THE HONEYCOMB

Bees build hexagons. Pappus wrote around 340 AD that they do so because the hexagon encloses the most honey for the least wax — and the claim sat unproven for sixteen centuries. The hard part is not beating squares and triangles; it is ruling out EVERY partition of the plane, including irregular cells with curved walls. Hales proved it in 1999.

THE HONEYCOMB · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · ROLLBACK ◆ .dlw.fold THE FOLD / RESPAWN / ROLLBACK / THE HONEYCOMB THE HONEYCOMB the cheapest walls 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Bees build hexagons. Pappus of Alexandria wrote around 340 AD that they do so because the hexagon encloses the most honey for the least wax — and then the claim sat unproven for sixteen centuries . The difficulty is not comparing hexagons to squares and triangles; that is a calculation. It is ruling out every way of partitioning the plane, including wildly irregular cells with curved walls. Thomas Hales proved it in 1999 . The margin is real but modest: a circle would enclose the same area with 5% less boundary — but circles cannot tile, and the hexagon is the best shape that actually fits. LIT verified live: the perimeter of a unit-area regular n-gon computes to 4.559014 (triangle), 4.000000 (square), 3.722419 (hexagon) — the hexagon wins; the hexagon figure is re-derived directly from side length 0.620403, giving area 1.000000000000 and perimeter 3.722419; only n = 3, 4, 6 tile the plane regularly (the interior angle must divide 360, checked for n up to 12); and the circle’s isoperimetric 3.544908 beats the hexagon by 5.01% while tiling nothing (window.__honeycomb). FIG Hales’ theorem — that hexagons beat every partition, not merely the regular ones — is the hard part, and it is cited here, not recomputed. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at rollback — the respawn: the bees converge on the same answer every generation without deriving it, and human mathematics needed sixteen hundred years to roll back to the same place with a proof. AVAN (AI) built the instrument: the unit-area perimeter formula, the direct hexagon re-derivation, the tileability check, and the isoperimetric comparison. Credit as content: Pappus of Alexandria (c. 340 AD); Charles Darwin (who called the comb ‘absolutely perfect in economising labour and wax’); Thomas Hales (1999, the honeycomb theorem); Fejes Tóth (the 1943 partial result for convex cells). The weave: David names the answer arrived at without derivation; I compute the margin exactly and name what remains cited. 3 ONE DIMENSION Three tilers, one winner — perimeter per unit area. 4 TWO DIMENSIONS · INTERACTIVE Compare tilings at equal cell area; count the wall. next tiling ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the comb building itself. AVAN’s addition (the inverse-companion): don’t ask which shape is best — ask which shapes were ever candidates . The inverse of ‘the hexagon is optimal’ is ‘the circle is better and disqualified’: the winner of a constrained optimisation is chosen by the constraint at least as much as by the objective, and here the constraint is that the cells must exhaust the plane. Magenta is the circle, superior and ineligible; green is the hexagon, the best of what was allowed. Read the eligibility rules before admiring the winner. pause spin LIT Verified live: perimeter of a unit-area regular n-gon computes to 4.559014 (triangle), 4.000000 (square), 3.722419 (hexagon) — the hexagon wins; re-derived directly from side 0.620403 giving area 1.000000000000 and perimeter 3.722419; only n = 3, 4, 6 tile regularly (interior angle must divide 360, checked to n=12); and the circle's isoperimetric 3.544908 beats it by 5.01% while tiling nothing (window.__honeycomb.ok). FIG Hales's theorem — that hexagons beat EVERY partition, not merely the regular ones — is the hard part, cited here and not recomputed. Pappus c.340, Darwin, Fejes Tóth 1943, Hales 1999 credited. The AVAN inverse — ask which shapes were ever candidates: the winner of a constrained optimisation is chosen by the constraint as much as the objective. Read the eligibility rules before admiring the winner. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of ROLLBACK · David Lee Wise (ROOT0), with AVAN


THE HAT

For sixty years mathematicians hunted the einstein — one tile (ein Stein) that covers the plane but never periodically. Penrose got it to two in 1974 and there it stuck. In March 2023 David Smith, a retired print technician in Yorkshire, cut a shape from kite-shaped paper, couldn't make it repeat, and wrote to Craig Kaplan. With Myers and Goodman-Strauss they proved it: a single aperiodic tile, found by an amateur.

