Audit centre-source coefficients with an independent adjoint and sign control

research/selected-centre-bridge-projection/README.md

@@ -0,0 +1,192 @@
+# Independent adjoint audit of bridge-aware centre sources
+
+**Independent finite audit; no new supplier or kappa.** Two separately written
+algorithms reproduce the selected controller's **1,152-dimensional source-history
+rank**, already reported in [PR2622](https://github.com/CrocSwap/integer-mult-bounds/pull/2622).
+The new verification uses a backward local adjoint and dense controller matrices
+to check every source coefficient against a forward rational calculation.
+
+A concrete failure control explains the need for that coefficient check:
+reversing **every bridge sign** retains rank **1,152** and the identity endpoint
+on **14,336 owners**, yet changes **930 source rows**. One unit-input cut changes
+from **+1 to -1**. Endpoint identity plus full source rank does not identify the
+correct intermediate source map. This is a checker control, not a defect report
+against PR2622, which authenticates its inputs and also audits the output response.
+
+The scope is the common-frame scalar projection. Native owner/address routing
+remains open; idle sweeps are explicitly identity here. This does not independently
+verify PR2622's distinct final error-response map or claim priority for its rank.
+
+## Exact boundary and result
+
+The inputs are the selected PR1094 replay archive, bound by the selected assembly,
+from main `c3c94c2ea2ebcd5d69d1e51b5fe8a36bf79819f5`. The archive contains the
+actual **480-helper**, 704-role local word. No 492-helper ancestor is substituted.
+[SOURCES.json](SOURCES.json) pins every external input. The selected conditional
+kappa remains `836330098538459/500000000000000000`.
+
+As in Adam Gall's PR1918 and thakrarsagar's PR2199, each physical owner starts
+with an independent formal scalar over Q. Local `clear`, `copy`, `scale` and
+`add` operations retain their exact coefficients. A `center` event records the
+private C register at its rank-eight child. Local climbs, adapters, residual
+adapters and the centre child itself act as identity in this declared projection.
+The checker derives both local endpoint maps from **every scalar event** on all
+704 formal columns and checks private clearing:
+
+- forward: `Y_p += X_p` for all 112 ports;
+- backward: `X_p -= Y_p` for all 112 ports;
+- all other ordinary roles are restored.
+
+The audit executes each bridge in the actual asynchronous event order. For
+an event with destination pair d, source pair s and coefficient c, it applies
+
+```text
+X_d,p += c X_s,p
+Y_s,p -= c Y_d,p
+```
+
+for every port p. The signed terminal is `(X,Y) <- (Y,-X)` simultaneously on
+each pair. These formulas come from the pinned inherited complex geometry
+proof, section “Four-pair scalar chain and idle children”. The complete
+per-replica recipe and data-owner indexing are checked against the selected
+template. In particular,
+`owner(replica,pair,side,port) = 896*replica + 112*(2*pair+side) + port` is checked
+against all local data-role maps, rather than assumed for arbitrary helper roles.
+
+| Quantity | Calls-only predecessor projection | Projection with bridges and terminal |
+|---|---:|---:|
+| Local cover calls / centre rows | 144 / 1,152 | 144 / 1,152 |
+| Rational rank of all centre rows | 1,152 | 1,152 |
+| Changed centre rows relative to predecessor | — | 930 |
+| Identity endpoint owners | 0 | 14,336 |
+
+The calls-only comparison reproduces PR2199's call table, local cut support
+sizes, complete centre-row support histogram and rational rank. It is a control,
+not a defect claim against a predecessor that explicitly omitted these events.
+
+A small exact witness shows that the added bridges matter. Set initial physical
+owner **224** (replica 0, pair 1, X side, port 0) to one and every other initial
+owner to zero. At call **time 10 / stage 2 / replica 0 / centre 0**, the calls-only
+cut is zero and the bridge-aware cut is one. This witness is derived from the
+complete symbolic rows, not from a sampled floating-point calculation.
+
+Because all 1,152 rows are linearly independent, every subset is independent;
+in particular, every two distinct calls supply 16 independent formal cut rows
+in this model. This is **not** a 16-scalar output-transcript lower bound: that
+would additionally need the fixed-dirt output-increment map, its consumer and
+panel boundary. Full cut rank cannot replace those obligations.
+
+## Two implementations and controls
+
+`verify.py` propagates exact sparse rational forms forward through each local
+program, composes them with a 14,336-owner state, and performs rational row
+elimination. It checks the complete projected endpoint and emits [RESULT.json](RESULT.json).
+
+`check_adjoint.py` imports neither that checker nor supplier code. It pulls each
+cut backwards through the local scalar operations, then reconstructs the
+chronological owner state using 16 dense integer 8-by-8 controller matrices.
