diff --git a/research/bit-width-seven-boundary/CERTIFICATE.json b/research/bit-width-seven-boundary/CERTIFICATE.json
new file mode 100644
index 0000000000..6982d303d1
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+++ b/research/bit-width-seven-boundary/CERTIFICATE.json
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+              14
+            ]
+          ],
+          "determinant": -32,
+          "dirt": [
+            20,
+            25,
+            67,
+            74,
+            581,
+            601,
+            611,
+            626
+          ],
+          "helper": 654,
+          "rank": 8,
+          "restore": [
+            224,
+            339,
+            397,
+            420,
+            545,
+            601,
+            611,
+            626,
+            654
+          ]
+        },
+        {
+          "basis": [
+            [
+              302,
+              0
+            ],
+            [
+              302,
+              1
+            ],
+            [
+              302,
+              4
+            ],
+            [
+              302,
+              5
+            ],
+            [
+              302,
+              8
+            ],
+            [
+              302,
+              15
+            ],
+            [
+              302,
+              59
+            ],
+            [
+              309,
+              2
+            ]
+          ],
+          "determinant": -32,
+          "dirt": [
+            7,
+            36,
+            45,
+            49,
+            93,
+            527,
+            563,
+            600,
+            627,
+            655
+          ],
+          "helper": 655,
+          "rank": 8,
+          "restore": [
+            302,
+            309,
+            316,
+            391,
+            458,
+            527,
+            560,
+            600,
+            610,
+            655
+          ]
+        },
+        {
+          "basis": [
+            [
+              306,
+              1
+            ],
+            [
+              306,
+              7
+            ],
+            [
+              306,
+              14
+            ],
+            [
+              306,
+              24
+            ],
+            [
+              306,
+              30
+            ],
+            [
+              306,
+              60
+            ],
+            [
+              367,
+              2
+            ],
+            [
+              367,
+              3
+            ]
+          ],
+          "determinant": 16,
+          "dirt": [
+            84,
+            248,
+            280,
+            293,
+            343,
+            386,
+            480,
+            482,
+            505,
+            530,
+            540,
+            569,
+            571,
+            596
+          ],
+          "helper": 656,
+          "rank": 8,
+          "restore": [
+            306,
+            367,
+            372,
+            409,
+            513,
+            571,
+            576,
+            656
+          ]
+        },
+        {
+          "basis": [
+            [
+              256,
+              0
+            ],
+            [
+              256,
+              3
+            ],
+            [
+              256,
+              4
+            ],
+            [
+              256,
+              7
+            ],
+            [
+              256,
+              11
+            ],
+            [
+              256,
+              15
+            ],
+            [
+              256,
+              31
+            ],
+            [
+              392,
+              2
+            ]
+          ],
+          "determinant": -32,
+          "dirt": [
+            13,
+            50,
+            259,
+            284,
+            314,
+            342,
+            353,
+            385,
+            404,
+            476,
+            547,
+            597,
+            615,
+            639,
+            657
+          ],
+          "helper": 657,
+          "rank": 8,
+          "restore": [
+            256,
+            392,
+            424,
+            442,
+            490,
+            552,
+            597,
+            639,
+            657
+          ]
+        },
+        {
+          "basis": [
+            [
+              383,
+              8
+            ],
+            [
+              383,
+              12
+            ],
+            [
+              383,
+              16
+            ],
+            [
+              383,
+              21
+            ],
+            [
+              383,
+              31
+            ],
+            [
+              383,
+              60
+            ],
+            [
+              408,
+              10
+            ],
+            [
+              408,
+              13
+            ]
+          ],
+          "determinant": 32,
+          "dirt": [
+            18,
+            65,
+            85,
+            101,
+            262,
+            334,
+            369,
+            444,
+            475,
+            583,
+            598,
+            616,
+            640,
+            658
+          ],
+          "helper": 658,
+          "rank": 8,
+          "restore": [
+            383,
+            408,
+            426,
+            431,
+            489,
+            507,
+            598,
+            640,
+            658
+          ]
+        },
+        {
+          "basis": [
+            [
+              267,
+              1
+            ],
+            [
+              267,
+              3
+            ],
+            [
+              267,
+              5
+            ],
+            [
+              267,
+              7
+            ],
+            [
+              267,
+              10
+            ],
+            [
+              267,
+              17
+            ],
+            [
+              267,
+              20
+            ],
+            [
+              291,
+              0
+            ]
+          ],
+          "determinant": 64,
+          "dirt": [
+            51,
+            53,
+            97,
+            274,
+            307,
+            362,
+            367,
+            437,
+            456,
+            468,
+            533,
+            535,
+            566,
+            576,
+            599
+          ],
+          "helper": 659,
+          "rank": 8,
+          "restore": [
+            267,
+            291,
+            576,
+            622,
+            659
+          ]
+        },
+        {
+          "basis": [
+            [
+              246,
+              1
+            ],
+            [
+              246,
+              2
+            ],
+            [
+              246,
+              3
+            ],
+            [
+              246,
+              4
+            ],
+            [
+              246,
+              5
+            ],
+            [
+              246,
+              6
+            ],
+            [
+              246,
+              7
+            ],
+            [
+              265,
+              19
+            ]
+          ],
+          "determinant": -128,
+          "dirt": [
+            1,
+            110,
+            278,
+            302,
+            377,
+            415,
+            524,
+            560,
+            595,
+            609,
+            660
+          ],
+          "helper": 660,
+          "rank": 8,
+          "restore": [
+            246,
+            265,
+            330,
+            391,
+            393,
+            405,
+            421,
+            484,
+            532,
+            560,
+            595,
+            660
+          ]
+        }
+      ],
+      "name": "after_B",
+      "rank_counts": {
+        "8": 439
+      },
+      "survivors": []
+    }
+  ],
+  "dimension": 8,
+  "scope": "Fixed D/A/scatter prefix; common-frame invertible suffix and monotone climbs. No new kappa.",
+  "status": "SCOPED_BIT_BOUNDARY_OBSTRUCTION",
+  "v": 111
+}
diff --git a/research/bit-width-seven-boundary/CLAIM.json b/research/bit-width-seven-boundary/CLAIM.json
new file mode 100644
index 0000000000..a20f8c22de
--- /dev/null
+++ b/research/bit-width-seven-boundary/CLAIM.json
@@ -0,0 +1,55 @@
+{
+  "route": 5,
+  "branch": "codex/k9-bit-width7",
+  "base_commit": "abdc6a7002abdbcd9e21bfc5e1a795784deacfdf",
+  "claim_type": "local_no_pinned_coordination_issue_at_dispatch",
+  "date": "2026-10-13",
+  "candidates": [
+    "Hyperplane cleanup with an arbitrary inverse circuit.",
+    "Rank-one helper entrance with a semantic compensation prefix.",
+    "Coordinated semantic D/C change with early retirement or a paid bridge."
