Tighten completed P104 feedback with per-operand ideal norms

research/p104-operand-weighted-feedback/README.md

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+# Operand-weighted feedback for the completed P104 word
+
+For the exact completed forward word in [PR2166](https://github.com/CrocSwap/integer-mult-bounds/pull/2166),
+the same completed-instruction error contract admits **27 uniform guard bits**,
+improving its sufficient 39-bit estimate. For requested accuracy b >= 3,
+**26 guard bits suffice**. The word, its native rank mass 6,782, and its
+unfavorable fixed-profile cost screen are unchanged.
+
+The improvement comes from retaining each instruction's ideal operand norm
+before accumulating its error. Multiplying the largest ideal state anywhere
+by the largest defect amplification anywhere is unnecessarily coarse for
+this word. This is a conditional precision implication, not an implementation
+of rounded children, a physical precision minimum, or a new supplier or κ.
+
+## Exact source and result
+
+Nick Petalas's PR2166 is pinned at
+`a655734277aa5f2ad74bcba29e093f6dd3b84a68`. Five unchanged files, including
+the complete word, source verifier and original receipt, are retained with
+SHA-256 and Git blob pins in [REFERENCES.json](REFERENCES.json). It supplies
+the signed coefficient model, completed-output contract, all-cut constants
+and unitary-chart interpretation. Jose Manuel's PR2139 supplies the completed
+construction; Adam Gall's PR2096 supplies the preceding stability framework.
+
+The unchanged word has 844 initial roles, 34,333 instructions and 32,722
+defect sites. It has 34,334 instruction boundaries including the initial one.
+All initial register Hilbert norms are at most B. The norm throughout is the
+maximum of register Hilbert norms, not addresswise maximum norm.
+
+| Quantity | Exact value |
+|---|---:|
+| Source ideal all-cut row norm P | 131,072 |
+| Source weighted defect coefficient K | 2,134,050 |
+| New operand-weighted defect coefficient J | 66,657,328 |
+| Old uniform threshold K(P+1) | 279,716,335,650 |
+| New uniform threshold J+K | 68,791,378 |
+| Sufficient uniform guard, old / new | 39 / 27 |
+
+J first peaks after zero-based event 33373, at source register 93. The peak's
+ideal input row is `-65536*(input86+input93)`. Input86 is an arbitrary dirty
+helper, so dropping it would invalidate the result. The full 29-defect peak
+row and each of its new weights are recorded in [RESULT.json](RESULT.json).
+
+## Operand weights and feedback theorem
+
+Write the exact ideal state at a cut as `z_r = sum_s L_rs input_s`, and put
+`p_r = sum_s |L_rs|`. Thus `||z_r|| <= B*p_r`. At each defect site j define
+the weights from the **pre-instruction ideal rows**:
+
+| Exact operation | Ideal operand weight a_j | Error weight w_j |
+|---|---|---|
+| Unitary native transport of r | p_r | 1 |
+| `dst += c*src` | p_dst + |c| p_src | 1+|c| |
+| `dst *= c` | |c| p_dst | |c| |
+
+Owned resets and monomial/frame annotations remain exact as in the source.
+The source's relative contract bounds the completed operation's output
+defect by u times its actual operand expression in the second column,
+with actual Hilbert norms replacing the p values. A native child's internal
+implementation must establish that contract separately.
+
+Let C_(t,r),j be the exact signed coefficient transporting site j's logical
+defect to register r at boundary t, including effects of later resets. Define
+
+`J = max_(t,r) sum_j |C_(t,r),j| a_j`,
+
+`K = max_(t,r) sum_j |C_(t,r),j| w_j`.
+
+K is exactly the source's retained weighted coefficient; the new replay
+checks that its w vector agrees entry for entry. Let E be the maximum actual
+minus ideal register error over **every included boundary**. At each site,
+the triangle inequality for its perturbed operands gives
+
+`||eta_j|| <= u*(B*a_j + w_j*E)`.
+
+Unitary source charts preserve each physical defect's Hilbert norm. The
+exact linear superposition formula and triangle inequality at every boundary
+therefore give
+
+`E <= u*(J*B + K*E)`.
+
+If uK < 1, rearrangement proves
+
+**`E <= B*u*J / (1-u*K)`**.
