diff --git a/.gitattributes b/.gitattributes index 314f2a9c94..79aa58f744 100644 --- a/.gitattributes +++ b/.gitattributes @@ -43,3 +43,7 @@ research/e8-complex-gauge-kernel/** -text -whitespace research/e8-lemma-a-lean/** -text -whitespace research/e8-lemma-g-lean/** -text -whitespace research/community-formal-20261012-audit/** -text -whitespace + +# Exact scoped static44 packing certificate and independent receipts. +research/static44-packing-ceiling/** -text -whitespace +research/static44-packing-ceiling-validation/** -text -whitespace diff --git a/.github/workflows/static44-packing-ceiling.yml b/.github/workflows/static44-packing-ceiling.yml new file mode 100644 index 0000000000..1d27144901 --- /dev/null +++ b/.github/workflows/static44-packing-ceiling.yml @@ -0,0 +1,32 @@ +name: Static44 exact packing ceiling +on: + push: + paths: + - 'research/static44-packing-ceiling/**' + - '.github/workflows/static44-packing-ceiling.yml' + pull_request: + paths: + - 'research/static44-packing-ceiling/**' + - '.github/workflows/static44-packing-ceiling.yml' + workflow_dispatch: +permissions: + contents: read +jobs: + verify: + runs-on: ubuntu-24.04 + timeout-minutes: 5 + strategy: + fail-fast: false + matrix: + python-version: ['3.11', '3.14'] + steps: + - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1 + with: + persist-credentials: false + - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97 + with: + python-version: ${{ matrix.python-version }} + - name: Verify both exact duals and rejection controls + run: python -B research/static44-packing-ceiling/verify.py "$RUNNER_TEMP/static44-packing-ceiling.json" + - name: Require unchanged tracked sources + run: git diff --exit-code diff --git a/research/static44-packing-ceiling-validation/CROSS-PLATFORM.json b/research/static44-packing-ceiling-validation/CROSS-PLATFORM.json new file mode 100644 index 0000000000..b386491275 --- /dev/null +++ b/research/static44-packing-ceiling-validation/CROSS-PLATFORM.json @@ -0,0 +1,9 @@ +{ + "status": "PASS_BYTE_IDENTICAL_COMPLETE_RECEIPTS", + "platforms": [ + "macOS arm64 Python 3.14.8", + "Linux aarch64 Python 3.11.15" + ], + "receipt_sha256": "bef2bf8d8b93a5373db8117248394e69e2ed577e4940e92c6c5f6267b633cd70", + "package_manifest_sha256": "ce9d4419a8b2941ba0887dd27411c44077bf26879e830643f7aea70cf9e47503" +} diff --git a/research/static44-packing-ceiling-validation/README.md b/research/static44-packing-ceiling-validation/README.md new file mode 100644 index 0000000000..4f340db7a9 --- /dev/null +++ b/research/static44-packing-ceiling-validation/README.md @@ -0,0 +1,7 @@ +# Independent verification receipts + +The complete package passed using macOS arm64/Python 3.14.8 and Linux +aarch64/Python 3.11.15. Both runs used only the standard library. Their entire +JSON receipts are byte-identical, including both exact interval/dual margins, +all pattern checks, both rejection controls and the package manifest hash. +CI is separate and is not claimed green by these receipts. diff --git a/research/static44-packing-ceiling-validation/linux-python311.json b/research/static44-packing-ceiling-validation/linux-python311.json new file mode 100644 index 0000000000..c0c5291a98 --- /dev/null +++ b/research/static44-packing-ceiling-validation/linux-python311.json @@ -0,0 +1,48 @@ +{ + "status": "PASS_EXACT_STATIC44_POOLED_HOSTING_CEILING", + "excluded_b": "1593251957517/1000000000000000", + "current_admitted_b": "31864911873350496687/20000000000000000000000", + "strict_kappa_ceiling": "1593251957517/1001593251957517", + "current_composed_kappa": "397677800014309/250000000000000000", + "kappa_headroom_upper": "1588431607148135139889247/250398312989379250000000000000000", + "relative_kappa_headroom_percent_upper": "158843160714813513988924700/398311400947642851864860110753", + "stock": { + "16": 3584, + "8": 5376, + "32": 