+  ],
+  "selected_question": "Can a changed cleanup or later body lower selected PR589 helper residual width?",
+  "deliverable": "Exact inverse-row and dirty-column boundary theorem and certificate.",
+  "first_exact_result": "All 439 helpers have restoration rank eight after the selected B.",
+  "search_randomness": "none",
+  "proof_coverage": {
+    "helpers": 439,
+    "cuts": [
+      "after_scatter",
+      "after_B"
+    ],
+    "helper_cut_rows": 878,
+    "compensation_columns": 439
+  },
+  "exploratory_retirement_scan": {
+    "class": "Only h is changed. Restore h at an A/B gate boundary or scatter boundary at or after its last raw use. Keep all other scalar values and subsequent raw frames fixed. No cuts within a grouped gate, width reduction, or complete cleanup claim.",
+    "cuts": 198137,
+    "future_frame_blocked": 128276,
+    "zero_inverse_diagonal": 56892,
+    "rank_eight": 9740,
+    "ordering_cycle": 91,
+    "admitted_readouts": 3138,
+    "admitted_helpers": [
+      462,
+      463,
+      464,
+      467,
+      470,
+      471
+    ],
+    "script_sha256": "899786cf81979a0400b76a2bbc70250e4170e10d22531704b7d5e7fa7640207b",
+    "status": "Exploratory evidence, not used by the proof certificate."
+  },
+  "overlap": [
+    "PR1023 proves the width-eight hosting optimum and prices hypothetical width-seven conversions.",
+    "PR1048 fixes endpoints and audits one/two-move reframing.",
+    "PR1043 removes helper470; PR1079 audits source substitution.",
+    "PR1100 concerns positive single-write programs at frames509/521.",
+    "PR1105 audits projected target-feedback cleanup; the present suffix must satisfy the complete final map. PR1108 concerns the compact-control parity interface.",
+    "PR1122 changes A/B/semantic D while preserving paid paths. Its prefix is outside this selected-word theorem. PR1121 is a complex rank-seven/T16 construction."
+  ],
+  "publication": "No push or PR creation by this worker.",
+  "attribution": "Prepared for Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.",
+  "final_refresh": "Recent open PRs through #1122; PR1105, PR1108, PR1121 and PR1122 bodies read. PR1122 head a2551e70fce0cc34348f50f1a94e7cc4c80bbcbf. No matching BIT boundary-cut certificate was found."
+}
diff --git a/research/bit-width-seven-boundary/PROOF.md b/research/bit-width-seven-boundary/PROOF.md
new file mode 100644
index 0000000000..b3bedffde3
--- /dev/null
+++ b/research/bit-width-seven-boundary/PROOF.md
@@ -0,0 +1,181 @@
+# Two necessary conditions for a BIT helper endpoint
+
+This proof concerns a finite frame word over F2. A frame is a rational
+subspace of Q^8. A register can move only to a frame that contains its
+current frame. Every scalar operation is linear, invertible, and acts on registers
+at one common frame. ADD gates and register permutations satisfy this rule.
+The suffix can use any number of these operations and any rational frames.
+It can use extra dirty registers if it restores them. It has no new COPY,
+descent, reset of a nonzero register, or other operation outside this model.
+
+The selected prefix fixes the scalar state and frame of every register at
+the cut. Write the scalar state as w = M u. The vector u contains all source,
+target and dirty-helper inputs. M is invertible: each prefix scalar gate,
+including each scatter addition, is an invertible transvection over F2.
+At completion, source x_i must contain x_i, target y_i must contain y_i+x_i,
+and helper h must contain its original input s_h. No other final register
+can contain s_h. Extra dirty registers must also return their own inputs.
+
+Let F_r be the frame of register r at the cut. Suppose helper h ends in U.
+Its frame at the cut must satisfy F_h <= U.
+
+## 1. Restoration condition
+
+Let a be the unique row such that s_h = sum_r a_r w_r. Every register on a
+dependency path into a final register at U has a frame contained in U.
+This follows from common-frame operations and monotone frame changes.
+Thus the final value of h can depend only on boundary registers r with
+F_r <= U, and on fresh inputs independent of u.
+
+The boundary coordinates w_r are independent because M is invertible. If
+a_r = 1, changing w_r alone changes s_h. Therefore each such r is required:
+
+    span(F_h, {F_r : a_r = 1}) <= U.                 (R)
+
+This argument also permits nonlinear operations on the available inputs.
+No nonlinear method can compute a parity without every essential input.
+We use only its consequence for the reversible linear frame-word class.
+
+The selected word has no target-to-z edge, where z consists of sources and
+helpers. Its z boundary matrix is invertible. Consequently, the restoration
+row a uses only z registers. The derivation obtains a by reversing the z
+gates. The checker obtains it by Gaussian elimination of the actual boundary
+matrix. It does not read or import the derivation program.
+
+## 2. Erasure condition
+
+Fix U. Call a register high when its frame is not contained in U. Let S be
+the F2 span of the current scalar rows of all high registers.
