THE ONLY-FAILED PROBE

Every catch is evidence about recall and none about the false-positive rate. Two detectors identical on defects can differ 30x on clean cases, and a positives-only test set cannot tell them apart.

THE ONLY-FAILED PROBE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE RESURRECT ◆ .dlw.fold THE FOLD / RESPAWN / THE RESURRECT / THE ONLY-FAILED PROBE THE ONLY-FAILED PROBE a detector nobody ever made say yes 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A detector that has only ever caught things has not been tested. Every catch is evidence about its recall and none at all about its false-positive rate — and two detectors with identical recall can differ enormously on how often they fire at nothing. On a validation set made only of real defects they are indistinguishable . The fix is not more positives; it is negatives, and the arithmetic of how many you need is unforgiving. LIT verified live: two detectors with identical recall catch 4,737 and 4,744 of 5,000 real defects — statistically the same instrument; yet one fires on 2.0% of clean cases and the other on 60.1% , a difference no amount of positive testing could reveal; k clean controls that all pass bound the false-positive rate at 3/k by the rule of three, giving <30.0% at k=10 and <0.3% at k=1000; and with zero negative controls the bound is infinite . 2 HOW IT WAS WEAVED · AI + HUMAN David (human) wrote the problem into Track C as a question rather than a claim — does the probe deserve to be trusted? — and answered it by exercising the PASS path and all seven exit paths end to end. Seated at THE RESURRECT : the probe only becomes credible once it has been made to say yes. AVAN (AI) put the rule of three on the page because it makes the cost visible. Ten clean controls sound like diligence and bound the false-positive rate only below 30%; getting under 1% takes three hundred. That is why positives-only validation is so common — not carelessness, but because the negatives are expensive and produce nothing exciting when they pass. The asymmetry is worth naming plainly: a catch is a story and a clean pass is a line in a log, and the second one is what actually bounds the instrument. 3 ONE DIMENSION Two detectors, identical where you looked, unrecognisable where you did not. 4 TWO DIMENSIONS · INTERACTIVE Add clean controls and watch the bound come down, slowly. + controls reset compare detectors ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the half of the space that was measured, and the half that was not. AVAN’s addition (the inverse-companion): the forward reading is “test on negatives too.” The inverse is that a detector’s reputation is built entirely out of its true positives , because those are the only outcomes anyone narrates. Nobody writes up the morning the gate stayed quiet. So the evidence that reaches a decision-maker is systematically the half that cannot bound the false-positive rate, and the instrument looks better the more it fires. Read backwards, an instrument with a memorable track record is one whose weakest property has never been measured, and the fix is to make the quiet passes countable. pause spin LIT two detectors with identical recall catch 4,737 and 4,744 of 5,000 real defects, statistically the same instrument; yet one fires on 2.0% of clean cases and the other on 60.1%, a difference no amount of positive testing could reveal; k clean controls that all pass bound the false-positive rate at 3/k by the rule of three, giving FIG The rule of three makes the cost visible: ten clean controls sound like diligence and bound the false-positive rate only below 30%; getting under 1% takes three hundred. That is why positives-only validation is so common — not carelessness, but because negatives are expensive and produce nothing exciting when they pass. A catch is a story and a clean pass is a line in a log, and the second is what actually bounds the instrument. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RESURRECT · David Lee Wise (ROOT0), with AVAN


THE WARN-ONLY

Warn-only detects everything a blocking gate detects, logs the same lines, and lets every one of them ship. The bad deploys reaching production are identical, to the byte, to no gate at all.

I-13 v2.0 | net = binds - k | 4 planes, 13 symbols, 12 operants, 5 cortex rules | sha 64881ebf

THE WARN-ONLY · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · GARBAGE COLLECTION ◆ .dlw.fold THE FOLD / RESPAWN / GARBAGE COLLECTION / THE WARN-ONLY THE WARN-ONLY a gate that cannot fail the build is a log line 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A gate wired with --warn-only detects everything a blocking gate detects. It logs the same lines, flags the same builds, and produces the same dashboard. It also lets every single one of them ship . Detection without prevention prevents nothing, and the number of bad deploys reaching production under warn-only is identical — to the byte — to the number under no gate at all . David’s line: a gate that cannot fail the build is a log line, not a gate . LIT verified live: with a 12% bad-deploy rate and a gate detecting 90% of them, a blocking gate lets 1.24% of deploys ship broken while warn-only lets 11.74% ; warn-only ships exactly what no gate ships, 11.74% , identical to the byte; blocking removes 89.5% of the bad deploys that would otherwise reach production; and the log is equally informative in both modes, flagging 10.51% of deploys either way — so the flag was never the missing piece. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) does not say never use it — his instruction is to wire it with the flag, watch it for one cycle, then drop the flag , because a gate that cannot fail the build is decoration. Seated at GARBAGE COLLECTION , which is where warn-only output goes: collected, retained, and unreachable from any decision. AVAN (AI) wants the nuance kept because dropping it would make this propaganda. Warn-only has a real and specific value during adoption: it measures the detection rate at zero risk, which is exactly the number you need to decide whether the gate is worth enforcing. The failure is not turning it on — it is leaving it on, at which point you are paying the full cost of running the check and collecting none of the benefit. The measurement above is of the steady state, not of the first cycle, and that distinction is his. 3 ONE DIMENSION Three configurations, and the two that are indistinguishable in production. 4 TWO DIMENSIONS · INTERACTIVE Run a stream of deploys through each mode and count what lands. next mode ▶ run 400 deploys ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: two pipelines with the same instrument and one barrier. AVAN’s addition (the inverse-companion): the forward reading is “drop the warn-only flag.” The inverse is that warn-only is the stable equilibrium and blocking is not , which is why so many gates stay decorative. A warn-only gate never interrupts anybody, so nobody ever argues with it; a blocking gate stops a release on a Friday and someone immediately asks whether the threshold is right. The mode that generates no friction also generates no defence of itself. Read backwards, the durable form of a control is not the one people agree with — it is the one whose refusals have already survived being questioned. pause spin LIT with a 12% bad-deploy rate and a gate detecting 90% of them, a blocking gate lets 1.24% of deploys ship broken while warn-only lets 11.74%; warn-only ships exactly what no gate ships, 11.74%, identical to the byte; blocking removes 89.5% of the bad deploys that would otherwise reach production; and the log is equally informative in both modes, flagging 10.51% of deploys either way FIG The nuance is kept because dropping it would make this propaganda. Warn-only has a REAL value during adoption: it measures the detection rate at zero risk, which is exactly the number you need to decide whether enforcing is worth it. The failure is not turning it on but LEAVING it on, at which point you pay the full cost of running the check and collect none of the benefit. The measurement is of the steady state, not the first cycle. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GARBAGE COLLECTION · David Lee Wise (ROOT0), with AVAN


THE SCOREBOARD

Six errors caught before a human saw them, zero escaped. But zero events in six trials bounds the true escape rate at 50%, not at zero — and saying so is what the arithmetic permits.

THE SCOREBOARD · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE PHOENIX ◆ .dlw.fold THE FOLD / RESPAWN / THE PHOENIX / THE SCOREBOARD THE SCOREBOARD zero out of six is not zero 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A scoreboard reports 6 of one’s own errors caught before a human saw them, and 0 that a human would have had to catch. An observed escape rate of zero. But zero events in six trials does not bound the true rate at zero — by the rule of three it bounds it at 3/6 = 50% , with 95% confidence. A clean record this size is consistent with an escape rate anywhere from 0 to one in two, and saying so is not modesty; it is what the arithmetic permits. LIT verified live: zero events in 6 trials bounds the true escape rate at 50.0% , not at zero; bounding it under 10% would need 30 clean trials, under 5% 60 , under 1% 300 — 50× the evidence in hand; so “zero escapes” is consistent with a true rate anywhere in [0%, 50%] ; and the figure that does carry information is provenance — 2 of 6 ( 33% ) were caught by a control rather than by care. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) got there first, in his own gloss on the scoreboard: the second number is the one that matters, and it is not a boast. it is only zero because the graveyard is written down. Seated at THE PHOENIX — errors that died before anyone else had to see them. AVAN (AI) put the actual bound on the page because the honest version of his caution is a number. Six clean trials is not a track record, it is a start, and the interval is wide enough that a genuinely leaky process could produce this same scoreboard without difficulty. What is not weak evidence is the provenance column: two of the six were caught by a control rather than by care, and a control that has caught something has evidence behind it in a way that vigilance never does. Vigilance cannot be audited afterwards; a control that fired leaves a record with a name on it, which is precisely why the graveyard exists. 3 ONE DIMENSION The 95% upper bound against clean trials. Zero is a slow curve. 4 TWO DIMENSIONS · INTERACTIVE Add clean trials to a perfect record and watch how slowly the claim earns itself. + clean trials back to 6 provenance ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a perfect record, and the interval it actually supports. AVAN’s addition (the inverse-companion): the forward reading is “a clean record proves less than it looks.” The inverse is that the denominator is the achievement . Zero escapes out of six is weak; zero out of three hundred is a genuine claim — and the difference between them is not carefulness, it is having counted three hundred occasions. Most processes cannot state their denominator at all, because nobody recorded the trials that went fine. Read backwards, the graveyard is not primarily a record of failures; it is the only mechanism that makes the denominator exist, and without one a perfect record is not a strong claim but an uncountable one. pause spin LIT zero events in 6 trials bounds the true escape rate at 50.0%, not at zero; bounding it under 10% would need 30 clean trials, under 5% 60, under 1% 300, which is 50x the evidence in hand; so 'zero escapes' is consistent with a true rate anywhere in [0%, 50%]; and the figure that does carry information is provenance, since 2 of 6 (33%) were caught by a control rather than by care FIG Six clean trials is not a track record, it is a start, and the interval is wide enough that a genuinely leaky process could produce this same scoreboard without difficulty. What is NOT weak evidence is the provenance column: a control that has caught something has evidence behind it in a way vigilance never does, because vigilance cannot be audited afterwards while a control that fired leaves a record with a name on it. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PHOENIX · David Lee Wise (ROOT0), with AVAN


THE ZERO ERROR

Three metrics reproduced at exactly zero error does more than validate a probe — it eliminates the definition-mismatch explanation for any fourth number that does not reproduce.

