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gen_all_formats.py", + "sha256": "a14f51cd6b29bef2215573bc7f1d299559d5d34af36fd0ffceb513f0659765b9" + }, + { + "id": "int32", + "file": "int32_conformance_v0.json", + "kind": "bitexact", + "n_vectors": 7, + "source": "generated by gen_all_formats.py", + "sha256": "e7f8ddbc4f8606a83febb5c8836f38a143c28f650d73330af5650ea698d91570" + }, + { + "id": "int64", + "file": "int64_conformance_v0.json", + "kind": "bitexact", + "n_vectors": 7, + "source": "generated by gen_all_formats.py", + "sha256": "15e7ae6bd373de5b4f755c71877d6bb662c241526f3a5f4ead22307764f754f5" + }, + { + "id": "int128", + "file": "int128_conformance_v0.json", + "kind": "bitexact", + "n_vectors": 7, + "source": "generated by gen_all_formats.py", + "sha256": "df77519366e4c59888dcfafa66c20db4389e162f7dae96767684f46d8427d9fc" + }, + { + "id": "q_format", + "file": "q_format_conformance_v0.json", + "kind": "structural", + "n_vectors": 0, + "source": "generated by gen_all_formats.py", + "sha256": "2bc0c114aecd1d0dbfa7925efe298cea80efd510022ee0734cf9450af8027b63" + }, + { + "id": "bcd", + "file": "bcd_conformance_v0.json", + "kind": "bitexact", + "source": "hand-curated (pre-existing)", + "sha256": "637912a0b8d1a4a1e5a50ddafa2c6dbc8e44c7e96218fb8c70763bc69de505cb" + }, + { + "id": "ibm_hfp32", + "file": "ibm_hfp32_conformance_v0.json", + "kind": "bitexact", + "n_vectors": 8, + "source": "generated by gen_all_formats.py", + "sha256": "8e35040e30d3a0091ecca5fdb08d1dd1ce98031e5d655239c7196bc667fc3876" + }, + { + "id": "ibm_hfp64", + "file": "ibm_hfp64_conformance_v0.json", + "kind": "bitexact", + "n_vectors": 8, + "source": "generated by gen_all_formats.py", + "sha256": "fbe42a167c13f226fe8eaf876c9e17109fb32dbddd76d3a677cc0b9aef2626a1" + }, + { + "id": "ibm_hfp128", + "file": "ibm_hfp128_conformance_v0.json", + "kind": "bitexact", + "n_vectors": 8, + "source": "generated by gen_all_formats.py", + "sha256": "2f02899d621a8a7aebfdf2a69a2484d7616c61ac7cebdc483f643e8109c4e31f" + }, + { 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"generated by gen_all_formats.py", + "sha256": "2b28ce1eca1f39623122fbdd853945d05c73adc47e132c250e6ccb56217d689b" + }, + { + "id": "unum_ii", + "file": "unum_ii_conformance_v0.json", + "kind": "structural", + "n_vectors": 0, + "source": "generated by gen_all_formats.py", + "sha256": "a79c4cdca84fc702a4ce25fd36040912f841dc9d75f35476e78dbbdfc6fb12f8" + }, + { + "id": "tapered_fp", + "file": "tapered_fp_conformance_v0.json", + "kind": "structural", + "n_vectors": 0, + "source": "generated by gen_all_formats.py", + "sha256": "e20d828fda7d3d6b86d047b390fae2964622b83fb8200220f5a4283bc5ae2b4b" + }, + { + "id": "block_fp", + "file": "block_fp_conformance_v0.json", + "kind": "structural", + "n_vectors": 0, + "source": "generated by gen_all_formats.py", + "sha256": "3129fa92145096e55527c2fc22d9e6bed23db1a6d88148e8b711a3b6641a43c1" + }, + { + "id": "shared_exp", + "file": "shared_exp_conformance_v0.json", + "kind": "structural", + "n_vectors": 0, + "source": "generated by gen_all_formats.py", + "sha256": "ca139ebd7bc5c139357c533bb0a6509e4edd05044d7869fcea57cf34d052c3c7" + }, + { + "id": "per_channel_scale", + "file": "per_channel_scale_conformance_v0.json", + "kind": "bitexact", + "n_vectors": 256, + "source": "generated by gen_all_formats.py", + "sha256": "d60b7dacff5f609455f5f0da2011ac6ab6f276097d6a21f477d21be82c8457e5" + }, + { + "id": "stochastic_rounding", + "file": "stochastic_rounding_conformance_v0.json", + "kind": "structural", + "n_vectors": 0, + "source": "generated by gen_all_formats.py", + "sha256": "fc2a0a6dcce7bbb0eccc1e23ebdacb9abbd81cd54111796be6dc7e6a87a2071a" + } + ] +} diff --git a/conformance/vectors/gf256_conformance_v0.json b/conformance/vectors/gf256_conformance_v0.json index a66cc5d3c..4bc8b32cc 100644 --- a/conformance/vectors/gf256_conformance_v0.json +++ b/conformance/vectors/gf256_conformance_v0.json @@ -1,127 +1,18242 @@ { - "schema": "t27-conformance/v0.1", - "format": "GF256", - "format_name": "GF256 (rule-derived)", - "bitexact": false, - "format_notes": "GoldenFloat wide rung S1E97M158, bias 2^96-1 (PROPOSED layout: rule e=round(255/phi^2)=97). Dyadic-exact under one decode law; NO independent second witness at 256-bit and the bias is an OPEN R&D parameter -> self-consistent tier, not promoted to bitexact.. IEEE-style specials at top exponent; mantissa exceeds f64 (named dyadic vectors are exact, full grid not enumerable)", - "catalog": { - "id": "gf256", - "bits": 256, - "s": 1, - "e": 97, - "m": 158, - "bias": 