Mister-Meeseeks · Oct 28, 2026 · Oct 28, 2026
diff --git a/research/two-helper-zero-frame-feedback/README.md b/research/two-helper-zero-frame-feedback/README.md
@@ -0,0 +1,146 @@
+# Two-helper zero-frame shears: no feedback gain at any target count
+
+This contribution was produced by Nick Petalas. Apache-2.0.
+
+**Claim type: scoped proof (every target count T) with finite corroboration.
+No new supplier, κ or selected-frontier change.**
+
+## Result
+
+Use the model of [bit-zero-frame-feedback](../bit-zero-frame-feedback/README.md)
+(PR #1306) and [bit-zero-frame-feedback-to-8-registers](../bit-zero-frame-feedback-to-8-registers/README.md)
+(PR #2567). Registers are two helpers `h0, h1` and `T` targets `y_1..y_T`. All
+inputs are arbitrary (dirty). A gate `(d, s)` is `x_d ^= x_s` at unit cost; there
+are no extra registers. A restricted word has no gate with a helper destination
+and a target source. The required map is the shear `(h, y) -> (h, y + K h)`
+over F2, where `K` is `T × 2`.
+
+Write `a`, `b` and `c` for the number of rows of `K` equal to `(1,0)`, `(0,1)`
+and `(1,1)`, and `r = a + b + c` for the number of nonzero rows.
+
+**Theorem.** For every `T ≥ 1` and every `K`, the restricted and unrestricted
+minimum word lengths are equal:
+
+```text
+d_P(K) = d_full(K) = a + b + c + min(c, 2) = r + min(c, 2).
+```
+
+So target-to-helper feedback never lowers the cost of a two-helper shear. This
+settles the 2×7 shape of Open next question 1 in PR #2567, and every other
+2×T shape. By that package's duality `cost_{H×T}(K) = cost_{T×H}(K^T)`, it also
+settles every T×2 shape, including 7×2. The 3×6 and 4×5 shapes stay open.
+
+## Proof
+
+**Upper bound (restricted).** Apply `y_i ^= h0` or `y_i ^= h1` for each pure
+row. If `c ≤ 1`, apply `y_i ^= h0; y_i ^= h1` for the `(1,1)` row. If `c ≥ 2`,
+apply `h0 ^= h1`, then `y_i ^= h0` for each `(1,1)` row, then `h0 ^= h1`. The
+pure reads come first, so they see the original `h0`. The length is
+`a + b + 2c` or `a + b + c + 2`, which is `r + min(c, 2)` in both cases. No gate
+reads a target into a helper.
+
+**Lower bound (feedback allowed).** Let `w` be any word, in the unrestricted
+alphabet, whose product is the shear. Track each register's content as a
+linear form in the initial values. The contents of all registers form an
+invertible matrix after every prefix, so no register ever holds zero.
+
+1. *Each nonzero row needs its target as a destination.* A register that is
+   never a destination ends with its initial content. A target with a nonzero
+   row of `K` must change, so at least `r` gates have a target destination.
+2. *A once-modified target gets a pure row if no helper is a destination.*
+   Suppose no gate has a helper destination. Then `h0` and `h1` hold their
+   initial values throughout. Every prefix product then has block form
+   `[[I, 0], [X, Y]]` with `Y` invertible, so each target register's content has
+   a nonzero target part. If target `y_i` is the destination of exactly one gate,
+   its final content is `y_i + v`, where `v` is the source content at that gate.
+   The final target part must be `y_i` alone, so `v` has zero target part. Hence
+   `v` is `h0` or `h1`, and the row of `y_i` is `(1,0)` or `(0,1)`. So every
+   `(1,1)` target is a destination at least twice.
+3. *A once-modified helper cannot end unchanged.* If helper `h_x` is the
+   destination of exactly one gate, it ends with `h_x + v` with `v ≠ 0`. The
+   shear requires it to end as `h_x`.
