Reduce Gaussian phase payload work by 25–50 percent

research/streaming-gaussian-phase-tariff/PROOF.md

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+# Exact phase-scan reduction
+
+This construction changes the payload execution of the retained PR2250
+Gaussian phase pass. The phase polynomial, tensor convention, address order,
+affine word and compact operations stay the same. The source proof remains
+the authority for those contracts; this proof covers the new scan and bill.
+
+## Domain
+
+A complete declared address rectangle has M records, each with an integral
+number of Gaussian coefficients. A coefficient is two b-bit two's-complement
+words, real first, in least-significant-bit order, with b>=2. Put V equal to
+the complete payload bit count. The source head initially has offset h from
+its origin. Phase controls depend only on the current address, with the
+retained tensor exponent f*p0+sum_j q(x_j) modulo four. For an inverse pass,
+negate this exponent on the recovered input address. Let V_odd be the total
+payload volume of records whose phase is odd.
+
+The result is exact modulo 2^b in each component. It is also exact on signed
+Gaussian integers when the usual symmetric headroom prevents overflow.
+The most negative two's-complement word is included in the modular theorem;
+it is not evidence that signed negation cannot overflow.
+
+## Scan and correctness
+
+For phase 0, copy both halves directly to the output. For phase 2, stream
+the negation of each half directly. Reset the one-bit carry before each
+half. The recurrence starts with carry=1 and emits the low bit of
+1-input_bit+carry, then retains its high bit. Induction on the consumed
+prefix proves two's-complement negation modulo 2^b.
+
+For an odd phase, copy the real half into one scratch tape. Stream the
+imaginary half directly to output, negating it for phase 1. Rewind scratch
+and append the real half, negating it for phase 3. Erase scratch before the
+next coefficient. The resulting pairs are (-imag,real) and (imag,-real).
+Together with the even cases, these are exactly multiplication by i^phase.
+
+Every branch consumes exactly one coefficient and appends exactly one
+coefficient. It cannot mix record boundaries. Induction over coefficients
+and records proves the complete endpoint. Finish and erase the consumed
+input, rewind output and return with empty scratch at its origin. No
+payload is addressed by an output index. Bulk copy abbreviates consecutive
+bit reads and writes; the charge includes every bit.
+
+The scan uses three payload roles in total: input, output and one real-half
+scratch tape. Scratch has peak extent b, independent of the number of
+records. The phase counter/arithmetic interface is inherited and separately
+charged. The complete monomial wrapper uses the unchanged affine word,
+compact operations and inverse chronology, so replacing only this phase
+pass preserves its endpoint and ordinary-call inventory.
+
+## Exact invoice
+
+The following counts include input erasure, scratch clearing and output
+rewind. They apply per R=2b coefficient when h=0.
+
+| Phase parity | Reads | Writes | Erasures | Moves | Total |
+|---|---:|---:|---:|---:|---:|
+| even | R | R | R | 6R | 9R |
+| odd | 3R/2 | 3R/2 | 3R/2 | 9R | 27R/2 |
+
+In the odd branch, the extra half is copied to scratch, rewound, read,
+rewound again and erased. These actions account for the complete extra
+9R/2. The final source erase contributes its head return as well as its
+forward erase and return. An initial rewind contributes h once.
+Therefore the total is
+
+```text
+new = 9V + (9/2)V_odd + h
+old = 18V + h
+saving = 9V - (9/2)V_odd.
+```
+
+The old formula follows by counting both half-scratch paths in the unchanged
+source. At h=0, the new payload bill saves at least one quarter and at most
+one half. A nonzero initial head adds the same h to both bills; do not apply
+the percentage to that extra charge. The number of control evaluations is
+still M. No lower bound over all phase implementations is claimed.
+
+For each actual event8814 right/left fixture, V=32768 and V_odd=16384.
+The phase bill decreases from 589824 to 368640, exactly 37.5 percent.
+This percentage is for the phase component, not a complete monomial,
+recursive caller, runtime benchmark or multiplication exponent. The unchanged
+affine/ordinary calls can dominate the full monomial cost.
+
+## Evidence and remaining costs
+
+The verifier checks all coefficient pairs of widths 2 through 6 in all four
+phases, 768 tensor/polynomial/inverse patterns with nonzero initial heads,
+the actual right/left recipes and both complete monomial directions. It also
+checks a two-column compact fixture in both directions with arbitrary dirty
+compact fields and identical ordinary-call inventories. An independent
+integer oracle checks phase values. All measured bills must match exactly.
+
+Three arithmetic discriminators distinguish inverse phase, missing carry and
+missing odd-phase sign. Three malformed extents are executed and rejected.
+Finite checks corroborate the scan proof; they are not its all-width basis.
+
+Descriptor production, encoded phase arithmetic, the complete physical
+caller, recursive child, precision and final supplier admission remain open.
+This local constant reduction does not change the complete kappa calculation.
+It does not replace the separate Word RAM cost model or Gaussian normalization
+kernel. New work: Ali Romman with substantial OpenAI Codex assistance,
+Apache-2.0. Jose Manuel and the retained source authors keep their credit.