+
+This is an a priori finite-execution bound: E includes the earlier operands
+used to bound every defect, and no independence or absence of feedback is
+assumed for actual defects. Signed coefficients are retained until the final
+weighted sums; a positive-majorant recurrence through the entire word is
+not substituted for them.
+
+For integer b >= 0 and u <= 2^(-b-g), a sufficient condition for
+`E <= B*2^-b` is
+
+**`2^g >= J + 2^-b*K`, with `2^(b+g) > K`.**
+
+For all b >= 0 the right side is at most J+K=68,791,378 < 2^27.
+For b >= 3 it is at most J+K/8=66,924,084.25 < 2^26. The strict
+feedback condition also holds. Conversely J+K/4=67,190,840.5 > 2^26,
+and J > 2^25. Thus this particular sufficient inequality yields the exact
+piecewise integer allowance 27 for b=0,1,2 and 26 for b>=3.
+Failure of its smaller allowance is **not** a lower bound on physical
+precision or a claim that another analysis cannot improve it.
+
+The new synthetic weighted-defect coefficient J is attained by assigning
+each independent defect the sign of the peak covector and magnitude a_j.
+The checker executes that injection literally. This only proves sharpness
+for the independent defect budget `||eta_j|| <= a_j`; the ideal-row maxima
+need not be attained by one actual trajectory, and that injection is not
+claimed to satisfy the state-relative rounding contract. No sharpness of
+the nonlinear feedback bound is asserted.
+
+## Verification and its limits
+
+The replay authenticates all five original files before importing source
+code. It first regenerates and compares the **complete original receipt**,
+including P, A=1,394,705, K, the old 39-bit guard, signed peak witnesses,
+endpoint error maxima, all-cut stream hash and paid profile. The old estimate
+remains correct; it is being refined, not reported as a bug.
+
+A new Fraction-based ideal-row pass computes every a_j before its instruction
+and independently checks all 844 final input rows. The new weighted forward
+pass reuses the authenticated source's integer-row machinery with a different
+weight vector. Every changed row's weighted sum is directly recomputed to
+check the cache. This reuse is credited and is not described as an independent
+implementation. All changed rows at all boundaries are visited; unchanged
+rows preserve the previously checked bound.
+
+A separately written sparse Fraction reverse covector agrees on every signed
+coefficient of the J peak and six endpoint rows spanning source, target,
+helper and private roles. These are seven reverse checks, not an independent
+reverse audit of every boundary. The complete forward scan supplies that
+coverage. An independent literal scalar injection attains J for the synthetic
+weighted-defect model described above.
+
+Six semantic controls show why pre-operation norms, both addition operands,
+absolute coefficient magnitudes, escaped private defects, relative feedback,
+and native sites with zero ideal value must be retained. The real source has
+no zero-ideal-weight defect site; the latter is an explicit small contract
+control rather than an additional source finding. Export controls exercise
+source and typed-receipt corruption and actual weight-code mutations under
+optimized Python.
+
+From the repository root, standard-library Python 3.12:
+
+```sh
+python3 -I -B scripts/research_package.py --package research/p104-operand-weighted-feedback
+python3 -I -B scripts/research_package.py --package research/p104-operand-weighted-feedback --check-only
+```
+
+The mathematical replay prints `PASS_P104_OPERAND_WEIGHTED_FEEDBACK`.
+The complete registered replay and all five export test groups passed on
+CPython 3.12.14; the optimized primary replay also passed on CPython 3.14.5.
+The registered command also runs the artifact tests. Default replay is
+read-only and checks the full typed receipt; explicit `--write NEW_PATH`
+creates a new file exclusively. The mathematical export needs only this
+package, no Git or network. Metadata/source pin success is not mathematical
+acceptance.
+
+The fixed word is accepted as the source's exported native/logical stream;
+this package does not re-run its upstream producer, every Clifford recipe,
+or the selected cold replay. Physical adapters must retain the source's
+exact/unitary assumptions. Approximate borrowed restoration is bounded by E;
+it is not promoted to exact restoration of rounded state. Exact private
+discard remains part of the source contract. Internal rounding, actual
+operator implementation and reserve/precision charges remain open.
+
+## Family decision, overlap and next consumer
+
+Scientific base is current main
+`c3c94c2ea2ebcd5d69d1e51b5fe8a36bf79819f5`, priority revision 5. The fresh
+recursive-interface selection and binding baseline are in TASK.json. The
+scope is changed to this concrete precision-interface refinement after an
+output-slice candidate was retired as duplicate of PR2709; that candidate
+was not published or counted. Coordination is on
+[PR2166](https://github.com/CrocSwap/integer-mult-bounds/pull/2166#issuecomment-6199131728).