3584, + "48": 3584 + }, + "helper_widths": { + "9": 292, + "8": 156, + "5": 56 + }, + "demands": { + "5": 4032, + "8": 11232, + "9": 21024 + }, + "patterns_checked": 145, + "moment_engines": { + "mercator": { + "minimum_pattern_slack": "1/100000000000000000000", + "local_excess_lower": "21140593396223135953906832552890247523214783536240703507721555436441192626953125/13654704984662271147909857662387020959275168646931586482961066094099431424", + "dual_objective": "9530393696685161049826055929050807445184549170211281202953974874805588894672480465895743891/162776768024710072850106449870908510199489219748158293759358717132800000000000000000000", + "strict_excess_lower": "3437770789862432253190368897338495516562218179777666790139458108539277122659962759/3906642432593041748402554796901804244787741273955799050224609211187200000000000000000000", + "moment_minus_one_lower": "3437770789862432253190368897338495516562218179777666790139458108539277122659962759/1283039594713037548128035358103647015451321452848448654435942400000000000000000000000000000000000" + }, + "base_two": { + "minimum_pattern_slack": "20347046494217742549806547672838900387454821410489124841761521281106427096999/2034709600308875910626330623386356377493615246851978671991983964160000000000000000000000000000000", + "local_excess_lower": "4848107152408127057387688105681270377649626277/3131391425937500000000000000000000000000", + "dual_objective": "9530393696685161049826055929050807445184549170211281202953974874805588894672480465895743891/162776768024710072850106449870908510199489219748158293759358717132800000000000000000000", + "strict_excess_lower": "20982487731093776313209552308417510801328205455089525616015104803886439663559018501722549/23844253128619639577652311992808863798753303674046625062406062080000000000000000000000000000000", + "moment_minus_one_lower": "20982487731093776313209552308417510801328205455089525616015104803886439663559018501722549/7831052213824692066211153308738079928291756914358207119360000000000000000000000000000000000000000000000" + } + }, + "negative_controls": [ + "current_admitted_b", + "inflated_stock_dual" + ], + "manifest_sha256": "ce9d4419a8b2941ba0887dd27411c44077bf26879e830643f7aea70cf9e47503", + "scope": "All fractional pooled whole-helper packings of this fixed local word, idle-child stock, nine-stage chain, eight replicas and helper widths. Chronology, owner identity and integer restrictions are relaxed. Other local words, stage chains, widths or compiler/outer architectures are not bounded." +} diff --git a/research/static44-packing-ceiling-validation/macos-python314.json b/research/static44-packing-ceiling-validation/macos-python314.json new file mode 100644 index 0000000000..c0c5291a98 --- /dev/null +++ b/research/static44-packing-ceiling-validation/macos-python314.json @@ -0,0 +1,48 @@ +{ + "status": "PASS_EXACT_STATIC44_POOLED_HOSTING_CEILING", + "excluded_b": "1593251957517/1000000000000000", + "current_admitted_b": "31864911873350496687/20000000000000000000000", + "strict_kappa_ceiling": "1593251957517/1001593251957517", + "current_composed_kappa": "397677800014309/250000000000000000", + "kappa_headroom_upper": "1588431607148135139889247/250398312989379250000000000000000", + "relative_kappa_headroom_percent_upper": "158843160714813513988924700/398311400947642851864860110753", + "stock": { + "16": 3584, + "8": 5376, + "32": 3584, + "48": 3584 + }, + "helper_widths": { + "9": 292, + "8": 156, + "5": 56 + }, + "demands": { + "5": 4032, + "8": 11232, + "9": 21024 + }, + "patterns_checked": 