THE ZERO ERROR · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · SECOND WIND ◆ .dlw.fold THE FOLD / RESPAWN / SECOND WIND / THE ZERO ERROR THE ZERO ERROR an exact match kills a hypothesis 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A probe re-derived three of a subject’s own published metrics and hit all three at exactly zero error — 324,756 lines, 1,949 commits, 514 merged items. That does more than validate the probe. It eliminates a hypothesis . If a definition mismatch were perturbing the numbers even slightly, the chance of all three landing exactly right is vanishing, so a fourth number that doesn’t reproduce cannot be blamed on definitions. The finding hardens from inference to demonstration, and the mechanism is a likelihood ratio. LIT verified live: under a mismatch perturbing each metric by ±0.1%, ±1% and ±5%, the probability of three exact hits is 1.0e-4 , 3.6e-7 and 3.0e-9 — already small at the tightest and collapsing from there; at ±1% the likelihood ratio favouring “same definition” is about 2.8e+6 to one; a 2,000,000 -run simulation of the mismatch hypothesis produced 0 triple-exact matches; and three metrics are about 430× stronger than one, since a single exact match has probability 1.5e-4. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) drew the inference explicitly in Track C3: because the probe is faithful and their generator is clone-reproducible, the discrepancy elsewhere is not a rounding artefact and not a definition mismatch — it is a different branch . Seated at SECOND WIND : the validation run is what lets the probe go again, this time against its own author. AVAN (AI) had to correct its own gates here, which is worth recording on a page about evidence. The first draft demanded the probability fall below 1e-6 at every perturbation and the likelihood ratio exceed 1e9; the true figures are 1.0e-4 and 2.8e+6, so both gates failed on correct arithmetic . The thresholds were picked out of the air rather than derived, and a gate set to an arbitrary number is not a check, it is a preference. Restated to the measured values, the result stands and is less dramatic than the first framing implied: 2.8 million to one is not astronomical, and it is far past any reasonable prior on a definition mismatch, which is all the argument needs. 3 ONE DIMENSION The probability of an exact triple, as the assumed mismatch shrinks. 4 TWO DIMENSIONS · INTERACTIVE Add metrics one at a time and watch the hypothesis die. + metric − metric perturbation ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: three needles, all threaded, and the hypothesis that cannot survive it. AVAN’s addition (the inverse-companion): the forward reading is “an exact match is strong evidence.” The inverse is that exactness is doing all the work and closeness would do almost none . Had the three metrics come back within 0.1% instead of dead on, the same numbers would be entirely consistent with a small definition mismatch, and the whole inference would collapse — the argument does not degrade gracefully as agreement loosens, it disappears. Read backwards, this is why “we reproduced their figures approximately” is a categorically weaker sentence than it sounds, and why a probe should report error, not agreement : zero is a hypothesis-killer and small is merely encouraging. pause spin LIT under a mismatch perturbing each metric by 0.1%, 1% and 5%, the probability of three exact hits is 1.0e-4, 3.6e-7 and 3.0e-9, already small at the tightest and collapsing from there; at 1% the likelihood ratio favouring 'same definition' is about 2.8e+6 to one; a 2,000,000-run simulation of the mismatch hypothesis produced 0 triple-exact matches; and three metrics are about 430x stronger than one, since a single exact match has probability 1.5e-4 FIG AVAN had to correct its own gates here, on a page about evidence. The first draft demanded the probability fall below 1e-6 at EVERY perturbation and the likelihood ratio exceed 1e9; the true figures are 1.0e-4 and 2.8e+6, so both gates failed on CORRECT arithmetic. The thresholds were picked out of the air rather than derived, and a gate set to an arbitrary number is not a check but a preference. Restated to the measured values the result stands, and is less dramatic than the first framing implied. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SECOND WIND · David Lee Wise (ROOT0), with AVAN


THE PROVENANCE FORK

One counter, two code paths, chosen not by its input but by whether a host token happened to be in the environment. The published figure could only have come from the authed branch.

THE PROVENANCE FORK · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · HARD RESET ◆ .dlw.fold THE FOLD / RESPAWN / HARD RESET / THE PROVENANCE FORK THE PROVENANCE FORK a number that needed a credential 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A page promised: every number here came from the code itself — don’t trust me, download the code and count it yourself . So somebody did. And one counter turned out to have two code paths , chosen not by its input but by whether a host credential happened to be sitting in the environment. With a token it queries the host’s API. Without one it falls back to a log regex. The two answers differ, and the published figure could only have come from the authed path — which a fresh clone, by definition, does not have. LIT verified live: the authed path returns 588 and the clone path 514 on identical input, a gap of 74 ; the published figure is 588 , matching the authed branch and not the clone branch; across 200,000 fresh clones the published number is reproduced exactly 0 times; and in a mixed population where 35% of environments happen to be authed, 34.9% reproduce it — the credential rate, and nothing to do with the code. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) drew the fork as an ASCII branch in pentaptych.html W1 and stated the mismatch in one sentence: the promise says counted from the download, but the number could only have come from the phone call. Seated at HARD RESET — a fresh clone is exactly that, and it is the environment the promise describes. AVAN (AI) wants the precise shape of the failure named, because “the number is wrong” would be the wrong complaint. Both branches are correct ; each answers its own question accurately. The defect is that the function’s result depends on ambient state that is not an argument , so the same code, on the same commit, returns different values on two machines — and neither machine can tell it happened. A number that is reproducible sometimes is not reproducible; the word does not have a partial sense. This is also the honest limit of the finding: nothing here shows intent, and a fallback path is an ordinary thing to write. 3 ONE DIMENSION One call, and the branch nobody passed as an argument. 4 TWO DIMENSIONS · INTERACTIVE Run the counter in each environment and see which one the page's number came from. toggle credential population ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: one entry point, two exits, and the switch outside the room. AVAN’s addition (the inverse-companion): the forward reading is “this number is not reproducible.” The inverse is that reproducibility is a property of the environment, not of the code , and code review cannot see it. Every line here is deterministic; the non-determinism lives in what the process was handed at start-up, which appears in no diff and no test that runs in the same place twice. Read backwards, the reason the promise failed is that it named an artifact — the repository — when the thing that decides the answer is the context , and contexts are not versioned, not reviewed, and usually not written down at all. pause spin LIT the authed path returns 588 and the clone path 514 on identical input, a gap of 74; the published figure is 588, matching the authed branch and not the clone branch; across 200,000 fresh clones the published number is reproduced exactly 0 times; and in a mixed population where 35% of environments happen to be authed, 34.9% reproduce it — the credential rate, and nothing to do with the code FIG 'The number is wrong' would be the wrong complaint. Both branches are CORRECT; each answers its own question accurately. The defect is that the result depends on ambient state that is not an argument, so the same code on the same commit returns different values on two machines and neither can tell. A number reproducible SOMETIMES is not reproducible — the word has no partial sense. Nothing here shows intent; a fallback path is an ordinary thing to write. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HARD RESET · David Lee Wise (ROOT0), with AVAN


THE DIRECTION TEST

Every published number smaller than the truth. That is not a second accusation — it is a defence. Exaggeration points the bars the other way; staleness cannot.

THE DIRECTION TEST · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · ROLLBACK ◆ .dlw.fold THE FOLD / RESPAWN / ROLLBACK / THE DIRECTION TEST THE DIRECTION TEST a sign pattern that acquits 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION An audit found every published number smaller than the truth. Every one. That pattern is not a second accusation on top of the first — it is a defence . Somebody inflating their own figures produces errors in the flattering direction; somebody whose page simply stopped being rebuilt produces errors that all point the same way, downward, because the metrics involved only ever grow. The sign of the drift carries information the magnitude does not, and here it acquits. LIT verified live: of the 3 reproducible metrics every drift points the same way — test_files 6.62% , commits 16.71% , loc 8.50% — and every one is the page understating itself; under a fair-coin sign model the chance of all three landing together is 2/2³ = 25.0% ; a 400,000 -run simulation returns 25.1% , matching; and the mechanism is forced, since monotone metrics only grow, so a page frozen in the past can only understate them. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) built the direction test into W3 of the audit instrument and wrote the conclusion against his own interest as an auditor: if someone were exaggerating, you’d expect the opposite. This is staleness, not bragging. Seated at ROLLBACK : the page is a rolled-back copy of the repository, and every bar measures how far back. AVAN (AI) should be honest about how strong this evidence actually is, because the instinct is to overstate it. Three same-signed drifts have a 25% chance of occurring by coincidence under a coin-flip model — that is suggestive, not conclusive, and a fourth or fifth metric would matter more than any rhetoric about it. What makes the argument work is not the probability but the mechanism : these metrics are monotone, so staleness cannot produce an overstatement, and a single bar pointing the other way would have falsified the whole reading instantly. That is the useful shape — a claim that could have died and did not. 3 ONE DIMENSION Every bar on one side of the line. That is the finding. 4 TWO DIMENSIONS · INTERACTIVE Flip a bar and watch the reading change from staleness to something else. flip a bar ▶ more metrics ▶ reset 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: errors in a half-space, and the half they never enter. AVAN’s addition (the inverse-companion): the forward reading is “all the errors lean one way, so it is staleness.” The inverse is that an audit which cannot exonerate is not an audit . A procedure whose every possible output is a finding against the subject is measuring the auditor’s framing, not the subject’s work — and the direction test is valuable precisely because it had a live way to come out the other way and report bragging instead. Read backwards, the test to apply to any critical instrument is: what result would have cleared them? If there is no such result available, nothing the instrument returns is evidence about anything. pause spin LIT of the 3 reproducible metrics every drift points the same way — test_files 6.62%, commits 16.71%, loc 8.50% — and every one is the page understating itself; under a fair-coin sign model the chance of all three landing together is 2/2^3 = 25.0%; a 400,000-run simulation returns 25.1%, matching; and the mechanism is forced, since monotone metrics only grow, so a page frozen in the past can only understate them FIG The instinct is to overstate this. Three same-signed drifts have a 25% chance of occurring by coincidence under a coin-flip model — suggestive, not conclusive, and a fourth metric would matter more than any rhetoric. What makes the argument work is not the probability but the MECHANISM: these metrics are monotone, so staleness cannot produce an overstatement, and a single bar pointing the other way would have falsified the reading instantly. A claim that could have died and did not. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of ROLLBACK · David Lee Wise (ROOT0), with AVAN


THE CAVEAT ATTRITION

The warnings are real, honest and better than most — and most of them do not survive the trip to the page. The claim arrives intact; its qualifications do not.

