From 405b06ae5b1aba1bf452ee6e356fe8e2b82815d2 Mon Sep 17 00:00:00 2001 From: Gareth Davidson Date: Tue, 13 Oct 2026 09:31:47 +0100 Subject: [PATCH 1/2] research: audit fractional guard depth of bidirectional Horner --- research/dyadic-fractional-grid/CLAIM.json | 103 + research/dyadic-fractional-grid/EXPECTED.json | 20696 ++++++++++++++++ research/dyadic-fractional-grid/README.md | 184 + research/dyadic-fractional-grid/SOURCES.json | 113 + research/dyadic-fractional-grid/VALIDATION.md | 94 + research/dyadic-fractional-grid/crosscheck.py | 90 + research/dyadic-fractional-grid/fixed_grid.py | 108 + research/dyadic-fractional-grid/rows.py | 87 + research/dyadic-fractional-grid/test_grid.py | 88 + .../vendor/pr837/lower.py | 92 + .../vendor/pr837/vendor/pr696_lower.py | 56 + research/dyadic-fractional-grid/verify.py | 252 + 12 files changed, 21963 insertions(+) create mode 100644 research/dyadic-fractional-grid/CLAIM.json create mode 100644 research/dyadic-fractional-grid/EXPECTED.json create mode 100644 research/dyadic-fractional-grid/README.md create mode 100644 research/dyadic-fractional-grid/SOURCES.json create mode 100644 research/dyadic-fractional-grid/VALIDATION.md create mode 100644 research/dyadic-fractional-grid/crosscheck.py create mode 100644 research/dyadic-fractional-grid/fixed_grid.py create mode 100644 research/dyadic-fractional-grid/rows.py create mode 100644 research/dyadic-fractional-grid/test_grid.py create mode 100644 research/dyadic-fractional-grid/vendor/pr837/lower.py create mode 100644 research/dyadic-fractional-grid/vendor/pr837/vendor/pr696_lower.py create mode 100644 research/dyadic-fractional-grid/verify.py diff --git a/research/dyadic-fractional-grid/CLAIM.json b/research/dyadic-fractional-grid/CLAIM.json new file mode 100644 index 0000000000..9c3b46916b --- /dev/null +++ b/research/dyadic-fractional-grid/CLAIM.json @@ -0,0 +1,103 @@ +{ + "assignment_seed": 60, + "drawn_at_utc": "2026-10-13T07:35:04.000978+00:00", + "claimed_at_utc": "2026-10-13T07:46:09.123523+00:00", + "track": "adversarial mathematical audit", + "base_commit": "f567884aeb8851b6e794af1184119a3bfbe1e631", + "candidate_selection": "60 modulo 3 = 0", + "candidates": [ + { + "object": "Fixed-point fractional-grid transport for PR837 bidirectional Horner", + "method": "Exact 2-adic valuations of every coefficient and whole-program primitive prefix; integer-grid counterexamples", + "chosen": true + }, + { + "object": "Numerical frame-search optimality in PR815", + "method": "Exact dual/interval proof audit", + "chosen": false, + "reason": "The source explicitly discloses missing exact dual certificates; a further solver audit would need a separate larger scope" + }, + { + "object": "Global replica-prefix accounting", + "method": "Maximum-replica norm induction", + "chosen": false, + "reason": "PR821 already supplies this refinement" + } + ], + "statement": "Exact endpoint equality and positive magnitude-prefix bounds do not alone establish preservation of the old fixed-point fractional grid. Quantify the extra grid depth for the actual PR837 choices and all six pinned programs. This audits an explicitly unresolved precision obligation, not a claim of a defect in the selected construction.", + "method": "Prove the necessary/sufficient denominator-depth criterion for dyadic linear prefix maps. Census all 962 coefficients, replay all six original and replacement programs and produce exact integer-rounding witnesses with corrective guard bits.", + "domain": "The scalar primitive projections of fixed #626 CK/JP8 and #741 SW4 programs bound in PR837, both orientations, arbitrary native/private input columns. Fixed dyadic grids; exact intermediate representability, not physical frame/phase evaluation or all-size numerical stability.", + "stopping_rule": "Complete this finite census and replay; no compiler integration, exponent repricing or exploratory search beyond the stated domain", + "expected_finish": "Approximately 90 minutes from the claim; final quarter for validation and handoff", + "coordination": "Local claim; only issue #650 requested Discussions at previous refresh; no sprint coordination issue available.", + "related_prs": { + "837": { + "title": "Research: bidirectional Horner saves 312 SW4 primitives per orientation (no new kappa)", + "head": "0a2977195efc8fb2a918fbfb694ec8d294387f08", + "author": "bitplane" + }, + "830": { + "title": "Research: transposed JP8 work saves 32,810 backward primitives (no new kappa)", + "head": "6a0b8d55bb8026f8b6d708ae07740453b41c872f", + "author": "Ryan-Haines" + }, + "821": { + "title": "Research audit: 286-to-160-bit prefix estimate for pinned #626 (no new kappa)", + "head": "6b832a6ba4dce8dd6e2fb0897dd4543364b932f8", + "author": "Brodee042" + }, + "815": { + "title": "Research: exact frame search certifies #741 and #626 providers frame-optimal (rediscovers #741; negative result)", + "head": "d8c8b40ab0a95b6d77389e6726d86dddb60fb53c", + "author": "Rohansguliani" + }, + "764": { + "title": "Research: sharp inverse-square rounding bound for cyclic scans", + "head": "983d66f0f2e48ee500ec7eee2da9244c001c5064", + "author": "cbethin" + }, + "751": { + "title": "Research: save 19,688 JP8 backward primitives with reverse Horner (no new kappa)", + "head": "7a1a3a319da359ec8b6e83230a7fcf7733fac8e5", + "author": "thakrarsagar" + }, + "734": { + "title": "Research: prefix-bounded signed-Horner circuits for CK and JP8 (no new kappa)", + "head": "ba76ba3321d000c8996b0b254bcf0d7260e1bbb6", + "author": "AllStars101-sudo" + } + }, + "initial_open_pr_count": 648, + "status": "validated; ready for submission", + "final_overlap_check": { + "at_utc": "2026-10-13T08:30:13.943767+00:00", + "open_pr_count": 659, + "latest_number": 864, + "upstream_head": "f567884aeb8851b6e794af1184119a3bfbe1e631", + "pr837_head_unchanged": true, + "finding": "No overlapping fractional-grid audit found; new cyclic-inverse precision, signed CK reuse and X4 frame-optimality work has distinct scope." + }, + "result": { + "statement": "CK forward scalar-projection fractional guard rises from 3 to 5 bits under PR837; shared forward/backward guards remain CK=8, JP8=27, SW4=19.", + "coefficient_count": 962, + "candidate_circuits_checked": 3842, + "extra_grid_coefficients": 18, + "complete_traces": 12, + "scalar_primitives_crosschecked": 1016110, + "local_undercharged_counterexamples": 24, + "whole_program_counterexample": "CK forward, native basis column 16: endpoint errors at guards 3 and 4, exact at guard 5.", + "new_kappa": false, + "precision_scope": "Fixed dyadic-grid representability of the scalar projection; physical frame/phase/recursive realizations and integer overflow outside scope." + }, + "validation": { + "python_versions": [ + "3.12.3", + "3.14.4" + ], + "unit_tests": 9, + "package_controls": 6, + "receipt": "EXPECTED.json", + "details": "VALIDATION.md", + "all_local_checks": "passed" + } +} diff --git a/research/dyadic-fractional-grid/EXPECTED.json b/research/dyadic-fractional-grid/EXPECTED.json new file mode 100644 index 0000000000..2b7c513f49 --- /dev/null +++ b/research/dyadic-fractional-grid/EXPECTED.json @@ -0,0 +1,20696 @@ +{ + "base_commit": "f567884aeb8851b6e794af1184119a3bfbe1e631", + "candidate_circuits_checked": 3842, + "coefficients": [ + { + "all_candidate_depths": { + "binary-forward": 0, + "binary-reverse": 24, + "naf-forward": 0, + "naf-reverse": 25 + }, + "bidirectional_depth": 0, + "chosen": "naf-forward", + "coefficient": "-67108862", + "costs_per_real_component": [ + 30, + 30 + ], + "extra_bits": 0, + "forward_depth": 0, + "witness": null + }, + { + "all_candidate_depths": { + 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"forward_best": { + "all_endpoints_match": true, + "depth_stream_sha256": "a635a0517e93be1a536661b5f4659741d5b4ec5ac8ca1b78589e02304c10d42e", + "independent_witness_value": "1/8", + "native_columns": 728, + "private_columns": 9, + "real_primitives": 142536, + "scalar_projection_guard_bits": 3, + "sharpness_witness": { + "denominator_exponent": 3, + "event": 19076, + "frame": 1, + "input_column": 4, + "macro": [ + "add", + [ + "L", + 300 + ], + [ + "L", + 240 + ], + 1, + 4 + ], + "method": "binary-forward", + "odd_numerator": 1, + "primitive_index": 48915, + "register": [ + "W", + 0 + ], + "step": 2 + } + } + } + }, + "scope": "Exact intermediate representability of scalar primitive projections on dyadic input grids. Frame/phase/child realizations and physical floating-point precision are separate obligations.", + "status": "finite fractional-grid audit; no new kappa", + "whole_program_counterexamples": { + "ck/forward": { + "corrected": { + "discarded_halves": 0, + "endpoint_errors": [], + "endpoint_matches": true, + "first_discarded_half": null, + "guard_bits": 5, + "input_column": 16 + }, + "original": { + "discarded_halves": 0, + "endpoint_errors": [], + "endpoint_matches": true, + "first_discarded_half": null, + "guard_bits": 3, + "input_column": 16 + }, + "undercharged": [ + { + "discarded_halves": 3, + "endpoint_errors": [ + { + "error": "-1/8", + "register": [ + "L", + 43 + ] + }, + { + "error": "1/8", + "register": [ + "L", + 422 + ] + }, + { + "error": "1/8", + "register": [ + "L", + 425 + ] + } + ], + "endpoint_matches": false, + "first_discarded_half": { + "event": 22014, + "register": [ + "W", + 0 + ], + "step": 4 + }, + "guard_bits": 3, + "input_column": 16 + }, + { + "discarded_halves": 2, + "endpoint_errors": [ + { + "error": "-1/8", + "register": [ + "L", + 43 + ] + }, + { + "error": "1/8", + "register": [ + "L", + 425 + ] + } + ], + "endpoint_matches": false, + "first_discarded_half": { + "event": 22014, + "register": [ + "W", + 0 + ], + "step": 5 + }, + "guard_bits": 4, + "input_column": 16 + } + ] + } + } +} diff --git a/research/dyadic-fractional-grid/README.md b/research/dyadic-fractional-grid/README.md new file mode 100644 index 0000000000..afd3af9cf9 --- /dev/null +++ b/research/dyadic-fractional-grid/README.md @@ -0,0 +1,184 @@ +# Fractional-grid audit of bidirectional Horner + +**Mathematical audit / finite precision-interface result; no new κ.