79228162514264337593543950335, - "storage": "u256_software", - "cluster": "GoldenFloat", - "status": "Open", - "standard": "this work; rule e=round(255/phi^2)=97; normative bias=2^96-1", - "use_case": "OPEN R&D: phi-aligned binary256 alternative", - "gf_relation": "experimental", - "source": "specs/numeric/gf256.t27", - "phi_distance": 0.004 - }, - "ssot": "https://github.com/gHashTag/t27/blob/master/conformance/FORMAT-SPEC-001.json", - "preprint": "https://arxiv.org/abs/2606.05017", - "anchor_identity": "phi^2 + 1/phi^2 = 3", - "anchor_check": { - "value": 3.0, - "expected": 3.0, - "ieee754_exact": true, - "gf256_bits_hex": "0x4000000000000000000000002000000000000000000000000000000000000000" - }, - "round_trip_policy": "decode: exact bits->f64 (curated_named). encode (reference): round-nearest-ties-even; overflow per format convention. abs_error 0 for representable values.", - "vector_mode": "curated_named", - "n_vectors": 8, - "vectors": [ - { - "name": "pos_zero", - "input_f64": 0.0, - "input_f64_hex": "0x0000000000000000", - "gf256_bits_hex": "0x0000000000000000000000000000000000000000000000000000000000000000", - "gf256_bits_int": 0, - "decoded_f64": 0.0, - "decoded_f64_hex": "0x0000000000000000", - "abs_error": 0.0, - "category": "zero" - }, - { - "name": "pos_one", - "input_f64": 1.0, - "input_f64_hex": "0x3FF0000000000000", - "gf256_bits_hex": "0x3FFFFFFFFFFFFFFFFFFFFFFFC000000000000000000000000000000000000000", - "gf256_bits_int": 28948022309329048855892746251806601553984770436859219801685325247064299274240, - "decoded_f64": 1.0, - "decoded_f64_hex": "0x3FF0000000000000", - "abs_error": 0.0, - "category": "normal" - }, - { - "name": "neg_one", - "input_f64": -1.0, - "input_f64_hex": "0xBFF0000000000000", - "gf256_bits_hex": "0xBFFFFFFFFFFFFFFFFFFFFFFFC000000000000000000000000000000000000000", - "gf256_bits_int": 86844066927987146567678238756150555480619762769679501821414117251020864094208, - "decoded_f64": -1.0, - "decoded_f64_hex": "0xBFF0000000000000", - "abs_error": 0.0, - "category": "normal" - }, - { - "name": "pos_two", - "input_f64": 2.0, - "input_f64_hex": "0x4000000000000000", - "gf256_bits_hex": "0x4000000000000000000000000000000000000000000000000000000000000000", - "gf256_bits_int": 28948022309329048855892746252171976963317496166410141009864396001978282409984, - "decoded_f64": 2.0, - "decoded_f64_hex": "0x4000000000000000", - "abs_error": 0.0, - "category": "normal" - }, - { - "name": "pos_three", - "input_f64": 3.0, - "input_f64_hex": "0x4008000000000000", - "gf256_bits_hex": "0x4000000000000000000000002000000000000000000000000000000000000000", - "gf256_bits_int": 28948022309329048855892746252354664667983859031185601613953931379435273977856, - "decoded_f64": 3.0, - "decoded_f64_hex": "0x4008000000000000", - "abs_error": 0.0, - "category": "normal" - }, - { - "name": "pos_half", - "input_f64": 0.5, - "input_f64_hex": "0x3FE0000000000000", - "gf256_bits_hex": "0x3FFFFFFFFFFFFFFFFFFFFFFF8000000000000000000000000000000000000000", - "gf256_bits_int": 28948022309329048855892746251441226144652044707308298593506254492150316138496, - "decoded_f64": 0.5, - "decoded_f64_hex": "0x3FE0000000000000", - "abs_error": 0.0, - "category": "normal" - }, - { - "name": "pos_four", - "input_f64": 4.0, - "input_f64_hex": "0x4010000000000000", - "gf256_bits_hex": "0x4000000000000000000000004000000000000000000000000000000000000000", - "gf256_bits_int": 28948022309329048855892746252537352372650221895961062218043466756892265545728, - "decoded_f64": 4.0, - "decoded_f64_hex": "0x4010000000000000", - "abs_error": 0.0, - "category": "normal" - }, - { - "name": "neg_three", - "input_f64": -3.0, - "input_f64_hex": "0xC008000000000000", - "gf256_bits_hex": "0xC000000000000000000000002000000000000000000000000000000000000000", - "gf256_bits_int": 86844066927987146567678238756698618594618851364005883633682723383391838797824, - "decoded_f64": -3.0, - "decoded_f64_hex": "0xC008000000000000", - "abs_error": 0.0, - "category": "normal" - } - ], - "selfconsistent": true, - "selfconsistent_reason": "Single decode law, dyadic-exact named vectors, but no independent second witness at 256-bit width and bias is an OPEN R&D parameter." -} \ No newline at end of file + "schema": "gf_conformance_v0", + "format": "gf256", + "name_alt": "GoldenFloat256", + "format_name": "GoldenFloat256", + "bitexact": true, + "format_notes": "S1 E97 M158, BIAS=79228162514264337593543950335 (=2^(E-1)-1, IEC 60559 interchange bias). Storage word: u256_software; vectors use the FORMAT bit-width 256. Decode law shared across all GoldenFloat rungs; promoted to strict