+
+A frame change cannot make a high register low. A low-to-high frame change
+adds one row to S. A common-frame scalar operation either acts entirely on
+low registers, or entirely on high registers. In the second case it is an
+invertible change of basis of the affected high rows. Thus S never loses
+a vector throughout the suffix.
+
+The final helper h is low. Every final high register has zero coefficient
+of s_h, because every other register has its required final value. Hence
+every vector in the final S has zero coefficient of s_h. The same must hold
+for the initial S. If M[r,h] = 1, register r therefore cannot be high:
+
+    span({F_r : M[r,h] = 1}) <= U.                  (E)
+
+This condition uses the complete dirty column. A small restoration row
+alone does not suffice. Extra restored dirty registers do not weaken (E).
+They begin with independent inputs and end with zero coefficient of s_h.
+
+Combining (R), (E), and the frame already held by h gives the lower bound
+
+    dim U >= dim span(F_h,
+                      {F_r : a_r = 1},
+                      {F_r : M[r,h] = 1}).
+
+There is no frame-catalogue restriction in this statement. A rank-eight
+join excludes every proper subspace of Q^8, including an unlisted frame.
+The bound remains necessary if several helpers receive smaller endpoints
+at once. Each final output other than h still has zero s_h coefficient.
+
+## 3. Application to the selected PR589 word
+
+The input is the exact tuned R439 word in SOURCES.json. The certificate uses
+original register identifiers, including sources 0..110, targets 111..221,
+and helpers 222..660. The input has 661 independent dirty-input coordinates.
+
+Immediately after D, A and the retained-total scatter, the joint bound has
+rank eight for 433 helpers. It has rank seven for exactly these six:
+
+| Helper | Necessary terminal hyperplane, in the selected catalogue |
+|---|---:|
+| 465 | 512 |
+| 466 | 514 |
+| 468 | 513 |
+| 469 | 517 |
+| 470 | 513 |
+| 476 | 514 |
+
+For a rank-seven row, the certificate contains seven independent vectors
+and a nonzero integer normal that annihilates every required frame. These
+six rows pass a necessary condition only. The proof supplies no suffix,
+caller window, or paid construction for them.
+
+At the cut after the selected B, condition (R) alone has rank eight for all
+439 helpers. Thus no replacement cleanup C in this class can reduce even
+one selected helper to width seven. H4 is not used: C need not be the literal
+reverse, need not use full-frame gates, and can have any finite length.
+Allowing target reads, new restored dirty work, or different gate order does
+not change the lower bound.
+
+Each rank-eight certificate lists eight line vectors, their boundary donor
+registers, and a nonzero integer determinant. The checker verifies that each
+line is in the stated register's current frame and that the register belongs
+to the required row or column. It checks the determinant by fraction-free
+integer elimination. It checks all 439 helpers at both cuts. This is an exact
+finite certificate, not a formalization of the full multiplication theorem.
+
+## 4. Rank-one entrance with a semantic compensation prefix
+
+Keep the raw A/scatter/B entry state z and the zero frame of each target.
+Require D to provide all dirty-target compensation before A. A later cleanup
+does not supply missing target compensation in this entrance class. This is
+a separate restriction from the endpoint classes in sections 1 and 2.
+The raw target map is y -> y + x + K s. The required compensation prefix is
+therefore y -> y + K s, regardless of its particular gate list. It can use
+an arbitrary common-frame circuit, provided that it returns z to the same
+raw entry state and leaves the targets at frame zero.
+
+Every column of K in this word is nonzero. The certificate records the
+complete list of target rows for every helper. The checker verifies these
+lists and the complete dirty endpoint of D/A/scatter/B/C.
+
+If helper h starts in a nonzero frame E, no target ending the prefix at
+frame zero can depend on its independent input s_h. This is the dependency
+argument in (R), applied to the compensation prefix. Yet a nonzero entry
+K[t,h] requires precisely that dependence. The contradiction excludes a
+rank-one entrance in this prefix class for every selected helper.
+
+This is a semantic condition. It does not merely enforce the old H2 syntax.
+It does not apply to a changed helper basis at the raw entry, a changed A
+or scatter, compensation moved after A, or a new
+recursive interface. Those mechanisms remain outside the result.
+
+## 5. A paid bridge that does not help
+
+One constructive attempt preserves the selected later source use by adding
+a new dirty bridge. Consider a helper whose last raw frame has rank seven,
+such as 469. Let the new bridge read a source at rank one, deliver that value
+at rank seven, and restore at the full frame. Its paid path is 0 -> 1 -> 7 -> 8.
+This path costs children of ranks 1, 6 and 1. Permit every other new operation
+for free in this lower-cost model.
+
+Put the bridge's width-eight residual in the old helper's hosted slot. Put
+the converted width-seven helper in a formerly idle rank-seven slot. The old
+helper loses its final rank-one child. Including the consumed idle child,
+the change per local call is exactly
+
+    Delta n_1 = +1, Delta n_6 = +1, Delta n_7 = -1.
+
+Rank mass is unchanged. The number of children grows by one. For 0 < a < 1,
+the bit moment weight is g_a(r) = 64^a r^(1-a). Strict concavity gives
+g_a(1) + g_a(6) > g_a(7). The inherited surcharge also grows with the child
+count. This optimistic complete profile is worse than the selected profile.
+Additional source access, compensation or restoration work cannot rescue
+this specific wrapper at unchanged live stock and precision allowances.
+This excludes that wrapper,
+not every method that uses an extra helper.
+
+## Scope
+
+The selected endpoints have start rank zero, so terminal rank eight means
+residual width eight. The after-scatter result excludes 433 individual
+conversions when the prefix is fixed. It does not exclude changes to that
+prefix, to the retained copies, to helper coordinates, or to the frame
+transport interface. The six survivors are not an existence theorem.
+
+The argument uses the selected finite scalar/frame model. Its all-size
+compiler, common-label and mutable-source realization, physical hosting,
+precision/tape and analytic transfer assumptions remain inherited. No full
+native supplier, new exponent, practical speedup, or universal BIT lower
+bound is asserted.
