From 3bceac90599a02a96e9b744736b3feeb00171504 Mon Sep 17 00:00:00 2001
From: Ali Romman <226877337+aliromman@users.noreply.github.com>
Date: Tue, 13 Oct 2026 19:53:55 -0500
Subject: [PATCH] Prove fixed R480 helper cleanup boundary obstruction

---
 .../CERTIFICATE.json                          | 18496 ++++++++++++++++
 .../r480-source-independent-cleanup/README.md |   478 +
 .../SOURCES.json                              |    66 +
 .../r480-source-independent-cleanup/TASK.json |    66 +
 .../VERIFY.json                               |    24 +
 .../r480-source-independent-cleanup/verify.py |   538 +
 6 files changed, 19668 insertions(+)
 create mode 100644 research/r480-source-independent-cleanup/CERTIFICATE.json
 create mode 100644 research/r480-source-independent-cleanup/README.md
 create mode 100644 research/r480-source-independent-cleanup/SOURCES.json
 create mode 100644 research/r480-source-independent-cleanup/TASK.json
 create mode 100644 research/r480-source-independent-cleanup/VERIFY.json
 create mode 100644 research/r480-source-independent-cleanup/verify.py

diff --git a/research/r480-source-independent-cleanup/CERTIFICATE.json b/research/r480-source-independent-cleanup/CERTIFICATE.json
new file mode 100644
index 0000000000..bb4914f883
--- /dev/null
+++ b/research/r480-source-independent-cleanup/CERTIFICATE.json
@@ -0,0 +1,18496 @@
+{
+  "status": "PASS_R480_SOURCE_INDEPENDENT_CLEANUP",
+  "coordinates": 712,
+  "cleanup_macros": 6680,
+  "cleanup_nonzeros": 3908,
+  "inverse_nonzeros": 10595,
+  "final_helpers": 402,
+  "final_helper_profile": {
+    "1": 260,
+    "2": 36,
+    "3": 64,
+    "4": 2,
+    "5": 40
+  },
+  "final_helper_mass": 732,
+  "shared_word_identical_directional_climbs": 3486,
+  "isolated_nonfinal_helpers": [
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+    275,
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+              "basis_and_inverse_adds": 8,
+              "basis_and_inverse_scales": 2
+            },
+            "helpers": {
+              "rank": 0
+            }
+          },
+          "direct_target_repair_rank": 5
+        },
+        {
+          "helpers": [
+            333
+          ],
+          "omitted_ranks": [
+            5
+          ],
+          "modes": {
+            "physical_X": {
+              "rank": 1,
+              "pivot": 22,
+              "u": {
+                "22": "1",
+                "109": "-1"
+              },
+              "v": {
+                "22": "1/2",
+                "34": "-1/2",
+                "50": "1/2",
+                "109": "-1/2"
+              },
+              "basis_and_inverse_adds": 8,
+              "basis_and_inverse_scales": 2
+            },
+            "helpers": {
+              "rank": 0
+            }
+          },
+          "direct_target_repair_rank": 5
+        },
+        {
+          "helpers": [
+            345
+          ],
+          "omitted_ranks": [
+            5
+          ],
+          "modes": {
+            "physical_X": {
+              "rank": 1,
+              "pivot": 86,
+              "u": {
+                "86": "1",
+                "93": "1"
+              },
+              "v": {
+                "28": "-1/2",
+                "58": "-1/2",
+                "86": "1/2",
+                "93": "1/2"
+              },
+              "basis_and_inverse_adds": 8,
+              "basis_and_inverse_scales": 2
+            },
+            "helpers": {
+              "rank": 0
+            }
+          },
+          "direct_target_repair_rank": 5
+        },
+        {
+          "helpers": [
+            349
+          ],
+          "omitted_ranks": [
+            5
+          ],
+          "modes": {
+            "physical_X": {
+              "rank": 1,
+              "pivot": 28,
+              "u": {
+                "28": "1",
+                "58": "-1"
+              },
+              "v": {
+                "28": "1/2",
+                "58": "-1/2",
+                "75": "-1/2",
+                "82": "1/2"
+              },
+              "basis_and_inverse_adds": 8,
+              "basis_and_inverse_scales": 2
+            },
+            "helpers": {
+              "rank": 0
+            }
+          },
+          "direct_target_repair_rank": 5
+        },
+        {
+          "helpers": [
+            361
+          ],
+          "omitted_ranks": [
+            5
+          ],
+          "modes": {
+            "physical_X": {
+              "rank": 1,
+              "pivot": 29,
+              "u": {
+                "29": "1",
+                "59": "1"
+              },
+              "v": {
+                "29": "1/2",
+                "30": "-1/2",
+                "59": "1/2",
+                "60": "-1/2"
+              },
+              "basis_and_inverse_adds": 8,
+              "basis_and_inverse_scales": 2
+            },
+            "helpers": {
+              "rank": 0
+            }
+          },
+          "direct_target_repair_rank": 5
+        },
+        {
+          "helpers": [
+            365
+          ],
+          "omitted_ranks": [
+            5
+          ],
+          "modes": {
+            "physical_X": {
+              "rank": 1,
+              "pivot": 30,
+              "u": {
+                "30": "1",
+                "60": "-1"
+              },
+              "v": {
+                "30": "1/2",
+                "60": "-1/2",
+                "73": "-1/2",
+                "84": "1/2"
+              },
+              "basis_and_inverse_adds": 8,
+              "basis_and_inverse_scales": 2
+            },
+            "helpers": {
+              "rank": 0
+            }
+          },
+          "direct_target_repair_rank": 5
+        },
+        {
+          "helpers": [
+            377
+          ],
+          "omitted_ranks": [
+            5
+          ],
+          "modes": {
+            "physical_X": {
+              "rank": 1,
+              "pivot": 8,
+              "u": {
+                "8": "1",
+                "26": "1"
+              },
+              "v": {
+                "8": "1/2",
+                "26": "1/2",
+                "66": "-1/2",
+                "95": "-1/2"
+              },
+              "basis_and_inverse_adds": 8,
+              "basis_and_inverse_scales": 2
+            },
+            "helpers": {
+              "rank": 0
+            }
+          },
+          "direct_target_repair_rank": 5
+        },
+        {
+          "helpers": [
+            381
+          ],
+          "omitted_ranks": [
+            5
+          ],
+          "modes": {
+            "physical_X": {
+              "rank": 1,
+              "pivot": 66,
+              "u": {
+                "66": "1",
+                "95": "-1"
+              },
+              "v": {
+                "3": "-1/2",
+                "66": "1/2",
+                "67": "1/2",
+                "95": "-1/2"
+              },
+              "basis_and_inverse_adds": 8,
+              "basis_and_inverse_scales": 2
+            },
+            "helpers": {
+              "rank": 0
+            }
+          },
+          "direct_target_repair_rank": 5
+        },
+        {
+          "helpers": [
+            393
+          ],
+          "omitted_ranks": [
+            5
+          ],
+          "modes": {
+            "physical_X": {
+              "rank": 1,
+              "pivot": 65,
+              "u": {
+                "65": "1",
+                "96": "1"
+              },
+              "v": {
+                "64": "-1/2",
+                "65": "1/2",
+                "96": "1/2",
+                "97": "-1/2"
+              },
+              "basis_and_inverse_adds": 8,
+              "basis_and_inverse_scales": 2
+            },
+            "helpers": {
+              "rank": 0
+            }
+          },
+          "direct_target_repair_rank": 5
+        },
+        {
+          "helpers": [
+            397
+          ],
+          "omitted_ranks": [
+            5
+          ],
+          "modes": {
+            "physical_X": {
+              "rank": 1,
+              "pivot": 64,
+              "u": {
+                "64": "1",
+                "97": "-1"
+              },
+              "v": {
+                "5": "-1/2",
+                "64": "1/2",
+                "69": "1/2",
+                "97": "-1/2"
+              },
+              "basis_and_inverse_adds": 8,
+              "basis_and_inverse_scales": 2
+            },
+            "helpers": {
+              "rank": 0
+            }
+          },
+          "direct_target_repair_rank": 5
+        }
+      ],
+      "independent_target_modes": 19,
+      "direct_off_block_rank": {
+        "off_block_equations": 10995,
+        "rank": 383,
+        "nullity": 19
+      },
+      "one_tail_cost_screen": {
+        "baseline_ranks": [
+          5,
+          2
+        ],
+        "replacement_local_ranks": [
+          4,
+          1,
+          1
+        ],
+        "deferred_helper_rank": 1,
+        "host_ranks_with_strictly_worse_moment": [
+          0,
+          1,
+          2,
+          3,
+          4,
+          5,
+          6,
+          7,
+          8
+        ],
+        "uniform_b_range": "0 < b < 1",
+        "native_child_max_rank": 8,
+        "all_hosted_remainder_ranks": {
+          "0": 59250,
+          "1": 9106,
+          "3": 754,
+          "7": 10
+        },
+        "unhosted_rank48_children": 4866,
+        "affected_pair_remainder_patterns": [
+          {
+            "ranks": [
+              0,
+              0,
+              0
+            ],
+            "instances": 110
+          },
+          {
+            "ranks": [
+              0,
+              0,
+              0,
+              0
+            ],
+            "instances": 59
+          },
+          {
+            "ranks": [
+              0,
+              0,
+              0,
+              1
+            ],
+            "instances": 1
+          },
+          {
+            "ranks": [
+              0,
+              0,
+              1
+            ],
+            "instances": 3
+          },
+          {
+            "ranks": [
+              0,
+              0,
+              1,
+              1
+            ],
+            "instances": 1
+          },
+          {
+            "ranks": [
+              0,
+              1,
+              0
+            ],
+            "instances": 2
+          },
+          {
+            "ranks": [
+              0,
+              1,
+              1
+            ],
+            "instances": 2
+          },
+          {
+            "ranks": [
+              1,
+              0,
+              0
+            ],
+            "instances": 3
+          },
+          {
+            "ranks": [
+              1,
+              0,
+              0,
+              0
+            ],
+            "instances": 1
+          },
+          {
+            "ranks": [
+              1,
+              0,
+              1,
+              1
+            ],
+            "instances": 1
+          },
+          {
+            "ranks": [
+              1,
+              1,
+              1
+            ],
+            "instances": 8
+          },
+          {
+            "ranks": [
+              1,
+              1,
+              1,
+              1
+            ],
+            "instances": 1
+          }
+        ],
+        "not_covered": "Joint multi-call encodings, changed owners, or changed native cleanup maps."