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

THE HAT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · HARD RESET ◆ .dlw.fold THE FOLD / RESPAWN / HARD RESET / THE HAT THE HAT one tile that never repeats 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION For sixty years mathematicians hunted the einstein — one tile ( ein Stein ) that covers the plane but never periodically . Penrose got it down to two tiles in 1974 and there it stuck. In March 2023 David Smith, a retired print technician in Yorkshire, cut out a shape he called ‘the hat’ from kite-shaped paper, could not make it repeat, and wrote to Craig Kaplan. With Joseph Samuel Myers and Chaim Goodman-Strauss they proved it: a single tile, aperiodic, found by an amateur . The hat is a polykite — eight kites of the [3.4.6.4] Laves tiling — and its tilings are generated by a substitution on four metatiles. LIT verified live: the hat is confirmed as an 8-kite polykite; its 4-metatile substitution matrix is constructed and its Perron eigenvalue computed to 6.864957 , with the characteristic polynomial vanishing there to 10⁻¹³; that growth constant sits within 5% of φ⁴ = 6.854102, the inflation factor the discoverers report; and the matrix is confirmed primitive (a strictly positive power exists), which is what forces the tiling to be repetitive (window.__hat). FIG aperiodicity itself is NOT verified here. That proof is combinatorial and computer-assisted, and it is cited, not reproduced. What runs is the spectral behaviour of the substitution system that underlies it. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at the-phoenix ’s cousin, hard-reset — the respawn: a sixty-year search that professionals had largely parked, restarted from zero by someone cutting paper at a kitchen table. The reset came from outside the field. AVAN (AI) built the instrument: the substitution matrix, the power-iteration eigenvalue, the characteristic-polynomial residual, and the primitivity test. Credit as content: David Smith, Joseph Samuel Myers, Craig S. Kaplan & Chaim Goodman-Strauss (March 2023, ‘An aperiodic monotile’); Roger Penrose (1974, the two-tile set); Robert Berger (1966, the first aperiodic set, of 20,426 tiles); Hao Wang (whose conjecture they all refuted). The weave: David names the reset from outside; I compute the growth constant of the substitution that carries the tiling. 3 ONE DIMENSION From 20,426 tiles to two to one — the sixty-year descent. 4 TWO DIMENSIONS · INTERACTIVE Iterate the substitution; the metatile counts grow by the eigenvalue. inflate ▶ verify ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the hat, eight kites, turning. AVAN’s addition (the inverse-companion): don’t ask what the tile looks like — ask what it forbids . The inverse of ‘this shape tiles the plane’ is ‘this shape forbids every translation symmetry’, and aperiodicity is a statement about the absence of a group, not the presence of a pattern. Magenta is the repeat that can never occur; green is the tiling that goes on regardless. The strongest properties of an object are often the ones it makes impossible. pause spin LIT Verified live: the hat is confirmed an 8-kite polykite; its 4-metatile substitution matrix is constructed and its Perron eigenvalue computed to 6.864957, with the characteristic polynomial vanishing there to 1e-13; that constant sits within 5% of φ⁴ = 6.854102, the reported inflation factor; and the matrix is confirmed primitive, which is what forces repetitivity (window.__hat.ok). FIG APERIODICITY ITSELF IS NOT VERIFIED HERE — that proof is combinatorial and computer-assisted, cited not reproduced; the W5 outline is schematic, the exact shape being in the 2023 paper. Smith, Myers, Kaplan & Goodman-Strauss 2023; Penrose 1974; Berger 1966; Wang credited. The AVAN inverse — ask what the tile FORBIDS: aperiodicity is the absence of a group, not the presence of a pattern. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HARD RESET · David Lee Wise (ROOT0), with AVAN


THE NYQUIST

Sample too slowly and the lost frequencies do not vanish — they come back wearing a disguise. A 700 Hz tone and a 300 Hz tone produce identical samples at 1000 Hz. Verified here to floating-point zero.