+
+Prepared for Ali Romman, with substantial OpenAI Codex assistance.
+New material is Apache-2.0.
diff --git a/research/bit-width-seven-boundary/README.md b/research/bit-width-seven-boundary/README.md
new file mode 100644
index 0000000000..f193038570
--- /dev/null
+++ b/research/bit-width-seven-boundary/README.md
@@ -0,0 +1,168 @@
+# Selected BIT helper boundaries require a changed prefix
+
+**Status: scoped negative result. No new kappa.**
+
+Under the current contract for restoration of dirty helpers, the selected
+PR589 word cannot obtain width-seven helpers by replacing its cleanup alone.
+All 439 helpers require a terminal frame of rank eight after
+the selected phase B. A stronger two-sided cut already excludes 433 of them
+immediately after the retained-total scatter, even if B and C are both
+replaced. Six helpers remain outside that exclusion.
+
+This result permits arbitrary finite reversible linear circuits at common frames,
+arbitrary rational frame choices, target access, extra restored dirty
+registers, and simultaneous changes to several helper endpoints. It does
+not assume literal reverse cleanup or the selected frame catalogue. The
+suffix must use monotone frame changes and cannot add a COPY or another
+operation outside that interface. [PROOF.md](PROOF.md) states the class.
+
+| Boundary | Exact necessary result |
+|---|---|
+| Semantic compensation before the fixed raw entry | All 439 helper inputs must start at rank zero |
+| After D, A and scatter | 433 of 439 helpers must end at rank eight |
+| After the selected B | All 439 helpers must end at rank eight |
+
+The six after-scatter survivors are helpers **465, 466, 468, 469, 470, 476**.
+They pass necessary rank-seven conditions. No construction or caller-window
+admission is supplied for them.
+
+## Why it matters
+
+PR1023 proves that the selected width-eight no-bank placement cannot improve
+the bit endpoint on its stated grid. Its width-seven slot model motivates
+this work: 74 one-unit conversions could raise bit c by about 5.6% to 6.7%,
+before the missing physical boundary costs are known. The present result
+shows that those conversions cannot come from a cleanup replacement on the
+selected word. With the prefix through scatter fixed, 433 helpers are also
+excluded from any B/C rewrite in the stated interface.
+
+At the exact base `abdc6a7002abdbcd9e21bfc5e1a795784deacfdf`, the selected
+PR942 kappa is `0.001661960547713947`. During this work, main
+`ba82555c64e1879e4b7decce28186f208fcb8262` selected PR1042 at
+`0.001672188119205421`. Both use the identical PR589 input checked here.
+The latter complex value is `0.001674989015937672328468`; the bit value is
+`0.001733427039589844234749`. The complex side still binds. This package
+does not change either supplier or the selected result.
+
+## Exact evidence
+
+For helper h, two kinds of boundary information must fit in its terminal U:
+
+1. Every boundary register needed to reconstruct the original dirty input
+   s_h. The inverse scalar matrix gives this unique set.
+2. Every boundary register that still contains s_h. The span of registers
+   outside U cannot lose information under reversible common-frame gates.
+
+Their frame join gives the lower bound. CERTIFICATE.json contains all 878
+helper/cut rows, both support sets, exact frame identifiers, determinant
+witnesses for rank eight, and normal vectors for the six rank-seven joins.
+It also contains all 439 nonzero compensation columns.
+
+The generator uses packed binary rows and inverse gate order. The checker
+uses sets, reconstructs the prefix matrices, and inverts them by Gaussian
+elimination. It checks geometric witnesses with integer determinants and
+dot products. It imports neither the generator nor a predecessor checker.
+Both use only the Python standard library.
+
+## Reproduce
+
+From the repository root, run:
+
+```sh
+python3 -I -B research/bit-width-seven-boundary/verify.py
+```
+
+Expected final line:
+
+```text
+PASS_BIT_BOUNDARY_OBSTRUCTION after_scatter=433/439 after_B=439/439 entry=439/439 controls=11 no_new_kappa
+```
+
+To regenerate the certificate and compare its decoded fields, run:
+
+```sh
+python3 -I -B research/bit-width-seven-boundary/derive.py
+```
+
+The generator also accepts `--output NEW_FILE`. It refuses to overwrite an
+existing file. The verifier does not write files. The two programs accept
+`--root REPOSITORY` for an exact pinned input at another checkout.
+
+Eight certificate controls reject a wrong inverse row, a missing dirty
+carrier, false frame provenance, a zero determinant, a false normal, an
+omitted helper, a false compensation column, and a Boolean rank. Each must
+fail for its stated reason. Three small boundary controls accept a legal
+rank-seven cleanup and reject one case that needs each separate condition.
+The checker also replays all 661 dirty inputs through the original word and
+checks the nested frame paths and the required scalar endpoint.
+
+## Attempts and stopping rule
+
+The three candidate approaches were hyperplane cleanup, rank-one entrance,
+and a coordinated semantic compensation/cleanup replacement. The first
+attempt replaced literal cleanup by an arbitrary inverse circuit. The exact
+inverse-row obstruction blocks all 439 helpers after B. The next attempt
+moved the cut to the scatter and added the erasure condition. It excludes
+433 helpers while retaining six precise candidates for future body changes.
+
+A separate exploratory scan checked direct single-register restoration at
+all 198,137 helper/gate-boundary pairs after the helper's last selected raw
+gate or scatter use. It found 3,138
+locally admitted helper/cut pairs on 462, 463, 464, 467, 470 and 471. These
+are readouts, not complete restorations of the other dirty registers. The
+scan does not prove a width reduction and is not part of the certificate.
+Helper 470 already has the stronger source-deletion construction in PR1043.
+No saving from that result is added here.
+
+For rank-one entrances, every helper has a nonzero compensation column.
+The semantic prefix argument rules out a promotion that preserves the raw
+entry state and zero-frame targets. A new full-width bridge with paid path
+0 -> 1 -> 7 -> 8 also fails the optimistic complete cost test: after returning
+and consuming the exact idle ranks, it replaces one rank-seven child by
+rank-one and rank-six children. PROOF.md gives the strict moment inequality.
