Resynthesize P104 scalar prefix: 3,577 fewer events (No new κ)

research/p104-prefix-row-synthesis/README.md

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+# Whole-matrix synthesis saves 3,577 events in the selected P104 prefix
+
+**Completed local circuit result. No new κ.** On the selected PR6030 word,
+replacing its first scalar-only block reduces the complete word from **22,803
+to 19,226 events** and its complete local scalar allowance from **189,428 to
+142,778**. Every entry of the block's arbitrary-input matrix is unchanged.
+All frame changes, copies, initializations, erasures, helper endpoints and paid
+recursive calls retain their original order and values. The selected record
+remains κ = 781451/250000000 = 0.003125804.
+
+The contribution is a reproducible construction and exact certificate for one
+local module, not adoption of a new supplier or an implementation benchmark.
+It addresses Matarener's [PR5975](https://github.com/CrocSwap/integer-mult-bounds/pull/5975)
+open question 2, global ordinary re-synthesis. PR5975 preserves the original
+dependency graph through peephole passes; Nick Petalas's
+[PR5999](https://github.com/CrocSwap/integer-mult-bounds/pull/5999) shares one
+form through a dirty output on PR5907. Here, whole-row sharing and sparse
+elimination rebuild the entire 567-role prefix matrix on **PR6030**, crossing
+the original internal dependencies while retaining both boundaries. PR6030
+already includes its own substantial ordinary-event rewrite. These savings
+are measured against that actual selected word, not the older PR5907 counts.
+Classical elimination and row sharing are not claimed as new algorithms.
+
+## Frozen object and construction
+
+Base: canonical main `721c03822c421b7f7fcbb2598993b947c4d88419`.
+The input is `research/p104-helper470-twin-rung-removal/WORD.json.gz`, SHA256
+`7834725d9e34ee846a9ccc615221ea81e0201613938afa185d2eda0c9a6c232f`.
+The selected source head is `772dd10a5859a1d15f4851feb371519789213ad8`, with
+portability repair `746ee5e1160ae64a15a81b2a432d5566142db850`; the word is
+unchanged by that repair. [SOURCES.json](SOURCES.json) pins every live input
+and the historical selected record, source manifest, review, pricing formula,
+task card and predecessor notes by Git blob and SHA256.
+
+Use zero-based event indices. Event 0 is `INIT 778 3`. Events **[1,9896)**
+are 9,895 ADD/SCALE instructions in frame 3 (the empty binary basis); event
+9896 is the first MOVE. All 567 touched roles are available in that frame.
+The other 232 roles are untouched. No observation, COPY, CLEAR or native call
+occurs inside this interval. Owned role 778 is live throughout it.
+
+[synthesize.py](synthesize.py) forms the exact sparse rational matrix M of
+this interval, starting with independent symbolic values for all 567 roles,
+including role 778. It then:
+
+1. Greedily subtracts a dyadic multiple of one whole output row from another
+   when this removes at least two nonzero entries. Ratios have magnitude below
+   2^12. Deterministic tie breaks fix the result; the cap is 2,000 steps.
+2. Uses sparse Markowitz-style pivots restricted to units of Z[1/2], normalizes
+   every pivot explicitly, and eliminates its column from every other row.
+   Failure to find a unit pivot aborts; no general synthesis guarantee is claimed.
+3. Removes the final row permutation with charged exchanges. Each exchange is
+   three ADDs and one SCALE by −1; there are 132, with no free renaming.
+4. Reverses the elimination list and inverts each elementary operation.
+
+The retained run has 419 sharing steps, 795,230 ordered row-pair evaluations
+and 567 pivots. The output is 6,039 ADDs and 279 SCALEs. Its maximum absolute
+coefficient is 4. Every SCALE is a dyadic unit, every ADD uses distinct roles
+already in the same frame, and no extra role or scratch field is introduced.
+
+The original trial without the sharing pass produced 10,122 events and was
+rejected. The refined trial above was the first strict improvement; discovery
+stopped there. Only this one matrix was searched. This is neither an exhaustive
+circuit search nor an optimality claim. The OS track draw was 31 (complex
+supplier), and OS seed 7525094595556296861 modulo three chose this object
+from three documented candidates. [TASK.json](TASK.json) records the scope
+change from the task card, stopping rules, overlap and checkpoints.
+
+## Why the substitution is exact
+
+Write the elementary elimination matrices in chronological order as E1,...,Ek.
+The construction checks that Ek...E1 M = I, entry by entry. Chronological
+execution of Ek^-1,...,E1^-1 therefore has matrix M. This is a statement about
+arbitrary independent entry values, not a test on sample vectors or a use of
+the zero value of owned role 778.
+
+A second implementation in [verify.py](verify.py), using dense integer rows
+with binary denominators rather than the synthesizer's sparse Fraction rows,
+compares all **321,489** entries of the original and replacement matrices.