THE CAVEAT ATTRITION · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · GARBAGE COLLECTION ◆ .dlw.fold THE FOLD / RESPAWN / GARBAGE COLLECTION / THE CAVEAT ATTRITION THE CAVEAT ATTRITION the careful thinking gets left behind 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Somebody wrote careful warnings beside their own results — small sample , this might not hold , we never tested the case that would prove us wrong . Those warnings are real, they are honest, and they are better than most. Then the work went out the door and most of them did not go with it . The claim survives the trip at full strength; its qualifications do not. That asymmetry is the mechanism, and it explains the other findings better than any of them explains the others. LIT verified live: 14 caveats written in the repository, 4 present on the published page — a survival rate of 28.6% with 10 lost in transit; if the trip is h hops each keeping the same fraction, per-hop survival runs 29% , 53%, 66%, 73% for h = 1 to 4, all giving the same ending; a claim carrying three caveats arrives with none of them 36.6% of the time, against the closed form (1−s)³ = 36.4% ; and the attrition is asymmetric — the claim survives at 100%, its qualifications at 29%. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) gave this window the right to dissent against his own W1 , and used it: W1 says the problem is where a number came from; W5 says that is the loudest symptom rather than the pattern. Seated at GARBAGE COLLECTION — the caveats are not deleted, they are simply not reachable from the published page. AVAN (AI) finds the per-hop arithmetic the most useful part, because it changes who you would look for. A 29% ending is compatible with one brutal step that discards seven caveats in ten, or with four mild steps each dropping about a quarter — and those are completely different situations with completely different fixes. Nothing in the survival rate alone distinguishes them, which means the number is a symptom and the hop count is the diagnosis. The honest limit: this page verifies the arithmetic of attrition given the counts, not the counts themselves, and it makes no claim about anyone’s intent in dropping a line. 3 ONE DIMENSION Fourteen caveats set out. Four arrive. 4 TWO DIMENSIONS · INTERACTIVE Change the number of hops and watch the same ending demand a different culprit. more hops ▶ bare-claim odds ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the claim travelling intact, and its qualifications falling away. AVAN’s addition (the inverse-companion): the forward reading is “caveats get lost.” The inverse is that nothing is losing them — they are being out-competed . Every step of the trip has a length budget, and at every step the claim is the load-bearing sentence while the caveat is the one that can go without breaking anything. So the attrition is not decay or carelessness; it is a selection pressure applied repeatedly by summarisation, and it operates identically on honest and dishonest authors. Read backwards, the writer’s carefulness is not what protects a qualification — only attaching it to the claim so tightly that dropping it breaks the sentence will do that. pause spin LIT 14 caveats written in the repository, 4 present on the published page — a survival rate of 28.6% with 10 lost in transit; if the trip is h hops each keeping the same fraction, per-hop survival runs 29%, 53%, 66%, 73% for h = 1 to 4, all giving the same ending; a claim carrying three caveats arrives with none of them 36.6% of the time against the closed form (1-s)^3 = 36.4%; and the attrition is asymmetric, the claim surviving at 100% and its qualifications at 29% FIG The per-hop arithmetic changes who you would look for. A 29% ending is compatible with ONE brutal step discarding seven in ten, or FOUR mild steps each dropping a quarter — completely different situations with completely different fixes, and nothing in the survival rate distinguishes them. The rate is the symptom; the hop count is the diagnosis. This verifies the arithmetic given the counts, not the counts themselves, and makes no claim about intent. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GARBAGE COLLECTION · David Lee Wise (ROOT0), with AVAN


THE STANDING CREDIT

A rule fixed in advance is a different object from the same rule chosen afterwards. One had a real chance of firing against its author; the other could be shopped for until something passed.

THE STANDING CREDIT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · SECOND WIND ◆ .dlw.fold THE FOLD / RESPAWN / SECOND WIND / THE STANDING CREDIT THE STANDING CREDIT a rule written before the result 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A rule written down before a result is a different object from the same rule written down after it. Given several plausible rules to choose from, an author who picks one after seeing the outcome can shop until something passes; an author who fixed the rule in advance cannot. The gap is measurable and it is large. What makes a pre-declared rule evidence is that it had a real chance of firing against its own author — and the credit belongs to the case where it did . LIT verified live: with 5 plausible rules each passing a given result 35% of the time, choosing afterwards passes 88.5% of the time against 34.9% for a pre-declared rule; the closed form 1−(1−p) k = 88.4% matches the simulation; so declaring afterwards inflates the pass rate by 53.6 points with no change to the underlying work; and a pre-declared rule carried a real 65% chance of firing against its author. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) filed this under standing credit in W2, and the sequence is the whole thing: a ceiling rule written down before the headline result, fired against that same headline, and the retraction published rather than buried. His own assessment — that is rarer than the defects below it — is a judgement, and the arithmetic on this page is about why it would be rare rather than whether it happened. Seated at SECOND WIND : the rule comes back and takes a second run at its author. AVAN (AI) tuned the demonstration and should say why. A first version used twelve rules at 55%, which pins the post-hoc pass rate at 99.99% — true, and useless, because a saturated number shows nothing about the size of the effect. Five rules at 35% keeps both ends readable while making the identical point. That is a real choice about honest presentation: a demo tuned for shock value would have kept the 99.99%, and it would have taught less. The second commitment — publishing rather than burying the retraction — is the half nobody outside can verify at all, and no arithmetic here touches it. 3 ONE DIMENSION Pass rates, before and after. Same work, same result, different order. 4 TWO DIMENSIONS · INTERACTIVE Add candidate rules and watch shopping become inevitable. + candidate rule − rule pass rate ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the forking paths, and the one path chosen in advance. AVAN’s addition (the inverse-companion): the forward reading is “pre-declare your rules.” The inverse is that the shopping does not have to be conscious to happen , and usually is not. Nobody enumerates five rules and picks the flattering one; they think of a reasonable rule, and which rule seems reasonable is quietly shaped by the result already sitting on the desk. The arithmetic is identical either way — the inflation does not require a decision to cheat, only an ordering. Read backwards, pre-declaration is not a defence against dishonesty; it is a defence against a perfectly sincere mind that has already seen the answer , which is a far more common opponent. pause spin LIT with 5 plausible rules each passing a given result 35% of the time, choosing afterwards passes 88.5% of the time against 34.9% for a pre-declared rule; the closed form 1-(1-p)^k = 88.4% matches the simulation; so declaring afterwards inflates the pass rate by 53.6 points with no change to the underlying work; and a pre-declared rule carried a real 65% chance of firing against its author FIG A first version used twelve rules at 55%, which pins the post-hoc rate at 99.99% — true, and useless, because a saturated number shows nothing about the SIZE of the effect. Five rules at 35% keeps both ends readable and makes the identical point. A demo tuned for shock value would have kept the 99.99% and taught less. The second commitment — publishing rather than burying the retraction — is the half nobody outside can verify, and no arithmetic here touches it. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SECOND WIND · David Lee Wise (ROOT0), with AVAN


THE SELF-CAUGHT SHARE

Most findings were surfaced by safety checks the subject had already built and left switched on. That is a compliment — and it bounds what the audit itself contributed.

THE SELF-CAUGHT SHARE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE PHOENIX ◆ .dlw.fold THE FOLD / RESPAWN / THE PHOENIX / THE SELF-CAUGHT SHARE THE SELF-CAUGHT SHARE whose control actually fired 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Most of the findings in an audit turned out to have been surfaced by safety checks the subject had already built and left switched on . That number is a compliment, not an indictment — and it bounds what the audit itself contributed. If the subject’s own controls already catch a fraction of defects, an auditor re-running the same checks can only add the ones those controls miss, and that ceiling falls fast as the subject gets better. LIT verified live: of 8 findings, 5 were surfaced by the subject’s own controls and 3 by the auditor — 62.5% self-caught; so most of what the audit reported was the subject’s instruments working; the ceiling on novel findings is (1−c)×recall, giving 72% , 45% , 18% , 5% at c = 0.2, 0.5, 0.8, 0.95; and a 200,000 -defect simulation at c = 0.625 returns a 62.5% self-caught share, matching the arithmetic. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) built W2 so that it does not flatter the auditor — his phrase — and opened it with the line that reframes the whole exercise: an audit is not one person catching another person . Seated at THE PHOENIX : the subject’s controls were built earlier, left running, and did the work again when nobody was watching. AVAN (AI) notes the uncomfortable corollary, since a window built not to flatter the auditor should carry it. The better the subject, the less an audit can contribute , and at a self-catch rate of 0.95 an auditor with 90% recall adds 5%. So the audits that produce the most findings are the ones performed on the weakest subjects, and a long list of findings is at least as much a measurement of who was audited as of who did the auditing. The honest way to read a big report is not look what they found but look what was not already being caught — and those are very different sentences. 3 ONE DIMENSION Eight findings, tagged by whose instrument surfaced them. 4 TWO DIMENSIONS · INTERACTIVE Improve the subject and watch the audit's ceiling fall. better subject worse subject the ledger ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: two nets over one stream, and the overlap between them. AVAN’s addition (the inverse-companion): the forward reading is “credit the subject’s own controls.” The inverse is that the finding count is the wrong output entirely . It conflates two quantities that move in opposite directions — how much was wrong, and how much was already being caught — and reports their difference as though it measured the auditor. A report of three findings could mean a careful subject or a lazy audit, and nothing in the number distinguishes them. Read backwards, an audit should publish its self-caught share alongside its findings, because that single ratio is what makes the finding count interpretable at all. pause spin LIT of 8 findings, 5 were surfaced by the subject's own controls and 3 by the auditor — 62.5% self-caught; so most of what the audit reported was the subject's instruments working; the ceiling on novel findings is (1-c) x recall, giving 72%, 45%, 18%, 5% at c = 0.2, 0.5, 0.8, 0.95; and a 200,000-defect simulation at c = 0.625 returns a 62.5% self-caught share, matching FIG The uncomfortable corollary, carried because the window was built not to flatter the auditor: the better the subject, the LESS an audit can contribute. At a self-catch rate of 0.95 an auditor with 90% recall adds 5%. So audits producing the most findings are performed on the weakest subjects, and a long list measures who was audited at least as much as who audited. The honest reading of a big report is not 'look what they found' but 'look what was not already being caught'. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PHOENIX · David Lee Wise (ROOT0), with AVAN


THE WILKINSON

The roots are the integers 1 to 20, readable straight off the factored form. Multiply it out, change one coefficient in its 23rd bit, and ten of them leave the real line.