** The +bidirectional Horner circuits in [#837](https://github.com/CrocSwap/integer-mult-bounds/pull/837) +preserve their exact endpoint maps and improve primitive counts. Their +intermediate fractional-grid requirement can nevertheless increase. This +package quantifies that previously unresolved precision obligation. + +For the complete **scalar primitive projections**, the exact minimum number +of fractional guard bits beyond the input grid is: + +| Program | Forward binary/NAF baseline | Bidirectional | Change | +|---|---:|---:|---:| +| CK forward | 3 | **5** | +2 | +| CK backward | 8 | 8 | 0 | +| JP8 forward | 4 | 4 | 0 | +| JP8 backward | 27 | 27 | 0 | +| SW4 forward | 3 | 3 | 0 | +| SW4 backward | 19 | 19 | 0 | + +These values are necessary and sufficient for **every intermediate to remain +exact for every input on the initial dyadic grid**. A shared guard covering +both orientations is unchanged: **8 for CK, 27 for JP8, 19 for SW4**. This is +not a counterexample to the selected #799 construction, a claim that its +physical precision reserve is inadequate, or a full compiler precision budget. + +At the individual coefficient interface, **18 of 962 coefficients** require +one or two additional bits under #837's chosen circuits. They occur in 21 CK, +4 JP8 and 5 SW4 additions per orientation. Local requirements do not add +across macros: actual source divisibility and other larger requirements can +absorb them, as the full traces demonstrate. + +## Concrete fixed-grid counterexamples + +For `t += (31/8)*s`, with `s=1`, `t=0`, forward NAF uses 12 real primitives +and needs three fractional bits. Reverse NAF uses 11 and needs five. On a +three-bit fractional grid, rounding each halving toward minus infinity gives +**4 instead of 31/8** with the reverse circuit. Five bits restore exactness. +All copies, additions, subtractions and doublings are exact in this model; +the integer part is unbounded. Arbitrary incoming coefficient work is cleared +by the same pinned circuits. + +There is also a complete-word witness. Set native input column 16 to 1 and +all other native/private inputs to 0 in CK forward. The baseline at three +guard bits has no discarded halves and the correct endpoint. The reversed +program on that grid has three discarded halves and these endpoint errors: + +| Register | Error | +|---|---:| +| source `L43` | -1/8 | +| dirty helper `L422` | +1/8 | +| dirty helper `L425` | +1/8 | + +Four bits still leave errors at `L43` and `L425`. Five bits produce the exact +endpoint with no discarded halves. The first loss occurs in event 22014's +`-31/8` coefficient work. This is a literal integer-grid execution of the +entire scalar projection, not an inference from a large denominator alone. +In general, an unrepresentable intermediate need not imply a wrong endpoint; +the explicit execution establishes that additional fact for this witness. + +## Reproduce + +Python 3.11+, standard library, offline, no writes by default: + +```sh +python3 -B -I research/dyadic-fractional-grid/verify.py +python3 -B -I research/dyadic-fractional-grid/crosscheck.py +python3 -B -I research/dyadic-fractional-grid/test_grid.py +``` + +Expected main marker: + +```text +PASS_DYADIC_FRACTIONAL_GRID programs=6 coefficients=962 extra_grid_coefficients=18 kappa=unchanged +``` + +The independent checker reports `PASS_FRACTION_GRID_CROSSCHECK` for all twelve +complete traces: six programs under each lowering. Nine focused tests run in +the last command. `verify.py --write /new/path.json` reconstructs the receipt +and refuses an existing destination. [EXPECTED.json](EXPECTED.json) contains +every coefficient, candidate depth, full-program trace digest, sharpness +witness and rounding counterexample. [VALIDATION.md](VALIDATION.md) records +the executed checks and limitations. + +## Exactness criterion and proof + +Let inputs lie in `2^-p Z^N`, and let every scalar primitive use the fixed +grid `2^-(p+g) Z`, with an unbounded integer part. Let `P_j` be the exact +linear map after primitive j, including arbitrary initial native and private +inputs. Clears are linear projections and require no invertibility. Define + +```text +D = max over all j, rows r and columns c of + log2(denominator of the reduced dyadic coefficient (P_j)[r,c]), +``` + +with zero and integer coefficients contributing zero. **Every prefix is +exactly representable for every input if and only if `g >= D`.** + +For sufficiency, write an input as `2^-p z`, `z in Z^N`. If every coefficient +denominator divides `2^g`, each coordinate of every `P_j 2^-p z` belongs to +the working grid. Induction over primitives then makes any rounding rule +that fixes grid points exact at every step. For necessity, take a coefficient +with reduced form `a/2^D`, a odd. Input `2^-p e_c` produces +`a/2^(p+D)` at that row and prefix, which is outside every grid with `g= 0, "bad guard") + unit = 2**guard + values = initial(word, column, unit) + discarded, first = 0, None + for event_id, event in enumerate(events): + if event[0] != "scalar": + continue + for step_id, op in enumerate(lowering.choose(event[1], reverse=reverse)[1]): + if step(op, values, integer=True): + discarded += 1 + if first is None: + first = {"event": event_id, "step": step_id, "register": list(op[1])} + expected = {reg: 0 for reg in values} + n = 2*word["v"]+word["R"] + if column < n: + expected["L", column] = unit + if orientation == "forward" and column < word["v"]: + expected["L", word["v"]+column] = unit + elif orientation == "backward" and word["v"] <= column < 2*word["v"]: + expected["L", column-word["v"]] = -unit + differences = [{"register": list(reg), "error": str(Q(value-expected[reg], unit))} + for reg, value in values.items() if value != expected[reg]] + return {"guard_bits": guard, "input_column": column, "discarded_halves": discarded, + "first_discarded_half": first, "endpoint_matches": not differences, + "endpoint_errors": differences} + + +def local(circuit, guard): + values = {("S", 0): 2**guard, ("T", 0): 0, ("W", 0): 7*2**guard} + losses = 0 + for op in circuit: + losses += step(op, values, integer=True) + need(values["S", 0] == 2**guard and values["W", 0] == 0, "fixed-grid source/private contract") + return str(Q(values["T", 0], 2**guard)), losses diff --git a/research/dyadic-fractional-grid/rows.py b/research/dyadic-fractional-grid/rows.py new file mode 100644 index 0000000000..e61a4e135c --- /dev/null +++ b/research/dyadic-fractional-grid/rows.py @@ -0,0 +1,87 @@ +"""Exact integer rows with a common binary denominator. + +Gareth Davidson (@bitplane), substantial OpenAI Codex assistance; Apache-2.0. +No Fraction or predecessor interpreter is used in this engine. +""" + +from math import gcd + + +def need(ok, message): + if not ok: + raise ValueError(message) + + +def normalized(values, exponent): + need(type(exponent) is int and exponent >= 0, "bad binary exponent") + values = {k: n for k, n in values.items() if n} + if not values: + return {}, 0 + if exponent: + removable = min(exponent, min((abs(n) & -abs(n)).bit_length()-1 for n in values.values())) + if removable: + values = {k: n >> removable for k, n in values.items()} + exponent -= removable + return values, exponent + + +def plus(left, right, sign=1): + a, ea = left + b, eb = right + e = max(ea, eb) + values = {k: n << (e-ea) for k, n in a.items()} + for k, n in b.items(): + values[k] = values.get(k, 0) + sign*(n << (e-eb)) + return normalized(values, e) + + +def multiply(row, numerator, denominator): + need(denominator != 0, "zero multiplier denominator") + divisor = gcd(numerator, denominator) + numerator, denominator = numerator // divisor, denominator // divisor + if denominator < 0: + numerator, denominator = -numerator, -denominator + need(denominator > 0 and denominator & (denominator-1) == 0, "non-dyadic multiplier") + values, e = row + return normalized({k: numerator*n for k, n in values.items()}, e+denominator.bit_length()-1) + + +def primitive(step, state): + tag, target = step[:2] + row, e = state[target] + if tag == "clear": + state[target] = {}, 0 + elif tag == "copy": + source, se = state[step[2]] + state[target] = dict(source), se + elif tag == "add" or tag == "subtract": + state[target] = plus(state[target], state[step[2]], 1 if tag == "add" else -1) + elif tag == "double": + state[target] = normalized(row, e-1) if e else ({k: 2*n for k, n in row.items()}, 0) + elif tag == "halve": + state[target] = normalized(row, e+1) + elif tag == "negate": + state[target] = {k: -n for k, n in row.items()}, e + else: + raise ValueError("unknown primitive") + return state[target][1] + + +def macro(op, state): + tag, target = op[0], tuple(op[1]) + if tag == "clear": + return {}, 0 + if tag == "copy": + values, e = state[tuple(op[2])] + return dict(values), e + if tag == "scale": + return multiply(state[target], op[2], op[3]) + need(tag == "add", "unknown macro") + return plus(state[target], multiply(state[tuple(op[2])], op[3], op[4])) + + +def witness(row): + values, exponent = row + need(exponent > 0, "no fractional witness required") + column = min(k for k, n in values.items() if n % 2) + return {"input_column": column, "odd_numerator": values[column], "denominator_exponent": exponent} diff --git a/research/dyadic-fractional-grid/test_grid.py b/research/dyadic-fractional-grid/test_grid.py new file mode 100644 index 0000000000..d2a250b5bf --- /dev/null +++ b/research/dyadic-fractional-grid/test_grid.py @@ -0,0 +1,88 @@ +"""Exactness witnesses and adverse grid/primitive controls; Apache-2.0. + +Gareth Davidson (@bitplane), substantial OpenAI Codex assistance. +""" + +import importlib.util +import unittest +from fractions import Fraction as Q +from pathlib import Path + +HERE = Path(__file__).resolve().parent +spec = importlib.util.spec_from_file_location("grid_verifier", HERE / "verify.py") +v = importlib.util.module_from_spec(spec) +spec.loader.exec_module(v) +lower = v.load(HERE / "vendor/pr837/lower.py", "lower") +grid = v.load(HERE / "fixed_grid.py", "grid") + + +class GridTests(unittest.TestCase): + def test_coefficient_grid_counterexample(self): + op = ["add", ["T", 0], ["S", 0], 31, 8] + old, new = lower.choose(op, reverse=False)[1], lower.lower(op) + self.assertEqual(v.coefficient_trace(Q(31, 8), old)[0], 3) + self.assertEqual(v.coefficient_trace(Q(31, 8), new)[0], 5) + self.assertEqual(grid.local(old, 3), ("31/8", 0)) + self.assertEqual(grid.local(new, 3)[0], "4") + self.assertEqual(grid.local(new, 5), ("31/8", 0)) + + def test_negative_rounding_direction(self): + values = {("W", 