+      }
+    }
+  },
+  "source_dependent_escape": {
+    "helper": 224,
+    "source": 0,
+    "coefficient": "1"
+  },
+  "existing_low_end_control": 273,
+  "negative_and_boundary_controls": [
+    "removed_source_coefficient_rejected",
+    "source_dependent_forward_escape_attained",
+    "nonidentity_forward_frame_rejected_by_source_independence",
+    "same_frame_rejected_by_backward_physical_X_target_outputs",
+    "backward_target_covariance_escape_attained",
+    "unequal_paired_frame_rejected_by_output_block_contract",
+    "isolated_existing_low_end_can_carry_a_frame",
+    "omitted_private_clear_detected_on_private_column"
+  ],
+  "private_clear_control_event": 12551,
+  "accounting": {
+    "local_profile": {
+      "1": 2412,
+      "2": 418,
+      "3": 432,
+      "4": 142,
+      "5": 82,
+      "8": 8
+    },
+    "local_mass": 5586,
+    "calls": 144,
+    "global_profile": {
+      "1": 349604,
+      "2": 60192,
+      "3": 62585,
+      "4": 20448,
+      "5": 11808,
+      "7": 7,
+      "8": 1152,
+      "48": 4866
+    },
+    "complex_lower": "67018507072324140917/40000000000000000000000",
+    "bit_lower": "1733427039589844234749/1000000000000000000000000",
+    "kappa": "836330098538459/500000000000000000",
+    "deferred_climbs_in_theorem_class": 0,
+    "new_kappa": false
+  }
+}
diff --git a/research/r480-source-independent-cleanup/README.md b/research/r480-source-independent-cleanup/README.md
new file mode 100644
index 0000000000..69dc6cc1dc
--- /dev/null
+++ b/research/r480-source-independent-cleanup/README.md
@@ -0,0 +1,478 @@
+# Source-independent boundary obstruction for the fixed R480 cleanup
+
+**Status: scoped mathematical obstruction for the actual selected supplier.
+No new kappa.** This is an all-batch, all-depth statement for the interface
+below, with exact finite checks of its source-bound matrix premises. It is
+not a proof that every persistent-workspace interface fails.
+
+The selected R480 word has **402 final helper climbs with total rank 732**.
+Each of those helper rows in the inverse cleanup has a nonzero coefficient
+on a real source input. Thus no nonidentity transform on any of these helpers
+can pass the unchanged cleanup as a source-independent helper frame. The
+output helper frame may mix all helpers. Real data outputs retain their
+canonical values. Forward target-only covariance is an optional extension;
+backward target covariance has 22 algebraic survivors. Those survivors do
+not give an edit to the selected shared-word provider, whose same final
+climbs must also pass the forward contract. Independent backward circuits
+with paid output repairs are treated as a separate extension below.
+
+The result closes a specific route from the sign/cyclic contracts of
+PR1227/PR1287 to the selected supplier. Those contracts return workspace as a
+function of its incoming workspace, independently of the current source.
+They do not provide the source-history interface needed to escape this result.
+
+## Declared class
+
+Write the forward full-call scalar contract as
+
+    (X, Y, H) -> (X, Y + X, H),
+
+with 112 sources, 112 targets and 480 arbitrary helper inputs. The private
+centre inputs are separate; this package preserves and checks all eight.
+The backward orientation is the inverse transpose of the persistent block.
+
+The class changes the placement of final helper climbs and the helper/target
+frame types at complete call boundaries. It has these requirements:
+
+1. Keep the factory, source reads, target reads, private centre paths, helper
+   inventory, and data geometry fixed. Keep the physical cleanup map
+   `C_o tensor I` in its original common FULL frame. It may use any scalar
+   factorization in that same frame. There are no native frame changes,
+   recursive children, or register-dependent transports inside the cleanup.
+2. Model any omitted or postponed part of a final climb, in full-frame
+   coordinates, by an invertible operator `T_h` on that helper's fiber. The
+   source and target climbs stay fixed. The operators can have any dimension
+   and need not commute. They are not limited to signs or permutations.
+3. At each complete call, helper state has a linear invertible type `V H`
+   that is independent of current sources and targets. `V` may mix every
+   helper and fiber coordinate, and can differ at each boundary. Both data
+   banks have canonical values. Helper-only adapters may occur before the
+   fixed prefix or after the cleanup; they are part of the boundary types
+   and are charged. No such adapter occurs between an omitted climb and
+   the cleanup. The types do not mix data or stored source history into
+   helper state.
+4. Each typed call must satisfy its contract on arbitrary source, target and
+   helper inputs. Internal readers receive the specified values. Correctness
+   is not required only at the last output of a specially correlated batch.
+
+This is the original claim's canonical-real-output interface with arbitrary
+source-independent helper types. It tests the workspace part of the cited
+sign and cyclic proposals on a real supplier. Their output covariance is
+examined separately below. A changed physical cleanup, source-history
+interface, or data/helper mixing is a change to this class. Arbitrary paid
+native cleanup redesign is not excluded.
+
+## The source-column lemma
+
+Let `V` be a nonzero vector space over a characteristic-zero field. Lift a
+scalar invertible matrix `C` to the register space tensored with `V`. Let `S`
+be source register indices and `J` the candidate helper indices. Put
+
+    D = diag(I on all other registers, T_h for h in J).
+
+Suppose an output map `E` fixes each source subspace pointwise and
+
+    C D = E C.
+
+If `(C^-1)[h,s] != 0` for some `s in S`, then `T_h = I`.
+
+**Proof.** For every `v in V`, apply the equivalent identity
+`D C^-1 = C^-1 E` to `e_s tensor v`. Since `E` fixes that vector, the helper
+`h` component is
+
+    (C^-1)[h,s] T_h v = (C^-1)[h,s] v.
+
+The scalar coefficient is nonzero. Cancel it to get `T_h v = v` for every
+`v`, and hence `T_h = I`. This applies to all candidate helpers at once.
+No independence of their source witnesses is assumed. One source may witness
+many helpers. No count of distinct source copies is used. QED.
+
+The lemma allows any invertible output map fixing the source axes. In
+particular, it allows dense helper mixing and even target/helper mixing at
+this one cleanup cut. A source-dependent output map need not fix those axes;
+Section "Source-history boundary" states the exact resulting exception.
+
+## Instantiation and composition
+
+Let `C` be the selected forward `full_frame_cleanup` map on all 704 ordinary
+registers. It has 3,908 nonzero entries. Its inverse has 10,595. The cleanup
+fixes every target row and column. It does not touch the private centres.
+All 402 helpers with a positive final climb have a source witness in `C^-1`.
+[CERTIFICATE.json](CERTIFICATE.json) gives a source index and nonzero exact
+rational coefficient for each helper. It also gives the current final frame
+and its paid rank.
+
+Use boundary coordinates before the fixed helper post-adapters that follow
+the scalar cleanup. These adapters are invertible and helper-only; changing
+these coordinates preserves the stated type class. Their retained charges
+remain in the bill. Let `F_o` be the full native endpoint on the 704
+persistent registers in those coordinates, in orientation `o`, and `P_o`
+its fixed prefix. The retained native interface
+has separate data and helper blocks: `F_o = A_o direct_sum K_o`, with
+invertible `K_o`. The scalar endpoint checked below is its logical row map;
+it must not replace `K_o` by I in a physical calculation. The fixed cleanup
+gives `P_o = C_o^-1 F_o`. Final climbs
+act on distinct helper registers. In common full-frame coordinates, their
+omission or postponement therefore inserts exactly the block-diagonal map
+`D` between `P_o` and `C_o`. This is the reduction to `C_o D`; it uses the
+physical common-frame restriction in item 1. A logical scalar refactorization
+at different native frames would not justify this reduction.
