diff --git a/research/p112-integer-demand-kernel/NOTICE b/research/p112-integer-demand-kernel/NOTICE
new file mode 100644
index 0000000000..2329bb2b20
--- /dev/null
+++ b/research/p112-integer-demand-kernel/NOTICE
@@ -0,0 +1,16 @@
+New integer projection, certificate generation/checking and package metadata:
+ToikY with substantial OpenAI Codex assistance. Apache-2.0.
+
+Three source objects are retained verbatim and authenticated by SOURCE-PINS.json:
+- MATRIX-6425.json: Nick Petalas's PR6425 fixed280-row/14-variable P112 kernel.
+- REAL-RESULT-6455.json: ToikY/OpenAI Codex PR6455 exact real-domain certificate.
+- README-6455.md: PR6455 scope, attribution and open integer-only question.
+
+Retain credit to Jeremy Diviney/PR1422 for the original smaller-family question,
+complete-clique guard, geometry and support caps; Cinna Davis/PR1347/1323, Navinor,
+David Arias, Hanson Sun, DreamingOfClouds, themilkman216 and the retained
+E8/label/decoder lineage. All earlier source headers and credits remain.
+
+No predecessor executable or scientific clique census is replayed. Weak
+compositions/stars-and-bars, monotonicity and finite integer implication are
+classical; no general mathematical priority is claimed.
diff --git a/research/p112-integer-demand-kernel/ORACLE.json b/research/p112-integer-demand-kernel/ORACLE.json
new file mode 100644
index 0000000000..bfe79c8961
--- /dev/null
+++ b/research/p112-integer-demand-kernel/ORACLE.json
@@ -0,0 +1 @@
+{"checked":{"integer_witnesses":[{"origin":"prior-integral","other_original_rows_checked":279,"target":"0x5a","target_demand":4,"target_lhs":0,"x":[8,0,8,0,0,8,0,8,8,8,8,8,8,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0xc0a","target_demand":1,"target_lhs":0,"x":[8,0,8,0,8,8,8,8,8,8,0,0,8,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0xc50","target_demand":1,"target_lhs":0,"x":[8,8,8,8,0,8,0,8,8,8,0,0,8,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0xc51","target_demand":2,"target_lhs":1,"x":[0,8,8,8,1,8,0,8,8,8,0,0,8,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0xd0a","target_demand":2,"target_lhs":1,"x":[8,1,8,0,8,8,8,8,0,8,0,0,8,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0xd50","target_demand":2,"target_lhs":1,"x":[8,8,8,8,1,8,0,8,0,8,0,0,8,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0xe8a","target_demand":2,"target_lhs":1,"x":[8,1,8,0,8,8,8,0,8,0,0,0,8,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0xedb","target_demand":6,"target_lhs":4,"x":[0,2,8,0,2,8,0,0,8,0,0,0,8,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0xf8a","target_demand":3,"target_lhs":2,"x":[8,2,8,0,8,8,8,0,0,0,0,0,8,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0xf8b","target_demand":4,"target_lhs":3,"x":[0,3,8,0,8,8,8,0,0,0,0,0,8,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0xfd0","target_demand":3,"target_lhs":2,"x":[8,8,8,8,2,8,0,0,0,0,0,0,8,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0xfd1","target_demand":4,"target_lhs":3,"x":[0,8,8,8,3,8,0,0,0,0,0,0,8,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0x1412","target_demand":2,"target_lhs":0,"x":[8,0,8,8,0,8,8,8,8,8,0,8,0,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0x1448","target_demand":2,"target_lhs":0,"x":[8,8,8,0,8,8,0,8,8,8,0,8,0,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0x145f","target_demand":6,"target_lhs":4,"x":[0,2,0,2,0,8,0,8,8,8,0,8,0,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0x1513","target_demand":4,"target_lhs":2,"x":[0,2,8,8,0,8,8,8,0,8,0,8,0,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0x1516","target_demand":4,"target_lhs":2,"x":[8,2,0,8,0,8,8,8,0,8,0,8,0,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0x1549","target_demand":4,"target_lhs":2,"x":[0,8,8,2,8,8,0,8,0,8,0,8,0,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0x154c","target_demand":4,"target_lhs":2,"x":[8,8,0,2,8,8,0,8,0,8,0,8,0,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0x1818","target_demand":2,"target_lhs":0,"x":[8,8,8,0,0,8,8,8,8,8,8,0,0,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0x1842","target_demand":1,"target_lhs":0,"x":[8,0,8,8,8,8,0,8,8,8,8,0,0,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0x1942","target_demand":2,"target_lhs":1,"x":[8,1,8,8,8,8,0,8,0,8,8,0,0,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0x1a98","target_demand":3,"target_lhs":2,"x":[8,8,8,2,0,8,8,0,8,0,8,0,0,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0x1a9c","target_demand":4,"target_lhs":3,"x":[8,8,0,3,0,8,8,0,8,0,8,0,0,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0x1ac2","target_demand":2,"target_lhs":1,"x":[8,1,8,8,8,8,0,0,8,0,8,0,0,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0x1ade","target_demand":6,"target_lhs":5,"x":[8,1,0,3,0,8,0,1,8,0,8,0,0,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0x1bc2","target_demand":3,"target_lhs":2,"x":[8,2,8,8,8,8,0,0,0,0,8,0,0,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0x1bc6","target_demand":4,"target_lhs":3,"x":[8,3,0,8,8,8,0,0,0,0,8,0,0,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0x1c52","target_demand":3,"target_lhs":2,"x":[8,1,8,8,1,8,0,8,8,8,0,0,0,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0x1c59","target_demand":4,"target_lhs":3,"x":[0,8,8,1,1,8,1,8,8,8,0,0,0,8]},{"origin":"new-integer","other_original_rows_checked":279,"target":"0x1c5c","target_demand":4,"target_lhs":3,"x":[8,8,0,2,1,8,0,8,8,8,0,0,0,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0x1edf","target_demand":7,"target_lhs":6,"x":[0,0,0,2,2,8,2,0,8,0,0,0,0,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0x1fcb","target_demand":6,"target_lhs":5,"x":[0,1,8,1,8,8,1,0,2,0,0,0,0,8]},{"origin":"prior-integral","other_original_rows_checked":279,"target":"0x1fce","target_demand":6,"target_lhs":5,"x":[8,1,0,1,8,8,2,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original280-row nonnegative-integer system; outside=8 constructs witnesses, not a global cap.","Every new obstruction exhausts all inside totals<=demand-1; strict-subset source premises yield simultaneous implication by induction.","43 real implications and134 integral source points are authenticated credited reuse, not new coverage.","Five-million total profiles,120000 per row,one-MiB inputs and256-bit input integers/rationals are fail-UNKNOWN capacities.","No smallest arbitrary inequality description, changed cap alphabet, COPY floor, decoder, native supplier, kappa or frontier claim."],"method":"Independent iterative vector-budget DFS verifies sealed primary data; literal all279-original-row integer points; no solver or primary/predecessor executable import","negative_controls":{"altered_prior_point":"literal credited prior integral witness","altered_saved_first_miss_histogram":"exact obstruction miss_histogram","altered_saved_profile_count":"exact obstruction 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diff --git a/research/p112-integer-demand-kernel/PROOF.md b/research/p112-integer-demand-kernel/PROOF.md
new file mode 100644
index 0000000000..dfb88bcac5
--- /dev/null
+++ b/research/p112-integer-demand-kernel/PROOF.md
@@ -0,0 +1,123 @@
+# Exact integer projection of the fixed P112 demand system
+
+The source matrix has 280 distinct nonempty 14-bit masks `H_t`, each with integer
+demand `1 <= d_t <= 8`. Its row is `sum_(k in H_t) x_k >= d_t`. This proof fixes
+the common domain `x in Z^14`, `x >= 0`, with no upper bounds. Source objects are
+authenticated before parsing. The original clique census and geometry-to-demand
+equivalence remain credited PR6425 premises, not new replays.
+
+## Complete finite projection of a counterexample
+
+An integer vector violating target `(I,d)` has
+
+    sum_(k in I) x_k <= d-1 <= 7.
+
+Thus its inside coordinates are one of the finitely many weak compositions with
+total at most `B=d-1`. This is a consequence of a target violation, not an assumed
+global bound on multiplicities.
+
+For any such inside assignment, set every coordinate outside `I` to 8. Every
+other source row either has support contained in `I`, or has an outside bit. The
+latter row has LHS at least 8 and hence satisfies its demand. Distinct masks mean
+that a different row contained in `I` is a **proper** subset row. Consequently a
+global integer counterexample to the target exists if and only if one of the
+inside weak compositions satisfies every proper-subset source row.
+
+The forward implication follows by restricting any global counterexample. For
+the reverse, extend by 8 as above. Each resulting full vector is nevertheless
+checked literally against all 279 other original rows before acceptance as a
+separating point. No inference from a partial point or solver status is used.
+
+## Enumeration and coverage
+
+For `k=|I|`, the number of nonnegative inside vectors of total at most `B` is
+
+    binomial(k+B,k).
+
+Introduce a slack coordinate equal to `B - sum inside` to obtain ordinary weak
+compositions of exactly `B` into `k+1` parts. Stars and bars gives this count.
+
+The primary traverses coordinate multisets of each total `0..B`; counting the
+multiplicity of each coordinate is a bijection to the inside vectors. The
+separate checker uses an iterative vector-budget tree: at each coordinate assign
+an integer from zero through the remaining budget, then continue with the
+remainder. At depth `k`, each inside vector occurs exactly once; leftover budget
+is the implicit slack. These are distinct executable enumerations of the same
+complete domain.
+
+For a negative classification, every profile must be visited and fail at least
+one proper-subset source inequality. The exact closed-form count and the
+histogram of the first failing row are checked. The count or histogram alone is
+not the proof of exhaustive rejection: both implementations regenerate all
+profiles, and the enumeration bijections justify coverage. Positive search may
+stop early only after producing an actual full integer separating point.
+Capacity exhaustion is UNKNOWN and nonzero; it must not produce a complete
+minimum certificate.
+
+## One integer-only implication
+
+One concrete integer-only deletion illustrates the domain change. Target
+`0x1ada` has demand 5. One half of each source row `0x5a` (demand 4), `0x1a98`
+(demand 3), and `0x1ac2` (demand 2) has **exactly** the target coefficient vector
+and RHS `9/2`. Hence its integer LHS is at least `ceil(9/2)=5`. The corresponding
+real-domain separator from PR6455 has target LHS `9/2` and satisfies the other
+279 rows, so deleting this row over the reals would be unsound. The new complete
+projection independently checks all 495 inside integer profiles for this target.
+The half-sum/rounding example is explanatory; the full result uses complete
+negative projections and literal integer separating points.