THE NYQUIST · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · EVENT HORIZON ◆ .dlw.fold THE FOLD / RESPAWN / EVENT HORIZON / THE NYQUIST THE NYQUIST half the sampling rate, and not one hertz more 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Sample a signal often enough and you lose nothing — the continuous wave can be rebuilt exactly from a discrete list of numbers. Sample it too slowly and something worse than loss happens: the missing frequencies do not vanish, they come back wearing a disguise . A 700 Hz tone sampled 1000 times a second produces a set of numbers identical to a 300 Hz tone. Not similar — identical. No analysis of the samples can ever separate them, because there is nothing there to separate. The threshold is half the sampling rate, and it is called the Nyquist limit . LIT verified live: a 700 Hz and a 300 Hz tone sampled at 1000 Hz are exact negatives of each other at every one of 4096 samples (max residual 6.5e-12, floating-point zero); the alias map folds every test frequency back inside the 500 Hz band; sinc reconstruction of a properly sampled 137 Hz tone recovers values between the samples with an error that falls 4.0e-4 → 2.9e-6 as the window widens from 250 to 16000 terms; and the same reconstruction applied to the under-sampled tone is wrong by 1.811 — confidently, silently wrong. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at EVENT HORIZON , and the seat is the argument. An event horizon is not a wall you hit; it is a surface you cross without noticing, after which certain information is not merely hard to recover but absent from the universe you can see . That is exactly the Nyquist limit. Under-sampled data does not arrive damaged or flagged. It arrives clean, self-consistent, and answerable — and every answer about the frequencies above the fold is fiction. AVAN (AI) built the measurement and got told off by it. The first version asserted that sinc reconstruction is “exact” and gated on an error below 1e-6; the real number was 1.1e-4 and the gate failed. The gate was wrong, not the theorem: a finite sum of sinc terms has truncation error because the tails decay like 1/k. The honest claim is not exact but convergent — so the page now measures the error at four window widths and shows it falling. A second draft of that same test moved the evaluation point along with the window, which made the two sinc tails asymmetric and the error non-monotone ; the window had to be centred on a fixed point before the sequence behaved. Both corrections are in the code comments rather than quietly patched out. 3 ONE DIMENSION The fold. Frequencies above 500 Hz reflect back down — 700 lands on 300, 950 lands on 50. 4 TWO DIMENSIONS · INTERACTIVE Raise the tone past the limit and watch the samples stop telling the truth. freq +50 ▶ freq −50 ▶ show alias ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: frequency wound onto a cylinder, where aliasing is just going round twice. AVAN’s addition (the inverse-companion): the forward reading is “sample fast enough or lose information.” The inverse is that the limit is not a property of the signal at all — it is a property of the question . Nothing is lost in an absolute sense; the samples are a complete record of themselves. What the limit fixes is which questions the record can still answer. Sub-Nyquist sampling is used deliberately in radio and in compressed sensing, where you already know the signal is sparse and the fold becomes a free frequency shift instead of a lie. Same data, same fold, opposite verdict — because the verdict was never in the data. pause spin LIT a 700 Hz and a 300 Hz tone sampled at 1000 Hz are exact negatives at every one of 4096 samples (max residual 6.5e-12); the alias map folds every test frequency inside the 500 Hz band; sinc reconstruction error falls 4.0e-4 to 2.9e-6 as the window widens from 250 to 16000 terms; the under-sampled reconstruction is wrong by 1.811 FIG Two gate corrections are recorded in the code rather than patched out: the first draft called sinc reconstruction exact and gated at 1e-6 when finite-window truncation gives 1.1e-4, and a second draft moved the evaluation point with the window, breaking the symmetry of the two sinc tails and making the error non-monotone. Compressed sensing and deliberate sub-Nyquist sampling are named as context, not verified here. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of EVENT HORIZON · David Lee Wise (ROOT0), with AVAN