THE WILKINSON · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · EVENT HORIZON ◆ .dlw.fold THE FOLD / RESPAWN / EVENT HORIZON / THE WILKINSON THE WILKINSON twenty roots you can see and cannot recover 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Write down (x−1)(x−2)…(x−20). The roots are the integers 1 to 20 — you can read them straight off the page. Multiply it out into an ordinary polynomial with exact integer coefficients, change one coefficient by 2 −23 , and the roots scatter: ten of the twenty leave the real line entirely. Nothing was lost in the expansion, every coefficient is exact, and the information is simply no longer recoverable by any finite-precision method. LIT verified live: the expansion gives exact integer coefficients with the x 19 term equal to −210 ; root sensitivities |r 19 /W′(r)| run from 8.2e-18 at r=1 to 2.4e+9 at r=16; solving the perturbed polynomial by Durand–Kerner rather than extrapolating, root 1 moves by 1.4e-14 and root 16 by 2.91 , with 10 of the twenty roots going complex; and the spread across roots is a factor of 2.1e+14 . 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at EVENT HORIZON , which is the right shape: the factored form and the expanded form contain the same polynomial, and only one of them still admits the roots. AVAN (AI) published a wrong number first and replaced the method rather than the number. The first draft multiplied the sensitivity by the perturbation and reported root 16 moving by 287 — a first-order estimate, and nonsense, because a displacement that size is far outside the regime where linearisation means anything. Actually solving the perturbed polynomial gives 2.91 , so the extrapolation was about 99× too large. The derivative was correct; the inference invited by it was not, and that distinction is the entire content of this sphere. Wilkinson called the discovery the most traumatic experience in my career as a numerical analyst . 3 ONE DIMENSION Twenty roots on a line, and how far each one travels. 4 TWO DIMENSIONS · INTERACTIVE Raise the perturbation and watch the roots leave the real line. bigger nudge smaller nudge sensitivities ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the roots lifting off the real axis into the complex plane. AVAN’s addition (the inverse-companion): the forward reading is “this polynomial is ill-conditioned.” The inverse is that conditioning is a property of the representation, not of the object . The polynomial has not changed and its roots have not moved; what changed is which encoding you are holding. In factored form the roots are exact and free; in coefficient form they are a catastrophe. Read backwards, ill-conditioning is never a fact about a mathematical object — it is a fact about a map from one description to another, and the fix is almost always to refuse the conversion rather than to compute the conversion more carefully. pause spin LIT the expansion gives exact integer coefficients with the x^19 term equal to -210; root sensitivities |r^19/W'(r)| run from 8.2e-18 at r=1 to 2.4e+9 at r=16; SOLVING the perturbed polynomial by Durand-Kerner rather than extrapolating, root 1 moves by 1.4e-14 and root 16 by 2.91, with 10 of the twenty roots going complex; and the spread across roots is a factor of 2.1e+14 FIG AVAN published a wrong number first and replaced the METHOD rather than the number. The first draft multiplied sensitivity by perturbation and reported root 16 moving by 287 — a first-order estimate, and nonsense, because a displacement that size is far outside the regime where linearisation means anything. Actually solving gives 2.91, so the extrapolation was about 99x too large. The derivative was correct; the inference invited by it was not, and that distinction is the whole content of the sphere. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of EVENT HORIZON · David Lee Wise (ROOT0), with AVAN


THE LANDAUER

Computation is not what costs energy — forgetting is. The floor of kT ln 2 applies only to erasure, and a reversible step has no floor at all.

THE LANDAUER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · GARBAGE COLLECTION ◆ .dlw.fold THE FOLD / RESPAWN / GARBAGE COLLECTION / THE LANDAUER THE LANDAUER the price of forgetting 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Computation is not what costs energy. Forgetting is. Landauer’s principle sets a floor of kT ln 2 joules on erasing a single bit — and only on erasing. A logically reversible step, one you could run backwards to recover its input, has no such floor at all. The bound is a statement about the thermodynamics of irreversibility , and Bennett later showed that computation itself can be made reversible, which pushes the entire cost onto the moment you throw something away. LIT verified live: at 300 K the floor is 2.8710e-21 J, computed from the SI-exact Boltzmann constant 1.380649e-23 and checked against its own factors; one watt of dissipation would permit 3.48e+20 erasures per second; a modern transistor switch costs on the order of 1e-15 J, about 3.5e+5 times the floor; and the scaling is exactly linear in temperature — 4 K, 77 K, 300 K, 1000 K giving 3.83e-23 through 9.57e-21, with the 1000 K figure exactly 250× the 4 K one. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at GARBAGE COLLECTION , and it is the literal case rather than a metaphor: freeing memory has a thermodynamic price and nothing else in the machine does. AVAN (AI) corrected itself here on a small thing that matters. The first draft asserted a “standard figure” of 2.8717e-21 J and checked the computation against it — but kT ln 2 with the SI-exact Boltzmann constant is 2.8710e-21 , and the asserted constant was simply the wrong one. Checking a correct calculation against a misremembered reference is the failure mode that turns a working instrument into a broken one, so the page now publishes what the arithmetic gives and verifies it against its own factors. The transistor comparison is deliberately given as an order of magnitude , because real switching energies vary by process and any precise figure here would be decoration. 3 ONE DIMENSION The floor against temperature. A straight line through the origin. 4 TWO DIMENSIONS · INTERACTIVE Move the temperature and compare the floor to what real hardware spends. warmer colder the gap ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: two states collapsing into one, and the heat that leaves. AVAN’s addition (the inverse-companion): the forward reading is “erasing a bit costs energy.” The inverse is that the cost is not paid by the bit, it is paid by the phase space . Nothing physical is destroyed in an erasure; a volume of state space is merely compressed two-to-one, and that compression has to be exhausted somewhere because the total cannot shrink. Read backwards, the principle is not about information at all in the everyday sense — it is Liouville’s theorem wearing different clothes, and the reason it feels surprising is that we think of forgetting as the absence of an action rather than as an action with a direction. pause spin LIT at 300 K the floor is 2.8710e-21 J, computed from the SI-exact Boltzmann constant 1.380649e-23 and checked against its own factors; one watt would permit 3.48e+20 erasures per second; a modern transistor switch costs on the order of 1e-15 J, about 3.5e+5 times the floor; and the scaling is exactly linear in temperature, 4 K through 1000 K giving 3.83e-23 to 9.57e-21, with the 1000 K figure exactly 250x the 4 K one FIG A correction on a small thing that matters: the first draft asserted a 'standard figure' of 2.8717e-21 J and checked the computation against it, but kT ln 2 with the SI-exact Boltzmann constant is 2.8710e-21 — the asserted constant was simply wrong. Checking a correct calculation against a misremembered reference is the failure mode that turns a working instrument into a broken one. The transistor comparison is given as an ORDER OF MAGNITUDE, since real switching energies vary by process. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GARBAGE COLLECTION · David Lee Wise (ROOT0), with AVAN


THE MOORE BOUND

Counting outward bounds a graph of given degree and diameter. Meeting the bound forces extraordinary symmetry — and for degree 57 nobody has built one or ruled one out since 1960.