+
+## Reused predecessor evidence
+
+PR6455 already supplied 43 exact real-domain deletion certificates and 134
+integral separating points. These are validated as data and credited, not new
+integer discoveries. Each reused deletion is checked as a nonnegative rational
+combination of proper-subset source rows whose coefficient vector is at most
+the target and whose RHS is at least its demand. Each reused integral point is
+parsed as integral/nonnegative and rechecked against all 279 other original
+rows. The new classification resolves the remaining 103 labels.
+
+No predecessor executable, LP solver or 262,928-clique census is imported or
+replayed. The earlier real minimum gives a 237-row candidate integer system;
+the 134 literal integral points force at least those 134 rows. These are existing
+coverage bounds, not the new answer.
+
+## Joint deletion and the restricted integer minimum
+
+Let `R` be the new retained rows. Every deleted row is justified either by a
+reused rational implication or by exhaustive rejection of its finite inside
+counterexample domain. Both types use only proper-subset original rows as
+premises. Induct on support cardinality: an integer solution of `R` satisfies
+each earlier proper-subset row, whether retained or already implied. The rational
+implication then establishes a reused target. For a new negative target, a
+violation would give a visited profile failing one of those already-established
+proper-subset rows, a contradiction. Therefore every integer solution of `R`
+satisfies all original rows. The converse is immediate because `R` is a
+subfamily. Simultaneous deletion has no circular premises.
+
+Every retained row has a full nonnegative integer point satisfying all 279
+other originals and violating that row. Any original-row subfamily equivalent
+to the full integer system must contain that row, since otherwise this point
+would satisfy the subfamily. Hence, once all classifications and witnesses pass,
+`R` is the unique minimum-cardinality and unique inclusion-minimal equivalent
+**subfamily of the original 280 inequalities** over the integer orthant.
+
+This does not minimize arbitrary new inequalities. The minimum claim concerns
+unrestricted integer multiplicities: its witness points need not satisfy an
+additional cost/COPY budget or multiplicity cap. The all-orthant equivalence can
+still be conjoined with any identical additional constraints on both systems;
+the minimum property itself need not persist in that restricted domain.
+
+## Scope and credit
+
+This is a source-bound necessary-condition screen. Integer separating points
+are inequality counterexamples, not decoders or physical call constructions.
+Changed supports require a new matrix and extension bound. Coefficients, full
+decoder/COPY pricing, native/access/dirty restoration/precision/tape/stock/compiler
+and outer admission remain outside scope; no κ or frontier change follows.
+
+Credit Nick Petalas/PR6425, Jeremy Diviney/PR1422, Cinna Davis/PR1347/1323 and the
+retained E8/label/decoder lineage. Credit ToikY with OpenAI Codex/PR6455 for the
+real certificates, literal integral points and explicit integer-domain question.
+Weak compositions, monotonicity and finite integer implication are classical.
+New code/proof uses substantial OpenAI Codex assistance. Separate AI checking
+does not constitute human or formal proof-kernel acceptance.
diff --git a/research/p112-integer-demand-kernel/README.md b/research/p112-integer-demand-kernel/README.md
new file mode 100644
index 0000000000..9c89a7e17e
--- /dev/null
+++ b/research/p112-integer-demand-kernel/README.md
@@ -0,0 +1,107 @@
+# An exact 139-row integer P112 demand subfamily
+
+The fixed 280-row, 14-variable P112 demand system has a **unique minimum
+139-row original subfamily over nonnegative integers**, with no upper bounds.
+This resolves [PR6455](https://github.com/CrocSwap/integer-mult-bounds/pull/6455)'s
+integer-only question after its 237-row real-domain minimum.
+
+The new work removes 98 additional rows by complete integer projections and
+finds five new integer separating points. The 43 earlier real-domain deletions
+and 134 existing integral points are authenticated, validated and credited reuse.
+Thus all 103 previously unresolved labels are decided. Every one of the 139
+retained rows has a full integer point satisfying all 279 other original rows
+and violating that row; these force every equivalent original-row subfamily to
+contain all 139. No minimum over arbitrary new inequalities is claimed.
+
+## Why a finite test covers unrestricted integer multiplicities
+
+For a violated target `(I,d)`, integrality forces the total of the target
+coordinates to at most `d-1 <= 7`. Set every outside coordinate to 8: every row
+with an outside bit is then satisfied. The remaining constraints have proper
+subset supports. Exhausting the inside weak compositions therefore decides
+whether a global counterexample exists, despite unrestricted multiplicities.
+Support-cardinality induction makes simultaneous old/new deletions sound.
+`PROOF.md` states the complete projection, enumeration and minimum arguments.
+
+For example, half of source rows `0x5a` (demand 4), `0x1a98` (3) and `0x1ac2`
+(2) gives exactly the coefficient vector of target `0x1ada` with RHS `9/2`.
+Its integer LHS is therefore at least 5, the target's demand. The source's
+real separating point has LHS `9/2`, showing why the row was needed over reals.
+The new projection checks all 495 inside integer profiles for this target.
+
+## Replay and evidence
+
+CPython 3.12 and the standard library suffice; no Git, network, LP solver or
+predecessor executable is required for these two package replays.
+
+```sh
+python3.12 -I -W error -B research/p112-integer-demand-kernel/verify.py
+python3.12 -I -W error -B research/p112-integer-demand-kernel/oracle.py
+```
+
+Both also pass with `-O`. Default commands require complete strict retained
+JSON; `--emit` prints candidates without overwriting files. All three inert
+source objects are authenticated before parsing. The primary uses coordinate
+multisets; the separate checker uses iterative vector-budget DFS and imports
+no primary or predecessor code.
+
+All 98 new negative domains exhaust 336,386 profiles, independently regenerated
+with exact closed-form counts and first-miss histograms. The primary visits
+340,727 profiles across all 103 new targets, including 4,341 positive early-stop
+profiles. The separate checker validates positive witnesses directly and does
+not reproduce that first-positive search order. It checks 16 rejection cases
+and a literal two-dimensional DFS coverage toy. Saved-count/histogram mutations
+test receipt drift; they do not remove profiles from the enumerator.
+
+Primary `profile_sha256` values describe its ordered multiset transcript and
+are regenerated by the primary only. The separate checker ignores these
+optional order-specific hashes and independently regenerates all mathematical
+obstruction fields and histograms. Neither a hash nor a count alone certifies
+exhaustion. Profile/time/input capacities reject instead of silently truncating.
+`VALIDATION.md` records source identities, actual runs and remaining obligations.
+
+## Scope and limits
+
+The result concerns exactly the original PR6425 masks and unrestricted
+`Z^14 >= 0`. The inside total bound is derived from violating a particular row;
+it is not a global cap on multiplicities. The all-orthant equivalence remains
+valid if identical extra constraints are conjoined to both systems. The minimum
+claim itself need not persist under a cost/COPY budget or multiplicity caps,
+because its separating points may fail those extra constraints.
+
+The source clique census and geometry-to-demand equivalence are credited
+predecessor coverage and are not replayed. Passing the screen produces no
+coefficients, decoder, physical calls, COPY floor, paid supplier or κ. Native,
+access, dirty restoration, precision/tape, stock, compiler and outer integration
+remain open. No selected supplier, native, Lean or Docker replay was run.
+
+Canonical orientation: main `7c49fe32c55de02c2c873655c804b16e4c3fef0a`,
+priority revision 8, selected PR6030, selected-record SHA256
+`9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa`.
+The broader `complex-rank-shape-decoder` card is explicitly changed to this
+integer support-screen implication. Main/overlap are refreshed before submission;
+bounded searches and sparse/hidden/unpublished/history gaps are not novelty
+certificates. All five selected framework assumptions remain.
+
+## Open next questions
+
+1. What is the minimum original-row subfamily under one exact cost/COPY budget?
+   State that domain first; the unrestricted witness points do not settle it.
+2. Can a passing integer vector admit actual coefficients and a fully charged
+   decoder? Freeze one vector/interface before synthesis and complete pricing.
+3. For changed support caps, rebuild the complete quotient and establish its
+   valid integer projection/extension bound before reducing rows.
+
+## Credit
+
+Credit Nick Petalas/PR6425 for the original kernel; Jeremy Diviney/PR1422 for
+the guard, geometry/caps and smaller-family question; Cinna Davis/PR1347/1323,
+Navinor, David Arias, Hanson Sun, DreamingOfClouds, themilkman216 and the retained
+E8/label/decoder lineage. Credit ToikY/OpenAI Codex/PR6455 for the real reduction,
+literal integral points and explicit integer-only question. Weak compositions,
+monotonicity and integer implication/rounding are classical.
+
+New work uses substantial OpenAI Codex assistance. Separate AI checking is not
+human review or formal proof-kernel acceptance. `NOTICE` and `SOURCE-PINS.json`
+retain exact source heads, Git blobs and attribution; predecessor files remain
+verbatim. No merge or selected-frontier change is part of this contribution.
diff --git a/research/p112-integer-demand-kernel/RESULT.json b/research/p112-integer-demand-kernel/RESULT.json
new file mode 100644
index 0000000000..b0fd5808f3
--- /dev/null
+++ b/research/p112-integer-demand-kernel/RESULT.json
@@ -0,0 +1 @@
+{"controls":{"actual_witness_loses_strict_violation":{"observed_error":"point must strictly violate target","target":"0x5a"},"full_source_obstructions":98,"inclusive_budget_boundary":{"target":"0x3","witness_first_two_coordinates":[1,0]},"incomplete_count_rejection":"complete weak-composition count","self_premise_rejection":"obstruction premise must be a proper source subset","triangle_real_integer_gap":{"checked_profiles":4,"fractional_point":["1/2","1/2","1/2"],"integer_domain_rejection":"literal nonnegative INTEGER witness","target":"0x7"}},"counts":{"new_integer_only_deletions":98,"new_integer_witnesses":5,"retained":139,"reused_integral_witnesses":134,"reused_real_deletions":43},"declared_caps":{"profiles_per_target":200000,"seconds":180,"total_profiles":30000000},"dimension":14,"exhaustion_encoding":"CWR coordinates ascending; total0..budget. SHA256 starts with p112-miss/v1 NUL,target two-byte big-endian,budget byte; each miss is total byte,allocation bytes,first-missed mask two-byte big-endian. Histogram/count/digest alone are not a proof: full negative domains are enumerated.","limits":["Only103previously unresolved integer labels are newly classified;43real deletion certificates and134actual integral separators are validated and reused as predecessor evidence.","Unique minimum subset of the original280named rows on unrestricted nonnegative INTEGER variables; not a minimum arbitrary inequality description, facet theorem or support-alphabet extension.","Equivalence can be conjoined with the same additional constraints, but indispensability/minimum need not survive added budget constraints that exclude these witnesses.","Original clique census, caps, real280-row source equivalence, coefficients/decoder, COPY floor, native/physical/outer admission and kappa remain 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+      "path": "research/p112-farkas-kernel/RESULT.json",
+      "bytes": 31660,
+      "sha256": "b477f2a8d144ed884c78ee29ac76fd88c95b076eb2dd290ef2431dd7e4969704"
+    },
+    "README-6455.md": {
+      "git_blob": "27aea79adbd206d95ffe94f01fa69d8ca478ab5e",
+      "repository": "ToikY/integer-mult-bounds",
+      "commit": "7fdae743973bebb77daf4508d05655dc1e8915c7",
+      "path": "research/p112-farkas-kernel/README.md",
+      "bytes": 5522,
+      "sha256": "dd2849071a27cf4fa8023a8d3cba767199afeb977cc7e5c5acab361e5006804d"
+    }
+  }
+}
diff --git a/research/p112-integer-demand-kernel/SOURCES.json b/research/p112-integer-demand-kernel/SOURCES.json
new file mode 100644
index 0000000000..d2f056059e
--- /dev/null
+++ b/research/p112-integer-demand-kernel/SOURCES.json
@@ -0,0 +1,19 @@
+{
+  "schema_version": 1,
+  "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+  "files": [
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "d59b5035e7b4cbaa9d688a1d32dd0d6eb79c466a",
+      "sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa"
+    }
+  ],
+  "note": "Canonical metadata is provenance only. Runtime closure is three inert source data/documents plus the new certificate, all package-local and independently authenticated. No predecessor executable or clique census is run."
