Save six more scalar units after P104 sign normalization (no new κ)

research/p104-selected-scalar-pair/README.md

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+# Six fewer scalar allowance units after P104 sign normalization
+
+One exact replacement removes an ADD macro from both the selected #5599
+word and its published #5750 sign normalization. It reduces the latter's
+retained real scalar allowance from **219258 to 219252**, adding six units
+to Sricharan Vagvala's credited 974-unit saving. It preserves all 748 public
+inputs, all helper targets, the eight owned-role slots and every one of the
+3447 paid MOVE occurrences. The complete paid and productive rank profiles
+are identical, so no new kappa is claimed.
+
+| Pinned word | Events before | Events after | Scalar before | Scalar after |
+| --- | ---: | ---: | ---: | ---: |
+| Selected #5599 | 25475 | 25474 | 220232 | 220226 |
+| Sign-normalized #5750 | 25253 | 25252 | 219258 | 219252 |
+
+Claim type: **exact finite circuit improvement with a scoped all-input
+algebraic proof**. Search stopped at this first verified improvement.
+Publication overlap checking then identified #5750. The same literal block
+survives there, allowing composition without another improvement search.
+
+## The replacement
+
+In `research/p104-helper-subspace-splits/WORD.json.gz`, replace zero-based
+events 9751 through 9754, inclusive. The identical block is at events 9627
+through 9630 in `research/p104-sign-gauge/WORD.json.gz`:
+
+```text
+225 += -234
+229 +=  234
+227 +=  234
+225 += -234
+```
+
+with:
+
+```text
+225 += -2 * 234
+229 +=      234
+227 +=      234
+```
+
+Every operation uses frame 267. The actual source prefix places all four
+roles in that frame. Role 234 remains unchanged, and neither intervening
+operation reads role 225. For arbitrary initial values a,b,c,d in roles
+225,229,227,234, both blocks therefore return
+
+```text
+(a - 2d, b + d, c + d, d).
+```
+
+This identity holds over the complex payload space at every tensor width.
+The prefix and suffix of the complete word are retained verbatim. Composing
+either block with those identical contexts gives the same complete endpoint,
+including arbitrary dirty inputs and every helper output. This argument
+permits arbitrary native maps outside the block; it does not need to replay
+or assume any new identity between those maps.
+
+The new value of each role at each macro boundary already occurs for that
+role at some old boundary. In particular, the earlier value a-2d is the old
+block's final value. Thus this replacement does not increase the maximum
+per-role magnitude at scalar macro boundaries. Internal primitive states,
+rounding sites and error propagation are separate questions.
+
+The executable constructor in [verify.py](verify.py) authenticates the source
+bytes and replaces only this literal block, in memory. The canonical JSON
+hash of the constructed sign-normalized word is
+`bc123d9989d919caa48086b19a342fa83d8470d6a56e6ceb8d73b43f4ee84f9d`.
+No regenerated gzip container is required.
+
+## Exact cost and unchanged recursive bill
+
+The retained `local.py` scalar tariff charges an ADD with nonzero dyadic
+coefficient q, numerator magnitude n and positive denominator d, by
+
+```text
+2 * (3 + bit_length(n) - 1 + popcount(n) + bit_length(d) - 1).
+```
+
+An ADD with coefficient -1 costs 8; coefficient -2 costs 10. The removed
+pair costs 16 and its replacement costs 10. The other two ADDs remain.
+The complete scalar allowance consequently falls by exactly six units
+for each pinned source. The verifier independently sums this tariff over
+the original and replacement in both cases,
+including their SCALE, COPY, INIT and CLEAR charges.
+
+Every non-ADD event is retained in its original order, with identical ports,
+frames and ownership flags. No additional native map, source acquisition,
+helper basis change or allocation is required. Setup and dispatch for a
+physical caller still need that caller's own accounting; six units is a
+saving in the stated scalar allowance, not a measured runtime saving.
+
+## Evidence and reproduction
+
+From the repository root, using CPython 3.12 and the standard library:
+
+```sh
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -I -B research/p104-selected-scalar-pair/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -I -O -B research/p104-selected-scalar-pair/verify.py
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/p104-selected-scalar-pair
+/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/p104-selected-scalar-pair --check-source-history
+```
+
+Another CPython 3.12 interpreter may replace the local path. The expected
+status is `PASS_P104_SELECTED_SCALAR_PAIR`, with `scalar_saving` equal to 6.
+The verifier writes no files. Its stdout receipt binds all five package
+files and each compressed and decompressed source hash.
+
+The checks include exact rational four-role matrices and every local prefix,
+the actual common-frame cut, identical contexts and non-ADD occurrences,
+and complete normalized 756-row coefficient propagation for each source
+and its replacement.
+Normalization treats MOVE as a chart identification. That last comparison
+corroborates the scalar endpoint; it does not compute native Gaussian actions
+or phases. The local identity and identical-context proof establish equality
+in the presence of the actual unchanged native operations.
+
+Eleven controls per source reject wrong coefficients, sign, source,
+destination or frame; a missing fanout; changed helper metadata or paid transport; an
+intervening source update or target observation; and a changed source block.
+All controls use explicit exceptions and remain active under `-O`.
+
+## Scope, provenance and credit
+
+The source base is canonical main
+`19fdb086ea96af9a9f87936a2b888e10af4ed810`, priority revision 8. Exact Git
+blobs and SHA256 pins are in [SOURCES.json](SOURCES.json). The selected
+record and catalogue are provenance, so subsequent priority updates do not
+change this fixed result. The two source words are the only runtime inputs.
+
+This is a changed `p104-side-correction` experiment aimed at the current
+complex supplier. Its result is a scalar circuit improvement. No later
+supplier word is optimized, and no full selected replay, compiler admission, native
+phase replay, rounded multiplier or proof-kernel build is performed.
+The selected construction's five framework assumptions remain. Exact
+algebraic equivalence does not assert equality of two rounded executions.
+
+This contribution was **produced by Nick Petalas**. Ali Romman supplied the
+selected #5599 independent-helper-split word. Pierre-Loic Doulcet supplied
+the #5565 commuting-contact lineage; David Arias the coalesced word; Gareth
+Davidson its composition; and Cinna Davis the private reuse and productive
+controller lineage. The source README retains the full P104, native,
+geometry, caller, compiler and pricing attribution.
+
+Cinna Davis's #4692 already supplied commuting-ADD compression and triangle
+rewrites for an older 632-public-input word. The commutation method keeps
+that credit. Sricharan Vagvala / theon199 supplied the published #5750
+sign-normalized word and its 974-unit saving. The new artifact is this
+literal six-unit reduction, including its composition with that word.
+No predecessor executable is imported or modified. Independent human and
+formal proof-kernel review are unclaimed.
+Apache-2.0.
+
+## Verification bounds and deviations
+
+All loops are bounded by the authenticated source: at most 25475 events,
+1089 frames, 756 symbolic rows and 756 columns per row. The local check uses
+four roles and at most four events; there are eleven fixed controls per source.
+Reads are capped at 256 KiB compressed and 4 MiB decompressed. No recursion,
+dynamic code loading, subprocess, network access or mutable shared state
+is used. Functions remain below 60 lines. Warning-strict normal and optimized
+execution, Ruff 0.17.2 checks and syntax and indentation diagnostics check
+the touched code.
+
+Power of Ten rule 3 deviation: Python and Fraction allocate during this
+bounded offline calculation. The exact source hash and the explicit bounds
+above constrain live collections; the verifier has no service loop or
+unbounded input domain. A fixed-capacity implementation would be needed
+before reusing this checker in a streaming or real-time component. The
+registered timeout is an additional execution guard, not a proof of a
+memory or arithmetic complexity bound.
+
+Rule 9 deviation: the rejection-control helper accepts one of three fixed
+checker functions, `local_identity`, `boundaries` or `construct`. This is the
+control substitution boundary; no source data selects a callable. Its fixed
+call sites and eleven rejection cases constrain and exercise every target.
+Replace it with explicit dispatch if the control interface ever accepts
+external definitions.
+
+## Open next questions
+
+1. Can a later scalar rewrite eliminate an actual paid frame transition?
+   Specify one different block's complete entry and exit frames, and price
+   every changed native occurrence. The present scalar saving does not
+   change the signed recursive moment; no later block was searched.
+2. Does an active successor retain this exact block? Locate it by roles,
+   coefficients and frame in one pinned successor, then establish its legal
+   boundary. Event indices alone cannot transfer the patch, and successor
+   helper bases may differ. No such transfer is supplied or reserved.