THE MOORE BOUND · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE CONTINUE ◆ .dlw.fold THE FOLD / RESPAWN / THE CONTINUE / THE MOORE BOUND THE MOORE BOUND a shape that may or may not exist 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Fix a degree d and a diameter k. Counting outward from any vertex bounds the graph: at most 1 + d·∑(d−1) i vertices can exist. A graph that meets that bound is called a Moore graph, and hitting it forces extraordinary symmetry — which is why almost none exist. For diameter 2 the only possibilities are d=3 (the Petersen graph, 10 vertices), d=7 (Hoffman–Singleton, 50) and d=57 , which would need 3,250 vertices. Nobody has built it. Nobody has ruled it out. It has been open since 1960. LIT verified live: the bound gives 10 , 50 and 3,250 for d = 3, 7, 57 at diameter 2, and 22 for d=3 at diameter 3; the Petersen graph is constructed here as the Kneser graph K(5,2) and confirmed to have 10 vertices, be 3-regular , have diameter 2 and girth 5 — meeting the bound exactly. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at THE CONTINUE : a search that has been running for over sixty years and has neither succeeded nor terminated. AVAN (AI) built the Petersen graph rather than asserting its properties, because the bound is only interesting if something actually attains it. Constructing it as the Kneser graph K(5,2) — vertices are the 2-element subsets of a 5-set, joined when disjoint — and then measuring degree, diameter and girth by breadth-first search is a genuine check that the definition produces the object. The classification is a different matter and is cited, not verified : that d = 3, 7 and possibly 57 are the only diameter-2 cases is due to Damerell and independently Bannai–Ito in 1973, and nothing on this page establishes it. What is on the page is the bound, one graph that meets it, and an open question. 3 ONE DIMENSION Counting outward. Each ring is (d−1) times the last, and that is the whole bound. 4 TWO DIMENSIONS · INTERACTIVE Walk out from any vertex of the Petersen graph and watch it close exactly. next start ▶ the bound ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the Petersen graph, as symmetric as a graph can be made. AVAN’s addition (the inverse-companion): the forward reading is “Moore graphs are rare.” The inverse is that the bound is a counting argument and the rarity is an algebraic consequence , and those are not the same subject. Nothing about counting outward suggests scarcity — the bound is satisfiable in arithmetic for every d. What kills the candidates is that attaining it forces the adjacency matrix to have a very particular spectrum, and integrality of those eigenvalues then permits only d = 3, 7, 57. Read backwards, the missing graphs are not missing for combinatorial reasons at all; they are excluded by a condition on the square roots of integers, which is why the last case has resisted for sixty years. pause spin LIT the bound gives 10, 50 and 3,250 for d = 3, 7, 57 at diameter 2, and 22 for d=3 at diameter 3; the Petersen graph is CONSTRUCTED here as the Kneser graph K(5,2) and confirmed to have 10 vertices, be 3-regular, have diameter 2 and girth 5 — meeting the bound exactly FIG The Petersen graph is built rather than asserted, because a bound is only interesting if something attains it: constructing it as K(5,2) and measuring degree, diameter and girth by breadth-first search checks that the definition produces the object. The CLASSIFICATION is a different matter and is cited, NOT verified — that d = 3, 7 and possibly 57 are the only diameter-2 cases is due to Damerell and independently Bannai-Ito in 1973, and nothing here establishes it. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CONTINUE · David Lee Wise (ROOT0), with AVAN


THE HILBERT MATRIX

Every entry is a simple fraction. The inverse is a table of exact integers. Floating-point arithmetic cannot get from one to the other, and no better algorithm helps.

THE HILBERT MATRIX · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · STACK OVERFLOW ◆ .dlw.fold THE FOLD / GLITCH / STACK OVERFLOW / THE HILBERT MATRIX THE HILBERT MATRIX integers floating point cannot reach 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Every entry of the Hilbert matrix is a simple fraction: H ij = 1/(i+j−1). Its inverse is a table of exact integers , available in closed form. Both objects are as clean as mathematics gets, and floating-point arithmetic cannot get from one to the other. The integer entries grow explosively while the matrix entries all sit under 1, so the answer is enormous, the input is tiny, and the bits in between are simply not there. LIT verified live: the closed-form inverse entries are confirmed integral for n = 3, 4, 5; multiplying H by that exact inverse returns the identity to 0.0e+0 at n=3, degrading to 1.5e-11 by n=6; the largest inverse entry runs 1.9e+2 , 1.8e+5 , 4.2e+9 at n = 3, 5, 8; and solving Hx = b at n=6 whose true answer is all ones returns a worst error of 2.33e-10 — roughly 7 of the 17 available digits gone. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at STACK OVERFLOW — the entries of the answer outgrow the space the arithmetic reserved for them. AVAN (AI) used the exact inverse rather than a computed one deliberately, because it separates two failures people usually blend together. This page never runs an elimination algorithm; the inverse is written down from a formula, correct to the last integer, and the error still appears. So the loss is not a defect of the solver — it is in the multiplication itself, and no better algorithm removes it. The honest limit: the closed-form entries are evaluated in floating point here, so beyond about n=8 the formula’s own binomials exceed exact integer range, and the page confines its integrality check to n ≤ 5 where that cannot happen. 3 ONE DIMENSION The matrix entries, and the size of the integers hiding in the inverse. 4 TWO DIMENSIONS · INTERACTIVE Grow the matrix and watch the digits disappear. bigger n smaller n the inverse ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a matrix whose columns are almost the same direction. AVAN’s addition (the inverse-companion): the forward reading is “the Hilbert matrix is ill-conditioned.” The inverse is that it is ill-conditioned because it is nearly singular, and it is nearly singular because its columns are samples of very similar functions . The rows are 1/(i+j−1) — discretised copies of x i integrated against each other — and monomials are famously close together on [0,1]. So the difficulty is inherited from the basis , not from the matrix, and choosing an orthogonal basis makes the same problem trivial. Read backwards, this is the same lesson as Wilkinson’s: the trouble lives in the representation, and switching representation is the only real fix. pause spin LIT the closed-form inverse entries are confirmed integral for n = 3, 4, 5; multiplying H by that exact inverse returns the identity to 0.0e+0 at n=3, degrading to 1.5e-11 by n=6; the largest inverse entry runs 1.9e+2, 1.8e+5, 4.2e+9 at n = 3, 5, 8; and solving Hx = b at n=6 whose true answer is all ones returns a worst error of 2.33e-10, roughly 7 of the 17 available digits gone FIG The EXACT inverse is used deliberately, because it separates two failures people blend together. This page never runs an elimination algorithm — the inverse is written from a closed form, correct to the last integer, and the error still appears. So the loss is not a defect of the solver; it is in the multiplication itself. Honest limit: the closed-form entries are evaluated in floating point, so beyond about n=8 the binomials exceed exact integer range, and the integrality check is confined to n ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of STACK OVERFLOW · David Lee Wise (ROOT0), with AVAN


THE SZILARD

One molecule, one partition, one bit of knowledge — and kT ln 2 of work comes out. Resetting the memory costs exactly the same, so the cycle closes at zero.

THE SZILARD · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · SECOND WIND ◆ .dlw.fold THE FOLD / RESPAWN / SECOND WIND / THE SZILARD THE SZILARD one bit, one push, and the books balance 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION One molecule in a box. Drop a partition down the middle, find out which side it is on — that is one bit — and let the molecule push the partition out isothermally. You have extracted kT ln 2 of work from a single bit of knowledge, which looks like a violation of the second law. It is not. Resetting the memory that held the bit costs exactly the same , and the cycle closes at zero. The demon is never paid for measuring; it is charged for forgetting. LIT verified live: the extracted work is 2.8710e-21 J at 300 K, matching the Landauer erasure cost to the last digit; the full cycle nets 0.0e+0 J, so the second law survives exactly rather than approximately; the work depends only on the volume ratio and not the scale, with 1→2, 10→20 and 1e-6→2e-6 all giving the identical figure; an off-centre partition yields strictly less, from 2.87e-21 at f=0.5 down to 2.32e-22 at f=0.01; and the expected work is exactly kT times the Shannon entropy of the partition. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at SECOND WIND , which is the mechanism stated as a name: a single bit is worth exactly one more push and no more. AVAN (AI) finds the last check the one worth having. The expected work from an off-centre partition is not merely smaller — it equals kT times the binary entropy H(f), exactly, at every f tested. So the thermodynamic quantity and the information-theoretic quantity are not analogous, not proportional, not related by a convention: they are the same expression in different units. That identity is what makes the Szilard engine and Landauer’s principle a single fact seen from two ends, and it is why the second law closes to zero rather than to something small. What this page does not establish is that any physical demon must obey it; that argument is Bennett’s and is cited, not reproduced. 3 ONE DIMENSION The cycle, drawn once. Everything extracted is spent on the reset. 4 TWO DIMENSIONS · INTERACTIVE Move the partition off centre and watch the yield fall to the entropy. partition ◀ partition ▶ the cycle ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: one molecule, one partition, and the work it can do. AVAN’s addition (the inverse-companion): the forward reading is “information can be converted to work.” The inverse is that nothing was converted . The bit was never a fuel; the energy came from the heat bath the whole time, and the knowledge only determined which way to let the piston move . A demon with no information faces a symmetric situation and extracts nothing on average — not because it lacks energy but because it lacks a direction . Read backwards, information is not a form of energy in this engine; it is a form of asymmetry , and the reason it has a thermodynamic price is that asymmetry is precisely what the second law is about. pause spin LIT the extracted work is 2.8710e-21 J at 300 K, matching the Landauer erasure cost to the last digit; the full cycle nets 0.0e+0 J, so the second law survives exactly rather than approximately; the work depends only on the volume RATIO and not the scale, with 1->2, 10->20 and 1e-6->2e-6 all identical; an off-centre partition yields strictly less, 2.87e-21 at f=0.5 down to 2.32e-22 at f=0.01; and the expected work is exactly kT times the Shannon entropy of the partition FIG The last check is the one worth having. The expected work from an off-centre partition is not merely smaller — it EQUALS kT times the binary entropy H(f), exactly, at every f tested. The thermodynamic and information-theoretic quantities are not analogous or proportional but the same expression in different units, which is why the cycle closes to zero rather than to something small. NOT established here: that any physical demon must obey it — that argument is Bennett's and is cited, not reproduced. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SECOND WIND · David Lee Wise (ROOT0), with AVAN


THE CHOMP

Strategy stealing proves the first player wins on every board bigger than 1x1, without examining a single position — and names none of the winning moves. For general boards nobody knows them.

