From 83298467291d3bd4b53f76faaef60dbf585be171 Mon Sep 17 00:00:00 2001 From: huxint Date: Fri, 9 Oct 2026 03:42:27 +0800 Subject: [PATCH 1/3] Add balanced coarse sums on anchored split frames with exact certificate --- .github/workflows/balanced-split.yml | 25 + research/balanced-split-reclamation/Makefile | 17 + research/balanced-split-reclamation/NOTICE | 28 + research/balanced-split-reclamation/README.md | 90 + .../balanced-split-reclamation/SOURCE.json | 64 + .../balanced-split-reclamation/arithmetic.py | 150 ++ .../balanced-split-reclamation/compiler.py | 353 ++++ .../balanced-split-reclamation/generate.py | 89 + research/balanced-split-reclamation/graph.py | 68 + research/balanced-split-reclamation/paper.pdf | Bin 0 -> 300255 bytes research/balanced-split-reclamation/paper.tex | 271 +++ research/balanced-split-reclamation/pin.py | 31 + .../profile_oracle.cpp | 90 + .../selected/audit.json | 10 + .../selected/certificate.json | 855 +++++++++ .../selected/compiled-23.json | 47 + .../selected/compiled-25.json | 49 + .../selected/config-23.json | 9 + .../selected/config-25.json | 9 + .../selected/profile-23.json | 38 + .../selected/profile-25.json | 40 + .../selected/replay-23.json | 34 + .../selected/replay-25.json | 36 + .../selected/transitions-23.json | 35 + .../selected/transitions-25.json | 37 + .../selected/word-23.json.gz | Bin 0 -> 2922697 bytes .../selected/word-25.json.gz | Bin 0 -> 3954276 bytes .../balanced-split-reclamation/support.py | 44 + .../test_witness.py | 86 + .../vendor/pr68-compiler.py | 275 +++ .../vendor/pr69-compiler.py | 93 + .../vendor/pr71-certificate.json | 1547 +++++++++++++++++ .../vendor/pr71-graph.py | 122 ++ research/balanced-split-reclamation/verify.py | 125 ++ 34 files changed, 4767 insertions(+) create mode 100644 .github/workflows/balanced-split.yml create mode 100644 research/balanced-split-reclamation/Makefile create mode 100644 research/balanced-split-reclamation/NOTICE create mode 100644 research/balanced-split-reclamation/README.md create mode 100644 research/balanced-split-reclamation/SOURCE.json create mode 100644 research/balanced-split-reclamation/arithmetic.py create mode 100644 research/balanced-split-reclamation/compiler.py create mode 100644 research/balanced-split-reclamation/generate.py create mode 100644 research/balanced-split-reclamation/graph.py create mode 100644 research/balanced-split-reclamation/paper.pdf create mode 100644 research/balanced-split-reclamation/paper.tex create mode 100644 research/balanced-split-reclamation/pin.py create mode 100644 research/balanced-split-reclamation/profile_oracle.cpp create mode 100644 research/balanced-split-reclamation/selected/audit.json create mode 100644 research/balanced-split-reclamation/selected/certificate.json create mode 100644 research/balanced-split-reclamation/selected/compiled-23.json create mode 100644 research/balanced-split-reclamation/selected/compiled-25.json create mode 100644 research/balanced-split-reclamation/selected/config-23.json create mode 100644 research/balanced-split-reclamation/selected/config-25.json create mode 100644 research/balanced-split-reclamation/selected/profile-23.json create mode 100644 research/balanced-split-reclamation/selected/profile-25.json create mode 100644 research/balanced-split-reclamation/selected/replay-23.json create mode 100644 research/balanced-split-reclamation/selected/replay-25.json create mode 100644 research/balanced-split-reclamation/selected/transitions-23.json create mode 100644 research/balanced-split-reclamation/selected/transitions-25.json create mode 100644 research/balanced-split-reclamation/selected/word-23.json.gz create mode 100644 research/balanced-split-reclamation/selected/word-25.json.gz create mode 100644 research/balanced-split-reclamation/support.py create mode 100644 research/balanced-split-reclamation/test_witness.py create mode 100644 research/balanced-split-reclamation/vendor/pr68-compiler.py create mode 100644 research/balanced-split-reclamation/vendor/pr69-compiler.py create mode 100644 research/balanced-split-reclamation/vendor/pr71-certificate.json create mode 100644 research/balanced-split-reclamation/vendor/pr71-graph.py create mode 100644 research/balanced-split-reclamation/verify.py diff --git a/.github/workflows/balanced-split.yml b/.github/workflows/balanced-split.yml new file mode 100644 index 0000000000..d244c1909e --- /dev/null +++ b/.github/workflows/balanced-split.yml @@ -0,0 +1,25 @@ +name: Balanced split finite witness + +on: + pull_request: + paths: + - 'research/balanced-split-reclamation/**' + - 'research/pair-assembly/**' + - 'scripts/experiments/binary_frame*' + - 