Classify port-dependent C_u chart pairs on F_2^4

research/p104-port-chart-pair/PROOF.md

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+# Port-dependent charts in the C_u family match a data line only when they are equal
+
+No new κ. The selected conditional exponent stays `60403/20000000`.
+
+## Changed premise
+
+PR #4136 shows that a uniform chart `C_u` preserves its own encoding, and that
+an exit `C_u` (`u ≠ 0`) does not match the single next entrance `C_0 = I`.
+The open data-line question is the changed premise: the next entrance may be
+any chart in the same family, not only `I`.
+
+The address space is `F_2^4`, with the character basis of size 16. For
+`u` in that space, `X_u` is diagonal,
+
+```text
+X_u |z⟩ = (-1)^{u·z} |z⟩,
+```
+
+and the published chart is
+
+```text
+C_u = ((1+i) I + (1-i) X_u) / 2.
+```
+
+Thus `C_u |z⟩ = |z⟩` when `u·z = 0`, and `C_u |z⟩ = i |z⟩` when `u·z = 1`.
+In particular `C_0 = I`, and `C_u^{-1}` multiplies the odd set of `u` by `-i`.
+
+An adjacent data line whose exit chart is `C_a` and whose next entrance chart
+is `C_b` matches every data character when `C_b C_a^{-1}` acts by `1` on every
+character.
+
+## Classification
+
+On a character `z`, the multiplier of `C_b C_a^{-1}` is:
+
+| `a·z` | `b·z` | multiplier |
+|---|---|---|
+| 0 | 0 | 1 |
+| 1 | 1 | 1 |
+| 0 | 1 | `i` |
+| 1 | 0 | `-i` |
+
+The two dot products agree for every `z` if and only if `a` and `b` are the
+same linear form, hence the same vector. Therefore
+
+```text
+C_b C_a^{-1} = I  ⟺  a = b.
+```
+
+If `a ≠ b`, write `g = a + b ≠ 0`. The characters where the forms disagree are
+exactly `{z : g·z = 1}`, a set of size 8. Every one of those characters is
+multiplied by `i` or by `-i`, so the pair fails on exactly half of the data
+line. The other eight characters, where the forms agree, are fixed.
+
+There are 16 charts and 256 ordered pairs. The 16 pairs with `a = b` match
+every character. The other 240 pairs fail on exactly eight characters. The
+15 pairs checked in #4136 are the column `b = 0`, `a ≠ 0` of this table.
+
+## What the phase is doing
+
+If the odd eigenvalue `i` is replaced by `1`, every `C_u` becomes `I` and every
+ordered pair matches. The obstruction is the relative phase, not the support
+of `X_u` alone. Forgetting that phase is a false witness; the verifier rejects
+it by counting 16 matches rather than 256.
+
+## Scope
+
+This is the data-line half of the changed chart premise. Helper roles are not
+in the model, and this note does not supply an intertwiner, a dirty
+restoration, a hosting bill, or a supplier. A different chart family could
+still match. The uniform pairs `a = b` do match; they are not port-dependent.