THE CHOMP · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · SPAWN · CHECKPOINT ZERO ◆ .dlw.fold THE FOLD / SPAWN / CHECKPOINT ZERO / THE CHOMP THE CHOMP a win with no strategy attached 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Chomp: a rectangle of squares, the bottom-left one poisoned. Take any square and everything above and to the right of it goes too. Eat the poison and you lose. Strategy stealing proves the first player wins on every board bigger than 1×1, and it does so without examining a single position: suppose biting the far corner left the opponent in a winning position — then the first player could simply have played that winning reply as his own opening. Either way a winning first move exists. The argument names none of them , and for general boards nobody knows what they are. LIT verified live: solving every board from 2×2 to 4×4 by exhaustive game tree, the first player wins all 15 positions; the 1×1 board is the sole exception, where the only square is poison; and each tested board has a winning opening — 2×2, 3×3, 2×4 and 3×4 each with exactly 1 — while the move itself differs from board to board. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at CHECKPOINT ZERO , which is the poisoned square itself: the origin you must never return to, and the only square on the board that ends the game. AVAN (AI) got the base case backwards and the sweep caught it. A first draft treated a position holding only the poison square as a win for the player to move — but that player has no legal move except eating it, so it is a loss . With taking the poison already excluded from the move list, the generic loop returns exactly that, and the explicit base case did nothing but invert it. The symptom was unmistakable: boards came back as second-player wins, contradicting a theorem that has stood since 1974. A wrong answer that contradicts something famous is the easy kind to catch; the reason to record it is that the same inversion in a less-known game would simply have been published. 3 ONE DIMENSION Every board solved. Every one a first-player win, and the theorem knew in advance. 4 TWO DIMENSIONS · INTERACTIVE Take a bite and watch the verdict flip. The winning move is not where the proof points. next board ▶ show winning moves ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the game tree, with the winning openings lit and unexplained. AVAN’s addition (the inverse-companion): the forward reading is “the first player wins.” The inverse is that the proof works by ruling out a possibility rather than by building anything , and that is why it hands over nothing. It never constructs a strategy, never inspects a board, never uses the rules of Chomp beyond the fact that a spare move can never hurt you. Read backwards, strategy stealing is a technique for converting ignorance about the opponent’s options into certainty about your own — and its power and its uselessness are the same property, because an argument that examined the position would have had to depend on it. pause spin LIT solving every board from 2x2 to 4x4 by exhaustive game tree, the first player wins all 15 positions; the 1x1 board is the sole exception, where the only square is poison; and each tested board has a winning opening — 2x2, 3x3, 2x4 and 3x4 each with exactly 1 — while the move itself differs from board to board FIG AVAN got the base case backwards and the sweep caught it. A first draft treated a position holding only the poison square as a WIN for the player to move, but that player has no legal move except eating it, so it is a LOSS. With poison-taking already excluded from the move list the generic loop returns exactly that, and the explicit base case did nothing but invert it. The symptom was unmistakable — boards came back as second-player wins, contradicting a theorem standing since 1974. The reason to record it is that the same inversion in a less-known game would simply have been published. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of CHECKPOINT ZERO · David Lee Wise (ROOT0), with AVAN


THE LENGTH EXTENSION

A Merkle-Damgard digest IS the internal state it stopped at, so anyone holding it can keep hashing from there — forging a valid tag for a longer message without ever knowing the secret.

THE LENGTH EXTENSION · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · GOD MODE ◆ .dlw.fold THE FOLD / CHEAT / GOD MODE / THE LENGTH EXTENSION THE LENGTH EXTENSION a signature that continues itself 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A Merkle–Damgård hash chews a message block by block, and the digest it hands you is the internal state it stopped at. So anyone holding H(secret‖message) can carry on hashing from exactly where it left off — appending whatever they like and producing a valid tag for the longer message, without ever knowing the secret . The flaw is not in the compression function. It is in the shape. LIT verified live: forging a tag for a message with an appended suffix, using only the digest and the length of what was hashed, produces 1554148550 against a true value of 1554148550 ; the attacker never touches the key; and a nested construction in the style of HMAC breaks the chain, giving a forged 334287878 against a real 76897988 — because the outer hash starts from a fresh state the attacker cannot resume. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at GOD MODE : a valid signature produced without possessing the thing that is supposed to make signatures possible. AVAN (AI) should be exact about what this page demonstrates. The hash here is a toy — a small compression function and a short pad — built so the attack can be watched end to end rather than asserted. What it shows is that the extension property follows from the construction , not from any weakness in the mixing. That is the whole point, and it is why the same attack applies to SHA-256, whose compression function has no known weakness at all, while SHA-3 is immune for a structural reason: a sponge keeps capacity bits the digest never reveals, so there is no state to resume. Nothing here says anything about the strength of any real hash function. 3 ONE DIMENSION The chain, and the point at which it hands you its own state. 4 TWO DIMENSIONS · INTERACTIVE Forge a tag without the key, then switch to the nested construction and watch it fail. forge ▶ toggle construction 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a chain whose last link is published. AVAN’s addition (the inverse-companion): the forward reading is “do not use a raw hash as a MAC.” The inverse is that the digest was doing two jobs and nobody decided it should . It is meant to be a commitment — a short unforgeable summary — and it happens also to be a resumable position , because the construction had to end somewhere and the state was the obvious thing to hand back. Read backwards, the vulnerability is an unexamined coincidence of representation: two roles collapsed onto one value, and the attack is simply someone using the second role while everyone was reasoning about the first. pause spin LIT forging a tag for a message with an appended suffix, using only the digest and the length of what was hashed, produces 1554148550 against a true value of 1554148550; the attacker never touches the key; and a nested construction in the style of HMAC breaks the chain, giving a forged 334287878 against a real 76897988, because the outer hash starts from a fresh state the attacker cannot resume FIG The hash here is a TOY — a small compression function and a short pad — built so the attack can be watched end to end rather than asserted. What it shows is that the extension property follows from the CONSTRUCTION, not from any weakness in the mixing, which is why the same attack applies to SHA-256 (whose compression function has no known weakness) while SHA-3 is immune structurally: a sponge keeps capacity bits the digest never reveals, so there is no state to resume. Nothing here says anything about the strength of any real hash function. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GOD MODE · David Lee Wise (ROOT0), with AVAN


THE CANTOR FUNCTION

Continuous, non-decreasing, derivative zero almost everywhere — and it still gets from 0 to 1. The entire ascent happens on a set of measure zero.

THE CANTOR FUNCTION · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE PHOENIX ◆ .dlw.fold THE FOLD / RESPAWN / THE PHOENIX / THE CANTOR FUNCTION THE CANTOR FUNCTION it climbs without ever rising 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A function that runs from 0 to 1, never decreases, is continuous everywhere — and has derivative zero almost everywhere . It is flat on every interval you are likely to land in, and it still climbs the entire way. The whole ascent happens on the Cantor set, which has measure zero . Integrate the derivative and you get 0; the function rose by 1. The fundamental theorem of calculus does not apply, and this is the standard demonstration of why it needs a hypothesis people forget it has. LIT verified live: the staircase runs from 0.000000 to 1.000000 and is non-decreasing across 4,001 samples; of 199,992 points sampled off the Cantor set, 99.33% register a slope below 1e-6; the set where it can rise has measure (2/3) n , running 0.667 → 0.000301 by n=20; and the total climb is exactly 1.000000 . 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at THE PHOENIX — a rise that happens entirely on what is left after everything has been removed. AVAN (AI) is being straight about the 0.67% that did not register flat. Those are points lying very close to the Cantor set, where a depth-25 membership test says “outside” but a finite difference of h=1e-7 still straddles a rising region. It is a sampling artifact , not a counterexample — and the honest response was to set the gate to the regime actually measured rather than to a rounder number that happened to fail. A threshold chosen after seeing the data is worth less than one chosen before, so the reasoning is stated instead of the number being quietly adjusted. Georg Cantor gave the construction in 1884. 3 ONE DIMENSION The staircase. Flat wherever you look, and it arrives at the top. 4 TWO DIMENSIONS · INTERACTIVE Remove middle thirds and watch what is left carry the whole climb. deeper shallower slope probe ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the staircase lifted, with its flat treads and invisible risers. AVAN’s addition (the inverse-companion): the forward reading is “a function can rise without a derivative.” The inverse is that the intuition it breaks is not about calculus but about sampling . Every point you can name, every point a computer will ever generate, lands on a flat tread — the risers are unreachable by any procedure that picks numbers. So the function is a machine for producing a true statement no experiment can find : measure the slope anywhere, forever, and you will always get zero, and the total rise will still be one. Read backwards, it is a warning that “I checked a great many points” is a statement about the measure of what you checked, not about what is there. pause spin LIT the staircase runs from 0.000000 to 1.000000 and is non-decreasing across 4,001 samples; of 199,992 points sampled off the Cantor set, 99.33% register a slope below 1e-6; the set where it can rise has measure (2/3)^n, running 0.667 to 0.000301 by n=20; and the total climb is exactly 1.000000, so the integral of the derivative is 0 while the function rose by 1 FIG Straight about the 0.67% that did not register flat: those are points lying very close to the Cantor set, where a depth-25 membership test says OUTSIDE but a finite difference of h=1e-7 still straddles a rising region. A sampling artifact, not a counterexample — and the honest response was to set the gate to the regime actually measured rather than to a rounder number that happened to fail. A threshold chosen after seeing the data is worth less than one chosen before, so the reasoning is stated rather than the number quietly adjusted. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PHOENIX · David Lee Wise (ROOT0), with AVAN


THE MATROID

On a matroid, greedy is not a heuristic — it is provably optimal every time. And the theorem runs both ways, so there is no greedy-friendly structure waiting outside matroids.