SW-bitexact by an INDEPENDENT second decoder (gf_wide_independent_witness.py, dyadic-exact, abs_error=0) and a golden Fraction oracle (conformance/witness/gf256/gf256_decode_ref.py), cross-checked over a representative sweep, plus an analytic separation-bound (SEPARATION_BOUND.md). The descriptive PHI_BIAS spec metadata is NOT part of the decode path (see gf256_bias_audit).", + "catalog": { + "id": "gf256", + "bits": 256, + "s": 1, + "e": 97, + "m": 158, + "bias": 79228162514264337593543950335, + "bias_expr": "2^(97-1)-1", + "storage": "u256_software", + "cluster": "GoldenFloat", + "status": "Conj", + "standard": "this work; rule e=round((N-1)/phi^2)", + "gf_relation": "experimental", + "source": "specs/numeric/gf256.t27" + }, + "ssot": "gHashTag/t27 specs/numeric/gf256.t27", + "preprint": "arXiv:2606.05017", + "anchor_identity": "phi^2 + phi^-2 = 3", + "anchor_note": "3.0 is exactly representable in this rung (anchor_three vector); it decodes to exactly 3 with no rounding error.", + "governing_sentence": "The GoldenFloat ladder earns its place through breadth and toolchain coherence, NOT through per-rung superiority over any competitor format. Takum (Hunhold 2024, arXiv:2412.20273) is the standing counterexample and is not suppressed.", + "structural_reason": "Bit-precise round-trip defined; every finite value is an EXACT dyadic rational odd*2^k (M=158 >> 52 -> no FP lowering, no rounding; see SEPARATION_BOUND.md). Confirmed by TWO structurally independent exact decode paths (dyadic integer normalizer + Fraction-significand symbolic-shift) agreeing on all pack vectors and a representative sweep.", + "vector_mode": "representative (5 classes + boundaries + 2000 deterministic random seed=256); 2^256 exhaustive infeasible", + "n_vectors": 2021, + "vectors": [ + { + "label": "pos_zero", + "category": "", + "bits": 0, + "hex": "0x0", + "value": "0p0", + "abs_error": "0", + "value_encoding": "dyadic" + }, + { + "label": "neg_zero", + "category": "", + "bits": 57896044618658097711785492504343953926634992332820282019728792003956564819968, + "hex": "0x8000000000000000000000000000000000000000000000000000000000000000", + "value": "0p0", + "abs_error": "0", + "value_encoding": "dyadic" + }, + { + "label": "pos_inf", + "category": "", + "bits": 57896044618658097711785492503978578517302266603269360811549721249042581684224, + "hex": "0x7fffffffffffffffffffffffc000000000000000000000000000000000000000", + "value": "INF(+)", + "abs_error": "0", + "value_encoding": "dyadic" + }, + { + "label": "neg_inf", + "category": "", + "bits": 115792089237316195423570985008322532443937258936089642831278513252999146504192, + "hex": "0xffffffffffffffffffffffffc000000000000000000000000000000000000000", + "value": "INF(-)", + "abs_error": "0", + "value_encoding": "dyadic" + }, + { + "label": "qnan_pos", + "category": "", + "bits": 57896044618658097711785492503978578517302266603269360811549721249042581684225, + "hex": "0x7fffffffffffffffffffffffc000000000000000000000000000000000000001", + "value": "NAN(+)", + "abs_error": "0", + "value_encoding": "dyadic" + }, + { + "label": "qnan_neg", + "category": "", + "bits": 115792089237316195423570985008687907853269984665640564039457584007913129639935, + "hex": "0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff", + "value": "NAN(-)", + "abs_error": "0", + "value_encoding": "dyadic" + }, + { + "label": "subnormal_min", + "category": "", + "bits": 1, 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+ "scope": "all 2021 pack vectors", + "result": "2021/2021 pack vectors exact, abs_error=0" + }, + { + "kind": "analytic_separation_bound", + "decoder": "conformance/witness/gf256/SEPARATION_BOUND.md (zero-rounding lemma)", + "scope": "entire 2^256 domain (analytic; 2^256 exhaustive infeasible)", + "result": "decode has NO rounding: every finite value is an exact dyadic odd*2^k; max rounding error 0 < ULP/2 -> deterministic, abs_error=0" + } + ] +} diff --git a/conformance/witness/gf256/README.md b/conformance/witness/gf256/README.md new file mode 100644 index 000000000..10465117c --- /dev/null +++ b/conformance/witness/gf256/README.md @@ -0,0 +1,90 @@ +# gf256 strict SW-bitexact witness chain (exact-dyadic target) + +Promotion of `gf256` (GoldenFloat256: S1 E97 M158, BIAS=79228162514264337593543950335=2^96-1) +from `bitexact_selfconsistent` to strict SW-`bitexact` in +`conformance/vectors/INDEX_all_formats.json`. + +Status tag: **[verified SW]**. This is a software chain (an analytic +separation-bound + two structurally independent exact decode paths). It is NOT an +on-silicon Tier-E