'references/frame-compiler/pr48/**' + - '.github/workflows/balanced-split.yml' + workflow_dispatch: + +permissions: + contents: read + +jobs: + verify: + runs-on: ubuntu-latest + timeout-minutes: 45 + steps: + - uses: actions/checkout@v4 + - uses: actions/setup-python@v5 + with: + python-version: '3.12' + - run: make -C research/balanced-split-reclamation verify diff --git a/research/balanced-split-reclamation/Makefile b/research/balanced-split-reclamation/Makefile new file mode 100644 index 0000000000..d030bb7e8c --- /dev/null +++ b/research/balanced-split-reclamation/Makefile @@ -0,0 +1,17 @@ +.PHONY: verify replay test note + +verify: + python3 -B verify.py --rebuild + python3 -B -m unittest discover -s . -p test_witness.py -v + +replay: + python3 -B verify.py + +test: + python3 -B -m unittest discover -s . -p test_witness.py -v + +note: + mkdir -p ../../build/balanced-split-reclamation/pdf + pdflatex -interaction=nonstopmode -halt-on-error -output-directory=../../build/balanced-split-reclamation/pdf paper.tex + pdflatex -interaction=nonstopmode -halt-on-error -output-directory=../../build/balanced-split-reclamation/pdf paper.tex + cp ../../build/balanced-split-reclamation/pdf/paper.pdf paper.pdf diff --git a/research/balanced-split-reclamation/NOTICE b/research/balanced-split-reclamation/NOTICE new file mode 100644 index 0000000000..be4f46cf39 --- /dev/null +++ b/research/balanced-split-reclamation/NOTICE @@ -0,0 +1,28 @@ +Balanced coarse sums on anchored split frames +Prepared for huxint with substantial OpenAI Codex assistance, October 9, 2026. +Distributed under the repository's Apache License 2.0. + +The selected finite network composes previously attributed constructions. +Preserve all repository notices and these source-specific attributions: + +eumemic: PR57 joint reversible frame compiler and independent physical replay; +PR69 balanced coarse column sums, with OpenAI Codex assistance. +Avi Eisenberg / ikeboy: PR62 interval strips and core-aware pair assembly, +with Anthropic Claude assistance. +Rohan Garg / rohangar1: PR59 split-pair operation, with OpenAI Codex assistance. +Rohan Arun: PR65 schedule/refined arithmetic and PR67 profile-cost compiler, +with OpenAI Codex assistance. +Dominik Scholz: PR63 ranked composition and PR68 compatible pending controls, +with OpenAI GPT-6 Astra / Codex assistance. +Chafik Boukhalfa: PR60 ranked reclamation and PR71 anchored split graph and +next-use scoring, with OpenAI Codex assistance. + +The original code comments and preserved vendor snapshots retain their own +attributions. The inherited framework also credits icekylinx, RaD / hipotures, +James Chang, Zhihao Chen / jacklightChen, Aurel Prosz / Paureel, Swapnil Jain, +Douglas Colkitt, OpenAI, David Harvey, Joris van der Hoeven, and other +contributors identified in the root NOTICE and inherited source files. + +The new package's finite checks are conditional evidence. They are not a +complete theorem formalization, an independent human referee report, or a +practical multiplication benchmark. diff --git a/research/balanced-split-reclamation/README.md b/research/balanced-split-reclamation/README.md new file mode 100644 index 0000000000..bd922144c1 --- /dev/null +++ b/research/balanced-split-reclamation/README.md @@ -0,0 +1,90 @@ +# Balanced coarse sums on anchored split frames + +The selected finite construction gives the conditional integer-multiplication +bound + +\[ +T(n)=O\bigl(n(\log n)^{1-\kappa}\bigr),\qquad +\kappa=\frac{12886963972497}{250000000000000000} +=5.1547855889988\times10^{-5}. +\] + +This is **0.2590891819% above PR71** at commit +`1bef94fd40a746452548c84a4a8f8834670a3113`. It is a comparison of conditional +asymptotic exponent bounds, with no global-optimality or measured-runtime claim. + +The construction combines eumemic's PR69 balanced coarse columns with Chafik +Boukhalfa's PR71 anchored `[1,1,2]` split graph. It uses the inherited joint +reversible frame compiler and compatible pending controls, and orders both +eligible live controls and retired dependency candidates by their actual +profile cost. The latter changes the temporary dependency basis, not just the +choice among the first basis's dependent rows. Pending controls are priced +against their known next use. Every selected clearing expression is emitted +as literal XORs, and all frame moves and final cleanup remain charged. + +| Quantity | Pinned PR71 | This construction | +|---|---:|---:| +| Roles at h=23 | 27,455 | 27,338 | +| Roles at h=25 | 36,015 | 35,939 | +| Physical width W | 135,075,665 | 134,671,969 | +| Complete rank mass | 77,666,660,475 | 77,434,535,275 | +| Rank deficit | 1,846,900 | 1,846,900 | + +The strict bit-saving witness is `51550513208569/10^18`. The full paid child +list of PR71 has a rigorous moment lower bound above one at this value. The +new list has a rigorous upper bound below one. Its next `10^-18` bit grid +point is rejected, as is the next kappa grid point in the unchanged balanced +assembly. All 47 strict inequalities and seven margins are checked exactly. + +## Reproduction + +From the repository root, with Python 3.11+ and a C++17 compiler: + +```sh +make -C research/balanced-split-reclamation verify +make -j1 verify +``` + +The first command regenerates both selected words byte-for-byte, independently +replays every input/target/dirty basis vector in both orientations, derives +all physical transitions from literal XOR incidences, recomputes every +fixed-basis profile with the inherited bounded-minor/CRT checks, rebuilds +the complete global profile and exact arithmetic, and runs twelve failure-mode +tests. A second logarithm/exponential evaluation uses a different range +reduction and Taylor degree. This is a second arithmetic implementation, +not independent expert review of the theorem. + +`selected/` contains the words, actual profiles, receipts and exact certificate. +`SOURCE.json` freezes consumed sources and deliverables; verification never +rewrites this manifest. `pin.py` is an explicit maintenance operation. +`generate.py --output ` produces fresh artifacts without modifying +the selected witness. The proof is in [paper.tex](paper.tex) and [paper.pdf](paper.pdf). + +The selected graph uses columns rather than a three-addition coarse chain, +split vector `[1,1,2]`, and the lowest intact pair first in each common-point +ordering. The h23 region order is `cover-core`; h25 uses `reverse-node`. +The oracle's integer entropy score is only a deterministic search heuristic. +The final complete physical profile, not the score, determines acceptance. + +## Proof boundary and attribution + +The original-envelope interpretation, all-size residual/frame compiler, +fixed finite-alphabet tape layout, semantic and analytic multiplication +reduction, routing, prime setup and exact recovery remain inherited +assumptions. The complex branch is unchanged. Exact finite checks do not +constitute a complete formal proof of the multiplication theorem. + +This composition, cost-ordered dependency basis experiment, independent +reproduction and package were prepared for **huxint with OpenAI Codex +assistance**. Credit eumemic for PR57's reversible region compiler and PR69's +coarse-column identity; Avi Eisenberg for PR62's interval/core-aware graph; +Rohan Garg for PR59's split operation; Rohan Arun for PR65's schedules and +PR67's cost oracle and numerical refinement; Dominik Scholz for PR63's +composition and PR68's compatible pending controls; and Chafik Boukhalfa for +PR60's ranked reclamation and PR71's anchored split/next-use composition. +All prior contributors retain their original notices. See [NOTICE](NOTICE). + +The preserved contributor files in `vendor/` are unmodified snapshots, used +for provenance and the complete PR71 comparison. The inherited repository +files are retained unchanged. No endorsement or completed expert review by +those contributors is implied. diff --git a/research/balanced-split-reclamation/SOURCE.json b/research/balanced-split-reclamation/SOURCE.json new file mode 100644 index 0000000000..cc93786f91 --- /dev/null +++ b/research/balanced-split-reclamation/SOURCE.json @@ -0,0 +1,64 @@ +{ + "files": { + ".github/workflows/balanced-split.yml": "9f45519e842d7412c2c68d349252b57872a1a73e25447ca2ac63c625eb3849d3", + "references/frame-compiler/pr48/SOURCE.json": "6ccc673f88d3650d75ed600b082fe2fc2866dd88a172d182a38c1d0cd27d24b3", + "references/frame-compiler/pr48/research/copied-fixed/balanced_assembly.py": "4d27575dcad2b45a281a34054d311409ecd954f373c0c3a62c29bcc843f06a86", + "references/frame-compiler/pr48/research/copied-fixed/certificate.json": "33b3373c04e18a7f4cee33ba68da6ed71934acc3e0179cca6197696be4befa22", + "references/frame-compiler/pr48/research/copied-fixed/changed_graph.py": "375a7ca6cac6d153c1880fd5ea5b4525e5dca9d4ee3018d24d679c219b59c5b3", + 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The 3/2 range +reduction follows the independently evaluated PR67 audit. Existing PR48 +balanced assembly and contributor attributions remain in their source. +Integration by huxint with OpenAI Codex assistance. Apache-2.0. +""" +from functools import lru_cache +from fractions import Fraction as Q +import json +from math import factorial +from support import HERE, ROOT, SELECTED, load, json_value +import binary_frame_math as inherited + +balanced = load(ROOT/'references/frame-compiler/pr48/research/copied-fixed/balanced_assembly.py', 