swap \(\Pi\), and the P104 reading of \(H\) and
+\(T\) are Chris R / Clear1nsight's PR #3954, from the handoff in discussion
+#1166 comment 19034066. That package retains the PR #3882 / #3940 residual
+boundary and the native P104 lineage (Rohan Arun, Chafik Boukhalfa, Adam Gall,
+and the contributors named there). The dimension-free expansion, the scalar
+obstruction, and these checks are new.
+
+Prepared by unpaidmoon with substantial AI assistance (Cursor Grok 4.7).
+Apache-2.0. No independent human or formal-kernel review is claimed.

research/p104-literal-slot-defect/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "bb4bccbf42844d2801487450727c2f63e10d1d0f",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "67f085acd86acd7d4e03922d9cf4b26bca66fed5",
+      "sha256": "47b42433f5a85809a3aadd29c2c339da2c6ed728e4f7762df6b88bccb7874cf0"
+    }
+  ]
+}

research/p104-literal-slot-defect/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "p104-side-correction",
+  "priority_revision": 6,
+  "base_commit": "bb4bccbf42844d2801487450727c2f63e10d1d0f",
+  "original_card": {
+    "id": "p104-side-correction",
+    "question": "Can the proposed P104 joint endpoint flags be realized by a chronological correction program with coefficients, dirty restoration and an affordable complete bill?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/p104-native-channel/README.md",
+      "research/p104-native-channel/PROOF.md",
+      "docs/research/community-efficient1000-review.md",
+      "research/p104-joint-unit-flags/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 revision-6 family record and #2981 static joint flags, alongside the reviewed #1991 channel. #2981 is an author-checked source archive, not an independently validated executable correction or a refutation of program-level lifetime floors.",
+      "scope": "Source-bound orientation only. Read the named package and review before relying on its local evidence; this command is not its replay or supplier admission."
+    },
+    "first_experiment": "After checking live handoffs, pin one named collection from #2981 and attach acquisition/delivery times, source dependencies and coefficient requirements. Test legal access and arbitrary dirty restoration; price any completion against the full channel/endpoint/hosting allowance. Stop with the exact missing implication or a scoped counterexample rather than expanding the static cover census.",
+    "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."
+    ],
+    "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"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #2981 reports static joint endpoint flags and a matching of 208 departure and arrival flags; archival intake is not independent verification.",
+        "evidence": "research/p104-joint-unit-flags/README.md",
+        "remaining": "Independent flag check and a chronological coefficient program with dirty restoration and a complete cost.",
+        "status": "author-checked",
+        "source_commit": "3d24b92d8ba057b57cb2d480170ec407520e6882",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Prove the dimension-free literal-slot commutator of the old correction P with A_T=Pi L_T, and show that a uniform scalar cannot cancel the first-bank defect when two N-port scales differ.",
+  "scope_change_reason": "The card asks for a chronological P104 correction program and a complete bill. This session does not build one. Eligible handoffs on #1166, after dropping delivered work and open PRs through #4123, number 17. Item index (21-1) mod 17 is the adjacent completed-call pair in comment 19034066. The pair itself stays with that item. This package refines it to the commutator any such pair must cancel if it reuses the old correction.",
+  "bottleneck_link": "The complex supplier is the binding side. An adjacent completed call was the proposed way to spend the P104 correction. A uniform scalar intertwiner fails when N-port scales differ, so that repair is not a paid-profile improvement. No new kappa is claimed.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop once the block commutators and the scalar obstruction are checked, including the two dropped hypotheses. Do not search for an adjacent pair or price a native tariff in this package.",
+  "outcome": "negative-result",
+  "evidence": "PROOF.md is the block multiplication. verify.py recomputes it over the rationals for unequal N-port scales, the constant-scale sharpness case, a noncommuting T, and a non-projection H.",
+  "author": "unpaidmoon",
+  "ai_assistance": "Cursor Grok 4.7 wrote the proof expansion, verifier, and tests.",
+  "open_questions": [
+    {
+      "question": "Does one adjacent pair that uses Pi P_enc Pi, rather than P, match every helper role as well as both data banks with a favorable bill?",
+      "why_it_matters": "The scalar obstruction only rules out the old correction and a uniform multiplier. The handoff still asks for one matched pair.",
+      "first_step": "Pin one helper entrance and exit from the PR 3954 endpoint and price Pi as a slot exchange before any root census.",
+      "blocked_by": "This identity has no helper schedule, and linear rank is not a native tariff."
+    },
+    {
+      "question": "Which portwise diagonal corrections other than K on the image of H cancel the first bank for every N-port amplitude?",
+      "why_it_matters": "The first-bank defect is exactly -H K a, so the correction on that image is forced. A different alphabet would be a new contract.",
+      "first_step": "State the allowed diagonal alphabet and test two distinct N-port scales.",
+      "blocked_by": "A new alphabet is not the old correction P, and no such alphabet is specified by comment 19034066."
+    }
+  ]
+}

research/p104-literal-slot-defect/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "General ordered commutator; literal commutator under a projection that commutes with T; no scalar when N-port scales differ; scalar -1/2 when they agree; dropped commutation and dropped idempotence both fail closed."
+    }
+  ]
+}