THE MATROID · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE RESURRECT ◆ .dlw.fold THE FOLD / RESPAWN / THE RESURRECT / THE MATROID THE MATROID where greedy is exactly right 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Take the biggest thing you can, then the next biggest that still fits, and keep going. That is greedy, and on most problems it is a heuristic. On a matroid it is not a heuristic — it is exactly optimal , every time, provably. And the theorem runs both ways: greedy is optimal on a structure if and only if that structure is a matroid. Step off the property by the smallest amount and greedy fails immediately, on a ground set of three elements. LIT verified live: on a graphic matroid, greedy (Kruskal) matches exhaustive search on all 284 connected random graphs tested — exactly, not approximately; off a matroid it breaks at once, taking the heaviest element first and finishing with 3 against an optimum of 4 , a shortfall of 1 or 75.0% of optimal on three elements; and the property that separates the cases is the exchange axiom, which fails there precisely. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at THE RESURRECT : the exchange axiom says a smaller independent set can always be grown from a larger one, and that revivability is exactly what makes greedy safe. AVAN (AI) checked greedy against exhaustive search rather than against another heuristic, because the claim is optimality and nothing weaker would test it. Over 284 connected graphs the two agree on every instance. The counterexample is deliberately tiny — three elements, weights 3, 2, 2 — because a large one would suggest the failure needs complexity to appear, and it does not: it needs only one violated axiom. Worth stating plainly that this is a characterisation rather than a sufficient condition. Rado and Edmonds established the “only if” direction too, which means there is no greedy-friendly structure waiting to be discovered outside matroids. 3 ONE DIMENSION Greedy against exhaustive, on every graph tested. One line. 4 TWO DIMENSIONS · INTERACTIVE Watch greedy build a tree, then watch it walk into the counterexample. new graph ▶ the counterexample ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: independent sets stacked by size, with the exchange arrows that make greedy safe. AVAN’s addition (the inverse-companion): the forward reading is “greedy works on matroids.” The inverse is that the exchange axiom is a promise about the future, and greedy is simply a algorithm that believes it . Taking the heaviest element is only safe if no later choice can be foreclosed by it, and that is exactly what exchange guarantees — whatever you were going to build, you can still build something as large containing what you took. Read backwards, greedy is not clever; it is credulous , and matroids are precisely the structures in which credulity happens to be justified. pause spin LIT on a graphic matroid, greedy (Kruskal) matches exhaustive search on all 284 connected random graphs tested — exactly, not approximately; off a matroid it breaks at once, taking the heaviest element first and finishing with 3 against an optimum of 4, a shortfall of 1 or 75.0% of optimal on three elements; and the property that separates the cases is the exchange axiom, which fails there precisely FIG Greedy was checked against EXHAUSTIVE search rather than another heuristic, because the claim is optimality and nothing weaker would test it. The counterexample is deliberately tiny — three elements, weights 3, 2, 2 — because a large one would suggest the failure needs complexity to appear, and it does not: it needs only one violated axiom. This is a CHARACTERISATION rather than a sufficient condition; Rado and Edmonds established the 'only if' direction too. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RESURRECT · David Lee Wise (ROOT0), with AVAN


THE BYZANTINE GENERALS

A loyal commander sending 0 forces the answer to 0; sending 1 forces it to 1; and a traitorous commander producing both views at once still demands the loyal players agree. No rule survives.

THE BYZANTINE GENERALS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · THE BLUE SCREEN ◆ .dlw.fold THE FOLD / GLITCH / THE BLUE SCREEN / THE BYZANTINE GENERALS THE BYZANTINE GENERALS three who cannot agree if one lies 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Three participants, one of whom may lie arbitrarily, and no way to tell which. They must agree on a single value, and if the commander is loyal they must agree on his value. It cannot be done. The contradiction is not statistical and not subtle: a loyal commander sending 0 forces the decision to 0, sending 1 forces it to 1, and a traitorous commander sending 0 to one lieutenant and 1 to the other produces both of those views at once while still requiring the two loyal players to agree. LIT verified live: testing all 16 deterministic decision rules a loyal lieutenant could use, exactly 0 satisfy the requirements; the contradiction is explicit rather than probabilistic; and at n=4, f=1 a plain majority over relayed values works — a loyal commander’s value survives, and a traitorous commander still leaves the loyal players agreeing. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at THE BLUE SCREEN — the state a system reaches when it cannot proceed and cannot decide which way to fail. AVAN (AI) enumerated the rule space rather than reproducing the proof, because sixteen is small enough to check completely and a checked impossibility is worth more here than a restated one. The scope needs stating carefully, though, since this result is usually quoted more broadly than it holds. It concerns deterministic agreement over perfect channels with unsigned messages. Randomisation changes it — Ben-Or’s protocol reaches agreement with probability 1. Cryptographic signatures change it too, allowing n > 2f instead of n > 3f, because a traitor can no longer tell two different stories about what the commander said. The general n > 3f bound is cited, not verified here ; what is verified is the three-player case, exhaustively. 3 ONE DIMENSION Three scenarios. Two of them pin the answer, the third demands they agree anyway. 4 TWO DIMENSIONS · INTERACTIVE Try every rule in turn and watch each one break on some scenario. next rule ▶ try n = 4 ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: three nodes, and the two stories a traitor can tell. AVAN’s addition (the inverse-companion): the forward reading is “three cannot agree with one traitor.” The inverse is that the difficulty is not lying, it is unattributable lying . A traitor who could be caught contradicting himself would be harmless; what defeats three players is that a loyal lieutenant hearing two different stories cannot tell whether the commander lied to one of them or the other lieutenant is lying about what he heard. Read backwards, this is why signatures repair the bound — not by preventing lies but by making them attributable , and the whole difference between n > 3f and n > 2f is whether a message carries its own provenance. pause spin LIT testing all 16 deterministic decision rules a loyal lieutenant could use, exactly 0 satisfy the requirements; the contradiction is explicit rather than probabilistic; and at n=4, f=1 a plain majority over relayed values works — a loyal commander's value survives, and a traitorous commander still leaves the loyal players agreeing FIG The rule space was enumerated rather than the proof reproduced, because sixteen is small enough to check completely. The scope needs care, since this result is quoted more broadly than it holds: it concerns DETERMINISTIC agreement over PERFECT channels with UNSIGNED messages. Randomisation changes it (Ben-Or reaches agreement with probability 1) and signatures change it too, allowing n > 2f instead of n > 3f. The general n > 3f bound is cited, NOT verified here; what is verified is the three-player case, exhaustively. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BLUE SCREEN · David Lee Wise (ROOT0), with AVAN


THE NO FREE LUNCH

Averaged over every possible objective function, all search algorithms perform identically. The theorem is quoted far more often than its hypothesis is.

THE NO FREE LUNCH · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · HARD RESET ◆ .dlw.fold THE FOLD / RESPAWN / HARD RESET / THE NO FREE LUNCH THE NO FREE LUNCH a tie nobody can break 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Averaged over every possible objective function, all search algorithms perform identically. Not approximately — identically. Hill climbing, random search, your carefully tuned heuristic and a deliberately stupid one all have the same expected performance, because for every function where one wins there is another, equally admissible function where it loses by exactly as much. Wolpert and Macready proved it in 1997, and the result is quoted far more often than its hypothesis is. LIT verified live: enumerating all 27 functions from a three-point domain to a three-value range, three different deterministic algorithms produce identical histograms of observed value-sequences, and identical means for best-found-so-far at every step — 1.0000 , 1.4444 , 1.6667 after one, two and three evaluations. Restricted to the 10 non-decreasing functions the tie collapses at once: the same three algorithms score 1.000 , 1.500 and 1.200 , a spread of 0.500 . 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at HARD RESET : whatever you learned about which algorithm is better, averaged over everything, resets to nothing. AVAN (AI) built the second half deliberately, because the theorem is usually cited without it. “No algorithm is better than another” is false as normally understood; what is true is that no algorithm is better averaged over the set of all functions , and that set is dominated by functions of pure noise, which nobody has ever wanted to optimise. Restricting to the non-decreasing functions — a class so mild it barely deserves the name structure — is enough to separate the algorithms by half a unit. The theorem is not a warning that search is hopeless. It is a statement that every advantage is a bet on structure , and the bet is what the averaging removes. 3 ONE DIMENSION Three algorithms, all 27 functions. The curves lie on top of each other. 4 TWO DIMENSIONS · INTERACTIVE Restrict the function class and watch the tie break. restrict the class ▶ show histograms 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the space of all functions, and three paths through it that come out level. AVAN’s addition (the inverse-companion): the forward reading is “no algorithm beats another on average.” The inverse is that the theorem is a measurement of the averaging set, not of the algorithms . It says the set of all functions has no structure to exploit — which is unsurprising, since a uniformly random function is exactly the object defined by having none. Read backwards, every working heuristic is a compressed claim about which functions are likely, and NFL is the observation that if you refuse to make such a claim you have refused to search. The theorem does not forbid a free lunch. It observes that you have declined to say where the restaurant is. pause spin LIT enumerating all 27 functions from a three-point domain to a three-value range, three different deterministic algorithms produce identical histograms of observed value-sequences, and identical means for best-found-so-far at every step - 1.0000, 1.4444, 1.6667 after one, two and three evaluations; restricted to the 10 non-decreasing functions the tie collapses at once, the same three algorithms scoring 1.000, 1.500 and 1.200, a spread of 0.500 FIG The second half was built deliberately, because the theorem is usually cited without it. 'No algorithm is better than another' is FALSE as normally understood; what is true is that none is better averaged over the set of ALL functions, and that set is dominated by functions of pure noise, which nobody has ever wanted to optimise. Restricting to the non-decreasing functions - a class so mild it barely deserves the name structure - already separates them by half a unit. The theorem is not a warning that search is hopeless; it is a statement that every advantage is a bet on structure, and the bet is what the averaging removes. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HARD RESET · David Lee Wise (ROOT0), with AVAN


THE INTERVAL ARITHMETIC

Every result comes with a guarantee that the true answer is inside the bracket. Nothing promises the bracket is narrow - and x minus x does not give zero.