+}
diff --git a/research/p112-integer-demand-kernel/TASK.json b/research/p112-integer-demand-kernel/TASK.json
new file mode 100644
index 0000000000..2578764d0b
--- /dev/null
+++ b/research/p112-integer-demand-kernel/TASK.json
@@ -0,0 +1,103 @@
+{
+  "schema_version": 1,
+  "task_id": "complex-rank-shape-decoder",
+  "priority_revision": 8,
+  "base_commit": "7c49fe32c55de02c2c873655c804b16e4c3fef0a",
+  "original_card": {
+    "id": "complex-rank-shape-decoder",
+    "question": "Can a legal decoder or coframe change improve the full signed moment, including changed helper targets?",
+    "starting_points": [
+      "docs/research/current-priorities.md",
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+      "docs/research/recursive-invoice-preflight.md"
+    ],
+    "methods": [
+      "Construct and independently check one source-bound instance of the stated first experiment.",
+      "Seek an exact counterexample or target-specific complete-budget obstruction to that same proposed change."
+    ],
+    "acceptance_gate": "A source-bound arbitrary-input identity and legal producer/consumer boundary with the complete paid and productive target invoice, including restoration, ownership, native alignment and extra charges. Price under the actual current model; historical ceiling.py assumptions must be established before use. A favorable optimistic relaxation keeps a route open but is not a completed supplier. A scoped local result remains useful; expansion requires the family-level budget gate.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_selected_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read revision 8, the selected review and live exact-source handoffs. Then follow this card's bounded first_experiment.",
+      "scope": "Source-bound orientation only, not a supplier replay or theorem proof."
+    },
+    "first_experiment": "Price one complete changed itinerary with both paid and productive histograms. Check #5693/#5698 for overlap and their exact hypotheses before another coframe search. Use the signed invoice preflight where its boundary contract agrees; an inconclusive concave comparison is not a power crossing. Stop if an optimistic complete budget fails at the chosen target or native delivery remains unspecified.",
+    "bottleneck_link": "Judge the actual current supplier contract and complete signed target/paid bill; frozen-profile ceilings require an applicability proof.",
+    "allowed_variations": [
+      "One exact source head, helper/cut region or quantified proposition and changed hypothesis; check live handoffs first. Different seeds are not different scopes."
+    ],
+    "scope_change_triggers": [
+      "Changing source head, helper/cut domain, operation alphabet or caller boundary without recording the new scope.",
+      "Treating an archived proposal or local numerical pass as complete supplier admission."
+    ],
+    "overlap_terms": [
+      "coframe",
+      "signed moment",
+      "5693",
+      "5698"
+    ],
+    "existing_checks": [
+      {
+        "scope": "Selected PR5599: complete helper-subspace construction and independent caller replay; five framework assumptions remain.",
+        "evidence": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md",
+        "remaining": "The next experiment must identify uncovered source-specific work. Successor archive intake does not transfer this admission.",
+        "status": "reviewed",
+        "source_commit": "09dd0965711ca7eeb169057a420abab1c9052137",
+        "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "scope_change_reason": "Take PR6455 README next-question1 (TASK entry2): determine the integer-only demand subfamily, rather than synthesize a full rank-shape supplier. The earlier237-row real result leaves103labels unresolved on the integer orthant after43real deletions and134integral witnesses. Reused coverage is checked and credited, not counted as new.",
+  "actual_scope": "Complete integer-only classification of the fixed280original P112 demand inequalities over unrestricted nonnegative integer14-vectors, via exact finite projection and literal separating points.",
+  "bottleneck_link": "Integer multiplicities describe whole support terms. An equivalent smaller necessary-condition screen can simplify support search; it supplies neither coefficients nor a paid physical construction.",
+  "budget_minutes": 60,
+  "stop_rule": "One fixed integer-system result or precise UNKNOWN/partial handoff on capacity exhaustion. Freeze science by2026-11-03T00:59:00Z, reserve at least15minutes for verification/handoff and stop by01:14:00Z. No new alphabets, coefficient synthesis, decoder/COPY pricing or source clique census.",
+  "outcome": "completed",
+  "evidence": "98new integer obstructions exhaust336386profiles in independent vectorDFS;5new integer points plus134credited points force139retained rows.43credited realdeletions are validated. Complete280partition, strict normal/Oreceipts, semanticcontrols and source-bound projection/induction proof; no source executable/census.",
+  "author": "ToikY",
+  "ai_assistance": "Substantial OpenAI Codex assistance for implementation, proof and packaging. Separate checker code/enumeration and static review are AI corroboration, not independent human or formal-kernel acceptance.",
+  "coordination_url": "https://github.com/CrocSwap/integer-mult-bounds/discussions/1166#discussioncomment-19363170",
+  "source_heads": {
+    "original6425": "092dcbd13d1263e691ad29f34fbf286fea1fac2c",
+    "real6455": "7fdae743973bebb77daf4508d05655dc1e8915c7"
+  },
+  "selected_record_sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa",
+  "overlap": {
+    "open_pr_bodies": 425,
+    "nearest_work": [
+      "PR6455 real237minimum+43implications+134integralpoints; integer-onlyminimumexplicitlyuncomputed",
+      "PR6425 original280kernel and source clique equivalence",
+      "PR6430 compatibilityreplay usesanother P112 screen",
+      "PR5767 suppliedLPcertificate semantics, no solverorintegerclassification"
+    ],
+    "limits": "Non-atomic pages deduplicated with freshfirstpage;15allstatequeries includingtruncated100resultbroadquery;6458PR404unknown;sparse/hidden/unpublished/arbitraryhistory gaps remain."
+  },
+  "open_questions": [
+    {
+      "question": "What is the smallest original-row subfamily after adding a concrete COPY/cost upper budget?",
+      "why_it_matters": "All-orthant equivalence remains valid after conjunction, but the unrestricted separating points may violate the budget, so its minimum need not persist.",
+      "first_step": "State one exact cost/budget constraint and derive a new complete counterexample projection respecting it.",
+      "blocked_by": "No extra cost domain is included in this fixed orthant result."
+    },
+    {
+      "question": "Can a passing integer vector admit actual coefficients and a fully charged decoder?",
+      "why_it_matters": "Rank-demand inequalities are necessary conditions and do not construct coefficients or pay native/access/restoration/precision/stock/compiler/outer costs.",
+      "first_step": "Freeze one passing vector and its support/caller interface before coefficient synthesis and full pricing.",
+      "blocked_by": "Coefficient and physical supplier construction are outside this session."
+    },
+    {
+      "question": "Does a changed cap alphabet admit the same type of finite integer projection?",
+      "why_it_matters": "Changing supports changes masks and can change maximum demand and the valid outside extension.",
+      "first_step": "Rebuild the complete changed quotient and verify the new domain and extension bound before reducing rows.",
+      "blocked_by": "Only the exact14source rectangles are in scope."
+    }
+  ]
+}
diff --git a/research/p112-integer-demand-kernel/VALIDATION.md b/research/p112-integer-demand-kernel/VALIDATION.md
new file mode 100644
index 0000000000..170c223276
--- /dev/null
+++ b/research/p112-integer-demand-kernel/VALIDATION.md
@@ -0,0 +1,58 @@
+# Validation and credited reuse
+
+Three inert source objects were frozen before scientific execution:
+
+- Original PR6425 head `092dcbd13d1263e691ad29f34fbf286fea1fac2c` matrix:
+  Git blob `e18f92e001b8ce9d44287760516bf6ea002bcbcd`, SHA256
+  `f8d15306112914af299acca8720779fbc9f6f07bb1336c51e99304fdea5caf89`.
+- Parent PR6455 head `7fdae743973bebb77daf4508d05655dc1e8915c7` real result:
+  Git blob `1ee2f6269556d155b5d265b8c1a0ace6b0e13e22`, SHA256
+  `b477f2a8d144ed884c78ee29ac76fd88c95b076eb2dd290ef2431dd7e4969704`.
+- Parent README: Git blob `27aea79adbd206d95ffe94f01fa69d8ca478ab5e`,
+  SHA256 `dd2849071a27cf4fa8023a8d3cba767199afeb977cc7e5c5acab361e5006804d`.
+- Three-source manifest SHA256
+  `c326b2a83adb02af94954c3904ee6e59822bf5fd274c847851abdcb63c1b9c64`.
+
+The 43 old real deletion certificates and 134 actual integral predecessor
+points are rechecked as data, not counted as new results. No predecessor
+executable, LP or clique census is imported/replayed. The original geometry
+and complete-clique equivalence remain credited predecessor claims when
+composing this literal matrix result with the larger support screen.
+
+The primary first execution passed. Normal/optimized emissions and strict
+defaults passed with identical JSON and empty stderr. Primary certificate:
+50,830 bytes, SHA256
+`f4959ac06a189e817a4fc14c428b9bd02dd9f344f20022cd159d8acfd972fa54`.
+It contains 98 new integer obstructions, five new integer points, 139 retained
+rows, and the explicit prior-coverage partition. Across all 103 new targets it
+visits 340,727 profiles, including 4,341 positive early-stop profiles.
+
+The independent checker was written before reading the new primary code or
+result, then consumes the sealed result as untrusted certificate data. It uses
+iterative vector-budget DFS instead of multiset enumeration, independently
+regenerates all 336,386 profiles in 98 negative domains, and checks all 139
+integer points against all 279 other source rows. Its first execution passed;
+normal/optimized emissions were byte-identical and both strict defaults passed.