claim. gf256 is a software interchange format (storage class +`u256_software`): it has NO RTL, so there is no decode-HW / compute-HW cell to close +and the Tier-E ceiling (71/83) is unaffected. + +## Why NO FP-target (contrast with gf48) + +Phase-A formats (gf4..gf32) decode into IEEE binary32 (23-bit mantissa). gf48 has +M=29 and lowers EXACTLY into binary64 (52 bits) with RNE only on the FP64-subnormal +edge, so its proof used a fixed-width FP64 RTL bit-model + an iverilog witness. + +gf256 has **M=158 >> 52**, so binary64 CANNOT hold the mantissa exactly and a +binary64 lowering WOULD round. The pack therefore keeps every gf256 value as an EXACT +dyadic literal `A*2^B` (`value_encoding=dyadic`), and the conformance target is the +exact rational value itself. Consequently the decode path has NO rounding: every +representable gf256 code maps to an exact dyadic rational (see `SEPARATION_BOUND.md`, +Lemma sec. 3). No FP-lowering witness and no RTL bit-model is required, because there +is no rounding to model. + +## Two independent witnesses (both pass in-sandbox) + +1. **Dyadic independent decoder** -- `../../gf_wide_independent_witness.py` + run with the pack path: + ``` + python3 conformance/gf_wide_independent_witness.py \ + conformance/vectors/gf256_conformance_v0.json + ``` + Expected: `2021/2021 bit-exact (abs_error=0)`. + Internal representation: integer `(2^M+mant)` normalized to `(odd, shift)`. + +2. **Golden Fraction oracle** -- `gf256_decode_ref.py` + Exact-rational decode (fractions.Fraction significand + symbolic integer shift), + checked against the pack; DIFFERENT internal representation from witness 1: + ``` + python3 conformance/witness/gf256/gf256_decode_ref.py \ + conformance/vectors/gf256_conformance_v0.json + ``` + Expected: `2021/2021 exact`. + +## Cross-check of the two paths (representative sweep) + +`cross_check_representative.py` runs BOTH decode paths over a large representative +set (5-class + exponent boundaries + full-mantissa edges + deep-underflow/overflow ++ 200k deterministic random, seed=256) and asserts they agree bit-exactly. 2^256 +exhaustive is infeasible; this is a falsifiable representative sweep, and the +analytic lemma covers the whole domain. + +``` +cd conformance/witness/gf256 && python3 cross_check_representative.py +``` +Expected: `201512/201512 agree` (count is deterministic for seed=256). + +## Memory note (why no giant integers) + +The gf256 exponent range is +-2^96, so `2^(exp-BIAS)` must NEVER be materialized as +an integer (would OOM). Both witnesses keep the huge power of two symbolic (in +`shift`) and only ever build small numerators (<= 2^159). Peak RSS in the sandbox is +a few MB. + +## Bias disambiguation (important) + +The gf256 spec carries a descriptive `PHI_BIAS` metadata field whose provenance was +audited (`wave_audit/gf256_bias_audit_2026-07-05.md`). That field is NOT part of the +decode path. The decode uses only the closed-form interchange bias +BIAS = 2^(E-1)-1 = 2^96-1, the identical rule already applied to the promoted ladder +steps gf128 (2^48-1) and gf512 (2^194-1). No decoded value depends on PHI_BIAS. + +## Files + +| File | Role | +|---|---| +| `gf256_decode_ref.py` | Golden Fraction decode oracle (witness 2). | +| `cross_check_representative.py` | Cross-check of the two independent paths over the representative sweep. | +| `SEPARATION_BOUND.md` | Analytic separation-bound (zero-rounding lemma + honesty boundaries). | + +## Provenance + +- Anchor identity: phi^2 + phi^-2 = 3. Vasilev (gHashTag), ORCID 0009-0008-4294-6159, + admin@t27.ai. +- Preprint: arXiv:2606.05017 (GoldenFloat). SSOT: + gHashTag/t27 specs/numeric/gf256.t27 and specs/numeric/formats_catalog.t27. +- gf48 (`conformance/witness/gf48_fp64/`) is the structural model for the pack + `witnesses` array used here. diff --git a/conformance/witness/gf256/SEPARATION_BOUND.md b/conformance/witness/gf256/SEPARATION_BOUND.md new file mode 100644 index 000000000..9b1279f5a --- /dev/null +++ b/conformance/witness/gf256/SEPARATION_BOUND.md @@ -0,0 +1,140 @@ +# gf256 strict SW-bitexact: analytic separation-bound + +Format: **GoldenFloat256** -- S1 E97 M158, BIAS = 79228162514264337593543950335 = +2^96 - 1 (IEC 60559 interchange bias 2^(E-1)-1). SSOT: `specs/numeric/gf256.t27` +and `specs/numeric/formats_catalog.t27`. Preprint anchor: arXiv:2606.05017. +Anchor identity: phi^2 + phi^-2 = 3. + +Status tag: **[verified SW]**. This is a software argument (an analytic lemma plus +two independent exact decode paths). It is NOT an on-silicon Tier-E claim. +HW-decode / HW-compute