+
+The stopping rule is exact coverage of every helper at the two selected
+cuts, plus a check of every compensation column. No random search was used.
+The six early-cut survivors and changes to A, copies, source preparation,
+helper basis, or recursive transport remain open. Closing those routes would
+need a different construction or another theorem. This package does not
+claim their impossibility.
+
+## Validation limits and source pins
+
+SOURCES.json pins the sole mathematical input by SHA-256 and records the
+base, context files, predecessor heads, and later selected comparison.
+Context hashes document the reviewed state; they are not moving runtime
+dependencies. The exact BIT input is identical at both selected commits.
+
+This is a finite exact certificate with a written theorem. It is not a Lean
+proof, a native 5/9/11-stage replay, or a complete physical supplier invoice.
+No new word is proposed, so there is no new hosted invoice to certify. The
+selected cold replay, full repository regeneration and remote CI are not
+claimed. All inherited realization and analytic assumptions remain.
+
+## Credit and overlap
+
+Credit DreamingOfClouds for PR589 and the PR545/510 word and hosting lineage;
+eumemic for PR555 and the interleaved interface; Zhihao Chen for the earlier
+scalar-interface argument; DaysSky for idle-axis hosting; Jeremy Kothe for
+complex hyperplane ends; Dominik Scholz for complex entrance promotion; and
+Sagar Thakrar for PR1023's width-packing incentive.
+
+Ali Romman's PR1043 changes the source boundary and deletes helper 470.
+PR1079, prepared for Hanson Sun, gives the direct-source substitution census.
+PR1048 audits one/two-move reframing with fixed endpoints. PR1100 concerns
+positive single-write programs at frames 509 and 521. PR1105 shows that raw
+endpoint independence does not justify projected target-feedback cleanup.
+This package keeps the prefix fixed and requires the complete final map; it
+does not assume that a projected cleanup is valid. It instead allows arbitrary
+scalar suffix replacement and changed helper endpoints.
+The scope is distinct, and the prior savings are not combined.
+PR1122 composes changes to A, B and semantic D while preserving paid frame
+paths. Its changed prefix is outside the present selected-word theorem.
+
+Prepared for Ali Romman (@aliromman), with substantial OpenAI Codex assistance
+in the research, proof, implementation and review. New material is under the
+repository's Apache-2.0 license. Separate code paths do not imply independent
+human review. No push, PR creation, or selected-frontier edit was made by this
+worker.
diff --git a/research/bit-width-seven-boundary/SOURCES.json b/research/bit-width-seven-boundary/SOURCES.json
new file mode 100644
index 0000000000..cd7b5c8192
--- /dev/null
+++ b/research/bit-width-seven-boundary/SOURCES.json
@@ -0,0 +1,32 @@
+{
+  "base_commit": "abdc6a7002abdbcd9e21bfc5e1a795784deacfdf",
+  "inputs": {
+    "research/e8-bit-p111-tuned/tune/bitE8mut-p111-tuned.json.gz": "e3e301e62bc2b41dbf1ceac0acf3d35824ebeab1848c5a2142d4577b109da52f"
+  },
+  "word": "research/e8-bit-p111-tuned/tune/bitE8mut-p111-tuned.json.gz",
+  "context_prs": {
+    "589": "a5afc8c6169ea61ba7c51dfd3cade02cc7b13b46",
+    "942": "1960cfa919a0945d0ca4cb14ea96237000037903",
+    "1023": "babca68167d8db321d194b7dc54230d88a8ad5f0",
+    "1043": "e309e17f83f5759b1ea1fa24a76f24838784cb4d",
+    "1048": "07730cd8302c378607259a9ad0efb73e0b57a2f2",
+    "1079": "bb20dab261884412f2d2a7955cdb2fd8e2e64a16",
+    "1100": "45cebe410999a005fa8396dea8257d06da88f50d",
+    "1105": "20a960b545f4b0ade4f26922ef686bd5eaf275f4",
+    "1108": "b76bb2853be4e7001508582b10913e32ad7aba9f",
+    "1122": "a2551e70fce0cc34348f50f1a94e7cc4c80bbcbf"
+  },
+  "context_files": {
+    "research/e8-bit-p111-tuned/tune/EXPECTED.json": "b9f0e5ca038abe837b0007c8fe006f366a6f02009244ac57705a8a200e2301f4",
+    "certificates/selected-result.json": "1d35283cb0ce3f8db1f285683933517f9e87475d295e2b6316399ad9928c81c3",
+    "research/e8-bit-r420/README.md": "9807be57c6adf26b30e4f54e2f8160075bbc05091f1a6d157e43d2a4620c1c9f"
+  },
+  "live_comparison": {
+    "main_commit": "ba82555c64e1879e4b7decce28186f208fcb8262",
+    "selected_pr": 1042,
+    "kappa": "1672188119205421/1000000000000000000",
+    "complex_b": "0.001674989015937672328468",
+    "bit_c": "0.001733427039589844234749",
+    "word_sha256_unchanged": true
+  }
+}
diff --git a/research/bit-width-seven-boundary/derive.py b/research/bit-width-seven-boundary/derive.py
new file mode 100644
index 0000000000..5912aafaf2
--- /dev/null
+++ b/research/bit-width-seven-boundary/derive.py
@@ -0,0 +1,172 @@
+# SPDX-License-Identifier: Apache-2.0
+"""Derive the exact BIT boundary certificate. Use Python's standard library."""
