Verify five rank-four BIT terminals and their fixed-controller cost bound

research/bit-five-rank4-terminals/CONTENT-SHA256.json

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+  "predecessor/OPERATOR-BOUNDARY.md": "b268114ee08a3cb0c6fb18928361528cc7385bd5f89b90bf247b2739a513dbac",
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+  "verify_word.py": "526fe4422f2d761b4eaee2f562e2b504eed2c17fa73307cafaeda7b1cd38fd75"
+}

research/bit-five-rank4-terminals/INDEPENDENT-FIVE-TERMINALS.json

@@ -0,0 +1,79 @@
+{
+  "checked_frames": 636,
+  "checked_nested_pairs": 2394,
+  "controls": [
+    "Declaring the retired helper FULL fails actual endpoint check.",
+    "An even integer dirty error passes F2 and fails exact helper restoration."
+  ],
+  "forward": {
+    "ADDs": 36005,
+    "COPYs": 9,
+    "H": {
+      "1": 1322,
+      "2": 1342,
+      "3": 79,
+      "4": 39,
+      "5": 102,
+      "6": 11,
+      "7": 18
+    },
+    "mass": 5101,
+    "positive_envelope": 8402567,
+    "prefix_row_L1": 1302
+  },
+  "geometry_sha256": "a353999bceac4ac957f540adfd3e5f8768297138b27132eeb83332e2ea4b1022",
+  "helper_inventory": {
+    "4": 5,
+    "8": 433
+  },
+  "inherited_precision_diagnostics": {
+    "104_bit_expression": "15105050522746917552142185212669952",
+    "192_bit_expression": "1737850264432662463887984248694818736482648196425093483769888768",
+    "bit_length_104": 114,
+    "bit_length_192": 211,
+    "both_strict_caps_pass": false
+  },
+  "inverse": {
+    "ADDs": 36005,
+    "COPYs": 9,
+    "H": {
+      "1": 1322,
+      "2": 1342,
+      "3": 79,
+      "4": 39,
+      "5": 102,
+      "6": 11,
+      "7": 18
+    },
+    "mass": 5101,
+    "positive_envelope": 630744,
+    "prefix_row_L1": 1446
+  },
+  "limits": [
+    "Independent finite word, dyadic projector and legal snapshot verification.",
+    "Generic exact-stream operator proof is separate from all-size physical owner/compiler admission.",
+    "Old H2/H4 syntax and 104/192-bit precision admission are not inherited; no outer exponent."
+  ],
+  "no_new_kappa": true,
+  "persistent_columns": 660,
+  "private_zero_slots": 1,
+  "roundtrip": {
+    "ADDs": 36005,
+    "COPYs": 9,
+    "H": {
+      "1": 1322,
+      "2": 1342,
+      "3": 79,
+      "4": 39,
+      "5": 102,
+      "6": 11,
+      "7": 18
+    },
+    "mass": 5101,
+    "positive_envelope": 630744,
+    "prefix_row_L1": 1253
+  },
+  "source_sha256": "526fe4422f2d761b4eaee2f562e2b504eed2c17fa73307cafaeda7b1cd38fd75",
+  "status": "PASS_INDEPENDENT_FIVE_TERMINAL_WORD_WITH_PRECISION_CAP_FAILURE",
+  "word_sha256": "3bca74a2a8427eb18e86566baa19de597f10d0bc40c30b7443febd499ee2189a"
+}

research/bit-five-rank4-terminals/LOCAL-PROVENANCE.json

@@ -0,0 +1,17 @@
+{
+  "scope": "Only file discovery, metadata loading, and receipt source binding changed; mathematical checks retained. Historical originals preserved.",
+  "non_main_ast_identical": true,
+  "source_files": [
+    {
+      "original_path": "/Users/ross/git/math/.research-automation/runs/kappa-four-coordinate-20261019T0717Z/root-bit-audit/OPERATOR-BOUNDARY.md",
+      "retained_path": "predecessor/OPERATOR-BOUNDARY.md",
+      "sha256": "b268114ee08a3cb0c6fb18928361528cc7385bd5f89b90bf247b2739a513dbac"
+    },
+    {
+      "original_path": "/Users/ross/git/math/.research-automation/runs/kappa-four-coordinate-20261019T0717Z/root-bit-audit/verify_word.py",
+      "retained_path": "predecessor/verify_word.py",
+      "sha256": "64f62ecdbc77518dd7705087f0aaa852412d607cf197ee3fa9b0a2a96683750b"
+    }
+  ],
+  "candidate_word_sha256": "c46ccd7d3eedabc6a7efbb1a91ec13f2928bad8ee0bac57c6e593a092d13d96c"
+}