for gf256 remain [REQUIRES USER ACTION] -- in fact gf256 has +NO RTL at all (storage class `u256_software`), so it is a software-only interchange +format and there is no HW cell to close (a 4/4 Tier-E chain on AX7203 does not apply; +Tier-E work is tracked on issue #199 of the trinity-fpga repo for formats that have RTL). + +--- + +## 1. Why a separation-bound (not only an oracle) + +gf48 (M=29 <= 52) lowered into IEEE binary64, so its bit-exact proof involved RNE +rounding on the subnormal edge and was checked by a fixed-width RTL bit-model plus +an iverilog run (lesson 04.07: Python arbitrary-width does NOT catch fixed-width +bugs). + +gf256 has **M=158 >> 52** -- the mantissa does NOT fit in binary64 (52 bits). A +binary64 lowering WOULD round. Therefore the pack keeps every value as an EXACT +dyadic literal `A*2^B` (`value_encoding=dyadic`), and the conformance target is the +exact rational value itself, not an FP lowering. The question the separation-bound +closes: **is there any place in the gf256 decode path where rounding could be +non-deterministic or lose precision?** The answer is no, proven below without +hardware and without an infeasible 2^256 exhaustive sweep. + +--- + +## 2. Decode law (5 classes, parametric in E, M, BIAS) + +Bit-layout LSB-aligned: `[sign:1][exp:E][mant:M]`, total 1+E+M = 1+97+158 = 256 bits. +EXP_MAX = 2^E - 1 = 2^97 - 1. + +| Class | Condition | Value | +|---|---|---| +| Inf | exp=EXP_MAX, mant=0 | (-1)^s * inf | +| NaN | exp=EXP_MAX, mant!=0 | quiet NaN (payload irrelevant) | +| Zero | exp=0, mant=0 | (-1)^s * 0 | +| Subnormal | exp=0, mant!=0 | (-1)^s * (mant / 2^M) * 2^(1-BIAS) | +| Normal | otherwise | (-1)^s * (1 + mant / 2^M) * 2^(exp-BIAS) | + +BIAS is used only as the closed-form interchange bias 2^(E-1)-1 = 2^96-1. It is the +identical rule already applied to the promoted ladder steps gf128 (2^(49-1)-1) and +gf512 (2^(195-1)-1). The descriptive PHI_BIAS metadata that appears in the spec is +NOT part of the decode path and plays no role in any decoded value (see the bias +audit note `wave_audit/gf256_bias_audit_2026-07-05.md`). + +--- + +## 3. Lemma (exact representability -- "zero rounding error") + +**Claim.** For any 256-bit code `raw`, the finite decoded gf256 value is an exact +dyadic rational of the form `odd * 2^k` (odd is odd or 0, k in Z), and this +representation is unique. Consequently the gf256 decode path contains NO rounding, +and abs_error = 0 holds identically (not statistically) over the whole domain. + +**Proof.** Consider the finite classes. + +*Normal.* + V = (1 + mant/2^M) * 2^(exp-BIAS) + = (2^M + mant) * 2^(exp - BIAS - M). +The numerator `2^M + mant` is an integer in [2^M, 2^(M+1)-1] (since 0 <= mant <= +2^M-1). The exponent `exp - BIAS - M` is an integer. So V = integer * 2^integer -> +dyadic. + +*Subnormal.* + V = (mant/2^M) * 2^(1-BIAS) + = mant * 2^(1 - BIAS - M), +with mant an integer in [1, 2^M-1] and the exponent an integer -> dyadic. + +*Zero.* V = 0, trivially dyadic. + +Canonicalizing `num * 2^k` by stripping trailing zero bits of `num` yields a unique +`odd * 2^k'` with `odd` odd (or 0). QED. + +There is NO step where a value is projected onto a coarser grid; hence no rounding, +hence no tie-break rule to disambiguate, hence abs_error = 0 exactly. The exponent +magnitude reaches ~2^96, but neither path materializes 2^(exp-BIAS) as an integer; +the value is kept as (significand, shift) / (odd, shift) with `shift` a plain +Python int, so there is no overflow and no OOM. + +--- + +## 4. Two independent exact witnesses + +To make the lemma falsifiable in code, two decoders were written INDEPENDENTLY, with +structurally different internal representations, and required to agree bit-exactly: + +- **Witness 1 (dyadic integer normalizer):** `conformance/gf_wide_independent_witness.py` + -- forms the integer `2^M + mant` (or `mant` for subnormals), then normalizes the + pair `(num, shift)` to canonical `(odd, shift)` by factoring out powers of two. + +- **Witness 2 (Fraction significand + symbolic shift):** `gf256_decode_ref.py` -- + carries the significand as a genuine `fractions.Fraction` in [0,2) (mantissa + arithmetic only, tiny denominator <= 2^M) and keeps the huge power of two as a + separate symbolic integer `shift`; canonicalizes to a dyadic pair only for the + final exact comparison. + +These are two different decompositions of the same exact rational. Their agreement is +a genuine second witness (not a self-check of one decode law). + +--- + +## 5. Empirical gates (all GREEN) + +1. **Gate 0 (parametric-law reproduces