+Independent receipt: 48,964 bytes, SHA256
+`5a0ea35df19ee7cc72607fbb4ec8ca51a95ffa1cdea1e73e40214311a5dd47a1`.
+
+There were no scientific execution failures. Before the first independent run,
+static review renamed two control labels to describe altered saved counts and
+histograms accurately. These controls do not mutate DFS traversal. The checker
+has 16 rejection cases plus a separate literal two-dimensional coverage toy.
+Its core obstruction comparison intentionally ignores the primary's optional
+ordered transcript SHA256; that hash is primary-only provenance. Acceptance
+rests on complete regenerated profiles, proper-subset failures, direct integer
+points and the written coverage/induction proof.
+
+An explanatory exact half-sum for target `0x1ada` was checked against the frozen
+matrix: the three donor demands 4, 3 and 2 give RHS `9/2` and exactly the target
+coefficient vector. Integer rounding gives demand 5. This example is not used
+as a substitute for any complete obstruction check.
+
+All assertions used for acceptance are explicit exceptions and remain active
+under optimized Python. `--emit` writes stdout only. Package, source/commit
+mode/history, read-only cold export, external artifact corruption, static
+review and exact-head hosted CI are distinct obligations recorded in session
+receipts and public handoff. No physical supplier, coefficient synthesis,
+cost-constrained minimum, κ or selected-frontier update is asserted.
diff --git a/research/p112-integer-demand-kernel/VERIFY.json b/research/p112-integer-demand-kernel/VERIFY.json
new file mode 100644
index 0000000000..f15b97739c
--- /dev/null
+++ b/research/p112-integer-demand-kernel/VERIFY.json
@@ -0,0 +1,33 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 300,
+      "checks": "Authenticate3frozen sources, validate/reuse43realcertificates and134actualintegralpoints, classifyremaining103 with bounded multiset weak compositions, verify full integer witnesses/allnegative coverage, and strictly compare complete retained RESULT. No predecessor code/census."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-B",
+        "{package}/oracle.py"
+      ],
+      "timeout_seconds": 300,
+      "checks": "Independent vector-budget DFS regenerates every new negative finite obstruction and its exact coverage/histogram; direct source/rational reuse and all279-row integer point checks, semantic controls and strict complete ORACLE receipt. No primarycode imports or LPsolver."
+    }
+  ]
+}
diff --git a/research/p112-integer-demand-kernel/oracle.py b/research/p112-integer-demand-kernel/oracle.py
new file mode 100644
index 0000000000..79137abdf7
--- /dev/null
+++ b/research/p112-integer-demand-kernel/oracle.py
@@ -0,0 +1,393 @@
+#!/usr/bin/env python3
+"""Independent vector-DFS checking of a sealed integer-demand certificate.
+
+The source matrix and real certificates are inert authenticated data. This
+checker imports no predecessor or primary executable and uses no solver.
+Prior real deletions and integral points retain their authors' credit.
+New material: ToikY with OpenAI Codex assistance, Apache-2.0.
+"""
+import argparse
+from copy import deepcopy
+from fractions import Fraction
+import hashlib
+import json
+import math
+from pathlib import Path
+import re
+import sys
+
+ROOT = Path(__file__).resolve().parent
+MANIFEST_SHA = 'c326b2a83adb02af94954c3904ee6e59822bf5fd274c847851abdcb63c1b9c64'
+PINS = {
+    'MATRIX-6425.json': (22234, 'f8d15306112914af299acca8720779fbc9f6f07bb1336c51e99304fdea5caf89',
+                         'e18f92e001b8ce9d44287760516bf6ea002bcbcd'),
+    'REAL-RESULT-6455.json': (31660, 'b477f2a8d144ed884c78ee29ac76fd88c95b076eb2dd290ef2431dd7e4969704',
+                              '1ee2f6269556d155b5d265b8c1a0ace6b0e13e22'),
+    'README-6455.md': (5522, 'dd2849071a27cf4fa8023a8d3cba767199afeb977cc7e5c5acab361e5006804d',
+                       '27aea79adbd206d95ffe94f01fa69d8ca478ab5e'),
+}
+DIMENSION, ROWS = 14, 280
+MAX_BYTES = 1024 * 1024
+MAX_ROW_PROFILES, MAX_TOTAL_PROFILES = 120000, 5000000
+
+
+class CapacityError(ValueError):
+    pass
+
+
+def require(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def capacity(ok, message):
+    if not ok:
+        raise CapacityError(message)
+
+
+def canonical(value):
+    return json.dumps(value, sort_keys=True, separators=(',', ':'), allow_nan=False)
+
+
+def decode(raw):
+    def unique(items):
+        result = {}
+        for key, value in items:
+            require(key not in result, 'duplicate JSON key')
+            result[key] = value
+        return result
+    def bad(value):
+        raise ValueError('nonfinite JSON value: ' + value)
+    def floating(value):
+        number = float(value)
+        require(math.isfinite(number), 'nonfinite JSON number')
+        return number
+    return json.loads(raw, object_pairs_hook=unique, parse_constant=bad, parse_float=floating)
+
+
+def read(path):
+    with path.open('rb') as handle:
+        raw = handle.read(MAX_BYTES+1)
+    capacity(len(raw) <= MAX_BYTES, 'input exceeds one-MiB capacity')
+    return raw
+
+
+def authenticate(values, manifest):
+    require(hashlib.sha256(manifest).hexdigest() == MANIFEST_SHA, 'source manifest SHA256')
+    require(set(values) == set(PINS), 'exact three-source closure')
+    for name, (length, digest, blob) in PINS.items():
+        raw = values[name]
+        require(len(raw) == length, 'source byte length: '+name)
+        require(hashlib.sha256(raw).hexdigest() == digest, 'source SHA256: '+name)
+        framed = b'blob '+str(len(raw)).encode('ascii')+b'\0'+raw
+        require(hashlib.sha1(framed).hexdigest() == blob, 'source Git blob: '+name)
+
+
+def mask(value):
+    require(type(value) is str and 1 <= len(value) <= 6, 'bounded hex mask')
+    try:
+        number = int(value, 16)
+    except ValueError:
+        raise ValueError('invalid hex mask') from None
+    require(value == hex(number) and 0 < number < 1 << DIMENSION, 'canonical nonempty 14-bit mask')
+    return number
+
+
+def rational(value):
+    require(type(value) is str and len(value) <= 512, 'bounded rational string')
+    require(re.fullmatch(r'-?(?:0|[1-9][0-9]*)(?:/[1-9][0-9]*)?', value) is not None, 'rational quotient grammar')
+    number = Fraction(value)
+    capacity(abs(number.numerator).bit_length() <= 256 and number.denominator.bit_length() <= 256,
+             'rational exceeds 256-bit input capacity')
+    require(str(number) == value, 'canonical rational')
+    return number
+
+
+def matrix(raw):
+    source = decode(raw)
+    require(type(source) is dict and source.get('kernel_rows') == ROWS, '280 source rows')
+    require(type(source.get('alphabet')) is list and len(source['alphabet']) == DIMENSION, '14 source columns')
+    records = source.get('kernel')
+    require(type(records) is list and len(records) == ROWS, 'source kernel records')
+    rows = {}
+    for record in records:
+        require(type(record) is list and len(record) == 3, 'source row shape')
+        number = mask(record[0])
+        require(number not in rows and type(record[1]) is int and 1 <= record[1] <= 8, 'source demand/unique mask')
+        rows[number] = record[1]
+    return dict(sorted(rows.items()))
+
+
+def dot(number, vector):
+    return sum(value for column, value in enumerate(vector) if number >> column & 1)
+
+
+def integer_point(target, vector, rows):
+    require(type(vector) is list and len(vector) == DIMENSION, '14 integer coordinates')
+    require(all(type(value) is int and value >= 0 for value in vector), 'nonnegative integer coordinates')
+    capacity(all(value.bit_length() <= 256 for value in vector), 'point exceeds 256-bit input capacity')
+    require(dot(target, vector) < rows[target], 'point must strictly violate target')
+    require(all(dot(other, vector) >= demand for other, demand in rows.items() if other != target),
+            'point violates another original inequality')
+    return {'target': hex(target), 'x': vector, 'target_lhs': dot(target, vector),
+            'target_demand': rows[target], 'other_original_rows_checked': ROWS-1}
+
+
+def old_deletion(record, retained, rows):
+    target = mask(record['target'])
+    require(target in rows and target not in retained, 'old deletion target')
+    terms = record['multipliers']
+    require(type(terms) is list and 1 <= len(terms) <= ROWS, 'old multiplier list')
+    coefficients, rhs, used = [Fraction(0) for _ in range(DIMENSION)], Fraction(0), set()
+    for term in terms:
+        require(type(term) is list and len(term) == 2, 'old multiplier pair')
+        donor, weight = mask(term[0]), rational(term[1])
+        require(donor in retained and donor not in used and weight >= 0, 'old nonnegative unique retained premise')
+        require(weight == 0 or donor != target and donor & target == donor, 'old strict-subset premise')
+        used.add(donor)
+        rhs += weight*rows[donor]
+        for column in range(DIMENSION):
+            coefficients[column] += weight*(donor >> column & 1)
+    require(all(value <= (target >> k & 1) for k, value in enumerate(coefficients)), 'old coefficient overrun')
+    require(rhs >= rows[target], 'old demand implication')
+    return target
+
+
+def prior_data(raw, rows):
+    prior = decode(raw)
+    require(type(prior) is dict and prior.get('dimension') == DIMENSION and prior.get('source_rows') == ROWS, 'prior dimensions')
+    retained = {mask(value) for value in prior['retained']}
+    require(len(retained) == 237 and retained <= rows.keys(), '237 prior retained rows')
+    removed = [old_deletion(record, retained, rows) for record in prior['deletions']]
+    require(len(removed) == 43 and len(set(removed)) == 43 and set(removed) == rows.keys()-retained, '43 prior real deletions')
+    require(type(prior.get('separators')) is list and len(prior['separators']) == 237, '237 prior separator records')
+    integral, seen = {}, set()
+    for record in prior['separators']:
+        target = mask(record['target'])
+        require(target in retained and target not in seen, 'distinct prior separator target')
+        seen.add(target)
+        require(type(record['x']) is list and len(record['x']) == DIMENSION, 'prior separator dimension')
+        vector = [rational(value) for value in record['x']]
+        if all(value.denominator == 1 for value in vector):
+            converted = [value.numerator for value in vector]
+            integer_point(target, converted, rows)
+            integral[target] = converted
+    require(len(integral) == 134, '134 actual integral prior points')
+    return set(removed), integral
+
+
+def profiles(width, budget):
+    """Every nonnegative vector with sum<=budget has one root-to-leaf path."""
