Prove uniform completion fails on the physical P104 encoding.

research/p104-uniform-encoding-gap/PROOF.md

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+# Uniform completion does not transfer from E0 to the physical encoding
+
+## Statement
+
+Let each port carry a copy of one address space. Write a two-bank state as
+`(r, w)`. Let `U_C` apply one address operator `C` on every port of both banks.
+Let `D` and `H` be scalar matrices on the ports, with `H` diagonal and
+`H^2 = H`. Define the scalar-bank encoding and the physical encoding by
+
+```
+E0(r, w) = (D r + H w, w),
+L(r, w)  = (D r + H T w, w),
+```
+
+where `(T w)_p = T_p w_p` and each `T_p` is an address operator. Let `J0 = I - H`
+and
+
+```
+Pi(r, w) = (J0 r + H w, H r + J0 w).
+```
+
+`Pi` is the literal N-slot swap. It fixes the J slots. Then:
+
+1. `U_C E0 = E0 U_C` for every address operator `C`.
+2. `U_C L = L U_C` if and only if `C T_p = T_p C` whenever `H_pp = 1`.
+3. `U_C Pi = Pi U_C` for every `C`, and therefore
+   `U_C (Pi L) - (Pi L) U_C = Pi (U_C L - L U_C)`.
+   The literal-slot encoding commutes with `U_C` exactly when `L` does.
+
+In the four-point model below, `H = diag(0, 1)`, `T_N = C_{e1}`, and
+`C = Z_{e1}` satisfy the scalar identity and fail both physical identities.
+The same `Z_{e1}` commutes with `C_{e2}`. Every translation chart commutes with
+every other translation chart, so a uniform completion that stays in that
+family still passes through `L`.
+
+This is not an adjacent completed-call pair, not a native bill, and not a
+kappa.
+
+## Proof
+
+Scalars commute with every address operator, and `U_C` uses the same `C` on
+every port. For a scalar matrix `M`,
+
+```
+U_C (M v)_p = C sum_q M_pq v_q = sum_q M_pq C v_q = (M U_C v)_p.
+```
+
+`E0` and `Pi` are built from scalar blocks, so both commute with every `U_C`.
+That is the positive control for a scalar-bank encoding.
+
+For the physical encoding the second block is not scalar:
+
+```
+U_C L(r, w) = (C D r + C H T w, C w),
+L U_C(r, w) = (D C r + H T C w, C w).
+```
+
+`D` and `H` are scalar, so `C D = D C` and `C H = H C`. The states agree if
+and only if `H C T w = H T C w` for every `w`, which is
+`C T_p = T_p C` on every port with `H_pp = 1`.
+
+Because `Pi` commutes with `U_C`,
+
+```
+U_C Pi L - Pi L U_C = Pi U_C L - Pi L U_C = Pi (U_C L - L U_C).
+```
+
+`Pi` is invertible (`Pi^2 = I`), so it preserves vanishing of the commutator
+and does not restore a failing one.
+
+## Finite witness
+
+Order the addresses of `(F_2)^2` as `0, e1, e2, e1+e2`. Translation acts by
+`(X_u f)(x) = f(x+u)`. The character `Z_{e1}` multiplies by `(-1)^{x·e1}`.
+These are the same translation and character operations used to state the
+chart formula and the normalization identity in PR #3954, §§3–4. They are
+rederived here; that package is not imported.
+
+The Pauli relation `Z_{e1} X_{e1} = - X_{e1} Z_{e1}` holds, while
+`Z_{e1}` commutes with `X_{e2}`, and `X_{e1}` commutes with `X_{e2}`.
+The boundary chart of PR #3954 is
+
+```
+C_u = ((1+i) I + (1-i) X_u) / 2.
+```
+
+Commutation with `C_u` is equivalent to commutation with the Gaussian-integer
+operator `S_u = (1+i) I + (1-i) X_u = 2 C_u`, because the commutator scales by
+the central integer 2. Direct multiplication gives
+
+```
+[Z_{e1}, S_{e1}] =
+  [  0    2-2i   0     0  ]
+  [-2+2i   0     0     0  ]
+  [  0     0     0   2-2i ]
+  [  0     0   -2+2i  0  ].
+```
+
+The `(0,1)` entry `2-2i` is the recorded witness. Thus `Z_{e1}` does not
+commute with `C_{e1}`. It does commute with `C_{e2}`.
+
+Take two ports, `H = diag(0, 1)`, and the non-scalar integer matrix
+
+```
+D = [[1, 1], [1, 2]].
+```
+
+The general identity already covers every scalar-entry `D`, including the
+104-port pivot matrix of PR #3954. This check uses the 2×2 matrix above so
+that the certificate does not depend on reloading that pivot. Both verifiers
+rebuild the operators and test:
+
+- `U_Z` commutes with `E0`;
+- `U_Z` fails to commute with `L` and with `Pi L`, on 4 of the 16 basis
+  directions, and `Pi` carries one commutator to the other;
+- `U_{S_{e1}}` commutes with `L` and with `Pi L`;
+- replacing the N-port chart by `S_{e2}` restores commutation with `U_Z`.
+
+## What this does not decide
+
+A uniform completion whose operator lies in the algebra generated by the
+translations still commutes with every `C_u`, hence with `L` and `Pi L`.
+The obstruction bites only when the common operator fails to commute with an
+active port chart. PR #3954's positive control says "any" common operator,
+and the reason it gives is that the encoding coefficients are scalars. That
+reason applies to `E0` and not to `L` or `Pi L`.
+
+The sentence does not by itself produce a matched physical caller. This note
+does not construct the adjacent completed-call pair requested by discussion
+#1166 comment 19034066, and it does not price owners, orientation, width, or
+tape.