145, + "moment_engines": { + "mercator": { + "minimum_pattern_slack": "1/100000000000000000000", + "local_excess_lower": "21140593396223135953906832552890247523214783536240703507721555436441192626953125/13654704984662271147909857662387020959275168646931586482961066094099431424", + "dual_objective": "9530393696685161049826055929050807445184549170211281202953974874805588894672480465895743891/162776768024710072850106449870908510199489219748158293759358717132800000000000000000000", + "strict_excess_lower": "3437770789862432253190368897338495516562218179777666790139458108539277122659962759/3906642432593041748402554796901804244787741273955799050224609211187200000000000000000000", + "moment_minus_one_lower": "3437770789862432253190368897338495516562218179777666790139458108539277122659962759/1283039594713037548128035358103647015451321452848448654435942400000000000000000000000000000000000" + }, + "base_two": { + "minimum_pattern_slack": "20347046494217742549806547672838900387454821410489124841761521281106427096999/2034709600308875910626330623386356377493615246851978671991983964160000000000000000000000000000000", + "local_excess_lower": "4848107152408127057387688105681270377649626277/3131391425937500000000000000000000000000", + "dual_objective": "9530393696685161049826055929050807445184549170211281202953974874805588894672480465895743891/162776768024710072850106449870908510199489219748158293759358717132800000000000000000000", + "strict_excess_lower": "20982487731093776313209552308417510801328205455089525616015104803886439663559018501722549/23844253128619639577652311992808863798753303674046625062406062080000000000000000000000000000000", + "moment_minus_one_lower": "20982487731093776313209552308417510801328205455089525616015104803886439663559018501722549/7831052213824692066211153308738079928291756914358207119360000000000000000000000000000000000000000000000" + } + }, + "negative_controls": [ + "current_admitted_b", + "inflated_stock_dual" + ], + "manifest_sha256": "ce9d4419a8b2941ba0887dd27411c44077bf26879e830643f7aea70cf9e47503", + "scope": "All fractional pooled whole-helper packings of this fixed local word, idle-child stock, nine-stage chain, eight replicas and helper widths. Chronology, owner identity and integer restrictions are relaxed. Other local words, stage chains, widths or compiler/outer architectures are not bounded." +} diff --git a/research/static44-packing-ceiling/CERTIFICATE.json b/research/static44-packing-ceiling/CERTIFICATE.json new file mode 100644 index 0000000000..dd09a99419 --- /dev/null +++ b/research/static44-packing-ceiling/CERTIFICATE.json @@ -0,0 +1,13 @@ +{ + "excluded_b": "1593251957517/1000000000000000", + "dual": [ + "1159414936640430059330462179261511384581206599102713283120543068094957725638359245042654471/125012557842977335948881753500857735833207720766585569607187494757990400000000000000000000", + "2318829873283164971988217906679184253746337572867493152947851815442934289442437325042654471/125012557842977335948881753500857735833207720766585569607187494757990400000000000000000000", + "1943441799180566052578162798685443161579219588483422203801769244374593538716498125042654471/125012557842977335948881753500857735833207720766585569607187494757990400000000000000000000", + "2516643747598956163026288651309469013630730117317870594961393098503011022804777165042654471/125012557842977335948881753500857735833207720766585569607187494757990400000000000000000000", + "-92272544327/10000000000", + "-92743877627/10000000000", + "-21062393899/10000000000" + ], + "discovery": "Floating LP nominated this vector; all proof inequalities are independently rechecked over exact rational intervals. No optimizer is imported by verify.py." +} diff --git a/research/static44-packing-ceiling/LICENSE b/research/static44-packing-ceiling/LICENSE new file mode 100644 index 0000000000..261eeb9e9f --- /dev/null +++ b/research/static44-packing-ceiling/LICENSE @@ -0,0 +1,201 @@ + Apache License + Version 2.0, January 2004 + http://www.apache.org/licenses/ + + TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION + + 1. Definitions. + + "License" shall mean the terms and conditions for use, reproduction, + and distribution as defined by Sections 1 through 9 of this document. + + "Licensor" shall mean the copyright owner or entity authorized by + the copyright owner that is granting the License. + + "Legal Entity" shall mean the union of the acting entity and all + other entities that control, are controlled by, or are under common + control with that entity. 