+
+Write incoming and outgoing types as `B_in = I_data direct_sum V_in` and
+`B_out = I_data direct_sum V_out`. A typed call must satisfy
+
+    C_o D C_o^-1 F_o B_in = B_out F_o.
+
+Conjugation by `F_o` preserves the helper-only block:
+
+    F_o B_in F_o^-1 = I_data direct_sum (K_o V_in K_o^-1).
+
+Cancel the invertible maps to obtain
+
+    C_o D = E C_o,
+    E = B_out F_o B_in^-1 F_o^-1.
+
+Thus E fixes every data axis and row. No commutation of the native helper
+map `K_o` with a dense helper type is assumed. The private centre inputs
+are cleared and are not part of this invertible persistent block.
+
+For a forward call the source-column lemma now forces all 402 `T_h = I`.
+The coefficient premise is also checked directly from the full emitted
+prefix using integer dyadic rows on all 712 columns, independently of the
+cleanup inversion.
+
+For a backward call, `C_o = C^-T`. Every candidate helper column has a
+nonzero row in the **physical X bank**, since
+`(C^-T)[s,h] = (C^-1)[h,s]`. Physical X is the backward **target**, not its
+logical source. Both real data banks are canonical in the declared class.
+Taking those X rows in the covariance identity gives
+`(C^-T)[s,h](T_h-I)=0`, and again all 402 `T_h = I`.
+
+The reduction holds for every incoming helper type, so it applies at every
+completed call in any batch and at every finite recursive depth. Equivalently,
+induct on the number of completed typed calls. The proof does not assume
+independent source copies between calls. It uses the universal arbitrary-input
+contract required at each call; source reuse does not weaken that contract.
+
+Forward target-only covariance can be added: on source-only input, all helper
+and target inputs are zero, and the cleanup fixes the target block. The
+non-target output must still be the original source with zero helpers. The
+same source-column argument applies. This extension does not transfer to the
+backward target bank; the exact surviving covariance is stated below.
+
+A nontrivial native climb must have a nonidentity missing operator for this
+statement to charge it. The theorem concerns this operator, not a scalar
+projection that replaces every recursive child by the identity. Its exact
+finite checks establish the matrix coefficients used in the uniform proof;
+they do not replay a native transform at every size.
+
+### Corollary for the selected shared-word provider
+
+The selected emitter produces the forward and backward programs from one
+canonical word by the same-order inverse-transpose rule. All 3,486 climb
+events agree exactly between those emitted programs. This includes all 402
+final helper climbs; the verifier checks the complete event lists.
+
+Suppose a change keeps that shared-word interface and changes only the final
+helper climbs. Suppose also that its forward output remains a
+source-independent helper frame, with optional target-only covariance as
+above. The forward theorem forces every omitted tail to be the identity.
+The identical climb edit therefore cannot give a backward-only saving in
+that provider, even if its backward target output may change frame.
+
+This is a corollary for the coupled provider. It is not an extra restriction
+on independently authored directional circuits. Such circuits, or a paid
+source-dependent repair of the forward word, change this interface.
+
+## Source-history boundary
+
+Drop source independence, keep **both real data banks canonical**, and allow a general
+helper output `Q H + L X`. At a cleanup cut the unique transported map is
+`E = C_o D C_o^-1`. Its physical X rows are canonical if and only if
+
+    C_o[X,h] (T_h - I) = 0 for every candidate helper h.
+
+This condition is componentwise in the input register index. Therefore the
+complete one-cut classification is:
+
+| Direction with canonical X and Y | Forced final helpers | Helpers that can carry X history | Rank mass of those helpers |
+|---|---:|---:|---:|
+| Forward | 61 | 341 | 579 |
+| Backward | 402 | 0 | 0 |
+
+The 341 forward helpers have profile `{1:243, 2:8, 3:64, 4:2, 5:24}`.
+This is a necessary and sufficient algebraic classification of the stated
+one-cut map, not a native or hosted construction. Choosing `T_h = 2 I` for
+one such helper gives an invertible example. The exact defect is
+
+    E - I = C[:,h] (T_h-I) C^-1[h,:].
+
+Its source rows vanish but its helper/source block does not. This is a
+concrete escape from the source-independent premise. The verifier checks it
+on the actual R480 matrix. Existing isolated low ends also admit a frame,
+but they have no final climb to defer.
+
+### Backward target covariance is a real escape
+
+Permit an arbitrary output frame on physical X, which is the backward target,
+and a separate arbitrary helper frame. The complete algebraic equation is
+
+    C^-T D = diag(U_X, I_Y, Q_H) C^-T.
+
+It has exactly 19 free operator components on 22 final helpers. They are
+`[234,237]`, `[250,253]`, `[266,269]`, and the 16 singletons
+`281,285,297,301,313,317,329,333,345,349,361,365,377,381,393,397`.
+Within a component the operators `T_h` must be equal. Every other final
+helper operator must be the identity. The free helpers have profile
+`{2:3,5:19}`, with mass 101. A common nonidentity scalar transform on the
+first pair is an exact, nontrivial block-output covariance control.
+
+Here is a complete linear-algebra classification, not a finite parameter
+search. For each output block X and H, take the corresponding columns of
+`C^-T`. Columns with input in X are fixed anchors because `D_X=I`. Choose a
+column basis, first from those anchors and then from helper columns. An
+output operator fixes every anchor basis vector and acts on a helper basis
+vector by that helper's `T_h`. Expand every other helper column in the basis.
+Each nonzero basis coefficient forces its `T_h` to equal the associated
+helper operator, or to equal I when the basis vector is an anchor. These
+conditions are necessary by uniqueness of basis coordinates and sufficient
+by defining the output operator on that basis. Merge the equalities from
+both output blocks. The exact certificate gives every basis and expansion;
+its graph has one fixed component and precisely the 19 free components above.
+The target block has 86 independent anchors, and the helper block has 58.
+
+The basis calculation uses the 402 final helpers. Each of the other 78
+helpers is an isolated identity row and column of C and its inverse, and D
+is the identity there. The verifier checks every such row and column and
+lists all 78 helpers. Thus those coordinates add no fixed-anchor constraint.
+Independently, the direct off-block equations use all 704 ordinary registers.
+Their rank is 383 on the 402 variable operators. The 19 component indicator
+vectors are independent solutions, so this second calculation also proves
+that the listed solutions exhaust the full boundary equation.
+
+These are arbitrary invertible operator covariances. The next section checks
+their native ranks and the cost of a single transported tail. The main
+all-batch theorem is not asserted for the larger class with paid target
+adapters and mixed helper modes.
+
+### Native ranks and mandatory output repairs
+
+Write `T_U` for the retained exact Clifford representative of a frame U,
+and F for its FULL representative. A missing tail after frame W is
+`T_W F^-1`. Its Fourier rank is `9-dim(W)`. This follows from the retained
+preimage-frame distance formula. Equality of two such exact operators
+forces equality of their frame labels, so unequal Fourier ranks cannot
+give equal missing operators.
+
+Each singleton has a rank-five final climb. Its transported projector has
+rank one on X and zero on helpers. Omitting this climb therefore moves the
+same rank-five operator onto one independent target mode. A direct native
+repair on that mode has rank five. The sixteen singleton target modes are
+independent. This direct repair keeps their child profile unchanged; it is
+not a lower bound against a different multi-stream recursive interface.
+
+Each paired component contains one rank-five and one rank-two climb. Full
+omission is impossible because covariance requires equal missing operators.
+For monotone partial deferral, both helpers must reach a common frame W
+that contains their current frames. The current dimensions are 4 and 7;
+their intersection has dimension 3 and their sum has dimension 8. Hence
+the only nontrivial common tail is rank one, after their sum frame. The
+replacement prefix ranks are 4 and 1. The transported operator has one
+target mode and one helper mode. Canonical target output requires the
+inverse rank-one target transport. One rank-one helper transport remains.
+
+The certificate gives all three sum frames and all nineteen target
+projectors. For each nonzero rank-one projector P, it gives `P=u v^T` with
+`v^T u=1`. An in-place scalar change of basis isolates that mode. Choose p
+with `v_p != 0`, put `a=v^T x`, store a in register p, then set
+`x_i <- x_i-u_i a` for `i != p`. The inverse scalar word restores the
+original basis. Conjugating one native T by these two words implements
+`I+P tensor (T-I)`. No fresh register is required. The scalar ADD and SCALE
+counts are in the certificate. This gives an exact native local output
+repair using the retained Clifford interface; it does not give a later
+host for the remaining helper transport. These projectors are expressed at
+the common FULL-frame cleanup cut. A later physical placement must include
+the fixed post-adapters and each owner's chart. Scalar covariance does not
+remove those operations.
+
+### A complete one-tail cost screen on the retained owners
+
+The favorable bill for one pair is
+
+    old: [5,2,q],       new: [4,1,1,q+1].