research/streaming-gaussian-phase-tariff/README.md

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+# Streaming Gaussian phase tariff
+
+**Local algorithmic improvement; no new kappa.** Use direct streaming for
+even Gaussian phases and one real-half scratch tape for odd phases. The
+complete payload bill is `9V+(9/2)V_odd+h`, instead of the retained source's
+`18V+h`. Here h is the initial head offset and V_odd is the volume with an
+odd phase. From origin, this saves 25 to 50 percent of phase payload work.
+
+The [proof](PROOF.md) covers all valid complete record layouts and coefficient
+widths. The unchanged phase polynomial, affine word and compact interpreter
+are composed in [phase.py](phase.py). No frozen source file is changed.
+The actual event8814 right and left phase passes each save 37.5 percent:
+589824 to 368640 transitions. [RESULT.json](RESULT.json) records these values.
+This is a component tape invoice, not an end-to-end runtime or kappa gain.
+
+## Replay
+
+Use CPython 3.12, standard library only, from the repository root:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/streaming-gaussian-phase-tariff
+```
+
+Expected status: `PASS_STREAMING_GAUSSIAN_PHASE_TARIFF_NO_NEW_KAPPA`.
+The registered hook runs normal and optimized modes and checks source hashes
+and a typed exact receipt. Evidence includes 21824 coefficient/phase cases,
+768 tensor phase patterns, four actual complete monomial cases, two compact
+tensor round trips, three arithmetic discriminators and three input rejections.
+Every new result agrees with the unchanged predecessor and an independent
+integer phase oracle. All source ordinary calls and control evaluations are
+retained. The official export permits a Git-free replay.
+
+No full supplier, native catalogue replay, encoded-control lowering, Lean,
+Docker or selected cold replay was run. Human and formal-kernel review are
+unclaimed. The written scan proof supplies the uniform result. The complete
+caller, recursive child, precision and supplier budget remain separate.
+
+## Attribution and scope
+
+Base `7c472563474e60185e6ca101a17622133d047e56`, priority revision 7.
+[SOURCES.json](SOURCES.json) pins consumed files; selected-result and catalogue
+files are historical provenance. Jose Manuel/jmbarrancoml supplied the source
+scan, affine/compact routines and proof in
+[PR #2250](https://github.com/CrocSwap/integer-mult-bounds/pull/2250).
+Douglas Colkitt's compact construction remains credited there. JoyboyBrian's
+[PR #5327](https://github.com/CrocSwap/integer-mult-bounds/pull/5327)
+binds the actual event8814 fixture to Cinna Davis's mixed caller and retained
+native/Clifford lineage. The Word RAM phase work in
+[PR #4538](https://github.com/CrocSwap/integer-mult-bounds/pull/4538)
+and Gaussian normalization in
+[PR #3127](https://github.com/CrocSwap/integer-mult-bounds/pull/3127)
+are related work with different operations and cost models, not consumed
+implementations or new results claimed here.
+
+Prepared by Ali Romman with substantial OpenAI Codex assistance. New code and
+proof: Apache-2.0. Existing source notices remain unchanged.
+
+## Open next questions
+
+Lower this scan onto the encoded physical coefficient/Count interface and
+charge the phase controls there. Use the exact component bill and preserved
+ordinary-call inventory as comparison points. The full caller and recursive
+child are required before any complete supplier gain can be claimed.