+
+Now suppose `c ≥ 1` and `|w| ≤ r + min(c, 2) - 1`. Step 1 gives `|w| ≥ r`. If `|w| = r`, all gates have target destinations. If
+`c ≥ 2` and `|w| = r + 1`, at most one gate is not target-destined, and step 3
+forbids a lone helper destination. In both cases no gate has a helper
+destination. Step 2 then needs at least `r + c` target-destined gates. That
+exceeds `|w|`, because `c ≥ 1` when `|w| = r` and `c ≥ 2` when `|w| = r + 1`.
+When `c = 0`, step 1 alone gives `|w| ≥ r`. Therefore
+`d_full(K) ≥ r + min(c, 2)`. Since `d_full ≤ d_P`, both minima equal the formula. ∎
+
+With `T = 1` we have `c ≤ 1`, and the formula reduces to `wt(K)`, as in the
+`T = 1` lemma of PR #2567.
+
+## Evidence
+
+`verify.py` uses only the standard library and imports no predecessor code. It
+checks four things:
+
+- **Witnesses:** for every `K` with `T = 1..8` (87,380 matrices), the explicit
+  restricted word above has length `r + min(c, 2)` and computes the shear
+  exactly. The gates are linear, so this covers all dirty inputs.
+- **Exact BFS:** for `T = 1, 2, 3` (84 matrices), it runs independent
+  breadth-first searches in the unrestricted and restricted alphabets. Both
+  minima equal the formula.
+- **Consistency with PR #2567:** the formula matches the pinned certified
+  minima for every `K` in 2×4, 2×5 and 2×6 (5,376 matrices). The certificate is
+  authenticated by SHA256. This is a calibration of the formula, not a replay
+  of PR #2567's search.
+- **Controls:** the verifier rejects a truncated witness and a wrong shared
+  gate, detects an inserted feedback gate, and rejects the weight formula
+  `wt(K)` on a 2×3 matrix with three `(1,1)` rows (BFS minimum 5, weight 6).
+
+From the repository root, with CPython 3.12:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/two-helper-zero-frame-feedback/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/two-helper-zero-frame-feedback/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/two-helper-zero-frame-feedback
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/two-helper-zero-frame-feedback --check-source-history
+```
+
+Any CPython 3.12 path works. Each direct run takes under one second and prints
+
+```text
+PASS_TWO_HELPER_ZERO_FRAME_FEEDBACK witnesses=87380 bfs=84 pr2567=5376 controls=4 feedback_gains=0 no_new_kappa
+```
+
+The checks use explicit exceptions, so they stay active under `-O`.
+
+## Scope and limits
+
+The theorem covers `H = 2` (and `T = 2` by duality), F2, PR #1306's two
+unit-cost single-source alphabets, zero frames and no extra registers. Larger
+helper counts, nonzero frames, multi-source gates, other costs and other fields
+are outside scope. Physical precision, hosting and supplier pricing are not
+built or checked. The finite checks corroborate the written proof. They do not
+replace it for `T > 8`. Independent human and formal proof-kernel review are
+unclaimed.
+
+Power of Ten rule 3 deviation: CPython allocates dynamically. Data sizes are
+bounded by the fixed limits `T ≤ 8` for witnesses and `T ≤ 3` for BFS. The
+largest BFS ball has fewer than 10^5 states.
+
+## Open next questions
+
+1. **Does feedback help for three helpers at nine registers (3×6)?** Step 2
+   of the proof needs a replacement: with three helpers, a shared helper sum
+   can serve several row types, and two-step gains can combine. A first step is
+   a closed form for `d_P` on 3×T using the row-type counts, checked against
+   PR #2567's 3×5 certificate.
+2. **Is there a row-type formula for every H?** For `H = 2` the cost depends
+   only on the row-type counts. For larger `H` the restricted cost becomes a
+   shared-sum covering problem. A closed form for `d_P` would give the target for
+   an unrestricted lower bound.