+
+The new rank-one child is the target repair. The remaining rank-one helper
+tail is granted free access to, and exact fusion with, a rank-q child.
+Every scalar adapter, copy, chart change, and cone check is granted free.
+These grants can only favor the proposed saving. A rank-zero host means
+that no host child exists and restoration is a separate rank-one child.
+
+For `s=1-b` with `0<b<1`, let `f(r)=r^s`. The moment difference before
+division by `m^s W` is
+
+    Delta = 2-2^s -(5^s-4^s) + ((q+1)^s-q^s).
+
+For `0 <= q <= 8`, convexity in s gives `2-2^s >= b`. Integration of
+`f'(x)=s x^-b` gives
+
+    5^s-4^s <= s 4^-b,
+    (q+1)^s-q^s >= s 9^-b.
+
+Also `4^-b-9^-b <= b log(9/4)` and `log(9/4)<1`. Thus
+
+    Delta >= b [1-s log(9/4)] > 0.
+
+This is a strict mathematical obstruction for every single-tail repair in
+this favorable host class, not a sampled moment test. The retained local
+child ranks are at most 8. Every idle interval that hosts any helper has
+remainder rank 0, 1, 3, or 7. All 4,866 rank-48 idle children belong to
+intervals that host no helper. On the actual affected helper support, the
+192 backward pair instances have only rank-zero or rank-one remainders.
+The verifier checks those complete finite owner lists.
+
+A larger host can change the conclusion. A numerical screen at the retained
+b gives a lower favorable bill at q=18 and q=48. Neither is a construction.
+Reassigning owners, carrying several independent modes, or transporting a
+dense helper encoding through other calls requires a separate operator and
+cost argument. The selected intervals have at most six backward helper
+calls, but that count alone does not exclude relaying an encoding between
+families. This package does not claim an all-batch cost lower bound for
+those protocols.
+
+Target/helper mixing before a later call can also put current source
+responses into helper storage. Source-history access, cross-block adapters,
+changed readers, and batch-only cancellation remain outside the proved
+class. They need their native phases, cones, restoration, precision and
+hosted children priced explicitly.
+
+The final overlap check also found PR1391 and PR1410. They implement and
+compose an order-four toy boundary, respectively. Neither instantiates the
+selected R480 source/reader interface. The source-independent theorem here
+concerns that actual interface. PR1420 instead concerns projected XOR cleanup
+for the bit supplier, and PR1416 concerns leaf-error growth. These scopes do
+not replace this classification. The check covered the latest 20 updated
+open PRs and the latest 10 coordination comments, in addition to the initial
+coordinator snapshot; it is not a claim of exhaustive literature priority.
+A final check of the ten latest updated PRs found PR1439's paid source-mutation
+connector and PR1442's delivery-cut optimum. Their stated scopes do not
+instantiate the selected supplier. They concern broader interfaces that this
+source-independent theorem leaves open; this package does not use their results.
+
+## Kappa accounting and stopping decision
+
+The exact optimum number of deferred final climbs in the declared class is
+**zero**, attained by the retained supplier. Its final-helper profile is
+`{1:260, 2:36, 3:64, 4:2, 5:40}`, with rank mass 732. The rest of the local
+word is fixed. Its full local profile remains
+`{1:2412, 2:418, 3:432, 4:142, 5:82, 8:8}`, with mass 5,586.
+
+Splitting a required native climb into smaller children cannot lower its
+moment. If its frame distance is r and a native factorization has ranks
+`r_1,...,r_k`, the distance triangle inequality gives `sum r_i >= r`.
+For `0<s<1`, subadditivity gives `sum r_i^s >= (sum r_i)^s >= r^s`.
+The retained direct child attains this lower bound. This also fixes the
+optimum if the declared class allows such factorizations before the cleanup.
+
+On the retained 144-call owner construction, the global profile remains
+`{1:349604, 2:60192, 3:62585, 4:20448, 5:11808, 7:7, 8:1152, 48:4866}`.
+The fixed supplier moment, bit supplier, and outer witness therefore do not
+improve. They retain complex endpoint
+`0.001675462676808103522925`, bit endpoint
+`0.001733427039589844234749`, and selected
+`kappa = 836330098538459/500000000000000000`.
+Any extra paid adapter adds nonnegative work. An adapter that performs the
+missing nonidentity child has restored that work rather than deferred it
+through this contract. This is not an optimality theorem for arbitrary new
+paid adapters or different owner maps.
+
+The initial favorable screen was PR1012's historical mass-neutral T37
+helper-deferral ceiling, `0.001708640379500299`; ideal rehosting reached the
+retained outer bit cap near `0.001730427469823602`. It was not a price of this
+selected R480 word. The actual source-bound obstruction was found before
+attempting a new native construction. Graph connectivity was an initial
+check, but the proof uses inverse source coefficients and permits dense
+helper mixing. The source-history escape was investigated and retained as
+a scope boundary. No failed search is used as evidence of impossibility.
+
+## Reproduce and verification limits
+
+From the repository root:
+
+```sh
+python3 -I -O -B research/r480-source-independent-cleanup/verify.py
+```
+
+Expected: `PASS_R480_SOURCE_INDEPENDENT_CLEANUP helpers=402 mass=732
+columns=712 forward_history_survivors=341 backward_canonical_history_survivors=0
+backward_target_covariance_helpers=22 no_new_kappa` on one line. The verifier
+uses the standard library.
+
+It reconstructs both exact cleanup matrices from the emitted orientations,
+checks both matrix inverse identities, independently replays both complete
+scalar programs with integer dyadic rows on all 712 inputs and outputs,
+checks all 402 source witnesses, and checks the full one-cut survivor sets.
+The backward covariance is checked twice: column-basis relations give its
+components, and direct off-block equations have rank 383 on 402 variables.
+The resulting nullity 19 agrees with those independent component vectors.
+The verifier also checks the native pair frames, target-mode independence,
+shared-word climb coupling, and all selected owner remainders.
+Controls reject a missing source coefficient and two actual transported
+frames at the wrong boundary, exhibit source-history and backward target-frame
+escapes and an isolated low end, and detect a removed
+private-centre CLEAR on its private input column.
+
+Source files and archive members are pinned in [SOURCES.json](SOURCES.json).
+The exact base is `bce54fb3f895cbe5cd4c0eb477c3afeed100190f`. The input is the
+selected PR1094 archive itself, at its recorded base bytes, with audited
+source head `0fc329d0bec93ca7cd288eaea1295172e8789810`. The source receipt
+contains the retained local programs and canonical word.
+
+No selected construction, frontier record, frozen source, or `upstream/`
+file changes. No native phase execution, complete cold replay, new hosting
+admission, runtime improvement, or proof-assistant verification is claimed.
+All-size compilation, mutable-source realization, whole-call hosting,
+precision/tape, analytic transfer and bit H1-H5 obligations remain inherited.
+
+The direct optimized replay passed on CPython 3.9.6 and 3.14.7. A relocated
+fixture with read-only source files passed; altered certificate and source
+bytes were rejected for their intended reasons. Package metadata and source
+pins passed the repository checker. The registered replay passed on CPython
+3.12. An independent reviewer also passed the direct optimized replay on
+CPython 3.12. Hosted CI was not run.
+
+## Open next questions
+
+Can the 341 forward source-history directions form a legal mixed boundary
+without paying back the saved moment? The first step is to realize one
+actual matrix `C D C^-1` with its native fiber operators, then derive its
+next-call cone and reader contract. The blocker is a priced source-history
+interface, not a missing scalar identity. The backward direction also has 19
+operator components when its target output may change frame. Can an
+independent backward circuit carry its three rank-one helper modes through
+actual calls and beat the complete restoration bill? The first step is one
+exact native stream, including the target repair, basis changes,
+post-adapters, and all source and helper outputs. A single tail has no gain
+with a host of rank at most eight. Joint encodings and changed owners remain
+open; no operator-level admission for them is claimed. The retained global recipe
+has five forward and four backward stages; forward-only improvements must
+use their actual counts, not a directional maximum or a presumed common
+profile. [TASK.json](TASK.json) records the structured question.
+
+## Credit
+
+Prepared by Ali Romman with substantial OpenAI Codex assistance in algebra,
+implementation, proof review, verification and writing. New work is
+Apache-2.0. AI-assisted checks are not independent human peer review.
