Research audit: 286-to-160-bit prefix estimate for pinned #626 (no new kappa)

research/replica-prefix-audit/CLAIM.md

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+# Sprint claim and result — seed 72
+
+Track: adversarial mathematical audit.
+Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`.
+OS randomness: `System.Security.Cryptography.RandomNumberGenerator`,
+uniform UInt32 rejection below 4294967200, modulo 100. Assignment: **72**.
+
+Started 2026-10-13 05:32:30 UTC (2026-10-12 22:32 PDT), with a 40-minute
+budget and the final quarter reserved for verification and handoff.
+Narrow scope was recorded locally during the initial orientation period.
+Expected finish: 2026-10-13 06:12:30 UTC / 2026-10-12 23:12 PDT.
+
+The three candidates were moment-profile input validation, global scalar
+growth accounting, and cyclic-inverse phase assumptions. The first and
+third had extensive adjacent current work. The chosen object is the
+selected `GLOBAL-PREFIX.json` and `invoice_allcold.py` allowance
+`101 + 37 + 72 + 72 + 4 = 286`. Seed 72 selected inspection of the
+complete 72-call interleaving and terminal boundaries rather than a new
+frontier word or local scalar rewrite. This was an audit, not a randomized
+search; there is no seed-dependent probability claim.
+
+Hypothesis: counters per replica can replace the global sweep-factor sum
+under the inherited arbitrary-dirt and normalized helper interfaces.
+Method: source-pinned schedule/owner checks, a norm induction, exact
+finite dirty-helper models and adverse controls. All delegated work
+stayed within this prefix-accounting scope. The stopping rule was to
+complete this one proof/certificate package after accepting or rejecting
+the hypothesis; no new supplier, owner search or kappa search was opened.
+
+There was no pinned sprint coordination issue; the open issue query found
+only #650 (Discussions). The claim and this single outcome update are
+therefore local, as permitted by the sprint instructions. No reservation
+issue or unsolicited comment was posted.
+
+## Outcome
+
+The qualified hypothesis is supported: **160 bits** suffice in the same
+conditional prefix model, a 126-bit reduction of the estimate. Physical
+helpers may have nonidentity familywise unitary residuals, so the proof
+does not assume physical identity restoration. It keeps the independent
+local prefix allowance and explicitly excludes arbitrary cross-family
+unitary mixing. No selected constant or kappa is changed.
+
+Completed coverage: 72 call bindings, 240 scheduled events, 35,424 helper
+roles, 12 schedule/admission controls, 8 exact signed gadgets, all six
+two-by-two-call interleavings with 24 completed boundaries and 384 exact
+norm sample comparisons, plus six adverse mathematical controls.
+The written induction supplies the dimension-independent result;
+finite coverage is not a formal theorem proof or full supplier replay.
+
+The final metadata scan at 05:54:42 UTC covered 621 open PRs through #802,
+with a final empty pagination page. Matching contexts were inspected,
+and #496's exact prior proof was read and credited. Its global norm
+method is foundational; this result adds per-replica counters for the
+selected asynchronous construction. #675 and #696/#734/#783 have distinct
+owner-boundary and local circuit/counting scopes. [OVERLAP.json](OVERLAP.json)
+records heads and search results. No overlapping 286-to-160 result was
+identified; this is not an exhaustive literature or code novelty claim.
+Before packaging, the 100 most recently updated open PRs were refreshed
+through #809; the newly updated titles/bodies produced no additional
+replica/global-prefix match. The refreshed heads are retained in the
+overlap record.
+
+The research and clean-patch handoff completed within the original budget.
+Publication resumed after the user supplied a writable fork. Upstream had
+meanwhile adopted a newer frontier; this contribution retains its original
+base and audits the now-historical #626 construction. It claims no estimate
+for the newer frontier. The final submission refresh is recorded in
+`OVERLAP.json`.
+
+Deliverable: [proof](PROOF.md), [checker](verify.py), and
+[exact receipt](EXPECTED.json), with source pins and validation details.
+Contributor: Brody Arbon, with substantial OpenAI Codex assistance.
+Apache-2.0; no independent human review is claimed.