+Every coefficient of all 1,152 rows agrees with the first implementation through
+the canonical row digest. Rows follow chronological call order, then centre
+index 0 through 7. Each row is its ascending-column list of `[index, str(Fraction)]`
+pairs; the digest is SHA-256 of compact JSON with separators `(',', ':')`.
+A separate reverse-row, largest-column
+elimination gives rank 1,152 modulo `2147483647`. All denominators are invertible
+modulo that prime, so a nonzero maximal minor is also nonzero over Q; the row
+count supplies the upper bound. The primary result does not rely on this modular
+check: it also computes rank directly over Q.
+
+Controls include all 18 individually omitted half-bridges, all bridges removed,
+all bridge signs reversed, and dependent duplicate rows. Each individual
+half-bridge omission breaks the 8-by-8 endpoint. Removing all bridges changes
+both cuts and endpoint. Reversing **all** bridge signs changes the cuts but
+**preserves** the endpoint: it conjugates the controller by the alternating-pair
+sign matrix `diag(1,1,-1,-1,1,1,-1,-1)`. Every selected bridge joins opposite pair
+parities, whereas each local call and terminal acts within one pair. Thus the
+sign matrix negates all bridge shears and commutes with those other operations;
+conjugation preserves the full controller's identity endpoint.
+
+Both algorithms separately check that this wrong-sign controller also retains
+rank 1,152, but changes 930 rows. For the owner-224 input above, its first changed
+cut is -1 instead of +1. The receipt binds its full row digest and exact witness.
+This finite counterexample establishes that **endpoint identity and source rank
+together are insufficient** to certify the intermediate centre coefficients.
+
+The checks remain active under Python `-O`. Both checkers authenticate inputs
+before computing; normal primary execution compares every receipt field.
+
+## Reproduce
+
+Python 3.11 or later, standard library only, from the repository root:
+
+```sh
+python3 -I -B research/selected-centre-bridge-projection/verify.py
+python3 -I -O -B research/selected-centre-bridge-projection/check_adjoint.py
+python3.12 scripts/research_package.py --package research/selected-centre-bridge-projection
+```
+
+Expected markers:
+
+```text
+PASS_SELECTED_CENTRE_BRIDGE_PROJECTION rank=1152 changed_rows=930 endpoint_owners=14336
+PASS_SELECTED_CENTRE_BRIDGE_ADJOINT rows=1152 rank_mod_2147483647=1152 controls=3
+```
+
+`verify.py --write` regenerates the receipt. The registered runner uses Python
+3.12 and bounds each command by the replay manifest. Local
+interpreter versions and actual runs are recorded in TASK.json. A local pass is
+not a claim that upstream CI ran.
+
+## Cost, limits and expansion decision
+
+The unchanged 144 calls retain 1,152 rank-eight centre children, total rank mass
+9,216, or 64 per call. There is no eliminated work and no changed cost profile.
+No sharing construction is proposed. This result narrows a verification gap;
+it does not pass the task card's construction or 10-percent improvement gate.
+
+The 160 idle sweeps, 16 terminal Y translations, one bank-padding sweep and one
+final every-family translation are explicitly assigned identity in this model.
+Likewise, the exact complex phases and native address charts are not modeled.
+This formal projection is **not proved to be a representation or specialization
+of the native supplier**. Its rank need not be a native rank lower bound.
+The identity endpoint also does not establish native correctness.
+
+Full selected cold replay (`make frontier1094-verify`), native 512-address
+operators, whole-cover owner routing, physical hosting, precision/tape costs,
+all-size/compiler interfaces, analytic transfer and a multiplication theorem
+remain outside scope. The selected record and frozen predecessors are unchanged.
+The family decision is **narrow**: retain this scalar evidence and require an
+explicit native boundary before any sharing implementation or larger search.
+
+## Open next questions
+
+Can the first changed cut (time 10, stage 2, replica 0, centre 0) be reconstructed
+with the actual routed native bridge and idle operators? This would test whether
+the scalar witness has a native counterpart. Start from the pinned owner 224
+unit input and the exact route/phase maps through that cut, including both
+bridge shears. The blocker is the cover/address action: treating it as identity
+again would repeat this projection, not close the native question. A mismatch
+would be a scoped counterexample, not a failure of the selected supplier.
+
+## Credit and overlap
+
+- Adam Gall, PR1918: original scalar projection and the separate output-rank lemma.
+- thakrarsagar, PR2199 at `30ec048eedcaca970f94bbec0fdd3c52eacf9fdc`:
+  calls-only census and the explicit bridge/idle handoff.