research/p104-selected-scalar-pair/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "19fdb086ea96af9a9f87936a2b888e10af4ed810",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "581f8d87008354eb5b869f7ed9de1317ad837ce8",
+      "sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec",
+      "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81"
+    },
+    {
+      "path": "research/p104-helper-subspace-splits/WORD.json.gz",
+      "usage": "input",
+      "git_blob": "e325091e3051afe37a2d9c82f85afe9028510edb",
+      "sha256": "bbb072bced5824a8d9094c49be2fc51fbfdd05fb2b7dab8106e1275053075099"
+    },
+    {
+      "path": "research/p104-helper-subspace-splits/local.py",
+      "usage": "provenance",
+      "git_blob": "e6e28bacdad7cb51400d827b9a6e02f2b605d357",
+      "sha256": "ef430511ce36fc658ab711bd84cd2d7b484b8af173238ff89cbfec6139fefbad"
+    },
+    {
+      "path": "research/p104-helper-subspace-splits/README.md",
+      "usage": "provenance",
+      "git_blob": "f6303a881379723e1f6512dfe153fafeecbf62dd",
+      "sha256": "a83c5c3ae7a31a5dfbc3c7c59669ee55cc15f966a0cb4405cb48cb8fa2a94f56"
+    },
+    {
+      "path": "research/scalar-trace-compression/README.md",
+      "usage": "provenance",
+      "git_blob": "c85993d7048b287cdf2b4a1852dfb2dc18eb320f",
+      "sha256": "5ccdab3e70502840fabcb4b234db88c024ee3f28ff9ca7e2e7e4dc3c3115a794"
+    },
+    {
+      "path": "research/p104-sign-gauge/WORD.json.gz",
+      "usage": "input",
+      "git_blob": "8d58e759a8650b0280f1f9ff82b60330035b6a24",
+      "sha256": "4df09a16721231d6329ebf48779319a6d2c4ff059e6fb939fd272af18f9dc674"
+    },
+    {
+      "path": "research/p104-sign-gauge/README.md",
+      "usage": "provenance",
+      "git_blob": "a13bbdfd90eb097ff4df18d032d90ecf5fb11eea",
+      "sha256": "467138707f6ea693d9f11c06e5fa214a73d00e8271f7d3e7a4804cd81400cad3"
+    },
+    {
+      "path": "research/p104-sign-gauge/NOTICE",
+      "usage": "provenance",
+      "git_blob": "8f9470a9e0b94599916e5029b74b062db0a379ef",
+      "sha256": "f1280483dc93b1219c867c4e9b963ef2ae2e90cc66ec364abcf1378422c2b122"
+    }
+  ]
+}