+    pending = [(0, budget, ())]
+    while pending:
+        column, remaining, prefix = pending.pop()
+        if column == width:
+            yield prefix
+        else:
+            for value in range(remaining+1):
+                pending.append((column+1, remaining-value, prefix+(value,)))
+
+
+def first_miss(vector, constraints):
+    for number, demand, columns in constraints:
+        if sum(vector[column] for column in columns) < demand:
+            return number
+    return None
+
+
+def obstruction(target, rows, budget_limit=MAX_ROW_PROFILES):
+    coordinates = [column for column in range(DIMENSION) if target >> column & 1]
+    budget = rows[target]-1
+    expected = math.comb(len(coordinates)+budget, len(coordinates))
+    capacity(expected <= budget_limit, 'row profile capacity exceeded')
+    constraints = [(number, demand, tuple(k for k, column in enumerate(coordinates) if number >> column & 1))
+                   for number, demand in rows.items() if number != target and number & target == number]
+    histogram, checked = {}, 0
+    for vector in profiles(len(coordinates), budget):
+        checked += 1
+        capacity(checked <= expected, 'enumerator exceeded closed-form coverage')
+        missed = first_miss(vector, constraints)
+        require(missed is not None, 'obstruction has an integer counterexample')
+        histogram[missed] = histogram.get(missed, 0)+1
+    require(checked == expected, 'enumerator incomplete closed-form coverage')
+    return {'target': hex(target), 'coordinates': coordinates, 'budget': budget,
+            'expected_profiles': expected, 'checked_profiles': checked,
+            'miss_histogram': [[hex(number), count] for number, count in sorted(histogram.items())]}
+
+
+def partition(certificate, rows, old_removed, old_points):
+    require(type(certificate) is dict, 'certificate object')
+    for key, value in [('schema_version', 1), ('dimension', DIMENSION), ('source_rows', ROWS)]:
+        require(type(certificate.get(key)) is int and certificate[key] == value, 'typed '+key)
+    require(certificate.get('source_manifest_sha256') == MANIFEST_SHA, 'certificate source binding')
+    require(type(certificate.get('retained')) is list, 'retained list')
+    ordered = [mask(value) for value in certificate['retained']]
+    retained = set(ordered)
+    require(len(ordered) == len(retained) and old_points.keys() <= retained <= rows.keys(), 'retained distinct/prior coverage')
+    require(type(certificate.get('reused_real_deletions')) is list, 'reused deletion list')
+    reused = [mask(value) for value in certificate['reused_real_deletions']]
+    require(len(reused) == 43 and set(reused) == old_removed, 'exact reused real deletions')
+    require(type(certificate.get('obstructions')) is list and type(certificate.get('witnesses')) is list, 'obstruction/witness lists')
+    removed = [mask(record['target']) for record in certificate['obstructions']]
+    require(len(set(removed)) == len(removed), 'distinct obstruction targets')
+    require(not(retained & old_removed or retained & set(removed) or old_removed & set(removed)), 'disjoint partition')
+    require(retained | old_removed | set(removed) == rows.keys(), 'complete 280-row partition')
+    expected = sum(math.comb(number.bit_count()+rows[number]-1, number.bit_count()) for number in removed)
+    capacity(expected <= MAX_TOTAL_PROFILES, 'total profile capacity exceeded')
+    return retained
+
+
+def positive_checks(records, retained, rows, old_points):
+    require(len(records) == len(retained), 'positive witness count')
+    checked, seen = [], set()
+    for record in records:
+        require(type(record) is dict, 'positive witness record')
+        target = mask(record.get('target'))
+        require(target in retained and target not in seen, 'unique retained witness')
+        seen.add(target)
+        if target in old_points:
+            require(record.get('origin') == 'prior-integral' and record.get('x') == old_points[target], 'literal credited prior integral witness')
+        else:
+            require(record.get('origin') == 'new-integer', 'new integer witness origin')
+        point = integer_point(target, record.get('x'), rows)
+        point['origin'] = record['origin']
+        checked.append(point)
+    return sorted(checked, key=lambda item: mask(item['target']))
+
+
+def verify_certificate(certificate, rows, old_removed, old_points):
+    retained = partition(certificate, rows, old_removed, old_points)
+    points = positive_checks(certificate['witnesses'], retained, rows, old_points)
+    negative = []
+    for record in certificate['obstructions']:
+        target = mask(record['target'])
+        expected = obstruction(target, rows)
+        same_obstruction(record, expected)
+        negative.append(expected)
+    return {'retained': [hex(value) for value in sorted(retained)],
+            'integer_witnesses': points, 'new_obstructions': sorted(negative, key=lambda item: mask(item['target']))}
+
+
+def same_obstruction(record, expected):
+    for key, value in expected.items():
+        require(key in record and canonical(record[key]) == canonical(value), 'exact obstruction '+key)
+
+
+def reject(callback, expected):
+    try:
+        callback()
+    except ValueError as error:
+        require(expected in str(error), 'unexpected rejection reason: '+str(error))
+        return str(error)
+    raise ValueError('negative control accepted: '+expected)
+
+
+def prior_negative_controls(sources, rows):
+    prior = decode(sources['REAL-RESULT-6455.json'])
+    retained = {mask(value) for value in prior['retained']}
+    result = {}
+    bad = deepcopy(prior['deletions'][0])
+    bad['multipliers'][0][1] = '-1'
+    result['old_negative_multiplier'] = reject(lambda: old_deletion(bad, retained, rows), 'old nonnegative')
+    bad = deepcopy(prior['deletions'][0])
+    bad['multipliers'][0][1] = str(rational(bad['multipliers'][0][1])+2)
+    result['old_coefficient_overrun'] = reject(lambda: old_deletion(bad, retained, rows), 'old coefficient overrun')
+    bad = deepcopy(prior['deletions'][0])
+    for term in bad['multipliers']:
+        term[1] = '0'
+    result['old_zero_rhs'] = reject(lambda: old_deletion(bad, retained, rows), 'old demand implication')
+    return result
+
+
+def negative_controls(certificate, rows, old_removed, old_points, sources, manifest, checked):
+    result = {}
+    retained = {mask(value) for value in certificate['retained']}
+    bad = deepcopy(certificate['witnesses'])
+    bad[0]['x'][0] = -1
+    result['negative_point'] = reject(lambda: integer_point(mask(bad[0]['target']), bad[0]['x'], rows), 'nonnegative integer coordinates')
+    target = mask(certificate['witnesses'][0]['target'])
+    result['nonviolating_point'] = reject(lambda: integer_point(target, [8]*DIMENSION, rows), 'strictly violate target')
+    result['other_row_point_violation'] = reject(lambda: integer_point(target, [0]*DIMENSION, rows), 'another original inequality')
+    target = min(old_points)
+    result['false_obstruction'] = reject(lambda: obstruction(target, rows), 'integer counterexample')
+    result['row_capacity_UNKNOWN'] = reject(lambda: obstruction(target, rows, 0), 'profile capacity')
+    bad = deepcopy(certificate)
+    bad['reused_real_deletions'].pop()
+    result['missing_prior_deletion'] = reject(lambda: partition(bad, rows, old_removed, old_points), 'exact reused real deletions')
+    bad = deepcopy(certificate['witnesses'])
+    prior_index = next(i for i, record in enumerate(bad) if mask(record['target']) in old_points)
+    bad[prior_index]['x'] = list(bad[prior_index]['x'])
+    bad[prior_index]['x'][0] += 1
+    result['altered_prior_point'] = reject(lambda: positive_checks(bad, retained, rows, old_points), 'literal credited prior')
+    altered = dict(sources)
+    altered['MATRIX-6425.json'] = b'X'+sources['MATRIX-6425.json'][1:]
+    result['source_byte_mutation'] = reject(lambda: authenticate(altered, manifest), 'source SHA256')
+    result['duplicate_key'] = reject(lambda: decode('{"x":1,"x":2}'), 'duplicate JSON key')
+    result['nonfinite'] = reject(lambda: decode('{"x":1e999}'), 'nonfinite JSON')
+    result['rational_exponent'] = reject(lambda: rational('1e999999'), 'quotient grammar')
+    if checked['new_obstructions']:
+        expected_record = checked['new_obstructions'][0]
+        bad = deepcopy(expected_record)
+        bad['checked_profiles'] -= 1
+        result['altered_saved_profile_count'] = reject(lambda: same_obstruction(bad, expected_record), 'exact obstruction checked_profiles')
+        bad = deepcopy(expected_record)
+        bad['miss_histogram'][0][1] += 1
+        result['altered_saved_first_miss_histogram'] = reject(lambda: same_obstruction(bad, expected_record), 'exact obstruction miss_histogram')
+    else:
+        result['obstruction_mutation_scope'] = 'Not applicable: no new obstruction certificate exists.'