research/replica-prefix-audit/EXPECTED.json

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+{
+  "assignment_seed": 72,
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "finite_models": {
+    "coordinates_per_replica_family": 2,
+    "families": [
+      "X",
+      "Y",
+      "H"
+    ],
+    "interleavings": {
+      "completed_boundaries": 24,
+      "distinct_final_matrices": 1,
+      "exact_squared_norm_checks": 384,
+      "sample_majorant": "1 + maximum calls completed in any one replica",
+      "sample_vectors": 16,
+      "schedules": [
+        [
+          0,
+          0,
+          1,
+          1
+        ],
+        [
+          0,
+          1,
+          0,
+          1
+        ],
+        [
+          0,
+          1,
+          1,
+          0
+        ],
+        [
+          1,
+          0,
+          0,
+          1
+        ],
+        [
+          1,
+          0,
+          1,
+          0
+        ],
+        [
+          1,
+          1,
+          0,
+          0
+        ]
+      ]
+    },
+    "negative_controls": [
+      {
+        "foreign_source_in_helper": true,
+        "name": "missing_restoration"
+      },
+      {
+        "foreign_helper_in_target": true,
+        "name": "missing_dirt_compensation"
+      },
+      {
+        "cases": [
+          {
+            "N": 2,
+            "endpoint_norm_squared": 1,
+            "first_prefix_norm_squared": 4
+          },
+          {
+            "N": 17,
+            "endpoint_norm_squared": 1,
+            "first_prefix_norm_squared": 289
+          },
+          {
+            "N": 1000000,
+            "endpoint_norm_squared": 1,
+            "first_prefix_norm_squared": 1000000000000
+          }
+        ],
+        "name": "restored_endpoint_unbounded_prefix",
+        "parameterized_identity": "endpoint independent of N; first-prefix norm = N",
+        "required_hypothesis": "retain the local primitive-prefix cap"
+      },
+      {
+        "global_l2_squared_before_and_after": 2,
+        "max_family_squared_after": "49/25",
+        "max_family_squared_before": "1",
+        "name": "unitary_mixes_families"
+      },
+      {
+        "global_l2_squared_before_and_after": 2,
+        "max_family_squared_after": "49/25",
+        "max_family_squared_before": "1",
+        "name": "global_unitarity_is_not_max_family_contraction"
+      },
+      {
+        "events_per_replica": 1,
+        "incorrect_independent_depth_upper_bound_squared": 4,
+        "individual_operator_norm_upper_bound": 2,
+        "input_max_family_squared": 1,
+        "name": "inter_replica_endpoint_mixing",
+        "output_max_family_squared": 9
+      }
+    ],
+    "norm": "maximum over (replica, family) of that family's two-coordinate l2 norm",
+    "replicas": 2,
+    "scope": "exact finite models; not a native supplier or eight-replica replay",
+    "signed_dirty_helper_gadgets": {
+      "borrow_and_target_signs": [
+        -1,
+        1
+      ],
+      "both_replica_directions": true,
+      "cases": 8,
+      "endpoint_matrix_equalities": "all inputs in the 12-dimensional model",
+      "residual": [
+        [
+          "3/5",
+          "-4/5"
+        ],
+        [
+          "4/5",
+          "3/5"
+        ]
+      ]
+    },
+    "status": "PASS"
+  },
+  "prefix_bits": {
+    "inherited_arithmetic_margin": 4,
+    "inherited_input_conversion": 37,
+    "inherited_local": 101,
+    "old": 286,
+    "reduction": 126,
+    "replica_bound": 160,
+    "retained_payload": 512,
+    "schedule_sensitive_bound": 160
+  },
+  "schedule": {
+    "calls": 72,
+    "cross_replica_helper_roles": 28404,
+    "endpoint_norm": "maximum of individual streamed-family L2 norms, grouped by owner replica",
+    "global_growth_factors": 144,
+    "growth_factors_per_replica": [
+      18,
+      18,
+      18,
+      18,
+      18,
+      18,
+      18,
+      18
+    ],
+    "helper_roles": 35424,
+    "local_and_controller_prefix_bits": 119,
+    "scheduled_replica_events": 240,
+    "source_events_per_replica": 30,
+    "trace_sha256": "acca13940ffbf9beeb60fd0877bec0c36c89659ba8d82369aabd5c7bb0f92a08"
+  },
+  "schedule_controls": [
+    "duplicate_call",
+    "out_of_order_replica",
+    "changed_provider",
+    "doubled_bridge",
+    "cross_replica_bridge",
+    "missing_bridge",
+    "unknown_event",
+    "data_owner_in_other_replica",
+    "wrong_helper_replica",
+    "aliased_helper_family",
+    "helper_on_active_data",
+    "missing_local_allowance"
+  ],
+  "scope": "Conditional precision-accounting refinement only; no new kappa, cold supplier replay, physical realization proof or production integration.",
+  "status": "PASS_REPLICA_PREFIX_AUDIT"
+}

research/replica-prefix-audit/NOTICE.md

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+# Attribution and scope
+
+Contribution by Brody Arbon. Substantial OpenAI Codex assistance was used
+for source inspection, mathematical derivation, implementation, testing,
+writing and separate AI reviews. No independent human review or formal
+proof-assistant certification is claimed. New material is Apache-2.0.
+
+The audited #626 owner repair is Chafik Boukhalfa / chafreaky's work,
+using Jeremy Kothe / jeremykothe2's #622/#623 K-triangle construction and
+dispatch. GamingPuzzled / Joel Pulikkan's owner-headroom lead, icekylinx's
+controller/emitter/hosting and scalar accounting lineage, DreamingOfClouds'
+cold helpers, Rohan Arun's #559 recurrence correction, eumemic's #543 bit
+supplier and the cumulative community contributions remain credited.
+The original repository's notices and contributor record are authoritative
+for the full lineage and are unchanged by this package.
+
+Specific prior analytic credit: eumemic's #496 at
+`00cf70efd6a8637531deafef698157ca83e249ee` already states the global
+maximum-stream norm, whole-family right-translation bijections and
+unitary-helper endpoint argument. This work adds individual replica
+counters for #626; it does not originate that norm or cover argument.
+
+Concurrent #675 (Tekalign / tekuuu) checks owner-window boundaries and
+bridge-omission scalar evidence. #696 (Sagar Thakrar), #734 (Chris Pagolu)
+and #783 (Rohan Guliani) study/recount local scalar circuits and their
+prefixes. Those are distinct from this global replica-norm accounting
+argument; no code from those pending packages is imported.
+
+The triangle inequality, norm invariance under unitary maps and induction
+on an interleaving are standard mathematical tools. No literature-priority
+claim is made. The contribution is their explicit application to the
+selected source schedule, including borrowed familywise unitary residuals,
+with executable source bindings and adverse controls.