'balanced_split_assembly') +GRID = 10**18 +BACKOFF = Q(1, 10**12) + + +def down(value, grid=10**40): + return Q(value.numerator*grid//value.denominator, grid) + + +def up(value, grid=10**40): + return -down(-value, grid) + + +def moment(profile, saving): + assert type(saving) is Q and 0 < saving < 1 + lower, upper = Q(), Q() + rows = {int(t): n for t, n in profile['child_multiplicities'].items()} + assert sum(t*n for t, n in rows.items()) == profile['total_rank'] + terms = {} + for t, n in sorted(rows.items()): + assert type(n) is int and n > 0 and 0 < t < profile['m'] + lo, hi = inherited.logs(Q(profile['m'], t)) + u, v = saving*lo, saving*hi + assert 0 <= u <= v < 1 + elo = sum((u**j/factorial(j) for j in range(9)), Q()) + ehi = sum((v**j/factorial(j) for j in range(9)), Q()) + ehi += v**9/factorial(9)/(1-v/10) + weight = Q(t*n, profile['m']*profile['W']) + lower += weight*down(elo) + upper += weight*up(ehi) + terms[t] = dict(weight=weight, log_lower=lo, log_upper=hi) + return dict(saving=saving, lower=lower, upper=upper, strict_gap=1-upper, terms=terms) + + +def bridge_for(profile): + source = ROOT/'references/frame-compiler/pr48/research/copied-fixed/certificate.json' + bridge = json.loads(source.read_text())['finite_bridge'] + bridge['bit']['W'] = profile['W'] + assert bridge['bit']['m'] == profile['m'] == 575 + assert bridge['bit']['maxchild'] == profile['maxchild'] == 529 + assert bridge['bit']['wire_bits'] == profile['W'].bit_length() == 28 + assert 2*529**9 < 575**9 and 2*529**8 >= 575**8 + assert bridge['bit']['halving_degree'] == 9 + return bridge + + +def certify(profile): + low, high = 1, 71700000000000 + assert moment(profile, Q(low, GRID))['upper'] < 1 + assert moment(profile, Q(high, GRID))['lower'] > 1 + while low+1 < high: + middle = (low+high)//2 + test = moment(profile, Q(middle, GRID)) + if test['upper'] < 1: + low = middle + elif test['lower'] > 1: + high = middle + else: + raise ArithmeticError('Inconclusive interval; no implicit tolerance is allowed') + saving = Q(low, GRID) + bridge = bridge_for(profile) + first = balanced.assembly(bridge, saving, Q(1, GRID), h=BACKOFF) + margin = first['minimum_margin'] + kappa = up(margin, GRID)-Q(1, GRID) + accepted = balanced.assembly(bridge, saving, kappa, h=BACKOFF) + assert len(accepted['constraints']) == 47 and len(accepted['margins']) == 7 + try: + balanced.assembly(bridge, saving, kappa+Q(1, GRID), h=BACKOFF) + except balanced.InvalidAssembly as error: + next_rejection = str(error) + else: + raise AssertionError('Next kappa grid point unexpectedly passed') + predecessor = json.loads((HERE/'vendor/pr71-certificate.json').read_text()) + old_profile = predecessor['bit'] + old_test = moment(old_profile, saving) + assert old_test['lower'] > 1, 'The complete PR71 profile must be excluded' + a, b = moment(profile, saving), moment(profile, Q(high, GRID)) + assert a['upper'] < 1 < b['lower'] + return dict(status='Finite conditional witness; inherited all-size and analytic interfaces remain assumptions', + kappa=kappa, bit_saving=saving, next_bit_saving=Q(high, GRID), bit=profile, + accepted_moment=a, rejected_moment=b, assembly=accepted, finite_bridge=bridge, + next_kappa_rejection=next_rejection, eventual_bounds=balanced.cutoffs(bridge, accepted), + comparison=dict(pr71_commit='1bef94fd40a746452548c84a4a8f8834670a3113', + pr71_kappa=Q(predecessor['kappa']), difference=kappa-Q(predecessor['kappa']), + ratio=kappa/Q(predecessor['kappa']), complete_pr71_moment_lower=old_test['lower'])) + + +@lru_cache(None) +def independent_log(x): + k = 0 + while x > Q(3, 2): + x /= Q(3, 2) + k += 1 + def expand(y): + z = (y-1)/(y+1) + power, total = z, Q() + for j in range(60): + total += power/(2*j+1) + power *= z*z + return 2*total, 2*total+2*power/(121*(1-z*z)) + lo, hi = expand(x) + blo, bhi = expand(Q(3, 2)) + return lo+k*blo, hi+k*bhi + + +def independent_moment(profile, saving): + lower, upper = Q(), Q() + for t, n in sorted((int(t), n) for t, n in profile['child_multiplicities'].items()): + lo, hi = independent_log(Q(profile['m'], t)) + u, v = down(lo*saving), up(hi*saving) + assert 0 < u <= v < Q(1, 1000) + tl = th = el = eh = Q(1) + for j in range(1, 13): + tl, th = tl*u/j, th*v/j + el, eh = el+tl, eh+th + eh += 2*th*v/13 + weight = Q(t*n, profile['m']*profile['W']) + lower += weight*down(el) + upper += weight*up(eh) + return lower, upper + + +def independent_audit(certificate): + p = certificate['bit'] + saving, successor = Q(certificate['bit_saving']), Q(certificate['next_bit_saving']) + accepted = independent_moment(p, saving) + rejected = independent_moment(p, successor) + assert accepted[1] < 1 < rejected[0] + actual = balanced.assembly(bridge_for(p), saving, Q(certificate['kappa']), h=BACKOFF) + assert json_value(actual) == certificate['assembly'] + assert all(x > 0 for x in actual['constraints'].values()) + assert all(x > Q(certificate['kappa']) for x in actual['margins'].values()) + return dict(status='PASS separately evaluated logarithm/exponential enclosure and assembly', + accepted_gap=1-accepted[1], rejected_gap=rejected[0]-1, + logarithm_base='3/2', logarithm_terms=60, exponential_degree=12, + constraints=47, margins=7) diff --git a/research/balanced-split-reclamation/compiler.py b/research/balanced-split-reclamation/compiler.py new file mode 100644 index 0000000000..87552341ba --- /dev/null +++ b/research/balanced-split-reclamation/compiler.py @@ -0,0 +1,353 @@ +# Live-anchor composition of PR67 all-rank cost selection with PR68. +# Dominik Scholz with substantial OpenAI GPT-6 Astra assistance. +# Next-use-aware scoring experiments by huxint with OpenAI Codex assistance. +# Based on Rohan Arun's PR67 cost selection and Dominik Scholz's PR68 +# compatible live controls (8778fdb52e834a2a14371590d80ea6e0c734a866). +# All-rank profile-cost selection: Rohan Arun, with OpenAI Codex assistance. +# Derived from Dominik Scholz PR63, Chafik Boukhalfa PR60, Avi Eisenberg PR62 +# and eumemic PR57. Apache-2.0; all inherited notices remain applicable. +"""Exact-envelope block synthesis on the pinned PR48 scalar graph. +Experimental finite compiler; no fixed-I+J profile or multiplication claim. +""" +# Derived from eumemic's PR57 compiler; original notices remain applicable. +# Descending-rank reclamation priority: Chafik Boukhalfa, with OpenAI Codex assistance. +import sys +if sys.flags.optimize: + raise ValueError('Assertions must remain enabled') +sys.dont_write_bytecode=True +from pathlib import Path +ROOT=Path(__file__).resolve().parents[2]/'references/frame-compiler/pr48' +from hashlib import sha256 +import json +manifest=json.loads((ROOT/'SOURCE.json').read_text()) +for name,digest in manifest['files'].items(): + assert sha256((ROOT/name).read_bytes()).hexdigest()==digest, 'Changed producer source: '+name +sys.path[:0]=[str(ROOT/'scripts'),str(ROOT/'research/copied-fixed')] +from changed_graph import graph +from collections import defaultdict,Counter,deque +import argparse,json,time,struct + +# Discovery choices only. Every accepted word must pass the independent replay. +CONFIG = {} + +def independent(rows,vec): + for row in rows: + vec=min(vec,vec^row) + return bool(vec) +def basis(rows): + out=[] + for vec in rows: + for row in out:vec=min(vec,vec^row) + if vec:out.append(vec) + return out + +def build(h): + c=graph(h);groups={};owner={};signal={};blocks=[] + for x in sorted(c.active): + key=(c.core[x],c.union[x]) + if key not in groups: + groups[key]=len(blocks);blocks.append(dict(nodes=[],frame=key,inputs=set(),uses=[])) + g=groups[key];owner[x]=g;blocks[g]['nodes'].append(x) + signal[x]=(signal[c.args[x][0]]^signal[c.args[x][1]])if c.args[x]else 1<<(x-1) + for x in sorted(c.active): + for y in c.args[x]or(): + if owner[y]!=owner[x]:blocks[owner[x]]['inputs'].add(y) + uses=[];value_uses=defaultdict(list) + for g,b in enumerate(blocks): + b['inputs']=sorted(b['inputs']);b['source']=not c.args[b['nodes'][0]] + for y in b['inputs']: + k=len(uses);uses.append((y,g,None));value_uses[y].append(k);blocks[owner[y]]['uses'].append(k) + for target,x in c.outputs.items(): + k=len(uses);uses.append((x,owner[x],target));value_uses[x].append(k);blocks[owner[x]]['uses'].append(k) + for g,b in enumerate(blocks): + b['rank']=1 if b['source']else b['frame'][1].bit_count()-b['frame'][0].bit_count() + mode=CONFIG.get('order','original') + if mode=='original':key=lambda g:(blocks[g]['rank'],min(blocks[g]['nodes'])) + elif mode=='reverse-node':key=lambda g:(blocks[g]['rank'],-min(blocks[g]['nodes'])) + elif mode=='cover-core':key=lambda g:(blocks[g]['rank'],blocks[g]['frame'][1],blocks[g]['frame'][0]) + elif mode=='core-cover':key=lambda g:(blocks[g]['rank'],blocks[g]['frame'][0],blocks[g]['frame'][1]) + else:raise ValueError(mode) + order=sorted(range(len(blocks)),key=key) + place={g:i for i,g in enumerate(order)} + for g,b in enumerate(blocks): + for x in b['inputs']:assert place[owner[x]]1 for b in blocks) + pending_points=defaultdict(set);pending_order={};pending_clock=0 + def point_bits(mask): + while mask: + bit=mask&-mask;yield bit;mask^=bit + def set_pending(s,g): + nonlocal pending_clock + assert s not in pending + pending[s]=g;pending_order[s]=pending_clock;pending_clock+=1 + for bit in point_bits(blocks[frames[s]]['frame'][0]):pending_points[bit].add(s) + def pop_pending(s): + for bit in point_bits(blocks[frames[s]]['frame'][0]):pending_points[bit].remove(s) + del pending_order[s] + return pending.pop(s) + def compatible_live(g): + if not CONFIG.get('live',True):return [] + core=blocks[g]['frame'][0];assert core + candidates=pending_points[core&-core] + return sorted((s for s in candidates if + contains(blocks[frames[s]]['frame'],blocks[g]['frame']) and + contains(blocks[g]['frame'],blocks[pending[s]]['frame'])),key=pending_order.