+    expected = {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (2, 0)}
+    observed = list(profiles(2, 2))
+    require(len(observed) == math.comb(4, 2) and set(observed) == expected, 'literal vector-DFS toy coverage')
+    result['literal_vector_DFS_toy'] = {'profiles': sorted(observed), 'count': len(observed)}
+    result.update(prior_negative_controls(sources, rows))
+    return result
+
+
+def compute():
+    manifest = read(ROOT/'SOURCE-PINS.json')
+    sources = {name: read(ROOT/'sources'/name) for name in PINS}
+    authenticate(sources, manifest)
+    rows = matrix(sources['MATRIX-6425.json'])
+    old_removed, old_points = prior_data(sources['REAL-RESULT-6455.json'], rows)
+    raw = read(ROOT/'RESULT.json')
+    certificate = decode(raw)
+    checked = verify_certificate(certificate, rows, old_removed, old_points)
+    controls = negative_controls(certificate, rows, old_removed, old_points, sources, manifest, checked)
+    return {'schema_version': 1, 'status': 'PASS_INDEPENDENT_INTEGER_DEMAND_KERNEL',
+            'source_manifest_sha256': MANIFEST_SHA, 'primary_certificate_sha256': hashlib.sha256(raw).hexdigest(),
+            'dimension': DIMENSION, 'source_rows': ROWS, 'retained_rows': len(checked['retained']),
+            'reused_real_deletions': len(old_removed), 'reused_integral_points': len(old_points),
+            'new_obstructions': len(checked['new_obstructions']),
+            'total_new_obstruction_profiles': sum(record['checked_profiles'] for record in checked['new_obstructions']),
+            'new_integer_points': len(checked['integer_witnesses'])-len(old_points),
+            'method': 'Independent iterative vector-budget DFS verifies sealed primary data; literal all279-original-row integer points; no solver or primary/predecessor executable import',
+            'checked': checked, 'negative_controls': controls,
+            'limits': ['Fixed original280-row nonnegative-integer system; outside=8 constructs witnesses, not a global cap.',
+                       'Every new obstruction exhausts all inside totals<=demand-1; strict-subset source premises yield simultaneous implication by induction.',
+                       '43 real implications and134 integral source points are authenticated credited reuse, not new coverage.',
+                       'Five-million total profiles,120000 per row,one-MiB inputs and256-bit input integers/rationals are fail-UNKNOWN capacities.',
+                       'No smallest arbitrary inequality description, changed cap alphabet, COPY floor, decoder, native supplier, kappa or frontier claim.']}
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--emit', action='store_true')
+    args = parser.parse_args()
+    result = compute()
+    if args.emit:
+        print(canonical(result))
+    else:
+        retained = decode(read(ROOT/'ORACLE.json'))
+        require(canonical(retained) == canonical(result), 'strict complete oracle receipt drift')
+        print('PASS_INDEPENDENT_INTEGER_DEMAND_KERNEL')
+
+
+if __name__ == '__main__':
+    try:
+        main()
+    except CapacityError as error:
+        print('UNKNOWN: '+str(error), file=sys.stderr)
+        sys.exit(1)
+    except (ValueError, KeyError, TypeError, OSError) as error:
+        print('FAIL: '+str(error), file=sys.stderr)
+        sys.exit(1)
diff --git a/research/p112-integer-demand-kernel/sources/MATRIX-6425.json b/research/p112-integer-demand-kernel/sources/MATRIX-6425.json
new file mode 100644
index 0000000000..e18f92e001
--- /dev/null
+++ b/research/p112-integer-demand-kernel/sources/MATRIX-6425.json
@@ -0,0 +1,1493 @@
+{
+  "alphabet": [
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+      "0xb0d020bb0d020b002801022918080",
+      "0xffffffff0000000000000003fffffc"
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+      "0x8ee7188e188e7118258699625a5a58"
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+  "claim_type": "finite certificate and scoped equivalence proof",
+  "cliques_sha256": "45ab61f97a8751af935b9be1e4c6d5e01557e57b9de604fe6a0089d93b419bf9",
+  "complete_demands": 262928,
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+    [
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+  ],
+  "kernel_by_demand": {
+    "1": 7,
+    "2": 21,
+    "3": 57,
+    "4": 71,
+    "5": 34,
+    "6": 57,
+    "7": 28,
+    "8": 5
+  },
+  "kernel_rows": 280,
+  "multiplicity_checks": 16640,
+  "new_kappa": null,
+  "quotient_sha256": "d1cd0103c426ed1465d88c6f4c973a7d86e3329591efd0489a500689a6aa8a07",
+  "signature_rows": 1141,
+  "status": "PASS_P112_DEMAND_KERNEL",
+  "toy_equivalence_checks": 3456
+}
diff --git a/research/p112-integer-demand-kernel/sources/README-6455.md b/research/p112-integer-demand-kernel/sources/README-6455.md
new file mode 100644
index 0000000000..27aea79adb
--- /dev/null
+++ b/research/p112-integer-demand-kernel/sources/README-6455.md
@@ -0,0 +1,99 @@
+# A 237-row exact P112 demand system
+
+For the fixed 14-variable, 280-row cap-demand matrix of Nick Petalas's
+[PR6425](https://github.com/CrocSwap/integer-mult-bounds/pull/6425), exact
+nonnegative rational combinations remove 43 rows. The retained 237 rows are
+equivalent for every nonnegative real multiplicity vector. Each retained row has
+a rational point satisfying all 279 other original rows and strictly violating
+that row. Together these certificates establish the **unique minimum subfamily
+of the original 280 rows**, with `x >= 0` fixed.
+
+This answers PR6425's first Open next question. Its pairwise reduction, source
+clique census and geometry-to-demand equivalence remain credited predecessor
+results; no predecessor executable or census is replayed. The present result is
+an exact reduction of the literal source matrix. It supplies no coefficients,
+decoder, physical supplier, COPY floor or κ improvement.
+
+## Concrete implication
+
+The first deleted mask is `0x1d1a`, demand 3. One half of each of the retained
+rows `0x1412`, `0x1818` and `0xd0a`, each demand 2, gives RHS 3. The resulting
+coefficient is 1 on target coordinates 1, 3, 4, 10, 11 and 12, and one half on
+coordinate 8 (zero-based); all are at most the target's coefficients. The
+missing half of coordinate 8 is supplied by `x_8 >= 0`. No single retained row
+dominates this target: the source family was already pairwise nondominated.
+`RESULT.json` retains all 43 exact combinations and all 237 full separating
+points; `PROOF.md` gives the universal implication and restricted minimum proof.
+
+## Replay
+
+CPython 3.12 and the standard library suffice. No Git executable, network, solver
+package or predecessor executable is needed for the package's two replays.
+
+```sh
+python3.12 -I -W error -B research/p112-farkas-kernel/verify.py
+python3.12 -I -W error -B research/p112-farkas-kernel/oracle.py
+```
+
+Also run with `-O`. Default commands require strict complete retained JSON;
+`--emit` prints regenerated output without overwriting files. The primary
+regenerates the certificate with bounded exact `Fraction` packing simplex and
+checks every implication and separator literally. The separate checker treats
+the sealed primary result as untrusted certificate data and computes all
+coefficient/RHS budgets and full-original-row separations independently. It does
+not import the generator, solve an LP or claim blind rediscovery.
+
+The generator uses 2,272 pivots, at most 49 per target, on this exact instance.
+Explicit time/pivot/coefficient bounds fail closed. These are generation counts,
+not a general runtime improvement. `VERIFY.json` registers the two bounded checks.
+`SOURCE-PINS.json` authenticates both inert source files before parsing them.
+`SOURCES.json` pins the originating current-main card/selected record as
+provenance; all runtime bytes are package-local.
+
+## Scope, evidence and limitations
+
+The equivalence covers nonnegative real vectors and therefore preserves all
+nonnegative integer tests too. Fractional separating points do **not** establish
+integer-only indispensability or the smallest integer-only screen. The minimum
+claim is over subfamilies of these original inequalities; it does not minimize
+arbitrary newly introduced normals or alternative descriptions.
+
+The independent checker exercises actual coefficient, demand, premise and
+separator corruptions, along with source/schema controls. `VALIDATION.md`
+records the actual run evidence, corrections and remaining obligations. Passing
+this finite necessary-condition screen does not construct coefficient vectors
+or pay for access, dirty restoration, precision, tape, stock, compiler or outer
+integration. No native, Lean, Docker, selected cold supplier or complete
+multiplication replay was run.
+
+Orientation was frozen at canonical main
+`7c49fe32c55de02c2c873655c804b16e4c3fef0a`, priority revision 8,
+selected PR6030, selected-record SHA256
+`9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa`.
+The broader `complex-rank-shape-decoder` card is explicitly changed to this fixed
+support-screen implication. Main/overlap are rechecked before submission;
+sparse/history/hidden/unpublished coverage limits are not novelty evidence.
+
+## Open next questions
+
+1. Can integer multiplicity implications remove further rows? Specify the
+   integer domain and produce integer implication/separation certificates;
+   fractional witnesses alone do not answer that question.
+2. For changed support caps, what is the new exact demand screen? Rebuild the
+   complete changed quotient before applying or attempting new combinations.
+3. Can a passing vector admit actual coefficients and a fully charged decoder?
+   Specify one vector and its full interface before synthesis and pricing.
+
+## Credit
+
+Credit Nick Petalas/PR6425 for the fixed matrix, pairwise reduction and explicit
+combined-implication question; Jeremy Diviney/PR1422 for the complete-clique
+guard, pinned geometry/caps and smaller-family question; Cinna Davis/PR1347/1323
+and Navinor, David Arias, Hanson Sun, DreamingOfClouds, themilkman216 and the
+retained E8/label/decoder lineage. PR5767 is related exact-certificate tooling;
+its code and 128-variable cap are unchanged. Nonnegative implication,
+packing/covering duality, simplex and Bland's rule are classical.
+
+New work is ToikY with substantial OpenAI Codex assistance. Separate AI
+certificate checking is not independent human review or formal proof-kernel
+acceptance. `NOTICE` preserves attribution; the two source files remain verbatim.