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Apache-2.0; see LICENSE. + +This contribution supplies a fixed-word pooled-hosting relaxation, an exact +rational dual certificate, a two-engine independent checker and explicit scope. +The discovery LP nominated the dual only; no solver result is a proof input. + +The unchanged static44 local construction originates in Dugongue's PR541 and +icekylinx's PR535/#531 lineage. Our PR544 provides the admitted owner plan, +complete paid profile and fresh finite receipts. Rohan Arun's PR547 supplies the +comparison composition with eumemic's R420 bit word from PR543. All contributor +and assistance notices in those sources remain unchanged. + +The moment engines are copied byte-for-byte from PR544's attributed PR531 +engine dependencies. Their original copyright, license and assistance notices +remain in each file. The balanced conversion inequality is inherited, not a new +outer theorem. Source paths, commit IDs, Git blob IDs and SHA256 hashes are in +SOURCES.json. diff --git a/research/static44-packing-ceiling/PROOF.md b/research/static44-packing-ceiling/PROOF.md new file mode 100644 index 0000000000..a15f9079d5 --- /dev/null +++ b/research/static44-packing-ceiling/PROOF.md @@ -0,0 +1,84 @@ +# Exact dual exclusion + +Let m=73, W=7168 and delta=2560. The unchanged local paid profile is multiplied +by 72 calls; write the resulting multiplicities as H. Let N_r be the number of +original idle children of rank r, as derived from the pinned owner-plan windows. +Every call requires 56 width-five, 156 width-eight and 292 width-nine helpers. +Thus the total demand is D_w=72 times this inventory. + +A relaxed pattern for a rank-r idle child is any nonnegative integer tuple +(n5,n8,n9) satisfying 5n5+8n8+9n9<=r. Its remainder is +u=r-5n5-8n8-9n9. Permit a nonnegative real number x_r,n of copies of each +pattern, subject only to + +``` +sum_n x_r,n = N_r, +sum_r,n n_w x_r,n = D_w. +``` + +Every actual whole-helper assignment induces such counts. Chronology, ownership, +per-call exclusion, charts and integrality impose extra restrictions, so deleting +them can only lower the optimal cost. The rank-mass identity is checked exactly: + +``` +sum_j j H_j + sum_r r N_r - sum_w w D_w = m W - delta. +``` + +For the proposed endpoint b=0.001593251957517, define phi_b(0)=0 and, for r>0, + +``` +phi_b(r) = [r((m/r)^b-1) + 32 m^2 m^b / 10^16] / b. +``` + +The second term is the unchanged per-child bad-class fallback used by PR544. +For any resulting profile P, the exact normalized moment satisfies + +``` +m W (Moment(P,b)-1) + = b [sum_j H_j phi_b(j) + sum_r,n x_r,n phi_b(u) - delta/b]. +``` + +The verifier recomputes rational lower and upper enclosures for phi with two +independent engines. It does not use a floating logarithm or the discovery LP. +The zero remainder is explicitly cost-free and has no child fallback charge. + +Let theta_r and mu_w be the seven frozen rational dual weights. For every +regenerated pattern, each engine independently proves + +``` +theta_r + sum_w mu_w n_w <= lower_phi_b(u). +``` + +Multiply by x_r,n>=0 and sum, then apply the stock and demand equalities. This +gives the lower bound + +``` +sum_j H_j lower_phi_b(j) + sum_r N_r theta_r + sum_w D_w mu_w - delta/b. +``` + +Both engines certify that this number is strictly positive. Consequently every +packing in the relaxation has Moment(P,b)>1. Each actual packing does as well. +The moment is strictly increasing