+
+Credit Rohan Arun, Jeremy Kothe, Hao Yan, Chafik Boukhalfa and the retained
+PR1094/PR1042 supplier lineage. Credit Sagar Thakrar's PR1012 for the
+coverage-preserving deferral budget and PR1227 for separate output frames;
+Ross Richardson's PR1287 for cyclic output covariance; Scratch-net's PR1198
+for the full 712-coordinate private-input coverage warning; and the
+PR1155/PR1199/PR1203/PR1205 reader-contract lineage. PR1165 excludes literal
+repeated-operand cleanup corridors; this result instead permits arbitrary
+scalar factorizations of the same physical FULL-frame cleanup map. Credit
+icekylinx and DreamingOfClouds for the retained Clifford frame and emitter
+interfaces. PR1240 prices the fixed-word packing
+ceiling. Exact predecessor heads are retained in the source metadata. No
+predecessor implementation is copied into this package.
diff --git a/research/r480-source-independent-cleanup/SOURCES.json b/research/r480-source-independent-cleanup/SOURCES.json
new file mode 100644
index 0000000000..83dd1b4c6e
--- /dev/null
+++ b/research/r480-source-independent-cleanup/SOURCES.json
@@ -0,0 +1,66 @@
+{
+  "schema_version": 1,
+  "base_commit": "bce54fb3f895cbe5cd4c0eb477c3afeed100190f",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "c05fafaa25ba95b2d5af86b1800427198e3d20eb",
+      "sha256": "89b5928bc92c81e2647b6d702a7ad85a6795b8f44e2eb267b2ff3145d8f2b2ce"
+    },
+    {
+      "path": "research/community-frontier1094-audit/ASSEMBLY.json",
+      "usage": "input",
+      "git_blob": "5fd676df060f3b450782fdf5ff5f6beb63a5eeb0",
+      "sha256": "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045"
+    },
+    {
+      "path": "research/community-frontier1094-audit/SELECTED-SOURCE.json",
+      "usage": "input",
+      "git_blob": "6a75493790b4eff9c4a98051ef8b01d6aad0d5a8",
+      "sha256": "79b3de5db3dbcb9829789ba58e9e45758926031e2a7d80667a08429d369be53b"
+    },
+    {
+      "path": "research/community-frontier1094-audit/selected-replay.tar.gz",
+      "usage": "input",
+      "git_blob": "c24daf067f9c543421d89ca2eeaa4eb78a57fb6f",
+      "sha256": "f1da80610bf33c58527f18ad63b4350d0b864e1d4486141ee17ac9bed4d529d6"
+    },
+    {
+      "path": "research/complex-sw4-frame8/local/hpe_emit_local.py",
+      "usage": "provenance",
+      "git_blob": "534fdc71f7641c704575efc038363d7eb206b2df",
+      "sha256": "7e4a6fbc38c8228157b6311457d1496bf460291393043932d2c365f004f77805"
+    },
+    {
+      "path": "research/native-complex-moment552/notes/general-clifford-frames.tex",
+      "usage": "provenance",
+      "git_blob": "0367eeb005298601f28a673861d6f71bf1d0ebbb",
+      "sha256": "c9bb9655b594583c6b4bab8cb485233f1d1e588f83858cf1e4c3530cee8fed54"
+    }
+  ],
+  "archive_members": {
+    "CANONICAL-WORD.json.gz": "16bc0ebb04b4be6c0ade74badc6a0c0cdba4215eda86e4ff9f13abf8ecebd219",
+    "LOCAL-PROGRAMS.json.gz": "241fde27d4e43f5a9cb6b128f9f76c5131a53d09f0371676163f914d0b0fef3a"
+  },
+  "predecessors": {
+    "1012": "4ceaf380ed41b28dadcc41d2f974f025ac4d8c03",
+    "1165": "be83373cb951064eeb43db9d127c553b4daa52d8",
+    "1198": "2abd7e92b7ad3cd769376244ce32dc7b3501c2b3",
+    "1227": "fea44a4d0d78570021d854881d15d9efe62e1713",
+    "1240": "1e136b8c4a552050820334ad113628775182f08c",
+    "1287": "8606df05eca3c65032361c74ee3dbc2630b0cdc4",
+    "1391": "6ce8017e4c27c8b220b6c3a27a53bb18f138212a",
+    "1410": "5b1c00b9f2770aabaa0085539405757f7818a14a"
+  },
+  "hosted_archive_member": {
+    "path": "candidate/strict/global/hosted/GLOBAL-TEMPLATE.json.gz",
+    "sha256": "2acf02b418078e37090495c601fe550448751d4f35912757b13a1f778efc7760"
+  }
+}
diff --git a/research/r480-source-independent-cleanup/TASK.json b/research/r480-source-independent-cleanup/TASK.json
new file mode 100644
index 0000000000..5ee9b944e6
--- /dev/null
+++ b/research/r480-source-independent-cleanup/TASK.json
@@ -0,0 +1,66 @@
+{
+  "schema_version": 1,
+  "task_id": "persistent-boundary",
+  "priority_revision": 2,
+  "base_commit": "bce54fb3f895cbe5cd4c0eb477c3afeed100190f",
+  "original_card": {
+    "id": "persistent-boundary",
+    "question": "Can a specified pair of consecutive calls eliminate a restoration that the next call reconstructs?",
+    "starting_points": [
+      "research/heterogeneous-owner-assembly/README.md",
+      "certificates/selected-result.json"
+    ],
+    "methods": [
+      "Write typed entry/exit contracts and a contracting multitype recurrence.",
+      "Derive a lower bound for the compulsory transitions of one proposed typed interface."
+    ],
+    "acceptance_gate": "Show actual eliminated work and legal composition; changing a scalar recurrence into a matrix is insufficient.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Find a concrete consecutive-call boundary in the selected review and list its entry/exit obligations.",
+      "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay."
+    },
+    "first_experiment": "Specify two call types and one operation actually removed by composition; derive the normalized recurrence.",
+    "bottleneck_link": "Eliminating compulsory boundary work can lower recursive cost beyond packing gains.",
+    "allowed_variations": [
+      "A two-type pilot or one explicit deferred restoration."
+    ],
+    "scope_change_triggers": [
+      "Rewriting the old scalar recurrence as a matrix without eliminating work."
+    ],
+    "overlap_terms": [
+      "persistent boundary",
+      "deferred restoration",
+      "multitype"
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "Prove that omitted final helper climbs cannot pass the fixed physical FULL-frame R480 cleanup with canonical data and arbitrary source-independent helper boundary types. Include all batch lengths and recursion depths; classify the larger backward target-covariance class separately.",
+  "scope_change_reason": "The original claim retains real source and target outputs. The proof now states the exact physical cleanup and adapter locations. Backward target covariance is a larger interface with genuine survivors, not part of the main no-saving theorem.",
+  "bottleneck_link": "The selected complex moment binds. Deferral can improve its rank moment only if final helper climbs pass the real source and helper boundary. The theorem forces all 402 climbs in the declared class.",
+  "budget_minutes": 60,
+  "stop_rule": "Complete the original boundary obstruction and the exact larger-interface classification. Do not claim that finite owner checks exclude multi-call encodings or changed native cleanup programs.",
+  "outcome": "negative-result",
+  "evidence": "Written all-dimension proof; 402 inverse-source witnesses; both full 712-coordinate scalar maps; complete covariance classification by two exact methods; identical 3486 directional climb events; native pair frames, independent target projectors and one-tail owner cost theorem.",
+  "author": "Ali Romman",
+  "ai_assistance": "Substantial OpenAI Codex assistance with algebra, implementation, proof review, verification and writing.",
+  "open_questions": [
+    {
+      "question": "Can the 341 forward source-history directions give a legal lower-cost mixed boundary?",
+      "why_it_matters": "They carry 579 units of final-helper rank per forward call outside the proved source-independent class.",
+      "first_step": "Realize one actual conjugated map C D C^-1 with native fiber operators, then derive the next-call cone and reader contract with the full bill.",
+      "blocked_by": "Source-history access, cross-block phase transport, final restoration and hosting are not supplied by a scalar covariance identity."
+    },
+    {
+      "question": "Can an independent backward circuit carry the three common rank-one pair tails through real native call boundaries with a lower complete moment?",
+      "why_it_matters": "The backward covariance has 19 operator components, but 16 only move rank-five work to targets. Each pair leaves a rank-one helper tail after a rank-one target repair.",
+      "first_step": "Give an exact physical stream for a joint encoding or a changed owner; account for its target repair, scalar basis changes, post-adapters and all restored source and helper inputs.",
+      "blocked_by": "A single tail cannot help with any host of rank at most eight. The retained rank-48 idle children host no helper. Multi-call encodings need an operator argument, not only an invoice."
+    }
+  ]
+}
diff --git a/research/r480-source-independent-cleanup/VERIFY.json b/research/r480-source-independent-cleanup/VERIFY.json
new file mode 100644
index 0000000000..895d53b73f
--- /dev/null
+++ b/research/r480-source-independent-cleanup/VERIFY.json
@@ -0,0 +1,24 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/community-frontier1094-audit/ASSEMBLY.json",
+    "research/community-frontier1094-audit/SELECTED-SOURCE.json",
+    "research/community-frontier1094-audit/selected-replay.tar.gz"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Exact cleanup inverse and shared-word coupling, both full 712-coordinate scalar maps, all 402 helper witnesses, complete one-cut covariance by two methods, native pair ranks and projectors, all retained owner remainders, and boundary controls."