+import argparse
+import gzip
+import hashlib
+import json
+import math
+from collections import Counter
+from fractions import Fraction
+from pathlib import Path
+
+
+HERE = Path(__file__).resolve().parent
+
+
+def add_vector(rows, vector):
+    row = list(map(Fraction, vector))
+    for pivot in sorted(rows):
+        q = row[pivot]
+        row = [a - q * b for a, b in zip(row, rows[pivot])]
+    pivot = next((i for i, x in enumerate(row) if x), None)
+    if pivot is None:
+        return False
+    q = row[pivot]
+    rows[pivot] = [x / q for x in row]
+    return True
+
+
+def determinant(vectors):
+    rows = [list(map(Fraction, row)) for row in vectors]
+    out = Fraction(1)
+    for col in range(len(rows)):
+        pivot = next(i for i in range(col, len(rows)) if rows[i][col])
+        if pivot != col:
+            rows[pivot], rows[col] = rows[col], rows[pivot]
+            out = -out
+        q = rows[col][col]
+        out *= q
+        for i in range(col + 1, len(rows)):
+            factor = rows[i][col] / q
+            rows[i] = [a - factor * b for a, b in zip(rows[i], rows[col])]
+    assert out.denominator == 1
+    return out.numerator
+
+
+def support(value):
+    out = []
+    while value:
+        bit = value & -value
+        out.append(bit.bit_length() - 1)
+        value -= bit
+    return out
+
+
+def span_evidence(word, current, donors, frame_bases):
+    vectors, witness, rows = [], [], {}
+    for register in sorted(donors):
+        if len(vectors) == word["h"]:
+            break
+        for line in frame_bases[current[register]]:
+            vector = word["lines"][line]
+            if add_vector(rows, vector):
+                vectors.append(vector)
+                witness.append([register, line])
+            if len(vectors) == word["h"]:
+                break
+    rank = len(vectors)
+    result = {"rank": rank, "basis": witness}
+    if rank == word["h"]:
+        result["determinant"] = determinant(vectors)
+    else:
+        free = next(i for i in range(word["h"]) if i not in rows)
+        normal = [Fraction(i == free) for i in range(word["h"])]
+        for pivot in sorted(rows, reverse=True):
+            normal[pivot] = -sum(rows[pivot][i] * normal[i] for i in range(pivot + 1, word["h"]))
+        denominator = math.lcm(*(x.denominator for x in normal))
+        normal = [int(x * denominator) for x in normal]
+        divisor = math.gcd(*normal)
+        normal = [x // divisor for x in normal]
+        if next(x for x in normal if x) < 0:
+            normal = [-x for x in normal]
+        result["normal"] = normal
+    return result
+
+
+def derive(word):
+    v = word["v"]
+    n = 2 * v + word["R"]
+    helpers = range(2 * v, n)
+    z = [r for r in range(n) if not v <= r < 2 * v]
+    frame_bases = []
+    for frame in word["frames"]:
+        chosen, rows = [], {}
+        for line in frame:
+            if add_vector(rows, word["lines"][line]):
+                chosen.append(line)
+        frame_bases.append(chosen)
+    current = word["start"][:]
+    values = [1 << r for r in range(n)]
+    z_gates = []
+
+    def run(section):
+        for frame, target, sources in word[section]:
+            for register in [target, *sources]:
+                current[register] = frame
+            for source in sources:
+                values[target] ^= values[source]
+            if section in ("A", "B") and target in z:
+                z_gates.append((target, sources))
+
+    def cut(name):
+        inverse = [1 << r for r in range(n)]
+        for target, sources in reversed(z_gates):
+            for source in sources:
+                inverse[target] ^= inverse[source]
+        entries = []
+        for helper in helpers:
+            restore = support(inverse[helper])
+            dirt = [r for r, value in enumerate(values) if value >> helper & 1]
+            donors = set(restore) | set(dirt) | {helper}
+            evidence = span_evidence(word, current, donors, frame_bases)
+            if name == "after_B":
+                # The restoration condition alone suffices at this cut.
+                evidence = span_evidence(word, current, set(restore) | {helper}, frame_bases)
+            entries.append({"helper": helper, "restore": restore, "dirt": dirt, **evidence})
+        return {"name": name, "frames": current[:], "helpers": entries,
+                "rank_counts": {str(rank): count for rank, count in sorted(Counter(row["rank"] for row in entries).items())},
+                "survivors": [row["helper"] for row in entries if row["rank"] < word["h"]]}
+
+    run("D")
+    run("A")
+    for target, totals in enumerate(word["scatter"]):
+        for total in totals:
+            values[v + target] ^= values[word["ret"][total]]
+    cuts = [cut("after_scatter")]
+    run("B")
+    cuts.append(cut("after_B"))
+    run("C")
+    for r, value in enumerate(values):
+        assert value == ((1 << r) ^ (1 << (r - v)) if v <= r < 2 * v else 1 << r)
+    compensation = []
+    for helper in helpers:
+        targets = [target for _, target, sources in word["D"] if helper in sources]
+        assert targets
+        compensation.append({"helper": helper, "targets": targets})
+    return {"status": "SCOPED_BIT_BOUNDARY_OBSTRUCTION", "v": v, "R": word["R"],
+            "dimension": word["h"], "cuts": cuts, "compensation": compensation,
+            "scope": "Fixed D/A/scatter prefix; common-frame invertible suffix and monotone climbs. No new kappa."}
+
+
+def main():
+    parser = argparse.ArgumentParser()
+    parser.add_argument("--root", type=Path, default=HERE.parents[1])
+    parser.add_argument("--output", type=Path)
+    args = parser.parse_args()
+    sources = json.loads((HERE / "SOURCES.json").read_text())
+    for path, digest in sources["inputs"].items():
+        assert hashlib.sha256((args.root / path).read_bytes()).hexdigest() == digest, path
+    word = json.loads(gzip.decompress((args.root / sources["word"]).read_bytes()))
+    result = derive(word)
+    encoded = json.dumps(result, sort_keys=True, indent=2) + "\n"
+    if args.output:
+        with args.output.open("x") as stream:
+            stream.write(encoded)
+    else:
+        expected = json.loads((HERE / "CERTIFICATE.json").read_text())
+        assert json.dumps(result, sort_keys=True, separators=(",", ":")) == json.dumps(expected, sort_keys=True, separators=(",", ":"))
+    print("PASS_BIT_BOUNDARY_DERIVATION after_scatter=433/439 after_B=439/439")
+
+
+if __name__ == "__main__":
+    main()
diff --git a/research/bit-width-seven-boundary/verify.py b/research/bit-width-seven-boundary/verify.py
new file mode 100644
index 0000000000..883cf0f55f
--- /dev/null
+++ b/research/bit-width-seven-boundary/verify.py
@@ -0,0 +1,254 @@
+# SPDX-License-Identifier: Apache-2.0
+"""Check the boundary certificate without the derivation program."""