+3. **Does the destination-counting argument extend to twisted exits?** The
+   contamination lemma of PR #4071 handles twisted exits at fixed small shapes.
+   Step 2 assumes the helper block stays fixed, so it would need the twisted
+   helper block's own cost.
+
+## Credits
+
+The model, the restricted/unrestricted question and the 2×4–2×6 certificates
+are from rtyu193-rgb's PR #2567, which builds on commdata2338's PR #1306,
+bbMisk's PR #1557 and rtyu193-rgb's PR #1706. The twisted-exit question cites
+thakrarsagar's PR #4071. The new contribution here is the all-T two-helper
+theorem and its proof.
diff --git a/research/two-helper-zero-frame-feedback/SOURCES.json b/research/two-helper-zero-frame-feedback/SOURCES.json
@@ -0,0 +1,24 @@
+{
+  "schema_version": 1,
+  "base_commit": "19fdb086ea96af9a9f87936a2b888e10af4ed810",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "581f8d87008354eb5b869f7ed9de1317ad837ce8",
+      "sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "path": "research/bit-zero-frame-feedback-to-8-registers/CERTIFICATE.json",
+      "usage": "input",
+      "git_blob": "9bda10dad1646ec2003b56dcd5fb9376d3c35e94",
+      "sha256": "c47b7a294fd5ac6a9379e8967ecda48f4ef1428b7efeddca93fb8c3b03b8adfc"
+    }
+  ]
+}
diff --git a/research/two-helper-zero-frame-feedback/TASK.json b/research/two-helper-zero-frame-feedback/TASK.json
@@ -0,0 +1,120 @@
+{
+  "schema_version": 1,
+  "task_id": "bit-new-register-model",
+  "priority_revision": 8,
+  "base_commit": "19fdb086ea96af9a9f87936a2b888e10af4ed810",
+  "original_card": {
+    "id": "bit-new-register-model",
+    "question": "Can an explicitly larger or nonabelian register model escape the scoped three-register obstruction?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "research/bit-three-register-lower-bound/README.md",
+      "certificates/selected-result.json",
+      "research/bit-gl3-mixed-targets-length10/README.md",
+      "docs/research/community-efficient1000-review.md"
+    ],
+    "methods": [
+      "Construct the smallest exact identity in a newly specified legal operation alphabet.",
+      "Derive a target-specific relaxation bound for that same explicit alphabet."
+    ],
+    "acceptance_gate": "Name the changed hypothesis, all counted roles and a uniform identity; finite column tests alone do not certify it. 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_selected_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Resolve the current selected review, separate complex/ordinary BIT/admitted BIT/outer budgets, and compare the card against live handoffs before starting. Historical ceilings require exact source/model equivalence.",
+      "scope": "Source-binding orientation only; not a scientific replay or all-size proof. Materialize selected inputs if this checkout is sparse."
+    },
+    "first_experiment": "First compare the alphabet, role count and length with the fully replayed #1974 census and #2027 mixed-root tariff. Identify a changed premise and a legal native implementation; then price one smallest completed instance before searching larger words.",
+    "bottleneck_link": "A small-width completed saving changes the bit primitive; a finite-address identity alone is insufficient.",
+    "allowed_variations": [
+      "Choose a fixed larger register count or a specified nonabelian operation family."
+    ],
+    "scope_change_triggers": [
+      "Remaining inside the already-excluded three-register abelian model.",
+      "Repeating the two-register GL3 census or a covered three-register length without changing a hypothesis; a formal group-algebra word without native conversion is a partial result."