+    }
+  ]
+}
diff --git a/research/r480-source-independent-cleanup/verify.py b/research/r480-source-independent-cleanup/verify.py
new file mode 100644
index 0000000000..f06d16c1ef
--- /dev/null
+++ b/research/r480-source-independent-cleanup/verify.py
@@ -0,0 +1,538 @@
+#!/usr/bin/env python3
+"""Check the exact premises of the R480 boundary theorem and its limits."""
+# SPDX-License-Identifier: Apache-2.0
+import argparse
+from collections import Counter
+from fractions import Fraction as Q
+import gzip
+import hashlib
+import json
+from pathlib import Path
+import tarfile
+
+PACKAGE = Path(__file__).resolve().parent
+ROOT = PACKAGE.parents[1]
+ARCHIVE = "research/community-frontier1094-audit/selected-replay.tar.gz"
+PREFIX = "candidate/strict/global/base/providers/gauge44/"
+ORDINARY = 704
+SOURCES = 112
+HELPERS = range(224, ORDINARY)
+
+
+def require(ok, message):
+    if not ok:
+        raise ValueError(message)
+
+
+def identity(n):
+    return [{i: Q(1)} for i in range(n)]
+
+
+def scalar_matrix(gates, n=ORDINARY):
+    rows = identity(n)
+    for gate in gates:
+        if gate[0] == "add":
+            _, target, source, numerator, denominator, _ = gate
+            require(target != source, "self addition")
+            coefficient = Q(numerator, denominator)
+            for column, value in rows[source].items():
+                total = rows[target].get(column, Q(0)) + coefficient * value
+                if total:
+                    rows[target][column] = total
+                else:
+                    rows[target].pop(column, None)
+        elif gate[0] == "scale":
+            _, target, numerator, denominator, _ = gate
+            coefficient = Q(numerator, denominator)
+            require(coefficient != 0, "singular scale")
+            rows[target] = {j: coefficient * x for j, x in rows[target].items()}
+        else:
+            raise ValueError("unknown cleanup gate")
+    return rows
+
+
+def inverse_gates(gates):
+    result = []
+    for gate in reversed(gates):
+        item = list(gate)
+        if item[0] == "add":
+            item[3] = -item[3]
+        else:
+            item[2], item[3] = item[3], item[2]
+        result.append(item)
+    return result
+
+
+def transpose(rows):
+    out = [{} for _ in rows]
+    for i, row in enumerate(rows):
+        for j, value in row.items():
+            out[j][i] = value
+    return out
+
+
+def multiply(left, right):
+    out = []
+    for row in left:
+        values = {}
+        for middle, a in row.items():
+            for column, b in right[middle].items():
+                values[column] = values.get(column, Q(0)) + a * b
+        out.append({j: value for j, value in values.items() if value})
+    return out
+
+
+def emitted_cleanup(events):
+    gates = []
+    active = False
+    for event in events:
+        if event[0] == "phase":
+            active = event[1] == "full_frame_cleanup"
+        elif active and event[0] == "scalar":
+            op = event[1]
+            operands = op[1:3] if op[0] == "add" else op[1:2]
+            require(all(register[0] == "L" for register in operands), "private cleanup operand")
+            if op[0] == "add":
+                gates.append(["add", op[1][1], op[2][1], op[3], op[4], event[2]])
+            elif op[0] == "scale":
+                gates.append(["scale", op[1][1], op[2], op[3], event[2]])
+            else:
+                raise ValueError("singular cleanup operation")
+    return gates
+
+
+def wire(register):
+    bank, index = register
+    require(bank in ("L", "C"), "unknown register bank")
+    return index if bank == "L" else ORDINARY + index
+
+
+def dyadic_replay(events, omitted=None):
+    """Use integer row numerators and one power-of-two denominator per row."""
+    rows = [({i: 1}, 0) for i in range(712)]
+    before_cleanup = None
+    for position, event in enumerate(events):
+        if event == ["phase", "full_frame_cleanup"]:
+            before_cleanup = [(dict(row), exponent) for row, exponent in rows]
+        if event[0] != "scalar" or position == omitted:
+            continue
+        op = event[1]
+        target = wire(op[1])
+        if op[0] == "clear":
+            rows[target] = ({}, 0)
+        elif op[0] == "copy":
+            row, exponent = rows[wire(op[2])]
+            rows[target] = (dict(row), exponent)
+        elif op[0] in ("add", "scale"):
+            source = wire(op[2]) if op[0] == "add" else target
+            numerator, denominator = op[-2:]
+            if denominator < 0:
+                numerator, denominator = -numerator, -denominator
+            require(denominator > 0 and denominator & (denominator - 1) == 0, "non-dyadic gate")
+            row, exponent = rows[source]
+            extra = denominator.bit_length() - 1
+            if op[0] == "scale":
+                rows[target] = ({j: numerator * value for j, value in row.items()}, exponent + extra)
+            else:
+                dst, old = rows[target]
+                new = max(old, exponent + extra)
+                total = {j: value << (new - old) for j, value in dst.items()}
+                for j, value in row.items():
+                    total[j] = total.get(j, 0) + numerator * (value << (new - exponent - extra))
+                rows[target] = ({j: value for j, value in total.items() if value}, new)
+        else:
+            raise ValueError("unknown scalar operation")
+    def convert(data):
+        return [{j: Q(value, 1 << exponent) for j, value in row.items()} for row, exponent in data]
+    require(before_cleanup is not None, "missing cleanup boundary")
+    return convert(rows), convert(before_cleanup)
+
+
+def profile(indices, final):
+    return {str(rank): count for rank, count in sorted(Counter(final[i]["rank"] for i in indices).items())}
+
+
+def rejection(function, text):
+    try:
+        function()
+    except ValueError as error:
+        require(text in str(error), "control failed for another reason")
+    else:
+        raise ValueError("control was not rejected: " + text)
+
+
+def transported_scale(cleanup, inverse, helper, scale):
+    changed = [dict(row) for row in inverse]
+    changed[helper] = {j: scale * value for j, value in changed[helper].items()}
+    return multiply(cleanup, changed)
+
+
+def fixes_source_axes(matrix):
+    return all(matrix[i].get(s, Q(0)) == (1 if i == s else 0)
+               for i in range(ORDINARY) for s in range(SOURCES))
+
+
+def covariance_components(inverse, helpers):
+    """Find all block-output covariances of the backward cleanup."""
+    adjacency = {h: set() for h in helpers}
+    adjacency[-1] = set()
+    blocks = {}
+    for name, output_rows in (("physical_X", set(range(SOURCES))), ("helpers", set(helpers))):
+        vectors = {j: {i: x for i, x in inverse[j].items() if i in output_rows}
+                   for j in list(range(SOURCES)) + helpers}
+        pivots, selected = {}, []
+
+        def coordinates(vector):
+            row, result = dict(vector), {}
+            for pivot, (basis, representation) in sorted(pivots.items()):
+                factor = row.get(pivot, Q(0))
+                if not factor:
+                    continue
+                for i, value in basis.items():
+                    value = row.get(i, Q(0)) - factor * value
+                    if value:
+                        row[i] = value
+                    else:
+                        row.pop(i, None)
+                for i, value in representation.items():
+                    value = result.get(i, Q(0)) + factor * value
+                    if value:
+                        result[i] = value
+                    else:
+                        result.pop(i, None)
+            return row, result
+
+        for j, vector in vectors.items():
+            row, coefficients = coordinates(vector)
+            if row:
+                pivot = min(row)
+                scale = row[pivot]
+                number = len(selected)
+                selected.append(j)
+                representation = {i: -value / scale for i, value in coefficients.items()}
+                representation[number] = Q(1) / scale
+                pivots[pivot] = ({i: value / scale for i, value in row.items()}, representation)
+        require(len(selected) == len(output_rows), "output projection does not have full row rank")
+        expansions = {}
+        for h in helpers:
+            remainder, coefficients = coordinates(vectors[h])
+            require(not remainder, "column outside output basis")
+            rebuilt = {}
+            for number, coefficient in coefficients.items():
+                for i, value in vectors[selected[number]].items():
+                    rebuilt[i] = rebuilt.get(i, Q(0)) + coefficient * value
+                other = -1 if selected[number] < SOURCES else selected[number]
+                adjacency[h].add(other)
+                adjacency[other].add(h)
+            require({i: value for i, value in rebuilt.items() if value} == vectors[h], "column expansion")
+            expansions[str(h)] = {str(selected[number]): str(value) for number, value in sorted(coefficients.items())}
+        blocks[name] = {"basis_columns": selected, "fixed_anchor_rank": sum(j < SOURCES for j in selected),
+                        "helper_column_expansions": expansions}
+    reached, queue = {-1}, [-1]
+    for h in queue:
+        for other in sorted(adjacency[h] - reached):
+            reached.add(other)
+            queue.append(other)
+    free = []
+    for h in helpers:
+        if h in reached:
+            continue
+        component = [h]
+        reached.add(h)
+        for other in component:
+            for neighbor in sorted(adjacency[other] - reached):
+                reached.add(neighbor)
+                component.append(neighbor)
+        free.append(sorted(component))
+    return {"blocks": blocks, "free_components": free,
+            "forced_helpers": sorted(set(helpers) - {h for component in free for h in component})}
+
+
+def binary_span(vectors):
+    pivots = {}
+    for vector in vectors:
+        while vector:
+            pivot = vector.bit_length() - 1
+            if pivot in pivots:
+                vector ^= pivots[pivot]
+            else:
+                pivots[pivot] = vector
+                break
+    return sorted(pivots.values(), reverse=True)
+
+
+def covariance_constraints(cleanup, inverse, helpers, components):
+    """Check the dimension by direct off-block equations, without column bases."""