__getitem__) + def new(g): + s=len(slots);slots.append(0);frames.append(g);hist[blocks[g]['rank']]+=1;events.append((s,-1,g));return s + def raise_(s,g): + old=frames[s];assert contains(blocks[old]['frame'],blocks[g]['frame']) + if s in pending: + for bit in point_bits(blocks[old]['frame'][0]&~blocks[g]['frame'][0]):pending_points[bit].remove(s) + hist[blocks[g]['rank']-blocks[old]['rank']]+=1;frames[s]=g;events.append((s,old,g)) + def xor(a,b,g): + assert a!=b;raise_(a,g);raise_(b,g);slots[a]^=slots[b];ops.append((a,b,g)) + import subprocess,struct + from binary_frame_math import logs + # Integer midpoint enclosures give a reproducible discovery score. + # Every candidate is independently checked by the complete exact screen. + if CONFIG.get('weight','entropy')=='entropy': + weights=[0]+[int(t*sum(logs(t))*10**30//2) for t in range(1,h+1)] + elif CONFIG['weight']=='moment': + from fractions import Fraction as Q + from math import factorial + saving=Q(51036365162009,10**18) + weights=[0] + for t in range(1,h+1): + x=-saving*sum(logs(t))/2 + # An exact rational discovery approximation, never an acceptance bound. + value=t*sum((x**j/factorial(j) for j in range(9)),Q()) + weights.append(-int(value*10**40)) + else:raise ValueError(CONFIG['weight']) + oracle_input=Path(ORACLE_INPUT) + with oracle_input.open('wb') as stream: + stream.write(struct.pack('<6I2Q',h,v,0,len(blocks)+2,0,0,h*(h-1),h*(h-1))) + stream.write(struct.pack('<2QI',0,0,0));stream.write(struct.pack('<2QI',0,0,h)) + for b in blocks:stream.write(struct.pack('<2QI',*b['frame'],b['rank'])) + oracle=subprocess.Popen([ORACLE_EXE,str(oracle_input)],stdin=subprocess.PIPE,stdout=subprocess.PIPE,text=True,bufsize=1) + oracles.append(oracle) + cache={} + def profile_entropy(a,b): + if (a,b) not in cache: + oracle.stdin.write(f'{a} {b}\n');oracle.stdin.flush() + answer=oracle.stdout.readline() + assert answer, 'profile oracle failed' + parts=list(map(int,answer.split()));assert len(parts)==h+1 + cache[a,b]=sum(n*w for n,w in zip(parts,weights)) + return cache[a,b] + def profile_cost(a,g): + old=frames[a]+2;target=g+2 + end=pending[a]+2 if CONFIG.get('horizon','cleanup')=='next-use' and a in pending else 1 + return profile_entropy(old,end)-profile_entropy(old,target)-profile_entropy(target,end) + def acquire(g,anchors): + if reclaim: + bb={} + def ins(s): + row=slots[s];expr=1<>s&1;e^=1<>i&1) + if j!=i: + # Synthesize swap as three XORs at the single common frame. + for a,z in ((i,j),(j,i),(i,j)):rr[a]^=rr[z];elim.append((a,z)) + for j in range(len(ins)): + if j!=i and rr[j]>>i&1:rr[j]^=rr[i];elim.append((j,i)) + assert rr==[1<>i&1:xor(s,value_slots[y],g) + assert slots[s]==signal[x];value_slots[x]=s + used=set() + for u in outuses: + x=uses[u][0];s=value_slots[x] + if x in used: + ss=acquire(g,list(value_slots.values()));xor(ss,s,g);s=ss + used.add(x);assert u not in assign;assign[u]=s;set_pending(s,uses[u][1]) + if step%5000==0:print('compile',h,step,len(slots),'reclaimed',stats['reclaimed'],'seconds',time.time()-t0,flush=True,file=sys.stderr) + output_slots=[];output_records=[];scatter=[0]*v;J=[] + for u,(x,g,target)in enumerate(uses): + if target is None:continue + s=assign[u];assert slots[s]==signal[x];raise_(s,g);output_slots.append(s) + common,triple=target;output_records.append((s,g,common,triple)) + targets=[i for i,t in enumerate(c.inputs)if common in t]if len(triple)==1 else[c.inputs.index(triple)] + for i in targets:scatter[i]^=slots[s];J.append((v+i,2*v+s)) + rank=blocks[g]['rank'] + if len(triple)==1:hist[rank]+=1;hist[h-rank]+=1 + else:hist[h-1-rank]+=1;hist[1]+=1 + assert len(set(output_slots))==len(output_slots) + for s in retired:hist[h-blocks[frames[s]]['rank']]+=1 + assert len(retired)+len(output_slots)==len(slots) + assert scatter==[1< 3 and choice: + index = {1: 0, 2: (len(partition)-1)//2, 3: len(partition)-1}[choice] + if len(partition[index]) == 2: + partition[index:index+1] = [[x] for x in partition[index]] + assert all(len(g) in (1, 2) for g in partition) + assert len(partition) < len(points) + assert sorted(x for g in partition for x in g) == sorted(points) + return partition + return SplitPair + + namespace['circuit_class'] = circuit_class + original_points = namespace['alternating_points'] + if anchor != 'none': + def