+    ],
+    "overlap_terms": [
+      "three-register",
+      "nonabelian",
+      "group algebra"
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Exact restricted and unrestricted minimum unit-XOR word length for every two-helper zero-frame dirty shear (h,y)->(h,y+Kh) over F2, every target count T, in the PR1306/PR2567 model.",
+  "scope_change_reason": "Answers the two-helper part (including 2x7) of Open next question 1 in PR2567, which asks whether target feedback ever helps beyond eight registers. This is a register-model side theorem, not a new supplier.",
+  "bottleneck_link": "Side result: closes the D-local target-feedback relaxation for every two-helper prefix, so a feedback saving cannot come from H=2 or T=2 shears. No κ effect.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop at the first verified all-T two-helper theorem; leave 3x6 and 4x5 open.",
+  "outcome": "completed",
+  "evidence": "README.md gives the written proof. verify.py checks 87,380 restricted witnesses (T=1..8), exact BFS for T=1..3 in both alphabets, agreement with PR2567's 2x4-2x6 minima (5,376 K) and four controls.",
+  "author": "Nick Petalas",
+  "ai_assistance": "Independent human and formal proof-kernel review are unclaimed.",
+  "open_questions": [
+    {
+      "question": "Does target feedback help for 3x6 at nine registers?",
+      "why_it_matters": "Completes the nine-register case of PR2567's question with 4x5.",
+      "first_step": "Find a row-type closed form for d_P on 3xT and test it on PR2567's 3x5 certificate.",
+      "blocked_by": "Step 2 of the proof assumes two helpers."
+    },
+    {
+      "question": "Is there a row-type formula for every H?",
+      "why_it_matters": "A closed restricted cost gives the target for a general lower bound.",
+      "first_step": "Formulate d_P as a shared helper-sum covering problem.",
+      "blocked_by": "No closed form for H>=3."
+    },
+    {
+      "question": "Does destination counting extend to twisted two-helper exits?",
+      "why_it_matters": "PR4071 covers twisted exits only at fixed small shapes.",
+      "first_step": "Price the twisted helper block and redo step 2 with a changing helper block.",
+      "blocked_by": "Step 2 needs fixed helpers."
+    }
+  ],
+  "claim_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19230512",
+  "playbook": "Investigation",
+  "citations": [
+    {
+      "pr": 2567,
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/2567",
+      "author": "rtyu193-rgb",
+      "contribution": "Model, open question and 2x4-2x6 certified minima.",
+      "usage": "Pinned certificate used as a formula calibration input."
+    },
+    {
+      "pr": 1306,
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1306",
+      "author": "commdata2338",
+      "contribution": "Original model and alphabets.",
+      "usage": "Cited context only."
+    },
+    {
+      "pr": 1557,
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1557",
+      "author": "bbMisk",
+      "contribution": "3x4 certificate.",
+      "usage": "Cited context only."
+    },
+    {
+      "pr": 1706,
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/1706",
+      "author": "rtyu193-rgb",
+      "contribution": "4x4 certificate.",
+      "usage": "Cited context only."
+    },
+    {
+      "pr": 4071,
+      "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/4071",
+      "author": "thakrarsagar",
+      "contribution": "Twisted-exit contamination lemma.",
+      "usage": "Cited for an open question only."
+    }
+  ],
+  "preserved_checkout": {
+    "branch": null,
+    "detached_head": "19fdb086ea96af9a9f87936a2b888e10af4ed810",
+    "prior_commits": []
+  }
+}
diff --git a/research/two-helper-zero-frame-feedback/VERIFY.json b/research/two-helper-zero-frame-feedback/VERIFY.json
@@ -0,0 +1,36 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/bit-zero-frame-feedback-to-8-registers/CERTIFICATE.json"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Restricted witnesses for every K at T=1..8 (87,380), exact restricted and unrestricted BFS for T=1..3 (84), formula against pinned PR2567 2x4-2x6 minima (5,376), four rejection controls."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Restricted witnesses for every K at T=1..8 (87,380), exact restricted and unrestricted BFS for T=1..3 (84), formula against pinned PR2567 2x4-2x6 minima (5,376), four rejection controls."
+    }
+  ]
+}