+    left, right = transpose(inverse), transpose(cleanup)
+    equations = {}
+    block = lambda i: 0 if i < SOURCES else 1 if i < 2 * SOURCES else 2
+    for i, row in enumerate(left):
+        for h in helpers:
+            if h not in row:
+                continue
+            for j, value in right[h].items():
+                if block(i) != block(j):
+                    equation = equations.setdefault((i, j), {})
+                    equation[h] = row[h] * value
+    pivots = {}
+    for equation in equations.values():
+        for component in components:
+            require(sum(equation.get(h, Q(0)) for h in component) == 0, "free component violates off-block equation")
+        row = dict(equation)
+        while row:
+            pivot = min(row)
+            factor = row[pivot]
+            if pivot not in pivots:
+                pivots[pivot] = {j: x / factor for j, x in row.items()}
+                break
+            for j, value in pivots[pivot].items():
+                value = row.get(j, Q(0)) - factor * value
+                if value:
+                    row[j] = value
+                else:
+                    row.pop(j, None)
+    require(len(pivots) + len(components) == len(helpers), "covariance nullity")
+    return {"off_block_equations": len(equations), "rank": len(pivots), "nullity": len(components)}
+
+
+def covariance_native_evidence(word, cleanup, inverse, final, covariance, hosted):
+    left, right = transpose(inverse), transpose(cleanup)
+    components = covariance["free_components"]
+    evidence, target_modes = [], []
+    for component in components:
+        middle = [dict(row) if i in component else {} for i, row in enumerate(right)]
+        projector = multiply(left, middle)
+        item = {"helpers": component, "omitted_ranks": [final[h]["rank"] for h in component], "modes": {}}
+        for name, indices in (("physical_X", range(SOURCES)), ("helpers", HELPERS)):
+            rows = {i: projector[i] for i in indices if projector[i]}
+            if not rows:
+                item["modes"][name] = {"rank": 0}
+                continue
+            p = next(i for i in rows if rows[i].get(i))
+            v = rows[p]
+            u = {i: row[p] / v[p] for i, row in rows.items() if row.get(p)}
+            require(sum(u.get(j, Q(0)) * value for j, value in v.items()) == 1, "projector pairing")
+            require(rows == {i: {j: value * u[i] for j, value in v.items()} for i in u}, "rank-one projector")
+            mode = {"rank": 1, "pivot": p, "u": {str(i): str(x) for i, x in u.items()},
+                    "v": {str(i): str(x) for i, x in v.items()},
+                    "basis_and_inverse_adds": 2 * (len(u) + len(v) - 2),
+                    "basis_and_inverse_scales": 0 if v[p] == 1 else 2}
+            item["modes"][name] = mode
+            if name == "physical_X":
+                target_modes.append((u, v))
+        require(item["modes"]["physical_X"]["rank"] == 1, "missing target response")
+        if len(component) == 2:
+            frames = [word["frames"][final[h]["from"]] for h in component]
+            union = binary_span(frames[0] + frames[1])
+            require([len(frame) for frame in frames] == [4, 7] and len(union) == 8, "pair frame dimensions")
+            require(item["modes"]["helpers"]["rank"] == 1, "missing helper response")
+            item["monotone_partial_tail"] = {"sum_frame": union, "intersection_rank": 3,
+                "prefix_ranks": [4, 1], "target_repair_rank": 1, "helper_tail_rank": 1}
+        else:
+            require(item["omitted_ranks"] == [5] and item["modes"]["helpers"]["rank"] == 0, "singleton response")
+            item["direct_target_repair_rank"] = 5
+        evidence.append(item)
+    for i, (_, v) in enumerate(target_modes):
+        for j, (u, _) in enumerate(target_modes):
+            require(sum(v.get(k, Q(0)) * x for k, x in u.items()) == (1 if i == j else 0), "independent target modes")
+    slots = hosted["hosting"]["pooled_slots"]
+    owned = {(p["family"], p["child"]) for p in slots}
+    remainder = {(p["family"], p["child"]): p["rank"] for p in hosted["idle_children"]}
+    slot_map = {(p["time"], p["helper"]): p for p in slots}
+    ranks = Counter(remainder.get((p["family"], p["child"]), 0) for p in slots)
+    require(set(ranks) == {0, 1, 3, 7}, "hosted remainder ranks")
+    rank48 = [p for p in hosted["idle_children"] if p["rank"] == 48]
+    require(len(rank48) == 4866 and all((p["family"], p["child"]) not in owned for p in rank48), "rank-48 owner exclusion")
+    affected = Counter()
+    for call in hosted["calls"]:
+        if call["orientation"] != "backward":
+            continue
+        for item in evidence:
+            if len(item["helpers"]) != 2:
+                continue
+            values = []
+            for h in item["modes"]["helpers"]["u"]:
+                slot = slot_map[call["time"], int(h) - 224]
+                values.append(remainder.get((slot["family"], slot["child"]), 0))
+            affected[tuple(values)] += 1
+    require(sum(affected.values()) == 192 and all(set(key) <= {0, 1} for key in affected), "affected owner remainders")
+    return {"components": evidence, "independent_target_modes": len(target_modes),
+            "direct_off_block_rank": covariance_constraints(cleanup, inverse, sorted(final), components),
+            "one_tail_cost_screen": {"baseline_ranks": [5, 2], "replacement_local_ranks": [4, 1, 1],
+                "deferred_helper_rank": 1, "host_ranks_with_strictly_worse_moment": list(range(9)),
+                "uniform_b_range": "0 < b < 1", "native_child_max_rank": max(map(int, word["local_profile"])),
+                "all_hosted_remainder_ranks": {str(r): n for r, n in sorted(ranks.items())},
+                "unhosted_rank48_children": len(rank48),
+                "affected_pair_remainder_patterns": [{"ranks": list(key), "instances": n} for key, n in sorted(affected.items())],
+                "not_covered": "Joint multi-call encodings, changed owners, or changed native cleanup maps."}}
+
+
+def load_inputs():
+    pins = json.loads((PACKAGE / "SOURCES.json").read_text())
+    for item in pins["files"]:
+        if item["usage"] == "input":
+            actual = hashlib.sha256((ROOT / item["path"]).read_bytes()).hexdigest()
+            require(actual == item["sha256"], "source digest: " + item["path"])
+    members = {}
+    with tarfile.open(ROOT / ARCHIVE) as archive:
+        for name, expected in pins["archive_members"].items():
+            raw = archive.extractfile(PREFIX + name).read()
+            require(hashlib.sha256(raw).hexdigest() == expected, "archive member digest")
+            members[name] = json.loads(gzip.decompress(raw) if name.endswith(".gz") else raw)
+        item = pins["hosted_archive_member"]
+        raw = archive.extractfile(item["path"]).read()
+        require(hashlib.sha256(raw).hexdigest() == item["sha256"], "hosted archive member digest")
+        members["HOSTED"] = json.loads(gzip.decompress(raw))
+    return members
+
+
+def calculate():
+    data = load_inputs()
+    word = data["CANONICAL-WORD.json.gz"]
+    programs = data["LOCAL-PROGRAMS.json.gz"]
+    require((word["v"], word["R"], word["h"]) == (112, 480, 9), "wrong supplier")
+    require(all(gate[-1] == 1 for gate in word["cleanup"]), "cleanup is not in the FULL frame")
+    climbs = {direction: [event for event in events if event[0] == "climb"]
+              for direction, events in programs.items()}
+    require(climbs["forward"] == climbs["backward"] and len(climbs["forward"]) == 3486, "shared-word climb coupling")
+    cleanup = scalar_matrix(word["cleanup"])
+    inverse = scalar_matrix(inverse_gates(word["cleanup"]))
+    require(multiply(cleanup, inverse) == identity(ORDINARY), "cleanup right inverse")
+    require(multiply(inverse, cleanup) == identity(ORDINARY), "cleanup left inverse")
+    matrices = {"forward": cleanup, "backward": transpose(inverse)}
+    final = {}
+    for direction, events in programs.items():
+        require(scalar_matrix(emitted_cleanup(events)) == matrices[direction], "emitted cleanup map")
+        current = {event[2]: {"from": event[3], "to": event[4], "rank": event[5]}
+                   for event in events if event[:2] == ["climb", "final"] and event[2] in HELPERS}
+        for item in current.values():
+            require(item["to"] == 1 and item["rank"] == len(word["frames"][1]) - len(word["frames"][item["from"]]), "final rank")
+        if final:
+            require(current == final, "directional final climbs")
+        final = current
+        endpoint, prefix = dyadic_replay(events)
+        expected = identity(712)
+        for i in range(SOURCES):
+            target, source, sign = (i + SOURCES, i, 1) if direction == "forward" else (i, i + SOURCES, -1)