research/bit-five-rank4-terminals/PROOF.md

@@ -0,0 +1,106 @@
+# Five rank-four terminals: exact stream boundary and remaining admission
+
+The independent finite audit passes for the word with uncompressed SHA256
+`3bca74a2a8427eb18e86566baa19de597f10d0bc40c30b7443febd499ee2189a` and geometry
+SHA256 `a353999bceac4ac957f540adfd3e5f8768297138b27132eeb83332e2ea4b1022`.
+This is a local mixed-terminal operator under the exact stream operations
+below. It is not an improved conditional kappa or a completed supplier admission.
+
+## Predecessors and changed interface
+
+The source is the pinned R438 transported word. The generic frame simulation
+argument is credited to PR540 FramedReplay/OwnerTransport and its source-bound
+PR589 extension in `research/pr589-five-helper-hosting/PROOF.md`, at canonical
+20ecf2891d80b5acf18d980b2164fb44ea7faa90. The new construction moves helper
+preconditioning to ZERO and replaces the old FULL literal A/B cleanup by an
+early SOURCE cleanup plus a different signed tail. It therefore must not claim
+that the old syntactic H4 checker passed unchanged.
+
+PR1194 (2e39c98238f543d988f9efc1b076590a4b780dca) and PR1311
+(a2875c1834b9c9ef42a84eda4890d5462cc98fa5) are relevant semantic-prefix
+predecessors. Their stated substitution preserves H4; they do not directly
+admit these changed terminal frames and cleanup. The contribution here is the explicit source-bound word. Elementary
+preconditioning and the generic simulation lemma are prior mechanisms. PR1256
+(`research/bit-early-retirement-cut`) already lowers helper470 from width8
+to7 and prices returned idle progress; its fixed-prefix exclusions have different
+premises from the ZERO preconditioning used here. No broad novelty claim is made. Existing complex-side heterogeneous restoration in
+`research/e8-prefix-restoration` is also prior art for that general mechanism.
+
+## Exact stream interpretation
+
+For a frame projector P on K^8 define the address involution
+
+    D_P(x,y)=((I-P)x+Py, Px+(I-P)y).
+
+Here K is any field of characteristic other than two. The independent checker
+constructs all636 actual rational projectors by Gram-Schmidt, verifies their
+dyadic entries and basis identities, and checks all2394 nested pairs used by
+the word in either orientation. Idempotence and symmetry follow directly from
+the orthogonal-basis formula. If P<=Q, their relative operation D_P D_Q is the
+partial swap D_(Q-P), with rank dim(Q)-dim(P). Its exact rank survives odd
+characteristic specialization: the dyadic characteristic polynomial of each
+idempotent is t^(8-r)(t-1)^r. This supplies exact address operations, not their
+physical compiler or tape implementation.
+
+Take660 arbitrary Boolean-valued input functions, one for every persistent
+source, target and helper. The one private function starts zero. For each row
+i let Gamma_i=D_Pi at its current frame and let the decoded function be
+c_i=f_i composed with Gamma_i, where f_i is the physical payload. These are
+bookkeeping bijections; no input is reset or assumed independent in value.
+
+A MOVE from Gamma_a to Gamma_b transports the physical payload by
+Gamma_a Gamma_b^-1, so it preserves c_i. An ADD at a common frame changes the
+decoded destination by the decoded source. Signed integer ADDs give the stated
+integer lift; their reductions are XOR for Boolean values. A fresh COPY
+transports the source into the private frame and copies its decoded function.
+Every ERASE in the candidate sees precisely that unchanged decoded source and
+the corresponding private snapshot; bijectivity yields the physical equality
+needed for erasure. The private row is zero before its next reuse.
+
+Induction over every literal event therefore gives the decoded matrix checked
+by `verify_word.py`. That independent implementation imports no producer or
+predecessor checker. It tests all660 independent persistent input columns,
+the PRIVATEZERO contract, exact integer source/helper restoration, the Boolean
+public shear, and both signed forward/inverse roundtrips. The statement is for
+all input functions, including arbitrarily correlated dirty helper contents.
+
+## Endpoint
+
+The source/target entrance and exit frames equal the old CLEAN R438 frames,
+and the decoded Boolean endpoint is x'=x, y'=y+x. Thus their physical public
+operator is unchanged. For helpers462,463,464,467,470 the decoded value is
+exactly its original arbitrary helper value, but its physical residual is
+D_E for the explicit terminal rank-four frame E. The other433 helpers retain
+the old rank-eight residual. No dirty output is discarded, and no initial
+native query is hidden in the1765 ZERO helper ADDs.
+
+In a reflected embedding, replace the local gauge by D_(I-P) times the shared
+exterior factor. The relative endpoint from frame0 to E is still D_E, since
+D_I D_(I-E)=D_E. The same induction and signed inverse apply. An actual owner
+must map each E residual into its assigned pending coordinates; rank alone
+does not construct the required orthogonal chart or chronological placement.
+
+The complete local histogram is
+
+    H={1:1322,2:1342,3:79,4:39,5:102,6:11,7:18}.
+
+Its rank mass5101 equals111*14+433*8+5*4+63. All36005 ADDs and nine rank-seven
+COPY calls remain paid. The lost20 helper ranks exactly equal the removed
+query mass; they are not an extra unearned deficit.
+
+## Gates still open
+
+The conservative forward/reverse positive envelopes are8402567 and630744.
+The inherited expressions64 F^3 R^2 and2^18 F^5 R^4 require114 and211 bits,
+respectively, and fail the old104/192-bit caps. Exact signed prefix row norms
+1302 and1446 are recorded diagnostics, not an automatic replacement for the
+old defect and COPY/ERASE error analysis.
+
+This proof closes the arbitrary-input endpoint in the stated exact stream
+model. The finite residual isometries and coordinate words are now supplied by
+`admission/README.md`. Whole-owner chronological hosting, physical scalar/address routes, rounded-child errors, precision charges,
+all-size dispatch, storage/tape costs and joint outer assembly remain separate.
+The checker is finite exact evidence; this document is a written induction,
+not a formal-kernel certificate. No new kappa is claimed.
+
+Apache-2.0; substantial OpenAI Codex assistance.