0): -1} + self.assertTrue(grid.step(("halve", ("W", 0)), values, integer=True)) + self.assertEqual(values["W", 0], -1) # Floor, not truncation to zero. + + def test_zero_does_not_acquire_fractional_depth(self): + state = {("W", 0): ({}, 0)} + for _ in range(100): + self.assertEqual(v.rows.primitive(("halve", ("W", 0)), state), 0) + + def test_cancellation_removes_denominator(self): + self.assertEqual(v.rows.plus(({0: 3}, 7), ({0: -3}, 7)), ({}, 0)) + self.assertEqual(v.rows.plus(({0: 1}, 3), ({0: 3}, 3)), ({0: 1}, 1)) + + def test_negative_and_unreduced_multiplier_denominators(self): + row = ({0: 1}, 0) + self.assertEqual(v.rows.multiply(row, -3, -6), ({0: 1}, 1)) + self.assertEqual(v.rows.multiply(row, 3, -6), ({0: -1}, 1)) + with self.assertRaisesRegex(ValueError, "non-dyadic"): + v.rows.multiply(row, 1, 3) + + def test_copy_then_halve_does_not_change_source(self): + state = {("S", 0): ({0: 1}, 0), ("T", 0): ({1: 1}, 0)} + v.rows.primitive(("copy", ("T", 0), ("S", 0)), state) + v.rows.primitive(("halve", ("T", 0)), state) + self.assertEqual(state["S", 0], ({0: 1}, 0)) + self.assertEqual(state["T", 0], ({0: 1}, 1)) + + def test_dirty_work_and_wrong_halving_are_detected(self): + word = lower.lower(["add", ["T", 0], ["S", 0], 31, 8]) + for bad in [word[1:], word[:1]+word[2:], word[:-1]]: + with self.assertRaises(ValueError): + v.coefficient_trace(Q(31, 8), bad) + + def test_every_recorded_global_witness_rejects_one_less_bit(self): + receipt = v.read(HERE / "EXPECTED.json") + for result in receipt["programs"].values(): + for mode in ("forward_best", "bidirectional"): + row = result[mode] + proof = row["sharpness_witness"] + g = row["scalar_projection_guard_bits"] + value = Q(proof["odd_numerator"], 2**proof["denominator_exponent"]) + self.assertEqual((value * 2**g).denominator, 1) + self.assertEqual((value * 2**(g-1)).denominator, 2) + + def test_wrong_global_witness_location_is_rejected(self): + result = v.read(HERE / "EXPECTED.json")["programs"]["ck/forward"]["bidirectional"] + proof = dict(result["sharpness_witness"]) + proof["step"] += 1 + import json + provider, _, parts = next(v.inputs()) + self.assertEqual(provider, "ck") + word = json.loads(parts["CANONICAL-WORD.json.gz"]) + events = json.loads(parts["LOCAL-PROGRAMS.json.gz"])["forward"] + with self.assertRaisesRegex(ValueError, "witness address"): + grid.check_witness(events, word, lower, True, proof) + + +if __name__ == "__main__": + unittest.main() diff --git a/research/dyadic-fractional-grid/vendor/pr837/lower.py b/research/dyadic-fractional-grid/vendor/pr837/lower.py new file mode 100644 index 0000000000..a7855d2a87 --- /dev/null +++ b/research/dyadic-fractional-grid/vendor/pr837/lower.py @@ -0,0 +1,92 @@ +"""Both Horner directions for a single nonzero dyadic scalar macro. + +Gareth Davidson (@bitplane), with substantial OpenAI Codex assistance. +Apache-2.0. Binary copy/unit lowering is Sagar Thakrar's verbatim PR696 +module. Forward NAF follows PR723 and our PR787; see README.md for pins. +""" + +import importlib.util +from fractions import Fraction as Q +from pathlib import Path + + +def load_reference(): + spec = importlib.util.spec_from_file_location( + "pr696", Path(__file__).resolve().parent / "vendor/pr696_lower.py") + module = importlib.util.module_from_spec(spec) + spec.loader.exec_module(module) + return module + + +reference = load_reference() + + +def digits(n, signed): + """Least-significant-first binary or non-adjacent digits.""" + if type(n) is not int or n <= 0: + raise ValueError("positive integer required") + result = [] + while n: + d = (2 - n % 4 if signed else 1) if n % 2 else 0 + result.append(d) + n = (n - d) // 2 + return result + + +def candidates(op, work=("W", 0)): + """Emit four circuits in stable tie order; each clears work on exit. + + Non-addition macros are inherited unchanged. No circuit writes source, + borrows a logical register, or crosses the original macro boundary. + """ + binary = reference.lower(op, work) # Also validates dyadics and aliases. + result = {"binary-forward": binary} + if op[0] != "add" or abs(Q(op[3], op[4])) == 1: + return result + target, source = tuple(op[1]), tuple(op[2]) + q = Q(op[3], op[4]) + a, k = abs(q.numerator), q.denominator.bit_length() - 1 + sign = 1 if q > 0 else -1 + + forward = [("copy", work, source)] + for d in reversed(digits(a, True)[:-1]): + forward.append(("double", work)) + if d: + forward.append(("add" if d > 0 else "subtract", work, source)) + forward.extend([("halve", work)] * k) + forward.extend([("add" if sign > 0 else "subtract", target, work), ("clear", work)]) + result["naf-forward"] = forward + + # Remove numerator trailing zeros so the initial reverse digit is +/-1. + odd, exponent = a, -k + while odd % 2 == 0: + odd //= 2 + exponent += 1 + for signed in (False, True): + ds = digits(odd, signed) + initial_sign = ds[0] + reverse = [("copy", work, source)] + for d in ds[1:]: + reverse.append(("halve", work)) + if d: + reverse.append(("add" if d * initial_sign > 0 else "subtract", work, source)) + shift = len(ds) - 1 + exponent + reverse.extend([("double" if shift > 0 else "halve", work)] * abs(shift)) + reverse.extend([ + ("add" if sign * initial_sign > 0 else "subtract", target, work), + ("clear", work), + ]) + result["naf-reverse" if signed else "binary-reverse"] = reverse + return result + + +def choose(op, work=("W", 0), reverse=True): + words = candidates(op, work) + if not reverse: + words = {k: v for k, v in words.items() if k.endswith("forward")} + name = min(words, key=lambda key: len(words[key])) + return name, words[name] + + +def lower(op, work=("W", 0)): + return choose(op, work)[1] diff --git a/research/dyadic-fractional-grid/vendor/pr837/vendor/pr696_lower.py b/research/dyadic-fractional-grid/vendor/pr837/vendor/pr696_lower.py new file mode 100644 index 0000000000..a2c2889e5b --- /dev/null +++ b/research/dyadic-fractional-grid/vendor/pr837/vendor/pr696_lower.py @@ -0,0 +1,56 @@ +"""Smaller exact dyadic circuits for the fixed PR626 complex scalar words. + +Sagar Thakrar, with substantial OpenAI Codex assistance. Apache-2.0. +Primitive vocabulary and the original Horner circuit follow icekylinx's +PR425/433/531 lineage, retained by Jeremy Kothe's PR622 supplier. +""" +from fractions import Fraction + + +def power_of_two(n): + return n > 0 and n & (n - 1) == 0 + + +def lower(op, work=("W", 0)): + """Return real-component primitives; work may be dirty on entry. + + Every addition leaves work zero. All operations use the macro's frame. + The complex circuit applies this word separately to real and imaginary parts. + """ + tag, target = op[0], tuple(op[1]) + if target == work: + raise ValueError("coefficient work aliases a scalar register") + if tag == "clear": + return [("clear", target)] + if tag == "copy": + source = tuple(op[2]) + if source == work: + raise ValueError("coefficient work used as a scalar source") + return [("copy", target, source)] + if tag == "scale": + q = Fraction(op[2], op[3]) + if not power_of_two(abs(q.numerator)) or not power_of_two(q.denominator): + raise ValueError("scale is not a signed power of two") + shift = abs(q.numerator).bit_length() - q.denominator.bit_length() + result = [("double" if shift > 0 else "halve", target)] * abs(shift) + if q < 0: + result.append(("negate", target)) + return result + if tag != "add": + raise ValueError("unknown scalar macro") + source = tuple(op[2]) + q = Fraction(op[3], op[4]) + if source in (target, work) or not q or not power_of_two(q.denominator): + raise ValueError("addition requires distinct registers and a nonzero dyadic") + accumulate = "add" if q > 0 else "subtract" + if abs(q) == 1: + return [(accumulate, target, source), ("clear", work)] + # Copy replaces clear + addition of the leading binary digit. + result = [("copy", work, source)] + for bit in bin(abs(q.numerator))[3:]: + result.append(("double", work)) + if bit == "1": + result.append(("add", work, source)) + result.extend([("halve", work)] * (q.denominator.bit_length() - 1)) + result.extend([(accumulate, target, work), ("clear", work)]) + return result diff --git a/research/dyadic-fractional-grid/verify.py b/research/dyadic-fractional-grid/verify.py new file mode 100644 index 0000000000..973136717a --- /dev/null +++ b/research/dyadic-fractional-grid/verify.py @@ -0,0 +1,252 @@ +"""Exact fractional-grid audit of the fixed PR837 Horner circuits. + +Gareth Davidson (@bitplane), substantial OpenAI Codex assistance; Apache-2.0. +""" + +import argparse +import gzip +import hashlib +import importlib.util +import json +import tarfile +from collections import Counter +from fractions import Fraction as Q +from pathlib import Path + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] +WORK = ("W", 0) + + +def load(path, name): + spec = importlib.util.spec_from_file_location(name, path) + module = importlib.util.module_from_spec(spec) + spec.loader.exec_module(module) + return module + + +rows = load(HERE / "rows.py", "integer_rows") +need = rows.need + + +def canonical(value): + return json.dumps(value, sort_keys=True, separators=(",", ":")) + + +def digest(value): + return hashlib.sha256(canonical(value).encode()).hexdigest() + + +def payload(path): + raw = path.read_bytes() + return gzip.decompress(raw) if path.suffix == ".gz" else raw + + +def read(path): + return json.loads(payload(path)) + + +def source_pins(root=ROOT): + pins = read(HERE / "SOURCES.json") + for item in pins["files"]: + path = root / item["path"] + raw = payload(path) if item["encoding"] == "gzip-payload" else path.read_bytes() + need(hashlib.sha256(raw).hexdigest() == item["sha256"], "source pin mismatch: " + item["path"]) + for item in pins["records"]: + record = read(root / item["path"]) + need(record["audited_source_head"] == item["head"], "checkpoint head mismatch") + manifest = read(root / record["source_manifest"]) + need(manifest["files"][item["spec"]] == hashlib.sha256((root / item["spec"]).read_bytes()).hexdigest(), + "checkpoint SPEC binding") + return pins + + +def inputs(root=ROOT): + old = "research/complex-k492-owner-repair/data/SPEC.json" + base = "research/community-k492-audit/pr626/replay/mixed-global/base/providers" + for provider in ("ck", "jp8"): + parts = {name: payload(root / base / provider / name) for name in + ("LOCAL-PROGRAMS.json.gz", "CANONICAL-WORD.json.gz")} + yield