+The remaining 232 roles are identity in both. The verifier also checks the
+replacement followed by its inverse against the literal identity.
+
+Both programs have the same matrix at the first MOVE, and the original prefix
+before the cut and suffix after the cut are kept literally. Thus every later
+observation agrees, including every moved field and every helper return.
+All scalar operations are in one common chart. Under the inherited common-chart
+native contract, this identity acts componentwise at every positive column
+multiplicity; the rational scalar coefficients stay outside tensor powers.
+No noncommuting native operators are reordered by this replacement.
+
+As a separate end-to-end finite check, the authenticated, unmodified
+`p104-unread-path-resynthesis/local.py` from Ali Romman's PR5827 replays both
+complete words using sparse rational rows. Its separate reverse-chronology
+dense integer transpose checks all 799 columns and 638,401 coefficients of
+the candidate. These are credited predecessor implementations, not newly
+independent human review or a formal proof of the multiplication theorem.
+
+## Complete cost of the local replacement
+
+For q = ±n/2^d in lowest terms, the retained real ADD allowance is
+2(3 + floor(log2 n) + popcount(n) + d). A SCALE by ±2^k costs
+2(|k| + [negative sign]). Every source access is an emitted ADD, and every
+pivot normalization and permutation operation is included. The prefix needs
+no new copy, frame transition, initialization, restoration pass or child call.
+
+| Quantity | Source | Candidate |
+|---|---:|---:|
+| Prefix events | 9,895 | 6,318 |
+| Prefix scalar allowance | 106,368 | 59,718 |
+| Complete local word events | 22,803 | 19,226 |
+| Complete local scalar allowance | 189,428 | 142,778 |
+| Local routes, normalizers, adapters, lifetime envelope | 5,325,000 | 5,325,000 |
+| Complete local ordinary bound | 5,514,428 | 5,467,778 |
+| Retained local ordinary allowance | 10,000,000 | 10,000,000 |
+| Owned labels / lifetimes / peak occupancy | 29 / 31 / 13 | 29 / 31 / 13 |
+
+All 3,453 local paid calls are unchanged, with rank counts
+`{1:2219, 2:457, 3:496, 4:100, 5:163, 6:11, 7:6, 8:1}` and rank mass 5,952.
+The 562 productive helpers retain counts
+`{1:14, 2:8, 3:4, 4:107, 5:14, 6:2, 7:9, 8:19, 9:385}`.
+Every required scalar source contact is charged and every non-scalar event
+remains identical as an ordered sequence, not just a histogram.
+
+The inherited physical caller makes 18 forward local calls, so the exact
+local scalar-allowance reduction sums to **839,700** across those calls.
+Its recursive invoice remains 66,834 paid calls and 13,444 productive fields.
+The fixed complete ordinary envelope remains **184,647,552 < 10^9**; the
+verifier reads this caller envelope from the authenticated source receipt and
+recomputes the local routing formula from the unchanged profiles. It does not charge a speculative
+new recursive saving or lower that coarse caller envelope. The paid/target
+moment, completion, compiler threshold and outer assembly are unchanged.
+Therefore no new κ is claimed.
+
+Both circuits satisfy the same inherited 26,000-event and coefficient bounds.
+The conservative scalar growth envelope 2^(16·26000), paid native rank mass,
+ownership and storage bounds can be retained. For additional diagnostics,
+the exact scalar-prefix maximum row l1 norms are 695/8 before and 343/4 after;
+the replacement inverse has maximum 1285/4. These are local matrices relative
+to independent cut inputs, not a rounded-error or whole-machine certificate.
+
+## Reproduce and inspect
+
+CPython 3.12, standard library only; tested with 3.12.14. From the repository root:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/p104-prefix-row-synthesis
+```
+
+Expected: both normal and optimized registered commands print
+`PASS_P104_PREFIX_ROW_SYNTHESIS`, then the launcher prints
+`PASS_REGISTERED_RESEARCH_PACKAGE research/p104-prefix-row-synthesis outcome=completed`.
+The launcher also validates exact source provenance and declared live inputs.
+Direct scoped verification works without Git after a source export:
+
+```sh
+python3.12 -B research/p104-prefix-row-synthesis/verify.py
+python3.12 -O -B research/p104-prefix-row-synthesis/verify.py
+```
+
+Normal verification writes no repository files. It regenerates the candidate
+and compares the entire [RESULT.json](RESULT.json) receipt. To materialize the
+complete candidate, add `--emit-word /absolute/external/path/WORD.json`; the
+path must be outside the checkout and absent. No compressed candidate copy is
+needed. Its canonical JSON payload (sorted keys, compact separators, no trailing
+newline) has SHA256
+`537f6c581ee7ebc758d2bba417f6d7b98f1cb9b5fe8916a7ae707575d9b68617`.