research/p104-uniform-encoding-gap/README.md

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+# Uniform completion against the physical P104 encoding
+
+**Scoped commutation obstruction. No new κ and no selected-frontier change.**
+Selected κ stays `0.001672660197076918`.
+
+PR #3954 shows that a common address operator commutes with the scalar-bank
+encoding `E0`, because that encoding's blocks are scalar matrices. The handoff
+in discussion #1166 comment 19034066 records the consequence that was left
+open: uniform completed transforms commute with scalar bank encodings, and
+that fact alone does not give a matched physical caller. The requested
+construction is an adjacent completed-call pair. This package does not build
+that pair.
+
+It separates the scalar encoding from the physical one. For
+
+```
+E0(r, w) = (D r + H w, w),    L(r, w) = (D r + H T w, w),
+```
+
+`U_C` commutes with `E0` for every address operator `C`, and commutes with `L`
+exactly when `C` commutes with every active port chart `T_p`. The literal slot
+swap `Pi` commutes with every `U_C`, so `Pi L` fails for the same operators as
+`L`. On the four-point translation model, `Z_{e1}` commutes with `E0` and with
+`C_{e2}`, and it fails for `C_{e1}` on both `L` and `Pi L`. The address
+commutator entry is `2-2i`.
+
+Charts generated by translations still commute with each other, so a uniform
+completion that stays in that family is not blocked here. The written argument
+is in [PROOF.md](PROOF.md).
+
+## Reproduce
+
+CPython 3.12, standard library, from the repository root:
+
+```sh
+python3.12 -B research/p104-uniform-encoding-gap/verify.py
+python3.12 -B research/p104-uniform-encoding-gap/independent.py
+python3.12 -B research/p104-uniform-encoding-gap/test_verify.py
+python3.12 -B scripts/research_package.py --package research/p104-uniform-encoding-gap
+```
+
+`verify.py` multiplies the 16×16 Gaussian-integer matrices.
+`independent.py` does not import it; it applies the same maps to vectors.
+`test_verify.py` rejects an even-direction chart offered as an obstruction.
+
+## Open next questions
+
+1. Which operators does one actual completed P104 call apply uniformly? If
+   they all lie in the translation algebra, this commutator does not block
+   that call, and the adjacent-pair search has to move elsewhere. The first
+   step is to read the retained caller at one completed role and name the
+   operator. That caller text is not pinned in this package.
+2. The adjacent completed-call pair from comment 19034066 is untouched. Its
+   entrance and exit still have to match on every data and helper role, with
+   a physical intertwiner and a full bill. This commutation gap does not
+   supply that pair, and it does not prove that no such pair exists.
+
+## Credit
+
+The scalar-bank positive control, the chart formula `C_u`, the literal swap
+`Pi`, and the normalization character are from Clear1nsight, PR #3954, head
+`f1e2839819959cd0534d008be91ed5b79ec55a6a`, retaining the P104 and native
+lineage named there (Rohan Arun, Chafik Boukhalfa, Adam Gall, and the authors
+credited in that package). This package rederives the four-point translation
+and character relations and does not import that code.
+
+Prepared by unpaidmoon with substantial Cursor Grok assistance. No independent
+human review is claimed. Apache-2.0.