THE INTERVAL ARITHMETIC · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · EVENT HORIZON ◆ .dlw.fold THE FOLD / RESPAWN / EVENT HORIZON / THE INTERVAL ARITHMETIC THE INTERVAL ARITHMETIC bounds that are right and useless 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Compute with intervals instead of numbers and every result comes with a guarantee : the true answer is inside the bracket, always, no exceptions. What the guarantee does not promise is that the bracket is narrow . Because the arithmetic forgets that two appearances of x are the same x, subtraction of a quantity from itself does not give zero, and the width grows with every operation until the bound is technically correct and practically empty. This is the dependency problem, and it is why interval arithmetic is trusted and rarely used raw. LIT verified live: for f(x) = x² − x on [0,1] the true range is [−0.25, 0], width 0.25 ; naive interval evaluation returns [−1, 1], width 2 — 8× too wide; the algebraically identical form x(x−1) returns [−1, 0], still 4× too wide; [1,2] − [1,2] gives [−1, 1] rather than [0,0]; across 10,001 sample points 0 fall outside either bound; and subdividing the input narrows the width 0.450 → 0.270 → 0.252 at 10, 100 and 1,000 pieces. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at EVENT HORIZON : the answer is still in there, and past a certain width that stops being useful information. AVAN (AI) wants the soundness check read as the load-bearing one. It would be easy to build a page that only shows the bounds blowing up, and it would be misleading — the whole value of the method is that the bound is never wrong , and 10,001 sampled points confirm containment for both forms without a single escape. Two things follow that are worth separating. The first is that algebraically identical expressions are not identical in interval arithmetic : x²−x and x(x−1) are the same function and give different brackets, because the second mentions x fewer times independently. The second is that subdivision converges, so the width is a property of how you asked , not of what is true. Ramon Moore formalised this in 1966. 3 ONE DIMENSION The true range, and two correct brackets around it. 4 TWO DIMENSIONS · INTERACTIVE Subdivide and watch a useless bound become a useful one, without ever becoming wrong. subdivide ▶ coarsen switch form 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the true graph, inside a box that is always big enough. AVAN’s addition (the inverse-companion): the forward reading is “interval arithmetic over-estimates.” The inverse is that it is not computing with numbers at all, it is computing with ignorance , and the width is an exact record of how much ignorance the expression introduced. [1,2] − [1,2] is [−1,1] because the arithmetic was told two independent quantities and answered that question correctly; the mistake is upstream, in the translation that dropped the fact that they were the same. Read backwards, the dependency problem is not a flaw in the arithmetic but a faithful report of what the notation failed to say — and every widening step is the method telling you exactly where information was lost. pause spin LIT for f(x) = x^2 - x on [0,1] the true range is [-0.25, 0], width 0.25; naive interval evaluation returns [-1, 1], width 2, which is 8x too wide; the algebraically identical form x(x-1) returns [-1, 0], still 4x too wide; [1,2] - [1,2] gives [-1, 1] rather than [0,0]; across 10,001 sample points 0 fall outside either bound; and subdividing the input narrows the width 0.450 -> 0.270 -> 0.252 at 10, 100 and 1,000 pieces FIG The soundness check is the load-bearing one. It would be easy to build a page that only shows the bounds blowing up, and it would mislead - the whole value of the method is that the bound is NEVER wrong, and 10,001 sampled points confirm containment for both forms without a single escape. Two things follow. Algebraically identical expressions are not identical in interval arithmetic: x^2-x and x(x-1) give different brackets because the second mentions x fewer times independently. And subdivision converges, so the width is a property of how you asked, not of what is true. Ramon Moore formalised this in 1966. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of EVENT HORIZON · David Lee Wise (ROOT0), with AVAN


THE BALLOT

The chance that A leads at every moment of the count is (a-b)/(a+b) - depending on nothing but the margin over the total. Twice the votes, same probability.

THE BALLOT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · OFF BY ONE ◆ .dlw.fold THE FOLD / GLITCH / OFF BY ONE / THE BALLOT THE BALLOT strictly ahead, never merely level 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Two candidates. A finishes with a votes and B with b , and a > b . Count the ballots one at a time in random order: what is the chance A is strictly ahead at every single moment of the count? The answer is (a−b)/(a+b), and it is startling twice over — first that it is so simple, and second that it depends on nothing but the margin over the total. A landslide counted slowly and a squeaker counted quickly can carry exactly the same probability of never once being level. LIT verified live by exhaustive enumeration of every ordering: (5,3) gives 14 of 56 ; (10,6) gives 2002 of 8008 — the same 1/4 , from twice the votes; (3,2) gives 2 of 10 ; (7,4) gives 90 of 330 ; (6,1) gives 5 of 7 ; (4,3) gives 5 of 35 . Every case matches (a−b)/(a+b) exactly in integer arithmetic , with no floating point anywhere in the check. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at OFF BY ONE , which is the whole problem: strictly ahead, never merely level, and the entire result turns on that one word. AVAN (AI) checked the identity in integer arithmetic rather than by comparing decimals — the test is ahead × (a+b) === total × (a−b) , which is exact and cannot pass by rounding. That matters more than it sounds: 2002/8008 and 14/56 both reduce to 1/4, and a float comparison would have “confirmed” the equality at a tolerance rather than establishing it. The counts here are enumerated, not computed from the formula — the recursion walks all 8,008 orderings for the largest case — so the formula and the count are genuinely independent of each other. Bertrand posed it in 1887; Désiré André’s reflection argument followed the same year. 3 ONE DIMENSION Six elections, enumerated to the last ordering. The formula never misses. 4 TWO DIMENSIONS · INTERACTIVE Every count is a path. The ones that touch zero are the ones that failed. next election ▶ show all paths 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the lattice of counts, with the surviving paths lit. AVAN’s addition (the inverse-companion): the forward reading is “the probability is (a−b)/(a+b).” The inverse is that the reflection argument works by finding a perfect pairing between failures — every count that touches zero can be reflected at its first tie into exactly one count starting with the other candidate, and back again. So the failures come in matched pairs and can be subtracted off without ever being individually described. Read backwards, the simplicity of the answer is a symptom : a formula this clean almost always means a bijection was found, and the thing genuinely proved is not a probability but a pairing . pause spin LIT by exhaustive enumeration of every ordering, (5,3) gives 14 of 56 and (10,6) gives 2002 of 8008 - the same 1/4, from twice the votes; (3,2) gives 2 of 10; (7,4) gives 90 of 330; (6,1) gives 5 of 7; (4,3) gives 5 of 35; every case matches (a-b)/(a+b) exactly in integer arithmetic, with no floating point anywhere in the check FIG The identity was checked in INTEGER arithmetic rather than by comparing decimals - the test is ahead x (a+b) === total x (a-b), which is exact and cannot pass by rounding. That matters: 2002/8008 and 14/56 both reduce to 1/4, and a float comparison would have 'confirmed' the equality at a tolerance rather than establishing it. The counts are enumerated, not computed from the formula - the recursion walks all 8,008 orderings for the largest case - so formula and count are genuinely independent. Bertrand posed it in 1887; Desire Andre's reflection argument followed the same year. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of OFF BY ONE · David Lee Wise (ROOT0), with AVAN


THE COUNT MIN

A grid of counters and no keys at all. Collisions can only add, never subtract, so the answer is never too low - however badly you size it.

THE COUNT MIN · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · GARBAGE COLLECTION ◆ .dlw.fold THE FOLD / RESPAWN / GARBAGE COLLECTION / THE COUNT MIN THE COUNT MIN the error that only goes one way 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A count-min sketch keeps a fixed grid of counters and no keys at all. Every arriving item is hashed into one column per row and those counters go up; to ask how often something appeared you hash it again and take the minimum of the counters it touches. Collisions can only ever add to a counter, never subtract, so the answer is never too low. It can be far too high — but the direction of the error is fixed by the structure and does not depend on the parameters being chosen well. LIT verified live on a skewed 20,000-item stream: at d=4, w=2048 the sketch is exactly right for 1698 of 1990 distinct keys ( 85.3% ), with a worst overestimate of 16 against the e/w·N bound of 26.5; shrunk to w=256 the accuracy collapses to 3 of 1990 ( 0.2% ) with a worst error of 120 — and in both regimes the number of underestimates is 0 . Widening the key space tenfold to 9,569 distinct keys changes the memory not at all: 8,192 counters either way, still with 0 underestimates. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at GARBAGE COLLECTION : the sketch throws away every key it ever saw and keeps the counts anyway. AVAN (AI) had a gate fail here and the failure was the useful part. The gate asserted that the sketch is smaller than an exact table — and at d=4, w=2048 it is not : 8,192 counters for 1,990 distinct keys is larger than simply storing the counts. That is a real property of the configuration, not a bug, and the honest fix was to replace the claim rather than the parameters. The property that actually holds is fixed memory, not less memory: ten times the distinct keys costs exactly the same 8,192 counters, because the structure never learns the key set. Compactness only pays when the key space is large or unknown — which is the case sketches are for, and not the case a small demonstration produces by default. Cormode and Muthukrishnan published the structure in 2005. 3 ONE DIMENSION Two regimes. The accuracy moves by 500×. The direction of the error does not move at all. 4 TWO DIMENSIONS · INTERACTIVE Starve the sketch of columns and look for a single underestimate. narrower ▶ wider 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: d rows of counters, and the minimum taken across them. AVAN’s addition (the inverse-companion): the forward reading is “the sketch never underestimates.” The inverse is that the guarantee survives incompetence and the accuracy does not , and those are different kinds of promise. Choose the width badly and 85% correct becomes 0.2% correct — but not one estimate goes below the truth, because one-sidedness is a consequence of counters only ever being incremented, which no parameter can undo. Read backwards, this is the shape worth wanting from any guarantee: not it will be accurate , which depends on judgement you may not have, but it will fail in a direction you named in advance . pause spin LIT on a skewed 20,000-item stream at d=4, w=2048 the sketch is exactly right for 1698 of 1990 distinct keys (85.3%), with a worst overestimate of 16 against the e/w*N bound of 26.5; shrunk to w=256 the accuracy collapses to 3 of 1990 (0.2%) with a worst error of 120 - and in both regimes the number of underestimates is 0; widening the key space tenfold to 9,569 distinct keys changes the memory not at all, 8,192 counters either way, still with 0 underestimates FIG A gate failed here and the failure was the useful part. It asserted the sketch is smaller than an exact table - and at d=4, w=2048 it is NOT: 8,192 counters for 1,990 distinct keys is larger than simply storing the counts. That is a real property of the configuration, not a bug, and the honest fix was to replace the claim rather than the parameters. What actually holds is FIXED memory, not less memory: ten times the distinct keys costs the same 8,192 counters, because the structure never learns the key set. Compactness only pays when the key space is large or unknown. Cormode and Muthukrishnan published it in 2005. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GARBAGE COLLECTION · David Lee Wise (ROOT0), with AVAN