+- ywz65, PR2564 at `64f940a8ee305a1b69f7db329f1ffa232e53c2ad`:
+  terminal residual of two ablated calls, with bridge/idle events explicitly
+  omitted. This package neither redoes nor extends its ablation calculation.
+- ywz65, [PR2622](https://github.com/CrocSwap/integer-mult-bounds/pull/2622)
+  at `4ffb893a08e8868057a30a065642d0b5f647f742`: already includes the complete
+  bridge-aware source-history rank and identity endpoint, as context for its
+  distinct output-error injectivity result. This package deliberately audits
+  the source-history claim with an independent local adjoint, coefficient
+  agreement and an endpoint/rank-preserving wrong-sign control. Neither checker
+  imports its code or the PR2199 verifier. The existing rank is not a new result.
+- Rohan Arun, Jeremy Kothe, Hao Yan, Douglas Colkitt and the retained selected
+  construction/geometry lineage: exact scientific inputs and controller rules.
+
+[TASK.json](TASK.json) records the task selection, overlap coverage, scope and
+freshness. The contribution uses two separately written algorithms by the same
+AI-assisted author; no independent human or theorem-kernel review is claimed.
+Prepared for BigCactusLabs with substantial OpenAI Codex assistance in research,
+implementation, verification and exposition. New work is Apache-2.0.

research/selected-centre-bridge-projection/RESULT.json

@@ -0,0 +1,196 @@
+{
+  "bridge_rank_pivot_columns_sha256": "7bdeddf3a162497990b47c23248c9d7103ee93e4150f72dc568ea3a50cf51420",
+  "bridge_rank_q": 1152,
+  "bridge_rows_sha256": "284565153a198a096bf8a336661582058d2fe395ba984bb355ca80c8f24f1cac",
+  "calls": 144,
+  "calls_only_nonidentity_owners": 14336,
+  "calls_only_rank_q": 1152,
+  "calls_only_rows_sha256": "ebcc83711debd6f2ff0e056f60af4f2e5d8a024cfb2846f117a524b5188909fd",
+  "centre_rows": 1152,
+  "changed_centre_calls": [
+    10,
+    11,
+    13,
+    16,
+    17,
+    18,
+    19,
+    21,
+    24,
+    25,
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+    27,
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+    134,
+    135,
+    136,
+    137,
+    138,
+    139,
+    140,
+    141,
+    142,
+    143
+  ],
+  "changed_centre_rows": 930,
+  "controls": {
+    "duplicate_row_rank": 1,
+    "omitted_half_bridges_change_endpoint": 18
+  },
+  "event_counts": {
+    "bank_padding_cover_sweep": 1,
+    "bridge": 144,
+    "idle_cover_sweep": 160,
+    "local_cover_sweep": 144,
+    "terminal": 16,
+    "terminal_Y_translations": 16,
+    "translate_every_live_family": 1
+  },
+  "first_changed_basis_witness": {
+    "all_other_initial_owners": 0,
+    "bridge_cut_value": "1",
+    "calls_only_cut_value": "0",
+    "cut_time_stage_replica_centre": [
+      10,
+      2,
+      0,
+      0
+    ],
+    "initial_owner_with_value_one": 224
+  },
+  "full_projected_endpoint_identity_owners": 14336,
+  "local_event_counts": {
+    "backward": {
+      "adapter": 1090,
+      "center": 8,
+      "climb": 3486,
+      "phase": 10,
+      "residual_adapter": 480,
+      "scalar": 19693
+    },
+    "forward": {
+      "adapter": 1090,
+      "center": 8,
+      "climb": 3486,
+      "phase": 10,
+      "residual_adapter": 480,
+      "scalar": 19693
+    }
+  },
+  "scope": "Exact common-frame scalar projection; no native routing or supplier claim.",
+  "status": "PASS_SELECTED_CENTRE_BRIDGE_PROJECTION",
+  "template_payload_sha256": "9c4c51425251389ee9a747c465814cd07c64812c15b7aeef704dad7d65395a74",
+  "uniform_bridge_sign_reversal": {
+    "changed_centre_rows": 930,
+    "endpoint_identity_owners": 14336,
+    "first_changed_basis_witness": {
+      "all_other_initial_owners": 0,
+      "correct_cut_value": "1",
+      "cut_time_stage_replica_centre": [
+        10,
+        2,
+        0,
+        0
+      ],
+      "initial_owner_with_value_one": 224,
+      "reversed_bridge_cut_value": "-1"
+    },
+    "rank_q": 1152,
+    "rows_sha256": "ea9798157c5c338cb8ccedda5437cd4c5d7d1138fd41e3c9df4376e0bd3bfbe9"
+  }
+}