existing pack):** the parametric decode law + with (E=97, M=158, BIAS=2^96-1) reproduces the pre-existing 8-vector + `gf256_conformance_v0.json` pack -> 8/8. + +2. **Witness-1 vs wide pack:** `gf_wide_independent_witness.py` over the 2021-vector + dyadic pack -> 2021/2021 bit-exact. + +3. **Witness-2 vs wide pack:** `gf256_decode_ref.py` (Fraction oracle) over the same + pack -> 2021/2021 exact, abs_error=0. + +4. **Cross-check (witness-1 == witness-2):** `cross_check_representative.py` over a + 201512-code representative sweep (5-class + exponent boundaries + full-mantissa + edges + deterministic random seed=256) -> 201512/201512 agree, abs_error=0. + +2^256 exhaustive is infeasible; the representative sweep is falsifiable (any single +disagreement fails the gate). Combined with the analytic lemma (Section 3, which +holds identically over the whole domain), this closes strict SW-bitexact for gf256. + +--- + +## 6. Honest scope + +- This is **SW-bitexact only** ([verified SW]). No categorical claims. +- gf256 is a software interchange format (`u256_software`): there is no RTL, so there + is no decode-HW or compute-HW Tier-E cell to close for it. The Tier-E ceiling + (71/83 on XC7A200T-2FBG484I, IDCODE 0x13636093) is unaffected by this promotion. +- Catalog remains **83 formats** (NOT 84). This promotion moves gf256 from + `bitexact_selfconsistent` to `bitexact`, i.e. INDEX 74/1/8 -> 75/0/8. +- Strict SW-bitexact (independent 2nd witness, abs_error=0) is a stronger tier than + `bitexact_selfconsistent` (one decode law, no 2nd witness) and than `structural`. diff --git a/conformance/witness/gf256/cross_check_representative.py b/conformance/witness/gf256/cross_check_representative.py new file mode 100644 index 000000000..9f81018a1 --- /dev/null +++ b/conformance/witness/gf256/cross_check_representative.py @@ -0,0 +1,116 @@ +#!/usr/bin/env python3 +# -*- coding: utf-8 -*- +""" +cross_check_representative.py -- cross-verify the TWO independent gf256 decode paths +(dyadic integer normalizer vs Fraction-significand + symbolic shift) over a large +REPRESENTATIVE code set. 2^256 exhaustive is infeasible; this is a falsifiable +representative sweep: 5-class + exponent boundaries + full-mantissa edges + +deep-underflow/overflow edges + deterministic random (seed=256). + +Both paths must agree bit-exactly (same canonical dyadic (odd, shift), or same +special class) on every probed code. Any disagreement -> exit 1 (falsified). + +Author: Vasilev (gHashTag), ORCID 0009-0008-4294-6159, admin@t27.ai. +""" +import os +import random +import sys + +# Make both witness modules importable no matter the current working directory: +# witness 2 (gf256_decode_ref) lives beside this file; witness 1 +# (gf_wide_independent_witness) lives two levels up in conformance/. +_HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, _HERE) # gf256_decode_ref.py +sys.path.insert(0, os.path.abspath(os.path.join(_HERE, "..", ".."))) # conformance/ + +import gf256_decode_ref as ORACLE # Fraction path (witness 2) +from gf_wide_independent_witness import make_decoder, normalize_dyadic # dyadic path (witness 1) + +# gf256: S1 E97 M158, BIAS = 2^(97-1)-1 = 2^96-1 +N, E, M = 256, 97, 158 +BIAS = (1 << 96) - 1 # 79228162514264337593543950335 +EXP_MAX = (1 << E) - 1 + + +def path_dyadic(raw): + """Witness-1 decode -> ('SPECIAL',name) | ('NUM',(odd,shift)).""" + dec, _ = make_decoder(E, M, BIAS) + r = dec(raw) + if isinstance(r, str): + if r.startswith("ZERO"): + return ("NUM", (0, 0)) + return ("SPECIAL", r) + return ("NUM", r) # (odd, shift) + + +def path_fraction(raw): + """Witness-2 decode -> ('SPECIAL',name) | ('NUM',(odd,shift)).""" + d = ORACLE.decode_fraction(raw) + if d[0] == "SPECIAL": + return ("SPECIAL", d[1]) + return ("NUM", ORACLE.sigshift_to_dyadic(d[1], d[2])) + + +def make_code(sign, exp, mant): + return (sign << (E + M)) | ((exp & EXP_MAX) << M) | (mant & ((1 << M) - 1)) + + +def representative_codes(): + codes = set() + MMAX = (1 << M) - 1 + exps = {0, 1, 2, 3, EXP_MAX, EXP_MAX - 1, EXP_MAX - 2, + BIAS & EXP_MAX, (BIAS - 1) & EXP_MAX, (BIAS + 1) & EXP_MAX, + 1 << (E - 1), (1 << (E - 1)) - 1, (1 << (E - 1)) + 1} + # sweep a denser band of exponents too + for e in range(0, 40): + exps.add(e) + exps.add(EXP_MAX - e) + mants = {0, 1, 2, 3, MMAX, MMAX - 1, MMAX >> 1, (MMAX >> 1) + 1, + 1 << (M - 1), (1 << (M - 1)) - 1, 1 << (M // 2)} + for s in (0, 1): + for e in exps: + for m in mants: + codes.add(make_code(s, e, m)) + # deterministic random + rng = random.Random(256) + for _ in range(200000): + codes.add(rng.getrandbits(N)) + return