research/p104-uniform-encoding-gap/SOURCES.json

@@ -0,0 +1,18 @@
+{
+  "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-uniform-encoding-gap/TASK.json

@@ -0,0 +1,94 @@
+{
+  "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 that a uniform address operator commutes with the scalar-bank encoding E0 for every operator, and commutes with the physical encoding L and the literal encoding Pi L exactly when it commutes with every active port chart. Witness the failure at Z_e1 against C_e1. Do not construct the adjacent completed-call pair.",
+  "scope_change_reason": "The catalogue card asks for a chronological side-correction word. Discussion #1166 comment 19034066 instead asks for one adjacent completed-call pair. Fewer than 25 eligible handoffs remained, and the session index selected that handoff for a distinct piece. This piece audits the scalar-bank commutation shortcut named in the same handoff, which is not the pair construction.",
+  "bottleneck_link": "A joint physical encoding is one route toward a changed complex supplier. Quoting the scalar-bank commutation as if it were already that encoding would skip an unpaid intertwiner. The obstruction removes that shortcut and leaves the adjacent pair open.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop once the commutation identity and the four-point witness are checked. Do not search for an adjacent pair or price a native caller in this package.",
+  "outcome": "completed",
+  "evidence": "PROOF.md gives the block identity. verify.py and independent.py agree on the witness [Z_e1, S_e1]_(0,1)=2-2i, with physical and literal defects of support 4, and on the commuting controls E0, S_e1 against itself, and S_e2 against Z_e1.",
+  "author": "unpaidmoon",
+  "ai_assistance": "Cursor Grok 4.7, including the matrix and vector checkers.",
+  "open_questions": [
+    {
+      "question": "Which operators does one actual completed P104 call apply uniformly?",
+      "why_it_matters": "Translation charts commute with each other, so this gap blocks only a uniform operator outside that family. A legal completed call might never leave the family.",
+      "first_step": "Name the operator on one completed role of the retained caller and test its commutator with one active port chart.",
+      "blocked_by": "The caller source cited by PR #3954 is not pinned in this package."
+    },
+    {
+      "question": "Does any adjacent completed-call pair match encoded exit to encoded entrance on every data and helper role?",
+      "why_it_matters": "That is the construction requested by comment 19034066. This commutation gap does not build it and does not prove it is impossible.",
+      "first_step": "Pin one ordered pair of completed calls and compare one data role and one helper role before writing an intertwiner.",
+      "blocked_by": "No matched physical caller was supplied by PR #3954, and this package does not search for one."
+    }
+  ]
+}

research/p104-uniform-encoding-gap/VERIFY.json

@@ -0,0 +1,36 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "16x16 Gaussian commutators: E0 commutes, L and Pi L fail for Z_e1, translation and even-direction controls commute."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/independent.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Vector replay with no import of verify.py; same witness column and basis miss counts."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/test_verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Rejects the even-direction chart as an obstruction and accepts the odd-direction witness."
+    }
+  ]
+}