research/streaming-gaussian-phase-tariff/RESULT.json

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+{
+  "status": "PASS_STREAMING_GAUSSIAN_PHASE_TARIFF_NO_NEW_KAPPA",
+  "coefficient_cases": 20,
+  "coefficient_pairs": 21824,
+  "pattern_cases": 768,
+  "actual_recipe_phase_receipts": {
+    "right": {
+      "volume": 32768,
+      "odd_volume": 16384,
+      "new_transitions": 368640,
+      "old_transitions": 589824,
+      "output_sha256": "9ef1f647a6725b96633d79231f30f617ff2719b1534628615474cec42b0d8a3e"
+    },
+    "left": {
+      "volume": 32768,
+      "odd_volume": 16384,
+      "new_transitions": 368640,
+      "old_transitions": 589824,
+      "output_sha256": "3b5c2bdc4689619fe654625df3a73264f30be479b531d5f8b0f85cc6b23183e5"
+    }
+  },
+  "actual_monomial_cases": 4,
+  "compact_tensor_cases": 2,
+  "controls": 6,
+  "worst_case_payload_saving": "1/4",
+  "even_phase_payload_saving": "1/2"
+}

research/streaming-gaussian-phase-tariff/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "7c472563474e60185e6ca101a17622133d047e56",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "c35482cd9035843b15a634cc8276a7e1e40e8395",
+      "sha256": "6fe0a099b89aa2aed796a13c3a21dafc63f3388ebf397464e44e0451893b9375"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    },
+    {
+      "path": "research/complex-four-base-packet/FIXTURE.json",
+      "usage": "input",
+      "git_blob": "4788ef4a4c74847432ad1bb928d736f71245db9d",
+      "sha256": "25b296e61180c5d2683fef88edb4244faaeb35898ff76c7a8368ca7063a1ca86"
+    },
+    {
+      "path": "research/complex-four-base-packet/README.md",
+      "usage": "input",
+      "git_blob": "2b6caf55cb15a9bdff4ecb2c576de628c93a71cd",
+      "sha256": "f5653a054a3caa44343aec4f20b180c6c745af7bf1ad7404d77626e7d28fce0e"
+    },
+    {
+      "path": "research/selected-complex-compact-adapters/PROOF.md",
+      "usage": "input",
+      "git_blob": "685efc529493e4b841c1ce8799d28968b4991d7e",
+      "sha256": "804446e22304540eb02a5f4da3f71464a217be830a33b11420ea31eea878782c"
+    },
+    {
+      "path": "research/selected-complex-compact-adapters/compact.py",
+      "usage": "input",
+      "git_blob": "60bc206c78150f23563a4775800b0dd32025843d",
+      "sha256": "0e1027a60f38c21cfd808c8ce44dcc2423810efdab0e15f02b07f261ad48fc3c"
+    },
+    {
+      "path": "research/selected-complex-compact-adapters/monomial.py",
+      "usage": "input",
+      "git_blob": "5fa4494167c4ba668993a2b1c902969518728a4c",
+      "sha256": "394f9a465ac9148af0c20844d375e4befbef17d92b2fcc840d83419480dd4740"
+    },
+    {
+      "path": "research/selected-complex-compact-adapters/streams.py",
+      "usage": "input",
+      "git_blob": "61542ce4c9eb51bfc113475b4f98649cb90a853e",
+      "sha256": "718a582a50aee8f1f04945ed942be1d4162962fe47dbf05c6ff37b7fb8c61257"
+    }
+  ]
+}