research/p104-port-chart-pair/README.md

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+# Port-dependent C_u charts match a data line only when they are equal
+
+**Partial negative result. No new κ.** The selected conditional exponent stays
+`60403/20000000` (`0.00302015`, #4470).
+
+## Result
+
+In the chart family used by the uniform-commutation gap, an adjacent data line
+matches every character if and only if the exit chart and the next entrance
+chart are the same chart.
+
+The address space is `F_2^4`. For each `u` in that space the chart is
+
+```text
+C_u = ((1+i) I + (1-i) X_u) / 2,
+X_u |z⟩ = (-1)^{u·z} |z⟩.
+```
+
+So `C_u` fixes the even characters of `u` and multiplies the odd characters by
+`i`. The changed premise, relative to PR #4136, is that the next entrance may
+be an arbitrary chart `C_b` in this family, not only the common chart `I`.
+The exit is `C_a`. These two charts transport every data character by `1` if
+and only if `a = b`.
+
+There are 16 charts and 256 ordered pairs:
+
+| Pairs | Count | Data-line outcome |
+|---|---:|---|
+| `a = b` | 16 | Match on all 16 characters |
+| `a ≠ b` | 240 | Fail on exactly 8 characters |
+
+The eight failures are the odd set of the nonzero form `a + b`. On that set
+the multiplier is `i` or `-i`. The 15 pairs in #4136, exit `C_u` for `u ≠ 0`
+against entrance `I`, are one column of this table.
+
+Replacing the odd eigenvalue `i` by `1` makes every chart the identity and
+every pair a match. The obstruction is that relative phase.
+
+The argument is in [PROOF.md](PROOF.md). `verify.py` evaluates `C_u` from the
+displayed formula, checks the four-case multiplier, and enumerates all 256
+pairs. It does not import the #4136 package.
+
+## Scope
+
+Task card `p104-side-correction`, priority revision 7, base
+`fbc9ea0249746acca4f504d4fda2b84e0e36cf0d`. Scope status is **refined**.
+
+The card asks whether a changed producer/consumer contract can escape the
+scoped chart obstruction with a favorable full bill. This package changes one
+hypothesis of the published chart obstruction: the entrance chart is any
+`C_b`, not only `I`. Inside this family the data-line mismatch survives for
+every port-dependent pair. Helper roles, dirty restoration, an intertwiner
+and a hosting bill are not in the model, so this is not a supplier and not a
+complete channel bill.
+
+Ordered eligible list for this session, after dropping obligations already
+delivered on main or by the listed unpaidmoon PRs. Order is the
+priority-revision-7 table in `docs/research/current-priorities.md`, with the
+first row expanded by the shortest unaudited list in
+`docs/research/pr4100-audit-coverage-map.md` and then by the successor
+questions those audits still leave open:
+
+1. PR2809 encoded-stack visited-space bound (open question of `bit-stock-recurrence-audit`).
+2. One PR4088 global program's frame-label cancellation (open question of `complex-cover-lift-audit`).
+3. QQ16 matched parent park/restore bill (open question of `qq16-caller-install`).
+4. R437 residual binding to uniform fixed-tape primitives (open question 1 of `r437-carrier-integration`).
+5. **This package.** The fifth remaining item is the P104 row. Its delivered
+   static obstructions (#4133, #4136, #4140, #4153, #4154, #4370 and the other
+   listed P104 PRs) are excluded. The distinct remaining piece is open
+   question 1 of #4136: an adjacent pair whose entrance chart may depend on
+   the port. The data-line half is answered here. The helper half is still open.
+
+## Reproduce
+
+Python 3.12, standard library only, from the repository root:
+
+```sh
+python3 scripts/research_package.py --package research/p104-port-chart-pair --check-only
+python3.12 -B scripts/research_package.py --package research/p104-port-chart-pair
+```
+
+Direct checks:
+
+```sh
+python3 -B research/p104-port-chart-pair/verify.py
+python3 -B research/p104-port-chart-pair/test_verify.py
+```
+
+Expected marker:
+
+```text
+PASS_P104_PORT_CHART_PAIR address=F2^4 charts=16 ordered_pairs=256 matching_pairs=16 failing_pairs=240 failures_per_unequal_pair=8 entrance_I_failures=15 data_match_iff_equal_charts=true helpers=out_of_model no_new_kappa
+```
+
+Not run: Lean, Docker, the selected cold replay, a helper-role model, and any
+hosting or moment calculation.
+
+## Open next questions
+
+1. Does one helper block, written in this same chart family, have an exit and
+   a next entrance that match on every helper role when the data charts already
+   differ? The first step is to name the helper roles and their charts. This
+   package has no helper roles, so it cannot answer that half.
+2. Is there a chart family outside `{C_u}`, still legal for the P104 encoding,
+   in which two different charts transport every data character by `1`? A
+   positive example would reopen the adjacent-pair search. This classification
+   does not search that family.
+
+## Credit
+
+The chart formula, the uniform-match statement, and the 15 entrance-`I`
+failures are PR #4136 (unpaidmoon), from the two-axis gate in PR #3954
+(Chris R / Clear1nsight). The request for one adjacent pair is discussion
+#1166 comment 19034066. Predecessor lineage named in #4136 remains theirs,
+including Rohan Arun, Chafik Boukhalfa, and Adam Gall.
+
+Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.
+No independent human or formal-kernel review is claimed.