with b because every nonzero child rank is +smaller than m and the fallback term is increasing. Therefore no admissible +rate for this fixed class reaches the excluded endpoint. + +Allowing a single merged remainder is optimistic even for implementations that +split it: for 00. Hence + +``` +kappa < q/(1+q) < b/(1+b) + < b_excluded/(1+b_excluded). +``` + +This is a supremum bound, not a newly achieved kappa. Subtracting the pinned +PR547 `079c86c` value gives a strict headroom below 6.344e-9, or below 0.000399% relative. +The all-size and analytic assumptions behind the finite suppliers remain +conditional. The distinct 1/33 and Gaussian 1/4 assembly-domain restrictions +are already documented in the repository's assembly-domain audit; they are not +new results of this package. diff --git a/research/static44-packing-ceiling/README.md b/research/static44-packing-ceiling/README.md new file mode 100644 index 0000000000..f7f85e3dac --- /dev/null +++ b/research/static44-packing-ceiling/README.md @@ -0,0 +1,47 @@ +# Exact packing ceiling for the fixed static44 complex word + +**No improved multiplication bound is claimed.** For PR544's fixed local word, +helper widths, nine-stage chain, eight replicas and retained bad-class charge, +every whole-helper packing in the stated model has + +``` +b < 0.001593251957517. +``` + +The admitted value is `0.001593245593667524834350`. Even allowing arbitrary +fractional pooling and ignoring chronology and owner identity, the retained +balanced conversion therefore gives + +``` +kappa < 0.001590717543676681. +``` + +Compared with PR547's pinned `079c86c` checkpoint, `0.001590711200057236`, the remaining packing headroom is +**less than 6.344e-9 in kappa, or 0.000399%**. This is a scoped reason to seek +another local construction or chain rather than expect a substantial gain from +repacking this word. It does not bound other constructions or integer +multiplication in general. + +Run the self-contained, standard-library verifier: + +```sh +python -B verify.py /absolute/new/receipt.json +``` + +The verifier uses no optimizer or network. It regenerates every relaxed packing +pattern, derives stock and demand from pinned PR544 artifacts, and checks a +frozen dual with two independent rational moment engines. Both must prove all +pattern inequalities and a strictly positive exclusion margin. The current +admitted rate and an inflated dual weight are negative controls and must reject. + +`CERTIFICATE.json` contains only the excluded endpoint and seven rational dual +weights. Their order is stock ranks `(8,16,32,48)`, then helper widths `(5,8,9)`. +`SOURCES.json` binds the original Git blobs; all inherited engines and notices +are unchanged. `MANIFEST.json` covers every file. See [PROOF.md](PROOF.md). + +The model partitions each original idle child into whole helper widths and a +remainder, serving exactly 72 copies of each local helper. Its relaxation drops +integer, call, owner, chart and temporal restrictions. Splitting a remainder +further cannot improve the retained moment at this endpoint. Changed local +programs, helper widths, stage count, replica count, banks, bad-class charges or +outer proofs are outside this ceiling. diff --git a/research/static44-packing-ceiling/SOURCES.json b/research/static44-packing-ceiling/SOURCES.json new file mode 100644 index 0000000000..68c47d6111 --- /dev/null +++ b/research/static44-packing-ceiling/SOURCES.json @@ -0,0 +1,46 @@ +{ + "files": [ + { + "local": "inputs/PR544-PLAN.json", + "commit": "4858bb38da10cd8a1cd25ef85e7c14f4e2b529dd", + "source": "research/static44-selective/data/PLAN.json", + "sha256": "f5e285823588889c69ef4d37c85179f038dfee95c56c5f5ad8aae28706efe8af", + "git_blob": "31cb8fc5b20868f712f5cb4e436687f424e7e86c" + }, + { + "local": "inputs/PR544-EXPECTED.json", + "commit": "4858bb38da10cd8a1cd25ef85e7c14f4e2b529dd", + "source": "research/static44-selective/EXPECTED.json", + "sha256": "8f48531cb1ccc6c1fe7c80ba5990f8aea4278dbe9394bcc4d3f471a3e8d75f1c", + "git_blob": "d6ca3ca276ffe69c0d6d226ed8489ec5a5adec7d" + }, + { + "local": "inputs/PR547-EXPECTED.json", + "commit": "079c86cd4a0667506f4d75e0d2584a51a481a3bb", + "source": "research/r420-static44-composition/EXPECTED.json", + "sha256": "e9a2c95c479fb9ef65aaad86d5819c14bb88191ae914eb6079258d8b8358c916", + "git_blob": "5b786e190a8d4c55d76a4a40ad2e4ca220e79af7" + }, + { + "local": "engines/moment.py", + "commit": "4858bb38da10cd8a1cd25ef85e7c14f4e2b529dd", + "source": "research/static44-selective/vendor/pr531/inputs/data-base/work/pr361/code/engines/moment.py", + "sha256": "81cbc864e106418fc7854322746f72180cb8110ff39f74450a3b8d55a82c1482", + "git_blob": "7d697cb45203fe72d97037403d6da41f71f3b7fd" + }, + { + "local": "engines/base_two_moment.py", + "commit": "4858bb38da10cd8a1cd25ef85e7c14f4e2b529dd", + "source": "research/static44-selective/vendor/pr531/inputs/data-base/work/pr361/code/engines/base_two_moment.py", + "sha256": "aee70c242a2f1d225528fabe2002e1564d1dc884797c07d28a831c7878ee9098", + "git_blob": "14a90fa0e788217856f070382c9e61bd7c603c61" + }, + { + "local": "inputs/PR544-RESULT.json", + "commit": "4858bb38da10cd8a1cd25ef85e7c14f4e2b529dd", + "source": "research/static44-selective-validation/macos-python314/RESULT.json", + "sha256": "d1429f7949ad7ac34a6dc594e539172ba3b897b1cc1cc25dd3173f85d15026f6", + "git_blob": "0d96fa29531bdbfc30e712faa68256fb7cd80b61" + } + ] +} diff --git a/research/static44-packing-ceiling/engines/base_two_moment.py b/research/static44-packing-ceiling/engines/base_two_moment.py new file mode 100644 index 0000000000..14a90fa0e7 --- /dev/null +++ b/research/static44-packing-ceiling/engines/base_two_moment.py @@ -0,0 +1,63 @@ +#!/usr/bin/env python3 +"""Maintainer arithmetic cross-check of the pinned PR39 certificate. + +Uses no producer/checker imports. Counts are inputs, not proved by this script. +80-term rational logarithms and 12-term exponentials independently enclose +the two moments. The accompanying written review addresses their realization. +""" +import argparse +from fractions import Fraction as Q +from hashlib import sha256 +import json +from pathlib import Path + +ROOT = Path(__file__).resolve().parents[1] +GRID = 10**30 + + +def require(condition, message): + if not condition: + raise ValueError(message) + + +def log_unit(x): + require(1 <= x <= 2, "logarithm range") + z = (x-1)/(x+1) + lo = 2*sum((z**(2*j+1)/(2*j+1) for j in range(80)), Q(0)) + return lo, lo+2*z**161/(161*(1-z*z)) + + +def log_bounds(x): + k = 0 + while x > 2: + x /= 2 + k += 1 + lo, hi = log_unit(x) + l2, u2 = log_unit(Q(2)) + lo, hi = lo+k*l2, hi+k*u2 + # Outward rational rounding keeps later certificates compact. + return Q((lo*GRID).__floor__(), GRID), Q((hi*GRID).__ceil__(), GRID) + + +def exp_bounds(x): + require(0 <= x < 1, "exponential range") + term = total = Q(1) + for j in range(1, 13): + term *= x/j + total += term + tail = term*x/13/(1-x/14) + return total, total+tail + + +def moment(m, W, entries, saving): + lo = hi = Q(0) + for t, count in entries: + t, count = int(t), int(count) + require(0 < t < m and count > 0, "improper child") + l, u = log_bounds(Q(m, t)) + lower = exp_bounds(saving*l)[0] + upper = exp_bounds(saving*u)[1] + lo += Q(t*count, m*W)*lower + hi += Q(t*count, m*W)*upper + return Q((lo*GRID).__floor__(), GRID), Q((hi*GRID).