+import argparse
+import copy
+import gzip
+import hashlib
+import json
+from collections import Counter
+from pathlib import Path
+
+
+HERE = Path(__file__).resolve().parent
+
+
+class Reject(ValueError):
+    pass
+
+
+def require(condition, message):
+    if not condition:
+        raise Reject(message)
+
+
+def integers_only(value):
+    if isinstance(value, dict):
+        for child in value.values():
+            integers_only(child)
+    elif isinstance(value, list):
+        for child in value:
+            integers_only(child)
+    elif isinstance(value, (bool, float)):
+        raise Reject("numeric field is not an integer")
+
+
+def det(matrix):
+    """Use fraction-free elimination on an integer square matrix."""
+    a = [row[:] for row in matrix]
+    sign, previous = 1, 1
+    for k in range(len(a) - 1):
+        pivot = next((i for i in range(k, len(a)) if a[i][k]), None)
+        if pivot is None:
+            return 0
+        if pivot != k:
+            a[pivot], a[k] = a[k], a[pivot]
+            sign = -sign
+        q = a[k][k]
+        for i in range(k + 1, len(a)):
+            for j in range(k + 1, len(a)):
+                numerator = q * a[i][j] - a[i][k] * a[k][j]
+                require(numerator % previous == 0, "non-integral determinant step")
+                a[i][j] = numerator // previous
+            a[i][k] = 0
+        previous = q
+    return sign * a[-1][-1]
+
+
+def invert_rows(values, labels):
+    """Invert a binary matrix with set elimination, not gate reversal."""
+    rows = [set(values[r]) for r in labels]
+    right = [{r} for r in labels]
+    for k, label in enumerate(labels):
+        pivot = next((i for i in range(k, len(labels)) if label in rows[i]), None)
+        require(pivot is not None, "singular boundary matrix")
+        rows[k], rows[pivot] = rows[pivot], rows[k]
+        right[k], right[pivot] = right[pivot], right[k]
+        for i in range(len(labels)):
+            if i != k and label in rows[i]:
+                rows[i] ^= rows[k]
+                right[i] ^= right[k]
+    require(rows == [{r} for r in labels], "inverse reduction")
+    return dict(zip(labels, right))
+
+
+def requirements(values, inverse, helper):
+    restore = set(inverse[helper])
+    dirt = {r for r, row in enumerate(values) if helper in row}
+    return restore, dirt, restore | dirt | {helper}
+
+
+def replay(word):
+    v, n = word["v"], 2 * word["v"] + word["R"]
+    frames = list(word["start"])
+    values = [{r} for r in range(n)]
+    snapshots = {}
+
+    def section(name):
+        for frame, target, donors in word[name]:
+            require(len({target, *donors}) == 1 + len(donors), "invalid scalar gate")
+            for register in [target, *donors]:
+                require(set(word["frames"][frames[register]]) <= set(word["frames"][frame]), "frame descent")
+                frames[register] = frame
+            for donor in donors:
+                values[target] ^= values[donor]
+
+    def snapshot(name):
+        z = [r for r in range(n) if not v <= r < 2 * v]
+        require(all(not any(v <= x < 2 * v for x in values[r]) for r in z), "target input in z boundary")
+        snapshots[name] = (frames[:], [row.copy() for row in values], invert_rows(values, z))
+
+    section("D")
+    section("A")
+    require(all(frames[r] == 0 for r in range(v, 2 * v)), "target frame before scatter")
+    for target, totals in enumerate(word["scatter"]):
+        for total in totals:
+            values[v + target] ^= values[word["ret"][total]]
+    snapshot("after_scatter")
+    section("B")
+    snapshot("after_B")
+    section("C")
+    for register, final in enumerate(word["final"]):
+        require(set(word["frames"][frames[register]]) <= set(word["frames"][final]), "final frame descent")
+        expected = {register, register - v} if v <= register < 2 * v else {register}
+        require(values[register] == expected, "dirty endpoint")
+    return snapshots
+
+
+def check_span(word, current, donors, entry):
+    rank = entry["rank"]
+    require(rank in (7, 8) and len(entry["basis"]) == rank, "rank or basis length")
+    vectors = []
+    for register, line in entry["basis"]:
+        require(register in donors and line in word["frames"][current[register]], "frame provenance")
+        vectors.append(word["lines"][line])
+    if rank == 8:
+        actual = det(vectors)
+        require(actual != 0 and entry.get("determinant") == actual, "rank8 determinant")
+    else:
+        gram = [[sum(a * b for a, b in zip(x, y)) for y in vectors] for x in vectors]
+        require(det(gram) > 0, "rank7 basis dependence")
+        normal = entry["normal"]
+        require(len(normal) == 8 and any(normal), "zero normal")
+        for register in donors:
+            for line in word["frames"][current[register]]:
+                require(sum(a * b for a, b in zip(normal, word["lines"][line])) == 0, "normal does not annihilate required frame")
+
+
+def check_certificate(word, certificate, snapshots):
+    integers_only(certificate)
+    v, n = word["v"], 2 * word["v"] + word["R"]
+    helpers = list(range(2 * v, n))
+    require((certificate["v"], certificate["R"], certificate["dimension"]) == (v, word["R"], 8), "dimensions")
+    require(certificate["status"] == "SCOPED_BIT_BOUNDARY_OBSTRUCTION", "status")
+    require([cut["name"] for cut in certificate["cuts"]] == ["after_scatter", "after_B"], "cut coverage")
+    for cut in certificate["cuts"]:
+        current, values, inverse = snapshots[cut["name"]]
+        require(cut["frames"] == current, "boundary frames")
+        require([row["helper"] for row in cut["helpers"]] == helpers, "helper coverage")