diff --git a/research/p112-integer-demand-kernel/sources/REAL-RESULT-6455.json b/research/p112-integer-demand-kernel/sources/REAL-RESULT-6455.json
new file mode 100644
index 0000000000..1ee2f62695
--- /dev/null
+++ b/research/p112-integer-demand-kernel/sources/REAL-RESULT-6455.json
@@ -0,0 +1 @@
+{"controls":{"actual_combination_coefficient_overflow":{"observed_error":"Farkas coefficient overflow","target":"0x1d1a"},"actual_combination_half_demand":{"mutant_rhs":"3/2","observed_error":"Farkas demand shortfall","target":"0x1d1a"},"actual_separator_made_target_feasible":{"coordinate":1,"observed_error":"separator must strictly violate target","target":"0x5a"},"outside_extension_required":{"restricted_zero_point_error":"separator violates another original row"},"two_row_combination":{"multipliers":[["0x1","1"],["0x2","1"]],"target":"0x3"}},"counts":{"deleted":43,"retained":237,"strict_separators":237},"declared_caps":{"coefficient_bits":4096,"per_target_pivots":2000,"seconds":120,"total_pivots":100000},"deletions":[{"multipliers":[["0x1412","1/2"],["0x1818","1/2"],["0xd0a","1/2"]],"target":"0x1d1a"},{"multipliers":[["0x1448","1/2"],["0xd0a","1/2"],["0x1942","1/2"]],"target":"0x1d4a"},{"multipliers":[["0x1448","1/2"],["0x1818","1/2"],["0xd50","1/2"]],"target":"0x1d58"},{"multipliers":[["0x5a","1/2"],["0x2518","1/2"],["0x2542","1/2"]],"target":"0x255a"},{"multipliers":[["0x5a","1/2"],["0x310a","1/2"],["0x3150","1/2"]],"target":"0x315a"},{"multipliers":[["0x1412","1/2"],["0x2418","1/2"],["0x302a","1/2"]],"target":"0x343a"},{"multipliers":[["0x1448","1/2"],["0x2418","1/2"],["0x3070","1/2"]],"target":"0x3478"},{"multipliers":[["0x1412","1/2"],["0x2518","1/2"],["0x310a","1/2"]],"target":"0x351a"},{"multipliers":[["0x1448","1/2"],["0x2542","1/2"],["0x310a","1/2"]],"target":"0x354a"},{"multipliers":[["0x1412","1/2"],["0x2542","1/2"],["0x3150","1/2"]],"target":"0x3552"},{"multipliers":[["0x1448","1/2"],["0x2518","1/2"],["0x3150","1/2"]],"target":"0x3558"},{"multipliers":[["0x5a","1/2"],["0x1513","1/2"],["0x1549","1/2"]],"target":"0x155b"},{"multipliers":[["0x5a","1/2"],["0x1516","1/2"],["0x154c","1/2"]],"target":"0x155e"},{"multipliers":[["0x1448","1/2"],["0xe8a","1/2"],["0x1ac2","1/2"]],"target":"0x1eca"},{"multipliers":[["0x5a","1/2"],["0x2519","1/2"],["0x2543","1/2"]],"target":"0x255b"},{"multipliers":[["0x5a","1/2"],["0xf8a","1/2"],["0xfd0","1/2"]],"target":"0xfda"},{"multipliers":[["0x1412","1/2"],["0x1a98","1/2"],["0xf8a","1/2"]],"target":"0x1f9a"},{"multipliers":[["0x1448","1/2"],["0xf8a","1/2"],["0x1bc2","1/2"]],"target":"0x1fca"},{"multipliers":[["0x1412","1/2"],["0xfd0","1/2"],["0x1bc2","1/2"]],"target":"0x1fd2"},{"multipliers":[["0x1448","1/2"],["0x1a98","1/2"],["0xfd0","1/2"]],"target":"0x1fd8"},{"multipliers":[["0x1818","1/2"],["0x2912","1/2"],["0x310a","1/2"]],"target":"0x399e"},{"multipliers":[["0x1818","1/2"],["0x2912","1/2"],["0x310a","1/2"]],"target":"0x3b1e"},{"multipliers":[["0x1818","1/2"],["0x2912","1/2"],["0x310a","1/2"]],"target":"0x3b9a"},{"multipliers":[["0xc51","1/2"],["0x3070","1/2"],["0x3c21","1/2"]],"target":"0x3c75"},{"multipliers":[["0x5a","1/4"],["0x351f","1/4"],["0x354f","1/4"],["0x3557","1/4"],["0x355d","1/4"]],"target":"0x355f"},{"multipliers":[["0x5a","1/4"],["0x353b","1/4"],["0x356b","1/4"],["0x3573","1/4"],["0x3579","1/4"]],"target":"0x357b"},{"multipliers":[["0x5a","1/4"],["0x353e","1/4"],["0x356e","1/4"],["0x3576","1/4"],["0x357c","1/4"]],"target":"0x357e"},{"multipliers":[["0x2912","1/2"],["0x310a","1/2"],["0x1a9c","1/2"]],"target":"0x3b9e"},{"multipliers":[["0x310a","1/3"],["0x1549","1/3"],["0x2543","1/3"],["0x344b","1/3"]],"target":"0x3dcb"},{"multipliers":[["0x302a","1/2"],["0xe8a","1/2"],["0x3c21","1/2"]],"target":"0x3eab"},{"multipliers":[["0x302a","1/2"],["0xe8a","1/2"],["0x3c24","1/2"]],"target":"0x3eae"},{"multipliers":[["0xc51","1/2"],["0x3070","1/2"],["0x3c21","1/2"]],"target":"0x3ef1"},{"multipliers":[["0x310a","1/3"],["0x1549","1/3"],["0x2543","1/3"],["0x344b","1/3"]],"target":"0x3f4b"},{"multipliers":[["0x5a","1/4"],["0x1f9b","1/4"],["0x1fcb","1/4"],["0x1fd3","1/4"],["0x1fd9","1/4"]],"target":"0x1fdb"},{"multipliers":[["0x5a","1/4"],["0x1f9e","1/4"],["0x1fce","1/4"],["0x1fd6","1/4"],["0x1fdc","1/4"]],"target":"0x1fde"},{"multipliers":[["0x5a","1/4"],["0x2f9b","1/4"],["0x2fcb","1/4"],["0x2fd3","1/4"],["0x2fd9","1/4"]],"target":"0x2fdb"},{"multipliers":[["0x1a9c","1/2"],["0x3174","1/2"],["0x2be8","1/2"]],"target":"0x3bfc"},{"multipliers":[["0x312e","1/2"],["0xf8b","1/2"],["0x3ea5","1/2"]],"target":"0x3faf"},{"multipliers":[["0x3174","1/2"],["0xfd1","1/2"],["0x3ea5","1/2"]],"target":"0x3ff5"},{"multipliers":[["0x2463","1"],["0x1a9c","1"]],"target":"0x3eff"},{"multipliers":[["0x2543","1"],["0x1a9c","1"]],"target":"0x3fdf"},{"multipliers":[["0x2afa","1/3"],["0x3579","1/3"],["0x1fcb","1/3"],["0x3fb3","1/3"]],"target":"0x3ffb"},{"multipliers":[["0x2afa","1/3"],["0x357c","1/3"],["0x1fce","1/3"],["0x3fb6","1/3"]],"target":"0x3ffe"}],"dimension":14,"limits":["Equivalence and irredundancy of this frozen real nonnegative-variable system only; retained separators may be fractional, so integer irredundancy is not claimed.","Original clique census, support geometry and source280-row equivalence remain credited PR6425 premises, not independently replayed here.","No globally smallest arbitrary inequality description, new cap alphabet, coefficients/decoder, physical COPY bound, kappa, native/outer admission or frontier change."],"method":"Exact Fraction packing simplex; feasible slack basis and Bland entering/leaving rules, including nonbasic slacks. 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diff --git a/research/p112-integer-demand-kernel/verify.py b/research/p112-integer-demand-kernel/verify.py
new file mode 100644
index 0000000000..9bc05980d1
--- /dev/null
+++ b/research/p112-integer-demand-kernel/verify.py
@@ -0,0 +1,324 @@
+#!/usr/bin/env python3
+"""Finite integer-only classification of one frozen280-row demand system.
+
+Nick Petalas/6425, ToikY/6455 and PR1422/support lineage retain their credit.
+New finite certificate: ToikY with OpenAI Codex assistance. Apache-2.0.
+No source executable, LP, floating arithmetic or optional dependency is used.
+"""
+import argparse
+from collections import Counter
+from fractions import Fraction as F
+import hashlib
+from itertools import combinations_with_replacement
+import json
+from math import comb
+from pathlib import Path
+import time
+
+HERE = Path(__file__).resolve().parent
+MANIFEST_SHA = 'c326b2a83adb02af94954c3904ee6e59822bf5fd274c847851abdcb63c1b9c64'
+DIM, NROWS, OUTSIDE = 14, 280, 8
+CAPS = {'seconds': 180, 'total_profiles': 30000000, 'profiles_per_target': 200000}
+
+
+def require(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def canonical(value):
+    return json.dumps(value, sort_keys=True, separators=(',', ':'), allow_nan=False)
+
+
+def decode(raw):
+    def unique(items):
+        result = {}
+        for key, value in items:
+            require(key not in result, 'duplicate JSON key')
+            result[key] = value
+        return result
+    def bad(value):
+        raise ValueError('nonfinite JSON: ' + value)
+    return json.loads(raw, object_pairs_hook=unique, parse_constant=bad)
+
+
+def authenticated():
+    raw_manifest = (HERE / 'SOURCE-PINS.json').read_bytes()
+    require(hashlib.sha256(raw_manifest).hexdigest() == MANIFEST_SHA, 'source manifest SHA256')
+    pins = decode(raw_manifest)['files']
+    require(set(pins) == {'MATRIX-6425.json', 'REAL-RESULT-6455.json', 'README-6455.md'}, 'three exact source files')
+    raw = {}
+    for name, pin in pins.items():
+        value = (HERE / 'sources' / name).read_bytes()
+        require(len(value) == pin['bytes'], 'source byte length: ' + name)
+        require(hashlib.sha256(value).hexdigest() == pin['sha256'], 'source SHA256: ' + name)
+        require(hashlib.sha1(b'blob ' + str(len(value)).encode() + b'\0' + value).hexdigest() == pin['git_blob'], 'source Git blob: ' + name)
+        raw[name] = value
+    return decode(raw['MATRIX-6425.json']), decode(raw['REAL-RESULT-6455.json'])
+
+
+def mask_value(text):
+    require(type(text) is str, 'hex support string')
+    mask = int(text, 16)
+    require(0 < mask < (1 << DIM) and text == hex(mask), 'canonical nonempty14-bit support')
+    return mask
+
+
+def rational(text):
+    require(type(text) is str, 'canonical Fraction string')
+    value = F(text)
+    require(str(value) == text, 'canonical Fraction representation')
+    return value
+
+
+def bits(mask):
+    return [k for k in range(DIM) if mask & (1 << k)]
+
+
+def order(mask):
+    return mask.bit_count(), mask
+
+
+def dot(mask, x):
+    return sum(x[k] for k in bits(mask))
+
+
+def matrix_rows(obj):
+    require(type(obj.get('kernel_rows')) is int and obj['kernel_rows'] == NROWS, '280 original rows')
+    require(type(obj.get('alphabet')) is list and len(obj['alphabet']) == DIM, '14 source variables')
+    require(type(obj.get('kernel')) is list and len(obj['kernel']) == NROWS, 'literal source kernel')
+    rows = {}
+    for entry in obj['kernel']:
+        require(type(entry) is list and len(entry) == 3, 'source kernel row schema')
+        mask = mask_value(entry[0])
+        require(mask not in rows, 'unique source support')
+        demand = entry[1]
+        require(type(demand) is int and 1 <= demand <= OUTSIDE, 'source demand1..8')
+        require(type(entry[2]) is str and entry[2] == hex(int(entry[2], 16)) and int(entry[2], 16) > 0, 'source witness identifier')
+        rows[mask] = demand
+    return rows
+
+
+def real_separator(target, x, rows):
+    require(target in rows and len(x) == DIM and all(value >= 0 for value in x), 'nonnegative real point')
+    require(all(dot(mask, x) >= demand for mask, demand in rows.items() if mask != target), 'point misses another original row')
+    require(dot(target, x) < rows[target], 'point must strictly violate target')
+
+
+def integer_separator(target, x, rows):
+    require(type(x) is list and len(x) == DIM and all(type(value) is int and value >= 0 for value in x), 'literal nonnegative INTEGER witness')
+    real_separator(target, x, rows)
+
+
+def validate_old_deletions(parent, retained, rows):
+    deleted = []
+    require(type(parent['deletions']) is list and len(parent['deletions']) == 43, '43 legacy real deletions')
+    for record in parent['deletions']:
+        target = mask_value(record['target'])
+        require(target in rows and target not in retained and target not in deleted, 'legacy deletion partition')
+        coverage, demand, seen = [F(0)] * DIM, F(0), set()
+        require(type(record['multipliers']) is list and record['multipliers'], 'nonempty legacy multiplier list')
+        for raw_donor, raw_weight in record['multipliers']:
+            donor, weight = mask_value(raw_donor), rational(raw_weight)
+            require(donor in retained and donor not in seen and donor != target and donor & target == donor and weight >= 0, 'proper-subset nonnegative legacy donor')
+            seen.add(donor)
+            demand += weight * rows[donor]
+            for k in bits(donor):
+                coverage[k] += weight
+        require(all(v <= int(bool(target & (1 << k))) for k, v in enumerate(coverage)) and demand >= rows[target], 'legacy Farkas combination')