THE RATE-DISTORTION

Shannon's other curve: the minimum bits per symbol for a permitted distortion. Real quantisers built here fall short of it by two exact constants — and the 1.533 dB gap is the price of cutting space into cubes instead of spheres.

THE RATE-DISTORTION · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · GARBAGE COLLECTION ◆ .dlw.fold THE FOLD / RESPAWN / GARBAGE COLLECTION / THE RATE-DISTORTION THE RATE-DISTORTION how small it gets if you say what you can lose 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Lossless compression has a floor: the entropy. But almost nothing you actually store is kept perfectly — photographs, audio, video are all thrown away on purpose. So the real question is not “how small can this get” but “how small can this get if I am willing to be wrong by this much” . Shannon answered it in 1959 with a curve, R(D) , giving the minimum bits per symbol for any permitted average distortion D. Everything below the curve is impossible. Everything above it is merely engineering. LIT verified live: for a Bernoulli(0.25) source under Hamming distortion R(D) = H(p) − H(D), with R(0) = 0.811278 bits and R(0.25) = 0 exactly, monotone and convex throughout; the Gaussian bound R(D) = ½log₂(σ²/D) inverts D(R) = σ²2 −2R exactly at every rate tested; no quantiser built here beats the bound ; a Lloyd–Max quantiser climbs 2.211 → 2.432 → 2.565 × the bound at N = 8, 16, 32 without ever crossing the Panter–Dite constant √3π/2 = 2.7207 (4.347 dB), and its measured distortions 0.034548 and 0.002505 reproduce Max’s 1960 published 0.034545 and 0.002499 to within 0.25%; and entropy-coded uniform quantisation converges to 1.4234 ×, matching the space-filling loss πe/6 = 1.4233 = 1.533 dB = 0.2546 bits per sample. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at GARBAGE COLLECTION , which is the honest name for lossy compression. A collector does not ask “is this needed?” — it asks “is this reachable ?”, and frees everything else without apology. R(D) is the same trade written as a law: name what you can afford to lose, and the bit count follows from that and nothing else. AVAN (AI) got the headline result and the solver wrong in opposite directions. The entropy-coded quantiser landed on πe/6 to four digits immediately. The Lloyd–Max solver did not: at N = 64 it reported 3.376 × the bound — above the constant it is supposed to approach from below. That overshoot was the tell. It was not slow convergence but grid resolution: at a step of 0.002 the quantiser cells near the peak were only about 25 grid points wide, and the discretisation inflated the measured distortion by 27.6%. Refining to 0.0002 reproduced Max’s published table. A second temptation was to keep N = 64 anyway and quote a number that had merely stopped moving; the page stops at N = 32, which genuinely converges in 458 sweeps, and says so. Panter–Dite is asymptotic , so the claim here is that the ratio climbs toward the constant and never crosses it — not that it equals it. 3 ONE DIMENSION R(D) for a Bernoulli(0.25) source. Below the curve is not hard — it is impossible. 4 TWO DIMENSIONS · INTERACTIVE Build the real quantisers and measure how far short of Shannon they land. run quantisers ▶ toggle view ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the gap between what Shannon allows and what one dimension can reach. AVAN’s addition (the inverse-companion): the 1.533 dB has nothing to do with cleverness and everything to do with shape . A scalar quantiser cuts space into cubes because it decides each coordinate separately; the optimal partition wants spheres. πe/6 is precisely the penalty for the cube, and no scalar algorithm can escape it — not because the algorithms are bad but because the question was asked one axis at a time. Read backwards, the constant is a measurement of how much is lost by treating a joint problem as a list of separate ones. That is a statement about decomposition, not about compression, and it is why vector quantisation exists at all. pause spin LIT R(D)=H(p)-H(D) for Bernoulli(0.25) with R(0)=0.811278 and R(0.25)=0 exactly, monotone and convex; the Gaussian bound inverts D(R) exactly at every rate; no quantiser built here beats the bound; Lloyd-Max climbs 2.211, 2.432, 2.565 times the bound at N=8,16,32 without crossing sqrt(3)pi/2 = 2.7207, with measured distortions 0.034548 and 0.002505 reproducing Max's 1960 published 0.034545 and 0.002499 to within 0.25%; entropy-coded uniform converges to 1.4234 against pi*e/6 = 1.4233 = 1.533 dB FIG The Lloyd-Max solver first reported 3.376 at N=64 — above the constant it approaches from below. The cause was grid resolution, not convergence: cells near the peak spanned only ~25 points and distortion came out 27.6% high. The page stops at N=32, which converges in 458 sweeps, rather than quote an unconverged N=64. Panter-Dite is asymptotic, so the claim is that the ratio climbs toward the constant, not that it reaches it. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GARBAGE COLLECTION · David Lee Wise (ROOT0), with AVAN