__ceil__(), GRID) + diff --git a/research/static44-packing-ceiling/engines/moment.py b/research/static44-packing-ceiling/engines/moment.py new file mode 100644 index 0000000000..7d697cb452 --- /dev/null +++ b/research/static44-packing-ceiling/engines/moment.py @@ -0,0 +1,113 @@ +"""Independent prospective ordinary five-stage cost, never supplier admission. + +Consumes a raw one-invocation ledger made from the actual source/internal/target +paths. No packed profile is used as an input. Missing physical/finite/compiler +obligations must remain explicit in the accompanying report. +""" +from collections import Counter +from decimal import Decimal, localcontext, ROUND_FLOOR +from fractions import Fraction as F +from functools import lru_cache +from math import factorial +import hashlib, json, sys +from pathlib import Path + +DYAD = 1 << 224 + +def floor(x): return F(x.numerator*DYAD//x.denominator, DYAD) +def ceil(x): return -floor(-x) + +@lru_cache(None) +def logarithm(x): + """Mercator bounds after binary reduction; ln(2) via two Mercator sums. + + For 1 <= y < 2, log(y) is an alternating series in z=y-1. + We instead reduce to 1 <= y < 3/2, and bound log(3/2) and + log(2) by the same alternating series at z=1/2 and z=1/3, + using log(2)=log(3/2)+log(4/3). All sums are exact rationals. + """ + assert x >= 1 + k = 0 + while x >= 2: x /= 2; k += 1 + j = int(x >= F(3, 2)) + if j: x /= F(3, 2) + def mercator(y): + z=y-1 + assert 0 <= z <= F(1,2) + # An even partial sum is below log; the next term bounds its tail. + t=z; s=F(0) + for n in range(1, 225): + s += (t if n & 1 else -t)/n + t *= z + return s, s+t/225 + a,b=mercator(x); c,d=mercator(F(3,2)); e,f=mercator(F(4,3)) + return floor(a+(k+j)*c+k*e), ceil(b+(k+j)*d+k*f) + +def exponential(l,u): + assert 0 <= l <= u < F(1,10) + def poly(x): + s=t=F(1) + for j in range(1,25): t=t*x/j; s+=t + return s,t + lo,_=poly(l);hi,last=poly(u) + return floor(lo),ceil(hi+last*u/25/(1-u/26)) + +def moment(H,m,W,a,bad=False): + lo=hi=F(0) + for r,n in H.items(): + assert 0 < r < m and n > 0 + l,u=logarithm(F(m,r)); e,f=exponential(a*l,a*u) + p=F(r*n,m*W);lo+=p*e;hi+=p*f + if bad: + # Existing envelope shape, only conditionally portable to the new cover. + l,u=logarithm(F(m));e,f=exponential(a*l,a*u) + p=F(32*m*sum(H.values()),10**16*W) + lo+=p*e;hi+=p*f + return floor(lo),ceil(hi) + +def decimal(x): + with localcontext() as c: + c.prec=70 + return str(Decimal(x.numerator)/Decimal(x.denominator)) + +def certify(H,m,W,bad=False): + with localcontext() as c: + c.prec=70 + terms=[(Decimal(r*n)/Decimal(m*W),(Decimal(m)/Decimal(r)).ln()) for r,n in H.items()] + fallback=Decimal(32*m*sum(H.values()))/Decimal(10**16*W) if bad else Decimal(0) + lm=Decimal(m).ln();lo=Decimal(0);hi=Decimal('.003') + for _ in range(210): + a=(lo+hi)/2 + if sum(p*(a*l).exp() for p,l in terms)+fallback*(a*lm).exp()<1:lo=a + else:hi=a + q=int(((lo+hi)/2*10**24).to_integral_value(rounding=ROUND_FLOOR)) + lo=F(q,10**24);hi=lo+F(1,10**24) + lv=moment(H,m,W,lo,bad);hv=moment(H,m,W,hi,bad) + assert lv[1]<1=0:patterns.append((r,counts,left)) +used_ranks=sorted(({left for r,counts,left in patterns}|set(H))-{0}) + +def prove(certificate): + b=Q(certificate['excluded_b']);dual=list(map(Q,certificate['dual'])) + assert 0=0,(name,'dual infeasible') + local=sum(n*values[r][0] for r,n in H.items()) + dual_objective=sum(stock[r]*theta[r] for r in ranks)+sum(demand[w]*mu[w] for w in ws) + gap=local+dual_objective-Q(deficit)/b + assert gap>0,(name,'no strict exclusion') + receipts[name]=dict(minimum_pattern_slack=str(min(slacks)),local_excess_lower=str(local),dual_objective=str(dual_objective),strict_excess_lower=str(gap),moment_minus_one_lower=str(b*gap/(m*W))) + return b,receipts + +b,receipts=prove(cert) +controls=[] +for name,changed in [('current_admitted_b',dict(cert,excluded_b=published['complex_lower'])),('inflated_stock_dual',copy.deepcopy(cert))]: + if name=='inflated_stock_dual':changed['dual'][0]=str(Q(changed['dual'][0])+1) + try:prove(changed) + except AssertionError:controls.append(name) + else:raise AssertionError('invalid certificate accepted: '+name) +current=Q(published['complex_lower']);k=Q(composition['kappa']);limit=b/(1+b) +assert current