provider, read(root / old)["providers"][provider], parts + archive_path = root / "research/community-frontier741-audit/composed-replay-evidence.tar.gz" + with tarfile.open(archive_path) as archive: + parts = {} + for name in ("LOCAL-PROGRAMS.json.gz", "CANONICAL-WORD.json.gz"): + members = [m for m in archive.getmembers() if m.name == "complex/local/"+name] + need(len(members) == 1 and members[0].isfile(), "ambiguous archive member") + parts[name] = gzip.decompress(archive.extractfile(members[0]).read()) + yield "sw4", read(root / "research/complex-sw4-frame8/data/SPEC.json")["providers"]["gauge44"], parts + + +def coefficient_trace(q, word): + target, source = ("T", 0), ("S", 0) + state = {target: ({0: 1}, 0), source: ({1: 1}, 0), WORK: ({2: 1}, 0)} + depth, proof = 0, None + for i, step in enumerate(word): + need(step[1] in (target, WORK), "source or foreign write") + e = rows.primitive(step, state) + if e > depth: + depth = e + proof = {"step": i, "register": list(step[1]), **rows.witness(state[step[1]])} + expected = rows.normalized({0: q.denominator, 1: q.numerator}, q.denominator.bit_length()-1) + need(state[target] == expected and state[source] == ({1: 1}, 0), "coefficient endpoint") + need(state[WORK] == ({}, 0), "dirty work remains") + return depth, proof + + +def depth_formula(q, method): + k = q.denominator.bit_length()-1 + if method.endswith("forward"): + return k + odd = abs(q.numerator) + odd >>= (odd & -odd).bit_length()-1 + h = odd.bit_length()-1 if method == "binary-reverse" else (3*odd).bit_length()-2 + return max(h, k) + + +def program_trace(events, word, spec, orientation, lowering, reverse): + n = 2*word["v"]+word["R"] + state = {("L", i): ({i: 1}, 0) for i in range(n)} + state.update({("C", i): ({n+i: 1}, 0) for i in range(len(word["centers"]))}) + state[WORK] = ({n+len(word["centers"]): 1}, 0) + peak, proof, count, macros = 0, None, 0, 0 + counts, paid = Counter(), Counter() + trace = hashlib.sha256() + for event_id, event in enumerate(events): + if event[0] in ("climb", "center"): + paid[event[5] if event[0] == "climb" else event[4]] += 1 + if event[0] != "scalar": + continue + _, op, frame = event + expected = rows.macro(op, state) + target = tuple(op[1]) + method, circuit = lowering.choose(op, reverse=reverse) + if op[0] == "add": + counts[str(Q(op[3], op[4]))] += 1 + for step_id, step in enumerate(circuit): + need(step[1] in (target, WORK), "foreign program write") + e = rows.primitive(step, state) + if e > peak: + peak = e + proof = {"event": event_id, "step": step_id, "primitive_index": count, + "register": list(step[1]), "frame": frame, "macro": op, + "method": method, **rows.witness(state[step[1]])} + trace.update(canonical([e, step[0], list(step[1])]).encode()+b"\n") + count += 1 + need(state[target] == expected, "macro endpoint mismatch") + if op[0] == "add": + need(state[WORK] == ({}, 0), "macro work not erased") + macros += 1 + for i in range(n): + expected = {i: 1} + if orientation == "forward" and word["v"] <= i < 2*word["v"]: + expected[i-word["v"]] = 1 + elif orientation == "backward" and i < word["v"]: + expected[i+word["v"]] = -1 + need(state["L", i] == (expected, 0), "arbitrary-column endpoint mismatch") + need(all(value == ({}, 0) for reg, value in state.items() if reg[0] != "L"), "private input leak") + need(dict(paid) == {int(k): v for k, v in spec["expected_profile"].items()}, "paid profile mismatch") + prior = spec["expected_local"][orientation] if "expected_local" in spec else spec["expected_local."+orientation] + need(len(events) == prior["events"] and macros == prior["macros"], "program inventory mismatch") + return {"scalar_projection_guard_bits": peak, "sharpness_witness": proof, + "real_primitives": 2*count, "native_columns": n, "private_columns": len(state)-n, + "all_endpoints_match": True, "depth_stream_sha256": trace.hexdigest()}, counts + + +def verify(root=ROOT): + pins = source_pins(root) + lowering = load(HERE / "vendor/pr837/lower.py", "pinned_pr837") + grid = load(HERE / "fixed_grid.py", "independent_fixed_grid") + programs, occurrences, full_counterexamples = {}, {}, {} + all_coefficients = set() + for provider, spec, parts in inputs(root): + for name, data in parts.items(): + need(hashlib.sha256(data).hexdigest() == spec["artifact_semantic_sha256"][name], "SPEC payload binding") + word = json.loads(parts["CANONICAL-WORD.json.gz"]) + for orientation, events in sorted(json.loads(parts["LOCAL-PROGRAMS.json.gz"]).items()): + need(digest(events) == spec["program_hashes"][orientation], "program digest mismatch") + key = provider+"/"+orientation + results = {} + for reverse, label in ((False, "forward_best"), (True, "bidirectional")): + results[label], counts = program_trace(events, word, spec, orientation, lowering, reverse) + results[label]["independent_witness_value"] = grid.check_witness( + events, word, lowering, reverse, results[label]["sharpness_witness"]) + old_depth = results["forward_best"]["scalar_projection_guard_bits"] + new_depth = results["bidirectional"]["scalar_projection_guard_bits"] + if new_depth > old_depth: + column = results["bidirectional"]["sharpness_witness"]["input_column"] + original = grid.program(events, word, orientation, lowering, False, old_depth, column) + wrong = [grid.program(events, word, orientation, lowering, True, g, column) + for g in range(old_depth, new_depth)] + corrected = grid.program(events, word, orientation, lowering, True, new_depth, column) + need(original["endpoint_matches"] and original["discarded_halves"] == 0, "original fixed-grid control") + need(all(not case["endpoint_matches"] for case in wrong), "counterexample failed to change endpoint") + need(corrected["endpoint_matches"] and corrected["discarded_halves"] == 0, "sufficient guard control") + full_counterexamples[key] = {"original": original, "undercharged": wrong, "corrected": corrected} + programs[key] = results + occurrences[key] = counts + all_coefficients.update(map(Q, counts)) + coefficients = [] + local_counterexamples = [] + for q in sorted(all_coefficients): + op = ["add", ["T", 0], ["S", 0], q.numerator, q.denominator] + old_name, old = lowering.choose(op, reverse=False) + name, new = lowering.choose(op) + old_depth, old_proof = coefficient_trace(q, old) + new_depth, proof = coefficient_trace(q, new) + candidate_depths = {} + for method, circuit in lowering.candidates(op).items(): + depth, _ = coefficient_trace(q, circuit) + need(depth == depth_formula(q, method), "candidate fractional-depth formula") + candidate_depths[method] = depth + k = q.denominator.bit_length()-1 + need(old_depth == k, "forward grid formula") + coefficients.append({"coefficient": str(q), "chosen": name, + "forward_depth": old_depth, "bidirectional_depth": new_depth, + "extra_bits": new_depth-old_depth, "witness": proof, + "all_candidate_depths": candidate_depths, + "costs_per_real_component": [len(old), len(new)]}) + if new_depth > old_depth: + before, old_losses = grid.local(old, old_depth) + undercharged = [{"guard": g, "result": grid.local(new, g)[0], "discarded_halves": grid.local(new, g)[1]} + for g in range(old_depth, new_depth)] + corrected, losses = grid.local(new, new_depth) + need(Q(before) == q and old_losses == 0, "local forward grid control") + need(all(Q(x["result"]) != q for x in undercharged), "local rounded counterexample") + need(Q(corrected) == q and losses == 0, "local additional-guard control") + local_counterexamples.append({"coefficient": str(q), "original_result": before, + "undercharged": undercharged, "corrected_result": corrected}) + extras = {r["coefficient"] for r in coefficients if r["extra_bits"] > 0} + for key, results in programs.items(): + results["extra_grid_occurrences"] = sum(n for q, n in occurrences[key].items() if q in extras) + paired = {p: {mode: max(programs[p+"/"+orientation][mode]["scalar_projection_guard_bits"] + for orientation in ("forward", "backward")) + for mode in ("forward_best", "bidirectional")} for p in ("ck", "jp8", "sw4")} + return {"base_commit": pins["base_commit"], "programs": programs, "coefficients": coefficients, + "paired_orientation_guard_bits": paired, + "candidate_circuits_checked": sum(len(r["all_candidate_depths"]) for r in coefficients), + "whole_program_counterexamples": full_counterexamples, + "local_rounding_counterexamples": local_counterexamples, + "extra_grid_coefficients": len(extras), "status": "finite fractional-grid audit; no new kappa", + "scope": "Exact intermediate representability of scalar primitive projections on dyadic input grids. Frame/phase/child realizations and physical floating-point precision are separate obligations."} + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--write", type=Path, help="new certificate path; refuses overwrite") + args = parser.parse_args() + if args.write: + need(not args.write.exists(), "output already exists") + result = verify() + if args.write: + with args.write.open("x") as output: + json.dump(result, output, indent=2, sort_keys=True) + output.write("\n") + else: + need(canonical(result) == canonical(read(HERE / "EXPECTED.json")), "certificate mismatch") + print("PASS_DYADIC_FRACTIONAL_GRID programs=6 coefficients="+str(len(result["coefficients"]))+ + " extra_grid_coefficients="+str(result["extra_grid_coefficients"])+" kappa=unchanged") + + +if __name__ == "__main__": + main() From 5ae580bf68516f1cbe3b2dd3cb646be55170288f Mon Sep 17 00:00:00 2001 From: Gareth Davidson Date: Tue, 13 Oct 2026 09:32:54 +0100 Subject: [PATCH 2/2] docs: record fractional-grid audit submission --- research/dyadic-fractional-grid/CLAIM.json | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/research/dyadic-fractional-grid/CLAIM.json b/research/dyadic-fractional-grid/CLAIM.json index 9c3b46916b..36c30137d9 100644 --- a/research/dyadic-fractional-grid/CLAIM.json +++ b/research/dyadic-fractional-grid/CLAIM.json @@ -68,7 +68,7 @@ } }, "initial_open_pr_count": 648, - "status": "validated; ready for submission", + "status": "submitted", "final_overlap_check": { "at_utc": "2026-10-13T08:30:13.943767+00:00", "open_pr_count": 659, @@ -99,5 +99,7 @@ "receipt": "EXPECTED.json", "details": "VALIDATION.md", "all_local_checks": "passed" - } + }, + "pull_request": "https://github.com/CrocSwap/integer-mult-bounds/pull/866", + "submitted_at_utc": "2026-10-13T08:32:40.046289+00:00" }