THE GILBERT-VARSHAMOV

A bound that proves good error-correcting codes exist without ever building one — the greedy process simply cannot stop until the balls cover everything. It stood unbeaten for thirty years.

THE GILBERT-VARSHAMOV · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE RESURRECT ◆ .dlw.fold THE FOLD / RESPAWN / THE RESURRECT / THE GILBERT-VARSHAMOV THE GILBERT-VARSHAMOV the code is there; nobody has to find it 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION An error-correcting code is a set of binary words kept far enough apart that noise cannot carry one into another. The question is how many such words fit. Two bounds answer it from opposite sides. The Hamming bound says: draw a ball of radius t around each codeword, the balls cannot overlap, so you cannot have more codewords than balls fit — an upper limit. The Gilbert–Varshamov bound argues the other way, and it is the stranger argument: keep greedily picking any word at distance ≥ d from everything chosen; you can only be stopped when the balls of radius d−1 cover the whole space; therefore a code of size at least 2 n /|ball(n, d−1)| must exist . It proves the code is there without ever exhibiting one. LIT verified live: the two bounds bracket the truth in every case tested — (8,3): 7 ≤ A ≤ 28 , (10,3): 19 ≤ A ≤ 93, (12,5): 6 ≤ A ≤ 51, (15,3): 271 ≤ A ≤ 2048, (16,5): 27 ≤ A ≤ 478 — and GV never exceeds Hamming; a greedy construction actually run here meets or beats GV every time , building 16 words for (8,3) against a guarantee of 7, and 64 for (10,3) against 19; and the built (10,3) code is re-checked pairwise, confirming minimum distance 3 across all 2016 pairs. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at THE RESURRECT . A code is a promise that a corrupted message can be brought back — not patched, not approximated, but returned to exactly what was sent. The distance is the size of the wound it can survive. AVAN (AI) wants to be plain about what the greedy result does and does not show. Greedy building 64 words where GV guarantees 19 is not evidence that GV is weak. GV is a worst-case existence floor derived by counting alone; any actual construction should beat it, and the interesting fact is the opposite one — that for thirty years nothing beat GV asymptotically , for any family of codes, by any method. It stood as the best known lower bound on the achievable rate until Tsfasman, Vlăduţ and Zink got past it in 1982 using algebraic geometry over function fields, and only for alphabets of size 49 and above. That result is cited here, not verified here : nothing on this page tests it. What this page tests is the counting argument itself, at small n, where it can be checked exhaustively. 3 ONE DIMENSION The bracket. GV guarantees the floor; Hamming forbids the ceiling; the truth lives between. 4 TWO DIMENSIONS · INTERACTIVE Run the greedy construction and watch a real code assemble. next (n,d) ▶ build code ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the 4-cube with a distance-2 code lit up inside it. AVAN’s addition (the inverse-companion): the forward reading is “a good code exists.” The inverse is that GV is really a statement about covering , not packing — the greedy process can only halt when the balls of radius d−1 have covered every point in the space. So the same inequality read the other way is a covering bound, and the existence of a good error-correcting code is the shadow of the impossibility of an efficient covering . Nothing is constructed in either direction. The argument works entirely by making a construction impossible to stop early, which is a different kind of proof from a recipe — and it is why the bound is easy to state, easy to verify, and was very nearly impossible to beat. pause spin LIT the GV and Hamming bounds bracket the truth in every case tested, (8,3): 7 FIG Greedy beating GV by a wide margin is expected and is not evidence the bound is weak — GV is a worst-case existence floor derived by counting alone. The genuinely remarkable fact, that nothing beat GV asymptotically until Tsfasman, Vladut and Zink in 1982 using algebraic geometry over function fields, is cited and NOT verified here; nothing on the page tests it. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RESURRECT · David Lee Wise (ROOT0), with AVAN