research/p104-port-chart-pair/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "fbc9ea0249746acca4f504d4fda2b84e0e36cf0d",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "c35482cd9035843b15a634cc8276a7e1e40e8395",
+      "sha256": "6fe0a099b89aa2aed796a13c3a21dafc63f3388ebf397464e44e0451893b9375"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    }
+  ]
+}

research/p104-port-chart-pair/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "p104-side-correction",
+  "priority_revision": 7,
+  "base_commit": "fbc9ea0249746acca4f504d4fda2b84e0e36cf0d",
+  "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": "In the F_2^4 chart family C_u of the uniform-commutation gap, an adjacent data line matches every character if and only if the exit chart and the next entrance chart are equal. All 240 port-dependent pairs fail on exactly 8 characters. Helper roles and a hosting bill are out of scope.",
+  "scope_change_reason": "The card asks whether a changed producer/consumer contract escapes the chart obstruction with a full bill. The delivered P104 obstructions, including #4136, are excluded. This package changes one hypothesis of #4136: the next entrance may be any chart C_b in the same family, not only I. That is the data-line half of #4136 open question 1. It stops when every port-dependent data pair fails, and it does not build a helper block or a channel bill.",
+  "bottleneck_link": "A port-dependent entrance was the stated way past the uniform chart obstruction. Inside this family it does not produce a data-line match, so that premise does not yet supply a cheaper P104 channel. No selected exponent is changed.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop once every ordered pair in this chart family is classified. Do not search a second chart family or price a hosting bill in this session.",
+  "outcome": "partial",
+  "evidence": "Exact classification. verify.py prints PASS_P104_PORT_CHART_PAIR with matching_pairs 16, failing_pairs 240, failures_per_unequal_pair 8 and entrance_I_failures 15. test_verify.py prints PASS_P104_PORT_CHART_PAIR_CONTROLS controls=counts,identity,half_fail,eigenvalues,closed_form,phase,odd_set. PROOF.md gives the multiplier table. No new kappa.",
+  "author": "unpaidmoon",
+  "ai_assistance": "Substantial Cursor Grok assistance.",
+  "open_questions": [
+    {
+      "question": "Does one helper block in this same chart family match on every helper role when the data charts already differ?",
+      "why_it_matters": "The card's adjacent pair has to match helper roles as well as data roles. A helper mismatch would extend the obstruction; a helper match would be the first positive half of the pair.",
+      "first_step": "Name the helper roles and write their exit and entrance charts in this C_u family.",
+      "blocked_by": "This model has no helper roles."
+    },
+    {
+      "question": "Is there a legal P104 chart family outside {C_u} in which two different charts transport every data character by 1?",
+      "why_it_matters": "A positive example would reopen the adjacent-pair search that this family closes.",
+      "first_step": "Write one alternative legal chart on the same 16 characters and test it against C_0.",
+      "blocked_by": "No second chart family is an input of this package."
+    }
+  ]
+}

research/p104-port-chart-pair/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": "All 256 ordered C_u pairs: the displayed eigenvalue formula agrees with the four-case multiplier, equal charts match all 16 characters, and each of the 240 unequal charts fails on exactly 8 characters. The entrance-I column recovers the 15 failures from PR #4136. Forgetting the odd phase i matches all 256 pairs and is not accepted as the theorem."
+    },
+    {
+      "argv": ["{python}", "-B", "{package}/test_verify.py"],
+      "timeout_seconds": 60,
+      "checks": "Independent controls: C_0 is the identity, C_1 sends an odd character to i, transport against I is -i, the disagreement set is the odd set of a+b, and the phase-forgotten witness is rejected."
+    }
+  ]
+}