points(dimension, common): + old = original_points(dimension, common) + pairs = [(a, a+1) for a in range(0, dimension-1, 2) if common not in (a, a+1)] + chosen = {'first': pairs[0], 'second': pairs[min(1, len(pairs)-1)], 'last': pairs[-1]}[anchor] + result = list(chosen)+[x for x in old if x not in chosen] + assert sorted(result) == [x for x in range(dimension) if x != common] + return result + namespace['alternating_points'] = points + result = namespace['graph'](h) + result.verify() + return result diff --git a/research/balanced-split-reclamation/paper.pdf b/research/balanced-split-reclamation/paper.pdf new file mode 100644 index 0000000000000000000000000000000000000000..41300a3f9e9f81b5b9336b7c8e2824a9c6b10f25 GIT binary patch literal 300255 zcmb5VQ*t z9EH*T7A!AwOS3AiEoJ@raLGG_ZB@#@gDvhGAg1J#i9I{TI`n;dQw8uXe~hiKL=(5a 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a=\frac{51550513208569}{10^{18}}. \tag{8} +\] +All logarithms are enclosed by a rational atanh series with explicit +remainder. For \(0\le u\le v<1\), use the degree-eight Taylor lower bound +at \(u\), and the upper bound +\[ + \sum_{j=0}^{8}\frac{v^j}{j!} + +\frac{v^9}{9!\,(1-v/10)} \tag{9} +\] +at \(v\). The geometric remainder follows from the ratio of successive +omitted terms. Directed rounding widens the intervals. The exact accepted +upper moment is below one; the lower moment at \(a+10^{-18}\) is above one. +A second evaluation uses powers of \(3/2\) for logarithm range reduction, +60 atanh terms and a degree-twelve exponential bound, and confirms both signs. + +The complete PR71 profile, with its own width, has a lower moment above one +at (8). Thus the new finite profile gives a strict structural improvement. +The comparison is to PR71 commit +\texttt{1bef94fd40a746452548c84a4a8f8834670a3113}. + +\begin{theorem}[Conditional on the inherited interfaces] +Assume the base integer-multiplication theorem and the all-size ordered +residual compiler, framed-word transfer, fixed finite-alphabet multitape +layout, analytic and semantic reductions, routing, eligible-prime setup and +exact-recovery interfaces used by the cited construction chain. Then +\[ + T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad + \kappa=\frac{12886963972497}{250000000000000000}. +\] +\end{theorem} +\begin{proof} +Use (4)--(8) as the new fixed bit-network interface in the inherited balanced +assembly. The complex saving remains \(717/10^7\), \(\beta=1/20\), and the +semantic guard remains one. Set \(\eta=10^{-12}\), +\(q=a(1-2\eta)\), \(\epsilon=(1-\eta)/(1+q)\). The controlling margin is +\(G=\epsilon q\). The certificate recomputes all 47 strict constraints and +seven margins and verifies \(\kappa, the carrier matching is read from a pinned file +// instead of Hopcroft-Karp. Every pinned edge must lie in the same admissible +// adjacency (causal order and frame inclusion); donors and uses are distinct. +// This replay-only adaptation omits the discovery edge dumper. The original +// optimizer and profiler are preserved in references/copied-fixed/pr44. +// Replay-only adaptation prepared with OpenAI Codex assistance. +// Prepared by Rohan Arun with Anthropic Claude assistance. Apache-2.0. +// Adapted at h=23,25; original PR35 by Dominik Scholz, PR32 by icekylinx. +// Dimension adaptation at 45/47; extra primes for all source-growth cases. +// Exact bounded-minor inequalities are checked by verify.py. Original credit below. +// Copyright 2026 icekylinx. Apache-2.0; AI-assisted handoff integration. +// Original round3_rankone_profiles.cpp. Requires GCC/Clang unsigned __int128. + +// Adapted input reader for complete serialized frame-compiler transitions. +// Actual-word fixed-I+J profiling, adapted from PR48 profiles.cpp. +// All projector, pivot and CRT formulas below are unchanged. +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +using U=uint32_t;using V=uint64_t; +#include "binary_io.hpp" +int main(int argc,char**argv){ +assert(argc==2);std::ifstream f(argv[1],std::ios::binary);U hdr[6];read_array(f,hdr); +U h=hdr[0],v=hdr[1],R=hdr[2],nf=hdr[3],nt=hdr[4];int64_t singles=hdr[5];assert(h==23||h==25); +V sum,loss;f.read(reinterpret_cast(&sum),8);f.read(reinterpret_cast(&loss),8); +struct Frame{V core,cover;U rank;};std::vectorframes; +for(U i=0;i(&a.core),8);f.read(reinterpret_cast(&a.cover),8);f.read(reinterpret_cast(&a.rank),4);frames.push_back(a);} +std::map,int64_t>transitions; +for(U i=0;i(&a),4);f.read(reinterpret_cast(&b),4);f.read(reinterpret_cast(&count),8);transitions[{a,b}]=count;} +assert(f);assert(sum==h*V(R)+loss); +V p=2305843009213693951ULL; +auto mul=[&](V a,V b)->V{return (__uint128_t)a*b%p;}; +auto power=[&](V a,V b)->V{V