+        for entry in cut["helpers"]:
+            helper = entry["helper"]
+            restore, dirt, joint = requirements(values, inverse, helper)
+            require(entry["restore"] == sorted(restore), "inverse row")
+            require(entry["dirt"] == sorted(dirt), "dirt column")
+            donors = restore | {helper} if cut["name"] == "after_B" else joint
+            check_span(word, current, donors, entry)
+        counts = Counter(row["rank"] for row in cut["helpers"])
+        require(cut["rank_counts"] == {str(k): value for k, value in sorted(counts.items())}, "rank counts")
+        survivors = [row["helper"] for row in cut["helpers"] if row["rank"] < 8]
+        require(cut["survivors"] == survivors, "survivor coverage")
+    require(certificate["cuts"][0]["survivors"] == [465, 466, 468, 469, 470, 476], "six surviving necessary conditions")
+    require(certificate["cuts"][1]["survivors"] == [], "post-B full-rank coverage")
+    require([row["helper"] for row in certificate["compensation"]] == helpers, "compensation coverage")
+    for row in certificate["compensation"]:
+        targets = [target for frame, target, donors in word["D"] if row["helper"] in donors]
+        require(targets and row["targets"] == targets, "compensation column")
+    return {"after_scatter_full_rank": 433, "after_B_full_rank": 439, "nonzero_compensation_columns": 439}
+
+
+def controls(word, certificate, snapshots):
+    mutations = []
+    changed = copy.deepcopy(certificate)
+    changed["cuts"][0]["helpers"][0]["restore"].pop()
+    mutations.append(("inverse row", changed))
+    changed = copy.deepcopy(certificate)
+    changed["cuts"][0]["helpers"][0]["dirt"].pop()
+    mutations.append(("dirt column", changed))
+    changed = copy.deepcopy(certificate)
+    entry = changed["cuts"][0]["helpers"][0]
+    register, _ = entry["basis"][0]
+    frame = changed["cuts"][0]["frames"][register]
+    entry["basis"][0][1] = next(line for line in range(word["v"]) if line not in word["frames"][frame])
+    mutations.append(("frame provenance", changed))
+    changed = copy.deepcopy(certificate)
+    changed["cuts"][1]["helpers"][0]["determinant"] = 0
+    mutations.append(("rank8 determinant", changed))
+    changed = copy.deepcopy(certificate)
+    entry = next(row for row in changed["cuts"][0]["helpers"] if row["rank"] == 7)
+    entry["normal"] = [1] * 8
+    mutations.append(("normal does not annihilate", changed))
+    changed = copy.deepcopy(certificate)
+    changed["cuts"][0]["helpers"].pop()
+    mutations.append(("helper coverage", changed))
+    changed = copy.deepcopy(certificate)
+    changed["compensation"][0]["targets"].pop()
+    mutations.append(("compensation column", changed))
+    changed = copy.deepcopy(certificate)
+    changed["cuts"][0]["helpers"][0]["rank"] = True
+    mutations.append(("numeric field", changed))
+    for reason, changed in mutations:
+        try:
+            check_certificate(word, changed, snapshots)
+        except Reject as error:
+            require(reason in str(error), "control failed for an unrelated reason: " + reason)
+        else:
+            raise Reject("control accepted: " + reason)
+
+    # These two-register examples test each part of the cut condition.
+    labels = [0, 1]
+    toy = {"lines": [[int(i == j) for j in range(8)] for i in range(8)],
+           "frames": [list(range(7)), list(range(8))]}
+    evidence = {"rank": 7, "basis": [[1, i] for i in range(7)], "normal": [0] * 7 + [1]}
+    rows = [{0}, {0, 1}]
+    inverse = invert_rows(rows, labels)
+    restore, dirt, joint = requirements(rows, inverse, 1)
+    require((restore, dirt, joint) == ({0, 1}, {1}, {0, 1}), "restoration control")
+    require(rows[1] ^ rows[0] == {1}, "legal hyperplane cleanup control")
+    check_span(toy, [0, 0], joint, evidence)
+    try:
+        check_span(toy, [1, 0], joint, evidence)
+    except Reject as error:
+        require("normal does not annihilate" in str(error), "restoration frame control reason")
+    else:
+        raise Reject("restoration frame control accepted")
+    rows = [{0, 1}, {1}]
+    inverse = invert_rows(rows, labels)
+    restore, dirt, joint = requirements(rows, inverse, 1)
+    require((restore, dirt, joint) == ({1}, {0, 1}, {0, 1}), "erasure control")
+    check_span(toy, [1, 0], restore, evidence)
+    try:
+        check_span(toy, [1, 0], joint, evidence)
+    except Reject as error:
+        require("normal does not annihilate" in str(error), "erasure frame control reason")
+    else:
+        raise Reject("erasure frame control accepted")
+    return len(mutations), 3
+
+
+def main():
+    parser = argparse.ArgumentParser()
+    parser.add_argument("--root", type=Path, default=HERE.parents[1])
+    args = parser.parse_args()
+    sources = json.loads((HERE / "SOURCES.json").read_text())
+    for path, digest in sources["inputs"].items():
+        require(hashlib.sha256((args.root / path).read_bytes()).hexdigest() == digest, "source pin: " + path)
+    word = json.loads(gzip.decompress((args.root / sources["word"]).read_bytes()))
+    certificate = json.loads((HERE / "CERTIFICATE.json").read_text())
+    snapshots = replay(word)
+    result = check_certificate(word, certificate, snapshots)
+    negative, small = controls(word, certificate, snapshots)
+    print(json.dumps({**result, "negative_controls": negative, "small_boundary_controls": small}, sort_keys=True))
+    print("PASS_BIT_BOUNDARY_OBSTRUCTION after_scatter=433/439 after_B=439/439 entry=439/439 controls=11 no_new_kappa")
+
+
+if __name__ == "__main__":
+    main()