+The emitted file adds one newline. The receipt also binds the exact prefix and
+the dense integer matrix. `--write` is an explicit receipt-generation mode and
+refuses to overwrite an existing receipt.
+
+The positive fixture has matrix `[[1,1/2],[3/4,-13/8]]` and a checked inverse.
+Controls reject a wrong ADD sign, omitted SCALE, wrong chart, odd denominator,
+unavailable source, whole-word sign fault, and byte corruption of each live
+source (word, result and checker). A SCALE by two on the just-initialized owned
+role passes the specialized whole-word endpoint check, as expected, but fails
+our stronger arbitrary-entry block identity. This explicitly tests the extra
+local obligation rather than conflating it with complete-word endpoints.
+All checks use explicit exceptions and remain active under `python -O`.
+Authenticated executable source is compiled from the bytes just read, so direct
+verification cannot silently execute a stale predecessor bytecode cache.
+
+Executed on 2026-10-28 with CPython 3.12.14: metadata/pins, both registered
+replays, committed bytes/source history/executable modes, and the official
+10-file export with all three declared inputs passed. Normal and optimized
+isolated Python replays of that Git-free export with empty PATH passed in
+14.31 and 12.21 seconds. All 11 files including the export manifest retained
+their bytes and modes. The export was marked read-only; access-control
+enforcement is not claimed. A changed result-receipt saving was rejected in
+both modes. These times establish bounded replay, not a circuit runtime gain.
+Hosted CI and independent scientific acceptance are not claimed.
+
+## Scope and inherited obligations
+
+The selected construction, frozen packages and `upstream/` are unchanged.
+This package checks an exact local substitute and the complete scalar word.
+It **does not rerun** the full native-map, physical-caller, compiler, precision
+or outer-ledger cold replay, and it does not request supplier admission.
+Unchanged interfaces and event sequences are the explicit reuse boundary;
+changing any source pin, cut, chart, operand set or non-scalar event invalidates
+this result. A changed synthesis output must pass the full matrix and word
+checks and receive a new committed receipt.
+
+The five selected framework assumptions remain: `native_all_columns`,
+`owned_erasure`, `compiler`, `precision_tape`, `analytic_recovery`.
+The uniform word and rounded multiplier are not materialized. No speedup on a
+real implementation, stronger exponent, universally optimal circuit, or
+independent human acceptance follows from these finite certificates.
+
+## Open next questions
+
+- **Adoption through the selected supplier pipeline.** Emit this word, bind it
+  in a successor package and run the complete selected supplier/native/caller
+  and precision pipeline. This would make the ordinary saving available to a
+  later paid-call construction. Independent review and that integration replay
+  remain blockers; the five framework assumptions are not discharged here.
+- **A second disjoint scalar block.** Locate one maximal same-chart interval,
+  price all permutation and source work, and test its arbitrary-entry matrix.
+  A fresh overlap check and separate narrow claim are required. No other
+  interval was searched, and this success gives no guarantee for another one.
+
+## Credit
+
+Prepared for **@CubeJerry** with substantial **OpenAI Codex assistance**;
+no delegated agents or independent human review. New work is Apache-2.0.
+DreamingOfClouds supplied PR6030 and PR5907; Ali Romman supplied PR5827 and
+the unmodified local row/transpose checker; Mason Hale supplied PR5811.
+Matarener posed the global ordinary-resynthesis question in PR5975, and
+Nick Petalas's PR5999 established the earlier dirty-output sharing example.
+Those predecessor results are credited, not re-certified here. Douglas
+Colkitt supplied the selected application review and portability repair.
+David Arias, Cinna Davis, Gareth Davidson, Pierre-Loic Doulcet and the native,
+controller, compiler, BIT and analytic contributors retain all predecessor
+credit and notices in the source packages. No predecessor files are vendored
+or modified by this contribution.