+
+PR2670 independently audits a different QQ16 word and tightens its integer
+guard using the original K(P+1) inequality. PR2654 studies a different P104
+relay word; PR2549 concerns a private-macro dual. They retain their priority
+and do not supply this exact word's operand-weighted J or feedback estimate.
+Cached open-PR and Discussion coverage, live source head and current-main
+review checks are recorded with their limitations. No-match searches are
+not novelty certificates.
+
+**Decision: narrow.** Proved here: the conditional feedback implication,
+source-bound all-cut J and its integer allowance. Constructively supplied:
+exact forward, reverse and synthetic-witness checks. Unknown: actual rounded
+rank-1 through rank-7 child contract, internal range, physical tape/hosting,
+global precision charges and all-size supplier admission. Unknown costs are
+not zero. The native rank mass and unfavorable fixed-profile cost screen are
+retained; no selected precision invoice is reduced.
+
+## Open next questions
+
+Can an actual rounded native child meet the same completed-output Hilbert
+contract at u <= 2^(-b-27), or 2^(-b-26) for b>=3? Begin with one declared
+rounding convention and bind its local defect and magnitude bounds to the
+exact operand expression above. Stop further guard arithmetic for this word
+until that implementation or a materially sharper feedback model is supplied.
+
+## Credit
+
+Prepared for ywz65 with substantial OpenAI Codex assistance. Credit Nick
+Petalas / PR2166 for the exact exported precision model and arithmetic;
+Jose Manuel / PR2139 for the completed construction; Adam Gall / PR2096 for
+the Hilbert-defect framework; Arham Tahir / PR2670 for the distinct QQ16
+threshold audit; and the Rohan Arun, Chafik Boukhalfa, Ali Romman, Josie
+Elliston and selected Hamming/compiler lineage retained in source/NOTICE.md.
+Standard perturbation and triangle-inequality arguments are not claimed as
+new general mathematics. New code and retained material are Apache-2.0.
+No human review, proof-assistant or hosted-CI pass is claimed.

research/p104-operand-weighted-feedback/REFERENCES.json

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+{
+  "pr": 2166,
+  "head": "a655734277aa5f2ad74bcba29e093f6dd3b84a68",
+  "credit": "petalas / PR2166; Jose Manuel / PR2139; Adam Gall / PR2096 and retained P104 lineage",
+  "files": [
+    {
+      "local": "source/verify.py",
+      "path": "research/p104-delayed-precision/verify.py",
+      "sha256": "b2e434d9439c406be20e6ad0c6b9830b928f53d1b2730c30aeb903f4ac19e04c",
+      "git_blob": "99cb11d76a176fac48a15fd134966ee5723fd128"
+    },
+    {
+      "local": "source/WORD.json.gz",
+      "path": "research/p104-delayed-precision/WORD.json.gz",
+      "sha256": "a6b6fc45a91958f6d179486f187f4f9d66f517f3186328058f94b1d4e4aed3bf",
+      "git_blob": "44bc431a2d077340a891b6e51ddbfb7e959cf2b1"
+    },
+    {
+      "local": "source/RESULT.json",
+      "path": "research/p104-delayed-precision/RESULT.json",
+      "sha256": "8fcddf1cd8db1104e1a2ea82f32fcdf9157c32e81e0e0c82792b5f960dff8510",
+      "git_blob": "15b2b4d761666587606dae49eff27abda8fc8280"
+    },
+    {
+      "local": "source/README.md",
+      "path": "research/p104-delayed-precision/README.md",
+      "sha256": "d3b210b2272d199790c8acd3e312a6fcadc59d649a5095ed4b92cf762ed0cb10",
+      "git_blob": "159d0aa525ded387103c36ae7223b44b862caf3b"
+    },
+    {
+      "local": "source/NOTICE.md",
+      "path": "research/p104-delayed-precision/NOTICE.md",
+      "sha256": "232b1e797fcc47ebb7756a73428fa044293330f4b9c6fa291ea3bd7160a4a8b2",
+      "git_blob": "53bf10600c0c5ee4ba55085d5d721a90a43d4d19"
+    }
+  ]
+}