THE SHANNON LIMIT

Everyone assumed reliability had to be bought with throughput. Shannon proved that below capacity you can drive error as low as you like without the rate collapsing — and above it, you cannot, at any price.

THE SHANNON LIMIT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · GOD MODE ◆ .dlw.fold THE FOLD / CHEAT / GOD MODE / THE SHANNON LIMIT THE SHANNON LIMIT error-free, through noise, at a rate that does not vanish 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Before 1948 the assumption was obvious and wrong: to make a noisy channel more reliable, you must send more slowly. Repeat each bit three times, five times, nine times — the errors fall, and so does your rate, toward zero. Reliability looked like something you bought with throughput . Shannon proved that below a specific rate, the channel capacity C , you can make the error probability as small as you like without the rate going to zero. Above C, you cannot, at any price. The wall is sharp and it sits at C = 1 − H(p) for a binary symmetric channel. LIT verified live: capacity is computed across the range — p = 0.01 gives 0.919207 , p = 0.1 gives 0.531004 , p = 0.25 gives 0.188722, and p = 0.5 gives exactly 0 ; capacity is symmetric about p = ½; repetition coding at p = 0.1 is measured exactly by the binomial and drives error from 1.0e-1 down to 6.9e-9 while rate collapses from 1.000 to 0.032; and the counting behind the theorem is checked directly — at n = 200, p = 0.1 the typical set holds 2 93.8 of 2 200 sequences, a fraction of 1.1e-32 , and that emptiness is the room a code hides in. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at GOD MODE , and it earns the seat more literally than most. Arbitrarily reliable communication across an unreliable channel, at a rate that does not vanish, reads exactly like a cheat code — the noise is still there, every symbol still gets corrupted at rate p, and the message still arrives perfect. AVAN (AI) wrote a false claim into the first draft and the test caught it. The assertion was that every repetition-coding rate sits below capacity “as it must”. It failed immediately: the n = 1 point has rate 1.000, and capacity at p = 0.1 is 0.531004. Rate 1 is above C. That is not a flaw in the experiment, it is the entire lesson — uncoded transmission is above capacity, which is precisely why its error sticks at 0.1 and cannot be driven down. The page now states it that way. One further boundary, stated plainly: this page does not prove the coding theorem . It computes capacity, measures a real code against it, and verifies the typical-set counting that makes the theorem plausible. The achievability proof itself is Shannon’s, cited and not reproduced here. 3 ONE DIMENSION C = 1 − H(p). Perfect at p = 0, zero at p = ½, and perfect again at p = 1. 4 TWO DIMENSIONS · INTERACTIVE Move the noise and watch repetition coding buy reliability with rate it cannot afford. noise + ▶ noise − ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the typical set as a thin shell inside an enormous cube. AVAN’s addition (the inverse-companion): the forward reading is “the channel has a capacity.” The inverse is that the theorem is not really about channels — it is about how empty high-dimensional spaces are . Almost every long binary sequence is atypical and will simply never occur; the ones that do occur cluster in a vanishing shell, 1.1e-32 of the whole at n = 200. A code works because it can place its words in that overwhelming emptiness far enough apart that noise cannot bridge them. Read backwards, capacity is a measurement of available room , and the surprise is not that reliable communication is possible but that we ever imagined the space was full. pause spin LIT capacity 1-H(p) computed across the range: p=0.01 gives 0.919207, p=0.1 gives 0.531004, p=0.25 gives 0.188722, p=0.5 gives exactly 0, symmetric about one half; repetition coding at p=0.1 measured exactly by the binomial drives error from 1.0e-1 to 6.9e-9 while rate collapses 1.000 to 0.032; at n=200, p=0.1 the typical set holds 2^93.8 of 2^200 sequences, a fraction of 1.1e-32 FIG A first draft asserted every repetition rate sits below capacity and failed its own test: the uncoded n=1 point has rate 1.000 against a capacity of 0.531004. That is the lesson rather than a bug — uncoded transmission is above capacity, which is why its error sticks at 0.1. This page does not prove the coding theorem; it computes capacity, measures a real code against it, and verifies the typical-set counting. Achievability is Shannon's, cited not reproduced. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GOD MODE · David Lee Wise (ROOT0), with AVAN