r=1;for(;b;b>>=1,a=mul(a,a))if(b&1)r=mul(r,a);return r;}; +auto sub=[&](V a,V b)->V{return (a+p-b)%p;}; +std::vector>matrix_cache(frames.size()); +auto matrix=[&](U id)->const std::vector&{auto&A=matrix_cache[id];if(!A.empty())return A;A.assign(h*h,0);if(id==0)return A;if(id==1){for(U i=0;i>i)&1);V zj=((f.core>>j)&1)?3*(h+1)-10:p-10;A[i*h+j]=mul(mul(wi,zj),factor);}return A;} + assert(c==1||c==2);V s=3-c,d=s*s+(c-1)*nn,den=3*(h+1)*d,inv=power(den,p-2); + for(U i=0;i>i)&1,oj=(out>>j)&1,wi=3+((f.core>>i)&1),zj=((f.core>>j)&1)?3*(h+1)-10:p-10; + V num=(mul(s*oi,zj)+3*(h+1)*s*wi*oj+mul(nn*wi,zj))%p;num=sub(num,3*(h+1)*(c-1)*oi*oj);A[i*h+j]=(V(i==j&&oi)+mul(num,inv))%p; + }return A;}; + +std::vectorblocks(h+1);blocks[1]=singles;V done=0,matrices=0,crt_matrices=0,crt_disagreements=0;U longest=0;std::mapcorrection_hist; +U aa,bb; +while(std::cin>>aa>>bb){ + assert(aa=frames[aa].rank); + if(rr<=2){blocks[1]=rr; for(U t=0;t<=h;t++)std::cout<1&&bb>1){ + assert(!(frames[bb].core&~frames[aa].core)); + assert(!(frames[aa].cover&~frames[bb].cover)); + U ca=popcount64(frames[aa].core),cb=popcount64(frames[bb].core); + assert(ca==3||frames[aa].core==frames[bb].core||(ca==2&&cb==1)); + } + const auto&A=matrix(aa);const auto&B=matrix(bb);std::vectorM(h*h);for(U x=0;x>pivots; + for(U i=0;i=0&&!M[i*h+j])j--;if(j<0)continue;pivots.push_back({i,U(j)});V inv=power(M[i*h+j],p-2);for(U k=i+1;k core-one use all three primes at these dimensions. +if(aa>1&&bb>1&&(popcount64(frames[aa].core)==3||(popcount64(frames[aa].core)==2&&popcount64(frames[bb].core)==1))){ + assert(h==23||h==25);crt_matrices++;matrix_cache[aa].clear();matrix_cache[bb].clear();std::vector>> all{pivots}; + for(V extra:{2147483647ULL,524287ULL}){p=extra;matrix_cache[aa].clear();matrix_cache[bb].clear();const auto&C=matrix(aa);const auto&D=matrix(bb);std::vectorY(h*h);for(U z=0;z> pp; + for(U i=0;i=0&&!Y[i*h+j])j--;if(j<0)continue;pp.push_back({i,U(j)});V inv=power(Y[i*h+j],p-2);for(U k=i+1;k corner((h+1)*(h+1));for(U i=0;i=j;best=std::max(best,rank);}corner[(i+1)*(h+1)+j]=best;} + pivots.clear();for(U i=0;i= j+1 proves persistence.", + "common_cutoff": 14000000000000, + "cutoff_log2_input": { + "alpha": 6021760001, + "compact": 24087040001, + "geometry": 3999931270706, + "guard": 4306678, + "log_p": 25, + "normalization": 14000000000000, + "period": 48174080001, + "phase_cell": 19, + "reservoir": 14, + "stopped_leaf": 252 + }, + "power_checkpoints": [ + { + "log2_input": 14000000000000, + "powers": { + "alpha": { + "exponent": 721649484, + "rhs": 112000000155264, + "rhs_bits": 47 + }, + "compact": { + "exponent": 721649484, + "rhs": 448000000620992, + "rhs_bits": 49 + }, + "leaf": { + "exponent": 666666666666, + "rhs": 3136, + "rhs_bits": 12 + }, + "period": { + "exponent": 721649484, + "rhs": 224000000310456, + "rhs_bits": 48 + }, + "rows": { + "exponent": 721649484, + "rhs": 112000000155264000, + "rhs_bits": 57 + } + } + }, + { + "log2_input": 28000000000000, + "powers": { + "alpha": { + "exponent": 1443298969, + "rhs": 224000000155264, + "rhs_bits": 48 + }, + "compact": { + "exponent": 1443298969, + "rhs": 896000000620992, + "rhs_bits": 50 + }, + "leaf": { + "exponent": 1333333333333, + "rhs": 3136, + "rhs_bits": 12 + }, + "period": { + "exponent": 1443298969, + "rhs": 448000000310456, + "rhs_bits": 49 + }, + "rows": { + "exponent": 1443298969, + "rhs": 224000000155264000, + "rhs_bits": 58 + } + } + }, + { + "log2_input": 56000000000000, + "powers": { + "alpha": { + "exponent": 2886597938, + "rhs": 448000000155264, + "rhs_bits": 49 + }, + "compact": { + "exponent": 2886597938, + "rhs": 1792000000620992, + "rhs_bits": 51 + }, + "leaf": { + "exponent": 2666666666666, + "rhs": 3136, + "rhs_bits": 12 + }, + "period": { + "exponent": 2886597938, + "rhs": 896000000310456, + "rhs_bits": 50 + }, + "rows": { + "exponent": 2886597938, + "rhs": 448000000155264000, + "rhs_bits": 59 + } + } + }, + { + "log2_input": 112000000000000, + "powers": { + "alpha": { + "exponent": 5773195876, + "rhs": 896000000155264, + "rhs_bits": 50 + }, + "compact": { + "exponent": 5773195876, + "rhs": 3584000000620992, + "rhs_bits": 52 + }, + "leaf": { + "exponent": 5333333333333, + "rhs": 3136, + "rhs_bits": 12 + }, + "period": { + "exponent": 5773195876, + "rhs": 1792000000310456, + "rhs_bits": 51 + }, + "rows": { + "exponent": 5773195876, + "rhs": 896000000155264000, + "rhs_bits": 60 + } + } + }, + { + "log2_input": 224000000000000, + "powers": { + "alpha": { + "exponent": 11546391752, + "rhs": 1792000000155264, + "rhs_bits": 51 + }, + "compact": { + "exponent": 11546391752, + "rhs": 7168000000620992, + "rhs_bits": 53 + }, + "leaf": { + "exponent": 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