research/streaming-gaussian-phase-tariff/TASK.json

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+{
+  "schema_version": 1,
+  "task_id": "uniform-transfer-audit",
+  "priority_revision": 7,
+  "base_commit": "7c472563474e60185e6ca101a17622133d047e56",
+  "original_card": {
+    "id": "uniform-transfer-audit",
+    "question": "Does one quantified step of the proposed #4100 all-size construction hold at its actual caller?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "research/unconditional-batched-crt/README.md"
+    ],
+    "methods": [
+      "Prove one missing implication with explicit quantified hypotheses.",
+      "Find an exact counterexample at the claimed interface boundary."
+    ],
+    "acceptance_gate": "Identify the proposition and what would change downstream; do not equate a finite replay with a universal proof.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read PROOF.md and AUDIT.md in #4100. Select one distinct obligation: CRT total repair, native/free-algebra lifting, precision/conditioning, or complete fixed-tape cost. This is an author-submitted proof, not an accepted unconditional theorem.",
+      "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately."
+    },
+    "first_experiment": "Check live handoffs; name one proposition, its caller, quantified domain and dependency closure. Independently prove the implication or exhibit a precise gap/counterexample. Prioritize CRT.md total repair or COMPLEX.md free-algebra lift only if unclaimed. Stop at the chosen obligation; do not substitute finite samples or root arithmetic for an all-size proof, or launch the complete package by default.",
+    "bottleneck_link": "Finite supplier improvements are useful only under the retained transfer obligations.",
+    "allowed_variations": [
+      "One CRT repair, native lift, precision/conditioning or tape-cost proposition, with its exact caller."
+    ],
+    "scope_change_triggers": [
+      "Replacing a quantified obligation with another finite smoke test.",
+      "Treating archive intake, an author PASS receipt or a finite composed fixture as discharge of the universal hypotheses."
+    ],
+    "overlap_terms": [
+      "uniform transfer",
+      "all-size",
+      "precision",
+      "4100",
+      "unconditional-batched-crt"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4100: proposed unconditional construction and finite checks, without independent acceptance.",
+        "evidence": "research/unconditional-batched-crt/README.md",
+        "remaining": "Independent proof audit of one caller-bound all-size implication; all other premises remain explicit.",
+        "status": "author-checked",
+        "source_commit": "337d04939571a637aeab54a0f58f54fde811663a",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "An exact lower-cost sequential Gaussian phase scan and source monomial composition.",
+  "scope_change_reason": "Replace two unconditional half-scratch paths with direct even phases and one odd-phase scratch path; a scoped local implementation improvement.",
+  "bottleneck_link": "Reduces paid payload work in the actual monomial adapter, without claiming a complete supplier or asymptotic gain.",
+  "budget_minutes": 60,
+  "stop_rule": "Finish exact scan correctness and tariff; do not expand into encoded control or full supplier work.",
+  "outcome": "completed",
+  "evidence": "PROOF.md, RESULT.json, exhaustive small coefficients, mixed tensor patterns, actual complete monomials and compact composition.",
+  "author": "Ali Romman",
+  "ai_assistance": "Substantial OpenAI Codex assistance; independent human and proof-kernel review unclaimed.",
+  "open_questions": [
+    {
+      "question": "Can this scan be lowered to encoded physical controls with the same bill?",
+      "why_it_matters": "The current address/control interface is inherited and separately charged.",
+      "first_step": "Bind one actual coefficient/Count traversal and charge its phase controls.",
+      "blocked_by": "Full caller, recursive child and complete supplier pricing."
+    }
+  ],
+  "coordination": {
+    "claim": "https://github.com/CrocSwap/integer-mult-bounds/discussions/2238#discussioncomment-19153519",
+    "overlap": "318 open PRs, 4648 Discussion records, all 7267 main notes and targeted all-state searches; arbitrary diffs and unpublished work not covered."
+  }
+}

research/streaming-gaussian-phase-tariff/VERIFY.json

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+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/complex-four-base-packet/FIXTURE.json",
+    "research/complex-four-base-packet/README.md",
+    "research/selected-complex-compact-adapters/PROOF.md",
+    "research/selected-complex-compact-adapters/compact.py",
+    "research/selected-complex-compact-adapters/monomial.py",
+    "research/selected-complex-compact-adapters/streams.py"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Exact phase values and invoices, source monomial endpoints, compact composition, source hashes and typed receipt."
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-O",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 120,
+      "checks": "Exact phase values and invoices, source monomial endpoints, compact composition, source hashes and typed receipt."
+    }
+  ]
+}