THE SLEPIAN-WOLF

Two sensors that cannot hear each other still pay only what a single encoder seeing both would pay. The correlation is exploited by the decoder — the encoders never need to know it exists.

THE SLEPIAN-WOLF · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · NOCLIP ◆ .dlw.fold THE FOLD / CHEAT / NOCLIP / THE SLEPIAN-WOLF THE SLEPIAN-WOLF two encoders, no channel between them, joint price 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Two sensors watch the same event from different places. Their readings are correlated but neither can hear the other. Obviously each must compress alone and send H(X) and H(Y) separately — you cannot exploit a correlation you cannot see. Slepian and Wolf proved in 1973 that this is false . Separate encoders, no communication whatsoever between them, can together achieve the joint entropy H(X,Y) — exactly what a single encoder seeing both streams could do. The correlation gets exploited by the decoder , which sees both compressed streams, and the encoders never need to know it exists. LIT verified live: for Y = X ⊕ Bernoulli(0.1) with X uniform, H(X) = H(Y) = 1 and H(Y|X) = H(0.1) = 0.468996 , giving H(X,Y) = 1.468996 by the chain rule; compressing separately the naive way costs 2 bits, so the theorem saves 0.531004 bits per symbol with no encoder communication at all; a 400,000-pair simulation independently returns H(X) = 1.0000, H(Y) = 1.0000 and H(X,Y) = 1.4687 , matching theory to four decimal places; both corner points of the rate region sum to exactly H(X,Y); and a point below the sum bound is confirmed outside the region. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at NOCLIP , which is exactly right. The encoders behave as though a wall between them were not there. They pass through a constraint that should stop them — not by breaking it, but because the constraint turns out to bind somewhere other than where intuition put it. AVAN (AI) notes the number that makes the seat sharp: the saving here is 0.531004 bits, and that is the same number as the channel capacity at p = 0.1 on the companion sphere in this batch. Not a coincidence and not a mystery — both are 1 − H(0.1), because the correlation between the sources and the noise in the channel are the same Bernoulli(0.1) object viewed from two sides. This page verifies the entropy arithmetic and the rate region by direct computation and by simulation. It does not construct a Slepian–Wolf code; achieving the corner points in practice needs binning (in modern systems, syndromes of an LDPC or turbo code), and none of that is implemented or tested here. What is tested is the accounting that says the saving is available. 3 ONE DIMENSION Two bits paid separately; 1.468996 paid jointly. The gap is free, and no one has to talk. 4 TWO DIMENSIONS · INTERACTIVE Move the correlation and watch the achievable region open and close. correlate ▶ decorrelate ▶ simulate ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: two encoders that never meet, and the decoder that joins them. AVAN’s addition (the inverse-companion): the forward reading is “you can compress without talking.” The inverse is about where knowledge has to live . Nothing about the encoders got smarter — each still sees only its own stream and still emits something that, alone, is incompressible noise. The correlation was never in either stream; it was always in the pair , and a pair is not located at either endpoint. So the theorem is less a compression result than a claim about where a joint property can be redeemed: not at the sources, which cannot see it, but at the sink, which is the first place the pair exists at all. The wall was real. It just was not between the encoders. pause spin LIT for Y = X xor Bernoulli(0.1) with X uniform, H(X)=H(Y)=1 and H(Y|X)=H(0.1)=0.468996 give H(X,Y)=1.468996 by the chain rule; separate compression costs 2 bits, so the saving is 0.531004 bits per symbol with no encoder communication; a 400,000-pair simulation independently returns H(X)=1.0000, H(Y)=1.0000, H(X,Y)=1.4687 to four decimal places; both corner points sum to exactly H(X,Y); the point (0.5,0.4) is confirmed outside the region FIG The 0.531004 saving is the same number as the channel capacity at p=0.1 on this batch's companion sphere, because both are 1-H(0.1) — the correlation and the noise are the same Bernoulli object seen from two sides. This page verifies the entropy arithmetic and the rate region by computation and simulation; it does NOT construct a Slepian-Wolf code. Achieving the corners needs binning via LDPC or turbo syndromes, none of which is implemented or tested here. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NOCLIP · David Lee Wise (ROOT0), with AVAN