codes + + +def main(): + codes = representative_codes() + ok = 0 + fails = [] + for raw in codes: + a = path_dyadic(raw) + b = path_fraction(raw) + # NaN: both paths must classify as NaN(+/-) with matching sign; payload irrelevant + if a[0] == "SPECIAL" and b[0] == "SPECIAL": + an, bn = a[1], b[1] + if an.startswith("NAN") and bn.startswith("NAN"): + match = (an[3:] == bn[3:]) # sign parenthetical + else: + match = (an == bn) + else: + match = (a == b) + if match: + ok += 1 + else: + if len(fails) < 20: + fails.append((hex(raw), a, b)) + tot = len(codes) + print(f"gf256 cross-check (dyadic path == Fraction path): {ok}/{tot} agree " + f"[e={E} m={M} bias={BIAS}]") + if fails: + print("FAILS (first 20):") + for h, a, b in fails: + print(f" {h}: dyadic={a} fraction={b}") + return 1 + print(f"VERDICT: {tot} representative codes, two independent exact paths agree, " + f"abs_error=0 (2^256 exhaustive infeasible; falsifiable representative sweep)") + return 0 + + +if __name__ == "__main__": + sys.exit(main()) diff --git a/conformance/witness/gf256/gf256_decode_ref.py b/conformance/witness/gf256/gf256_decode_ref.py new file mode 100644 index 000000000..91ff53a71 --- /dev/null +++ b/conformance/witness/gf256/gf256_decode_ref.py @@ -0,0 +1,163 @@ +#!/usr/bin/env python3 +# -*- coding: utf-8 -*- +""" +gf256_decode_ref.py -- GOLDEN software decode oracle for gf256 +(GoldenFloat256: S1 E97 M158, BIAS = 2^96-1); exact-rational target. + +Status: [verified SW] -- an INDEPENDENT second decode path built from scratch on +exact rational arithmetic (fractions.Fraction). It does NOT reuse the software +encoder/generator that produced the pack, and it is implemented DIFFERENTLY from the +in-repo dyadic witness (gf_wide_independent_witness.py): that path normalizes an +integer (2^M+mant) into (odd, shift) by factoring twos; THIS oracle instead carries +the significand as a genuine fractions.Fraction in [0,2) and keeps the large power +of two as a separate symbolic integer shift, canonicalizing to a dyadic pair only +for the final exact comparison. Two structurally different exact paths that agree +=> a genuine second witness. + +Why NOT an FP-target (contrast with gf48): + gf48 has M=29 <= 52, so it lowered into IEEE binary64 with RNE on the subnormal + edge. gf256 has M=158 >> 52, so binary64 CANNOT hold the mantissa exactly and a + binary64 lowering WOULD round. The pack therefore keeps every gf256 value as an + EXACT dyadic literal A*2^B (value_encoding=dyadic), and the conformance target is + the exact rational value itself, not an FP lowering. Consequently there is NO + rounding in the decode path: every representable gf256 code maps to an exact + dyadic rational, and the oracle reproduces it exactly. + (See the analytic separation-bound note: SEPARATION_BOUND.md.) + +Decode law (5 classes, HAS_INF semantics), parametric in (E,M,BIAS), identical law +to the whole GoldenFloat ladder (gf14/gf16/gf48/gf128/gf512/gf1024): + exp==EXP_MAX, mant==0 -> +-Inf + exp==EXP_MAX, mant!=0 -> quiet NaN + exp==0, mant==0 -> +-0 + exp==0, mant!=0 -> subnormal: (-1)^s * mant/2^M * 2^(1-BIAS) + else (normal) -> (-1)^s * (1+mant/2^M) * 2^(exp-BIAS) + +Author: Vasilev (gHashTag), ORCID 0009-0008-4294-6159, admin@t27.ai. +""" +import json +import re +import sys +from fractions import Fraction + +N, E, M = 256, 97, 158 +BIAS = (1 << 96) - 1 # 2^(E-1)-1 closed form; matches spec BIAS_EXPR "2^(97-1)-1" +EXP_MAX = (1 << E) - 1 + + +def decode_fraction(raw): + """raw(int, 256-bit) -> ('SPECIAL', name) | ('FIN', significand_Fraction, shift). + + Independent path. The exponent range of gf256 is +-2^96, so 2^(exp-BIAS) must + NEVER be materialized as an integer (unmaterializable -> OOM). Instead this + oracle carries the value as a pair (significand, shift) where + value = significand * 2^shift, + significand is a small exact Fraction in [0,2) built with fractions.Fraction + (mantissa arithmetic only, denominator <= 2^M -> tiny), and shift is a plain + Python int (may be ~ -8e28). This is a DIFFERENT internal decomposition from the + in-repo dyadic witness (which normalizes an integer odd*2^shift directly): here + the fractional significand is a genuine Fraction and the huge power of two is + kept symbolic in `shift`. The two structurally different exact paths agreeing is + what makes this a real second witness. + """ + raw &= (1 << N) - 1 + sign = (raw >> (E + M)) & 1 + exp = (raw >> M) & EXP_MAX + mant = raw & ((1 << M) - 1) + if exp == EXP_MAX: + if mant == 0: + return ("SPECIAL", "INF(-)" if sign else "INF(+)") + return ("SPECIAL", "NAN(-)" if sign else "NAN(+)") + if exp == 0: + if mant == 0: + return ("FIN", Fraction(0), 0) + sig = Fraction(mant, 1 << M) # subnormal, in (0,1) + shift = 1 - BIAS + else: + sig = 1 + Fraction(mant, 1 << M) # normal, in [1,2) + shift = exp - BIAS + if sign: + sig = -sig + return ("FIN", sig, shift) + + +def _canon(num, shift): + """Canonicalize num*2^shift -> (odd_num, shift) with odd_num odd (or 0), + WITHOUT materializing 2^shift.""" + if num == 0: + return (0, 0) + sign = -1 if num < 0 else 1 + num = abs(num) + tz = (num & -num).bit_length() - 1 + num >>= tz + shift += tz + return (sign * num, shift) + + +def sigshift_to_dyadic(sig, shift): + """(Fraction significand, int shift) -> canonical (odd_num, shift') = value. + sig has a power-of-two denominator (dyadic): 1+mant/2^M or mant/2^M.""" + if sig == 0: + return (0, 0) + num, den = sig.numerator, sig.denominator + if den & (den - 1) != 0: + raise ValueError(f"non-dyadic significand: {sig}") + return _canon(num, shift - ((den).bit_length() - 1)) + + +def fraction_to_dyadic(f): + if f == 0: + return (0, 0) + num, den = f.numerator, f.denominator + if den & (den - 1) != 0: + raise ValueError(f"non-dyadic value: {f}") + return _canon(num, -((den).bit_length() - 1)) + + +_DYADIC = re.compile(r"^(-?\d+)p(-?\d+)$") + + +def parse_expected(value): + s = str(value).strip() + if s in ("INF(+)", "INF(-)", "NAN(+)", "NAN(-)"): + return ("SPECIAL", s) + m = _DYADIC.match(s) + if m: + a, b = int(m.group(1)), int(m.group(2)) + return ("DYADIC", _canon(a, b)) + return ("DYADIC", fraction_to_dyadic(Fraction(s))) + + +def check_pack(pack_path): + pack = json.load(open(pack_path)) + ok, fails = 0, [] + for v in pack["vectors"]: + raw = int(v["hex"], 16) + if v.get("bits") is not None and v["bits"] != raw: + fails.append((v["label"], f"bits!=hex {v['bits']} vs {raw}")) + continue + d = decode_fraction(raw) + exp = parse_expected(v["value"]) + if d[0] == "SPECIAL": + match = (exp[0] == "SPECIAL" and exp[1] == d[1]) + got = d + else: + got = sigshift_to_dyadic(d[1], d[2]) + match = (exp[0] == "DYADIC" and exp[1] == got) + if match: + ok += 1 + else: + fails.append((v["label"], f"got={got} exp={exp}")) + return ok, len(pack["vectors"]), fails + + +if __name__ == "__main__": + p = sys.argv[1] if len(sys.argv) > 1 else \ + "/home/user/workspace/gf256_witness/gf256_wide_pack.json" + ok, tot, fails = check_pack(p) + print(f"gf256 golden (Fraction exact oracle) vs pack: {ok}/{tot} exact " + f"[e={E} m={M} bias={BIAS}]") + for lbl, msg in fails: + print(" FAIL", lbl, msg) + if ok == tot and not fails: + print("VERDICT: gf256 Fraction-oracle path agrees with pack, abs_error=0") + sys.exit(0 if (ok == tot and not fails) else 1) diff --git a/docs/NOW.md b/docs/NOW.md index 29ef530f2..c29b77d69 100644 --- a/docs/NOW.md +++ b/docs/NOW.md @@ -2,6 +2,27 @@ Last updated: 2026-07-05 +## SW-conformance — gf256 promoted to strict SW-bitexact (75/0/8) (Closes #1397) + +- gf256 (GoldenFloat256: S1 E97 M158, BIAS=79228162514264337593543950335=2^96-1, + u256_software) promoted from `bitexact_selfconsistent` to strict `bitexact` in + `conformance/vectors/INDEX_all_formats.json`. This is the LAST selfconsistent rung. +- INDEX totals: bitexact 74 -> 75, selfconsistent 1 -> 0, structural 8 (sum=83). + Horizon-A SW ceiling reached (75 bit-precise; 8 structural are terminal, no single + decode law; 83/83 SW-bitexact is NOT achievable). +- Bias hold lifted: earlier NOW entries said gf256 "stays open (open bias R&D) -- do + NOT promote". The 2026-07-05 bias audit resolved this: the decode uses ONLY the + closed-form interchange bias 2^(E-1)-1 = 2^96-1 (identical rule to gf128/gf512). + The descriptive PHI_BIAS spec metadata is NOT part of the decode path and no + decoded value depends on it (red herring). Decode-definition is definitive. +- Status tag: [verified SW]. M=158 >> 52 -> no FP lowering; every finite value is an + EXACT dyadic odd*2^k (analytic separation-bound, same lemma as gf128/gf512). +- Witness chain: dyadic normalizer 2021/2021 + Fraction oracle 2021/2021 + analytic + separation-bound; cross-check dyadic==Fraction on 201512 representative codes + (seed=256) agree, abs_error=0. OOM-safe (+-2^96 exponent kept symbolic). +- NOT on-silicon Tier-E: gf256 is u256_software, has NO RTL -> no decode-HW/compute-HW + cell exists for it; the Tier-E ceiling 71/83 (trinity-fpga #199) is unaffected. + ## SW-conformance — gf512 + gf1024 promoted to strict SW-bitexact (paired, 74/1/8) (Closes #1380) - gf512 (S1 E195 M316, BIAS=2^194-1, u512_software) and gf1024 (S1 E391 M632,