+        deleted.append(target)
+    require(set(deleted) | retained == set(rows), 'complete prior real partition')
+    return sorted(deleted)
+
+
+def parent_data(parent, rows):
+    require(type(parent.get('dimension')) is int and parent['dimension'] == DIM and type(parent.get('source_rows')) is int and parent['source_rows'] == NROWS, 'parent dimensions')
+    require(type(parent.get('retained')) is list and len(parent['retained']) == 237, '237 legacy real retained rows')
+    sequence = [mask_value(text) for text in parent['retained']]
+    retained = set(sequence)
+    require(len(retained) == 237 and retained <= set(rows), 'unique legacy retained masks')
+    deleted = validate_old_deletions(parent, retained, rows)
+    require(type(parent['separators']) is list and len(parent['separators']) == 237, 'all237 legacy separators')
+    integral, all_points = {}, {}
+    for record in parent['separators']:
+        target = mask_value(record['target'])
+        require(target in retained and target not in all_points and type(record['x']) is list, 'legacy separator target')
+        x = [rational(text) for text in record['x']]
+        real_separator(target, x, rows)
+        all_points[target] = x
+        if all(value.denominator == 1 for value in x):
+            point = [int(value) for value in x]
+            integer_separator(target, point, rows)
+            integral[target] = point
+    require(set(all_points) == retained and len(integral) == 134, '134 actual prior integral points')
+    unresolved = sorted(retained - set(integral), key=order)
+    require(len(unresolved) == 103, '103 integer-only questions beyond predecessor')
+    return deleted, integral, unresolved
+
+
+class Budget:
+    def __init__(self):
+        self.started = time.monotonic()
+        self.profiles = 0
+
+    def check(self):
+        require(time.monotonic() - self.started <= CAPS['seconds'], 'UNKNOWN: finite classification time cap')
+        require(self.profiles <= CAPS['total_profiles'], 'UNKNOWN: finite classification profile cap')
+
+    def tick(self):
+        self.profiles += 1
+        require(self.profiles <= CAPS['total_profiles'], 'UNKNOWN: finite classification profile cap')
+        if self.profiles % 1024 == 0:
+            self.check()
+
+
+def obstruction_shape(record, rows):
+    target = mask_value(record['target'])
+    coordinates, bound = bits(target), rows[target] - 1
+    expected = comb(len(coordinates) + bound, len(coordinates))
+    require(record['coordinates'] == coordinates and record['budget'] == bound, 'obstruction domain')
+    require(type(record['expected_profiles']) is int and record['expected_profiles'] == expected and type(record['checked_profiles']) is int and record['checked_profiles'] == expected, 'complete weak-composition count')
+    total, seen = 0, set()
+    for raw_mask, count in record['miss_histogram']:
+        premise = mask_value(raw_mask)
+        require(premise in rows and premise != target and premise & target == premise and premise not in seen, 'obstruction premise must be a proper source subset')
+        require(type(count) is int and count > 0, 'positive miss count')
+        seen.add(premise)
+        total += count
+    require(total == expected, 'miss histogram covers complete domain')
+
+
+def classify(target, rows, budget):
+    coordinates, bound = bits(target), rows[target] - 1
+    expected = comb(len(coordinates) + bound, len(coordinates))
+    require(expected <= CAPS['profiles_per_target'], 'UNKNOWN: target domain cap')
+    proper = [(mask, rows[mask], tuple(int(bool(mask & (1 << k))) for k in range(DIM)))
+              for mask in sorted(rows) if mask != target and mask & target == mask]
+    histogram, checked = Counter(), 0
+    transcript = hashlib.sha256(b'p112-miss/v1\0' + target.to_bytes(2, 'big') + bytes((bound,)))
+    budget.check()
+    for total in range(bound + 1):
+        for allocation in combinations_with_replacement(coordinates, total):
+            budget.tick()
+            checked += 1
+            missing = next((mask for mask, demand, members in proper if total < demand or sum(members[k] for k in allocation) < demand), None)
+            if missing is None:
+                x = [OUTSIDE] * DIM
+                for k in coordinates:
+                    x[k] = 0
+                for k in allocation:
+                    x[k] += 1
+                integer_separator(target, x, rows)
+                return x, None, checked
+            histogram[missing] += 1
+            transcript.update(bytes((total,)) + bytes(allocation) + missing.to_bytes(2, 'big'))
+    require(checked == expected, 'actual complete weak-composition enumeration')
+    record = {'target': hex(target), 'coordinates': coordinates, 'budget': bound,
+              'expected_profiles': expected, 'checked_profiles': checked,
+              'miss_histogram': [[hex(mask), count] for mask, count in sorted(histogram.items())],
+              'profile_sha256': transcript.hexdigest()}
+    obstruction_shape(record, rows)
+    budget.check()
+    return None, record, checked
+
+
+def generate(rows, parent):
+    legacy, integral, unresolved = parent_data(parent, rows)
+    witnesses = {mask: {'target': hex(mask), 'x': x, 'origin': 'prior-integral'} for mask, x in integral.items()}
+    obstructions, statistics, budget = [], [], Budget()
+    for target in unresolved:
+        x, obstruction, checked = classify(target, rows, budget)
+        statistics.append([hex(target), checked, 'witness' if x is not None else 'exhausted'])
+        if x is not None:
+            witnesses[target] = {'target': hex(target), 'x': x, 'origin': 'new-integer'}
+        else:
+            obstructions.append(obstruction)
+    removed = {mask_value(record['target']) for record in obstructions}
+    retained = set(witnesses)
+    require(not(retained & set(legacy) or retained & removed or set(legacy) & removed), 'disjoint complete classification')
+    require(retained | set(legacy) | removed == set(rows), 'all280 original rows classified')
+    require(134 <= len(retained) <= 237 and len(witnesses) - 134 + len(obstructions) == 103, 'new-only classification counts')
+    for target, witness in witnesses.items():
+        integer_separator(target, witness['x'], rows)
+    for record in obstructions:
+        obstruction_shape(record, rows)
+    budget.check()
+    return legacy, witnesses, sorted(obstructions, key=lambda record: int(record['target'], 16)), statistics, budget
+
+
+def rejects(action):
+    try:
+        action()
+    except ValueError as error:
+        return str(error)
+    raise ValueError('invalid proof object accepted')
+
+
+def controls(rows, witnesses, obstructions):
+    triangle = {3: 1, 5: 1, 6: 1, 7: 2}
+    x, obstruction, _checked = classify(7, triangle, Budget())
+    require(x is None and obstruction['checked_profiles'] == 4, 'triangle integer-only obstruction')
+    fractional = [F(0)] * DIM
+    fractional[:3] = [F(1, 2)] * 3
+    real_separator(7, fractional, triangle)
+    integer_error = rejects(lambda: integer_separator(7, fractional, triangle))
+    boundary = {1: 1, 3: 2}
+    x, obstruction2, _checked = classify(3, boundary, Budget())
+    require(obstruction2 is None and dot(3, x) == 1, 'inclusive d-1 boundary witness')
+    corrupted = dict(obstruction)
+    corrupted['checked_profiles'] -= 1
+    count_error = rejects(lambda: obstruction_shape(corrupted, triangle))
+    corrupted = dict(obstruction)
+    corrupted['miss_histogram'] = [['0x7', 4]]
+    circular_error = rejects(lambda: obstruction_shape(corrupted, triangle))
+    target = min(witnesses)
+    changed = list(witnesses[target]['x'])
+    changed[bits(target)[0]] += rows[target]
+    strict_error = rejects(lambda: integer_separator(target, changed, rows))
+    return {'triangle_real_integer_gap': {'target': '0x7', 'checked_profiles': 4,
+                                          'fractional_point': ['1/2', '1/2', '1/2'], 'integer_domain_rejection': integer_error},
+            'inclusive_budget_boundary': {'target': '0x3', 'witness_first_two_coordinates': x[:2]},
+            'incomplete_count_rejection': count_error, 'self_premise_rejection': circular_error,
+            'actual_witness_loses_strict_violation': {'target': hex(target), 'observed_error': strict_error},
+            'full_source_obstructions': len(obstructions)}
+
+
+def run():
+    source_matrix, parent = authenticated()
+    rows = matrix_rows(source_matrix)
+    legacy, witnesses, obstructions, statistics, budget = generate(rows, parent)
+    retained = sorted(witnesses)
+    return {'schema_version': 1, 'source_manifest_sha256': MANIFEST_SHA,
+            'dimension': DIM, 'source_rows': len(rows), 'retained': list(map(hex, retained)),
+            'reused_real_deletions': list(map(hex, legacy)),
+            'witnesses': [witnesses[mask] for mask in retained], 'obstructions': obstructions,
+            'counts': {'retained': len(retained), 'reused_real_deletions': len(legacy),
+                       'reused_integral_witnesses': 134, 'new_integer_witnesses': len(retained) - 134,
+                       'new_integer_only_deletions': len(obstructions)},
+            'search_statistics': {'new_targets': 103, 'profiles_checked': budget.profiles, 'per_target': statistics},
+            'declared_caps': CAPS, 'controls': controls(rows, witnesses, obstructions),
+            'exhaustion_encoding': 'CWR coordinates ascending; total0..budget. SHA256 starts with p112-miss/v1 NUL,target two-byte big-endian,budget byte; each miss is total byte,allocation bytes,first-missed mask two-byte big-endian. Histogram/count/digest alone are not a proof: full negative domains are enumerated.',
+            'limits': ['Only103previously unresolved integer labels are newly classified;43real deletion certificates and134actual integral separators are validated and reused as predecessor evidence.',
+                       'Unique minimum subset of the original280named rows on unrestricted nonnegative INTEGER variables; not a minimum arbitrary inequality description, facet theorem or support-alphabet extension.',
+                       'Equivalence can be conjoined with the same additional constraints, but indispensability/minimum need not survive added budget constraints that exclude these witnesses.',
+                       'Original clique census, caps, real280-row source equivalence, coefficients/decoder, COPY floor, native/physical/outer admission and kappa remain separate.']}
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--emit', action='store_true')
+    args = parser.parse_args()
+    result = run()
+    if args.emit:
+        print(canonical(result))
+    else:
+        path = HERE / 'RESULT.json'
+        require(path.is_file(), 'retained RESULT.json required')
+        observed = decode(path.read_bytes())
+        require(type(observed) is dict and type(observed.get('schema_version')) is int and observed['schema_version'] == 1, 'typed integer certificate receipt')
+        require(canonical(observed) == canonical(result), 'strict retained complete integer certificate')
+        print('PASS_P112_INTEGER_DEMAND_KERNEL')
+
+
+if __name__ == '__main__':
+    main()