+            expected[target][source] = Q(sign)
+        for i in range(ORDINARY, 712):
+            expected[i] = {}
+        require(endpoint == expected, "full 712-coordinate scalar endpoint")
+        if direction == "forward":
+            for i in range(ORDINARY):
+                if SOURCES <= i < 2 * SOURCES:
+                    continue
+                require({j: x for j, x in prefix[i].items() if j < SOURCES} ==
+                        {j: x for j, x in inverse[i].items() if j < SOURCES}, "independent source-column replay")
+    require(len(final) == 402, "final helper inventory")
+    nonfinal = sorted(set(HELPERS) - set(final))
+    require(len(nonfinal) == 78, "non-final helper inventory")
+    for h in nonfinal:
+        require(cleanup[h] == {h: Q(1)} and inverse[h] == {h: Q(1)}, "non-final helper cleanup row")
+        require(all(not row.get(h) for i, row in enumerate(cleanup) if i != h), "non-final helper cleanup column")
+        require(all(not row.get(h) for i, row in enumerate(inverse) if i != h), "non-final helper inverse column")
+    witnesses = []
+    for helper in sorted(final):
+        columns = sorted(j for j in inverse[helper] if j < SOURCES)
+        require(columns, "helper has no source witness")
+        source = columns[0]
+        witnesses.append({"helper": helper, "source": source, "coefficient": str(inverse[helper][source]), **final[helper]})
+    visible = {direction: sorted(h for h in final if any(matrix[s].get(h) for s in range(SOURCES)))
+               for direction, matrix in matrices.items()}
+    survivors = {direction: sorted(set(final) - set(indices)) for direction, indices in visible.items()}
+    require(len(visible["forward"]) == 61 and not survivors["backward"], "source-dependent boundary classification")
+    covariance = covariance_components(inverse, sorted(final))
+    free_helpers = [h for component in covariance["free_components"] for h in component]
+    require(len(free_helpers) == 22 and len(covariance["free_components"]) == 19, "backward target-covariance class")
+    covariance["free_profile"] = profile(free_helpers, final)
+    covariance["free_mass"] = sum(final[h]["rank"] for h in free_helpers)
+    covariance["native_ranks_and_cost"] = covariance_native_evidence(word, cleanup, inverse, final, covariance, data["HOSTED"])
+    require(covariance["native_ranks_and_cost"]["one_tail_cost_screen"]["native_child_max_rank"] == 8, "local child rank cap")
+    for i in range(SOURCES, 2 * SOURCES):
+        require(cleanup[i] == {i: Q(1)} and inverse[i] == {i: Q(1)}, "target cleanup row")
+        require(all(not row.get(i) for j, row in enumerate(cleanup) if j != i), "target cleanup column")
+    defects = []
+    for witness in witnesses:
+        h, s = witness["helper"], witness["source"]
+        column = {i: row[h] * inverse[h][s] for i, row in enumerate(cleanup) if h in row}
+        require(column, "zero omission defect")
+        defects.append({"helper": h, "source": s, "nonzero_output_rows": len(column),
+                        "source_rows_changed": sum(i < SOURCES for i in column)})
+    controls = []
+    bad = [dict(row) for row in inverse]
+    bad[witnesses[0]["helper"]].pop(witnesses[0]["source"])
+    rejection(lambda: require(multiply(cleanup, bad) == identity(ORDINARY), "inverse mismatch"), "inverse mismatch")
+    controls.append("removed_source_coefficient_rejected")
+    h = survivors["forward"][0]
+    source = min(j for j in inverse[h] if j < SOURCES)
+    escape = transported_scale(cleanup, inverse, h, Q(2))
+    require(all(escape[s] == {s: Q(1)} for s in range(SOURCES)), "escape changes source outputs")
+    require(any(row.get(source) for row in escape[224:]), "escape lacks source history")
+    controls.append("source_dependent_forward_escape_attained")
+    rejection(lambda: require(fixes_source_axes(escape), "source-independent boundary defect"), "source-independent boundary defect")
+    controls.append("nonidentity_forward_frame_rejected_by_source_independence")
+    backward_defect = transported_scale(matrices["backward"], transpose(cleanup), h, Q(2))
+    rejection(lambda: require(all(backward_defect[s] == {s: Q(1)} for s in range(SOURCES)), "backward physical-X target defect"), "backward physical-X target defect")
+    controls.append("same_frame_rejected_by_backward_physical_X_target_outputs")
+    first_free = covariance["free_components"][0]
+    changed = [dict(row) for row in transpose(cleanup)]
+    for helper in first_free:
+        changed[helper] = {j: 2 * value for j, value in changed[helper].items()}
+    output = multiply(matrices["backward"], changed)
+    for i, row in enumerate(output):
+        block = 0 if i < SOURCES else 1 if i < 2 * SOURCES else 2
+        require(all((0 if j < SOURCES else 1 if j < 2 * SOURCES else 2) == block for j in row), "target covariance mixes blocks")
+    require(any(output[i] != {i: Q(1)} for i in range(SOURCES)), "target covariance control is trivial")
+    controls.append("backward_target_covariance_escape_attained")
+    unequal = transported_scale(matrices["backward"], transpose(cleanup), first_free[0], Q(2))
+    rejection(lambda: require(all(not value for i, row in enumerate(unequal) for j, value in row.items()
+                                  if (i < SOURCES) != (j < SOURCES)), "unequal paired frame mixes target and helpers"),
+              "unequal paired frame mixes target and helpers")
+    controls.append("unequal_paired_frame_rejected_by_output_block_contract")
+    ended = next(h for h in HELPERS if h not in final and cleanup[h] == {h: Q(1)} and inverse[h] == {h: Q(1)})
+    require(all(not row.get(ended) for i, row in enumerate(cleanup) if i != ended), "low-end isolation")
+    controls.append("isolated_existing_low_end_can_carry_a_frame")
+    first_clear = next(i for i, event in enumerate(programs["backward"])
+                       if event[0] == "scalar" and event[1] == ["clear", ["C", 0]])
+    dirty, _ = dyadic_replay(programs["backward"], omitted=first_clear)
+    require(any(row.get(ORDINARY) for row in dirty[:ORDINARY]), "private-input control did not fail")
+    controls.append("omitted_private_clear_detected_on_private_column")
+    assembly = json.loads((ROOT / "research/community-frontier1094-audit/ASSEMBLY.json").read_text())
+    local = {str(k): v for k, v in sorted((int(k), v) for k, v in word["local_profile"].items())}
+    require(sum(int(r) * count for r, count in local.items()) == 5586, "local rank mass")
+    require(assembly["provider_counts"] == {"gauge44": 144}, "selected call count")
+    return {"status": "PASS_R480_SOURCE_INDEPENDENT_CLEANUP", "coordinates": 712,
+            "cleanup_macros": len(word["cleanup"]), "cleanup_nonzeros": sum(map(len, cleanup)),
+            "inverse_nonzeros": sum(map(len, inverse)), "final_helpers": len(final),
+            "final_helper_profile": profile(final, final),
+            "final_helper_mass": sum(item["rank"] for item in final.values()),
+            "shared_word_identical_directional_climbs": len(climbs["forward"]),
+            "isolated_nonfinal_helpers": nonfinal,
+            "inverse_source_witnesses": witnesses, "rank_one_defects": defects,
+            "source_dependent_survivors": {direction: {"helpers": indices, "profile": profile(indices, final),
+                "mass": sum(final[h]["rank"] for h in indices)} for direction, indices in survivors.items()},
+            "backward_separate_target_covariance": covariance,
+            "source_dependent_escape": {"helper": h, "source": source, "coefficient": str(inverse[h][source])},
+            "existing_low_end_control": ended, "negative_and_boundary_controls": controls,
+            "private_clear_control_event": first_clear,
+            "accounting": {"local_profile": local, "local_mass": 5586, "calls": 144,
+                "global_profile": assembly["complex_profile"], "complex_lower": assembly["complex_lower"],
+                "bit_lower": assembly["bit_lower"], "kappa": assembly["kappa"],
+                "deferred_climbs_in_theorem_class": 0, "new_kappa": False}}
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument("--write-certificate", action="store_true")
+    args = parser.parse_args()
+    result = calculate()
+    path = PACKAGE / "CERTIFICATE.json"
+    if args.write_certificate:
+        path.write_text(json.dumps(result, indent=2) + "\n")
+    else:
+        require(result == json.loads(path.read_text()), "certificate mismatch")
+    print("PASS_R480_SOURCE_INDEPENDENT_CLEANUP helpers=402 mass=732 columns=712 forward_history_survivors=341 backward_canonical_history_survivors=0 backward_target_covariance_helpers=22 no_new_kappa")
+
+
+if __name__ == "__main__":
+    main()