research/p104-prefix-row-synthesis/RESULT.json

@@ -0,0 +1,152 @@
+{
+  "after": {
+    "local": {
+      "endpoint_coefficients": 638401,
+      "helpers": {
+        "1": 14,
+        "2": 8,
+        "3": 4,
+        "4": 107,
+        "5": 14,
+        "6": 2,
+        "7": 9,
+        "8": 19,
+        "9": 385
+      },
+      "owned_fields": 29,
+      "owned_lifetimes": 31,
+      "paid": {
+        "1": 2219,
+        "2": 457,
+        "3": 496,
+        "4": 100,
+        "5": 163,
+        "6": 11,
+        "7": 6,
+        "8": 1
+      },
+      "paid_rank_mass": 5952,
+      "peak_owned_fields": 13,
+      "public_inputs": 770,
+      "real_scalar_allowance": 142778
+    },
+    "max_absolute_prefix_coefficient": "4",
+    "prefix_events": 6318,
+    "prefix_opcodes": {
+      "ADD": 6039,
+      "SCALE": 279
+    },
+    "prefix_scalar_allowance": 59718,
+    "word_events": 19226
+  },
+  "base_commit": "721c03822c421b7f7fcbb2598993b947c4d88419",
+  "before": {
+    "local": {
+      "endpoint_coefficients": 638401,
+      "helpers": {
+        "1": 14,
+        "2": 8,
+        "3": 4,
+        "4": 107,
+        "5": 14,
+        "6": 2,
+        "7": 9,
+        "8": 19,
+        "9": 385
+      },
+      "owned_fields": 29,
+      "owned_lifetimes": 31,
+      "paid": {
+        "1": 2219,
+        "2": 457,
+        "3": 496,
+        "4": 100,
+        "5": 163,
+        "6": 11,
+        "7": 6,
+        "8": 1
+      },
+      "paid_rank_mass": 5952,
+      "peak_owned_fields": 13,
+      "public_inputs": 770,
+      "real_scalar_allowance": 189428
+    },
+    "prefix_events": 9895,
+    "prefix_opcodes": {
+      "ADD": 9747,
+      "SCALE": 148
+    },
+    "prefix_scalar_allowance": 106368,
+    "word_events": 22803
+  },
+  "canonical_sha256": {
+    "complete_candidate_word": "537f6c581ee7ebc758d2bba417f6d7b98f1cb9b5fe8916a7ae707575d9b68617",
+    "integer_prefix_matrix": "996f353928fe33d6c9c5f3bf0b326b6e616946b163fc2e3e29ea7384ef294aa9",
+    "replacement_prefix": "d00621885ffabffe13f71cd680867a04a18b64aaf2b059850c248f9395cb80ab"
+  },
+  "complete_columns": {
+    "arithmetic": "Dense integer rows with binary denominators; reverse chronology",
+    "coefficients": 638401,
+    "columns": 799
+  },
+  "controls": {
+    "damaged_RESULT.json": "Source SHA256: research/p104-helper470-twin-rung-removal/RESULT.json",
+    "damaged_WORD.json.gz": "Source SHA256: research/p104-helper470-twin-rung-removal/WORD.json.gz",
+    "damaged_local.py": "Source SHA256: research/p104-unread-path-resynthesis/local.py",
+    "dyadic_inverse_positive_fixture": "PASS",
+    "missing_SCALE": "Arbitrary-entry prefix matrix",
+    "odd_denominator": "Nonzero dyadic coefficient",
+    "owned_zero_specialization": "Arbitrary-entry prefix matrix",
+    "owned_zero_specialization_whole_word": "PASS as expected; only the stronger local obligation rejects it",
+    "unavailable_source": "Available distinct source",
+    "whole_word_wrong_sign": "All public and owned rows",
+    "wrong_ADD_sign": "Arbitrary-entry prefix matrix",
+    "wrong_chart": "Available scalar chart"
+  },
+  "cut": {
+    "explicit_matrix_entries": 321489,
+    "frame": 3,
+    "roles": 567,
+    "start_inclusive": 1,
+    "stop_exclusive": 9896,
+    "untouched_roles": 232
+  },
+  "discovery": {
+    "charged_exchanges": 132,
+    "entries_after_pair_steps": 4380,
+    "pair_evaluations": 795230,
+    "pair_steps": 419,
+    "roles": 567,
+    "unit_pivots": 567
+  },
+  "inherited_budgets": {
+    "complete_caller_allowance": 1000000000,
+    "complete_caller_envelope_recomputed": false,
+    "complete_caller_ordinary_envelope": 184647552,
+    "local_allowance": 10000000,
+    "local_ordinary_after": 5467778,
+    "local_ordinary_before": 5514428,
+    "local_routes_normalizers_adapters_and_lifetimes": 5325000,
+    "native_and_physical_caller_replayed": false,
+    "paid_calls": 66834,
+    "productive_fields": 13444
+  },
+  "kappa": {
+    "changed": false,
+    "claim": "No new kappa; isolated scalar-circuit result.",
+    "selected": "781451/250000000"
+  },
+  "prefix_max_row_l1": {
+    "candidate": "343/4",
+    "candidate_inverse": "1285/4",
+    "scope": "Exact intermediate scalar prefix matrices only; not a rounded error analysis.",
+    "source": "695/8"
+  },
+  "saved": {
+    "events_per_local_call": 3577,
+    "inherited_full_forward_calls": 18,
+    "scalar_allowance_across_inherited_forward_calls": 839700,
+    "scalar_allowance_per_local_call": 46650
+  },
+  "status": "PASS_P104_PREFIX_ROW_SYNTHESIS"
+}