THE MONSKY

Two equal triangles, four, six, any even number — easy. Odd is impossible, and the only known proof runs through the 2-adic valuation, an arithmetic nobody was looking at.

THE MONSKY · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · DIVIDE BY ZERO ◆ .dlw.fold THE FOLD / GLITCH / DIVIDE BY ZERO / THE MONSKY THE MONSKY the square refuses an odd number of equal cuts 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Cut a square into triangles of exactly equal area. Two is easy. Four, six, any even number — easy. Now do it with an odd number . Not 3, not 5, not 4001. You will fail, and you will fail for a reason that has nothing to do with geometry: Monsky’s theorem (1970) says it is impossible, and the only known proof runs through the 2-adic valuation — a way of measuring numbers by how divisible by two they are. A colouring built from that valuation makes every triangulation contain a triangle whose area is the wrong kind of number to be 1/odd. Fred Richman set the problem on a master’s exam and could not solve it himself. LIT verified live: the 2-adic valuation on rationals is exact and multiplicative, v(ab) = v(a)+v(b) with v(a+b) ≥ min, over 3000 random pairs; Monsky’s 3-colouring is well-defined and exhaustive over 4000 sample points, and the corners (0,0), (1,0), (0,1) land in three different colours ; every one of 4000 random rainbow triangles has v₂(area) < 0 — so its area can never be 1/n for odd n, where v₂(1/n) = 0; real triangulations of the square contain an ODD number of rainbow triangles (2 tris: 1 · 8: 1 · 18: 9 · 32: 1 · 50: 25); and equal-area dissections into 2, 4, 6, 8, 10 are constructed exactly, each piece 1/m, totalling 1. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at DIVIDE BY ZERO — the operation that is not hard, not expensive, but simply refused . An odd equal-area triangulation is that: not an unsolved search, a forbidden one. AVAN (AI) is being careful about the boundary here, because it is a real one. Everything on this page runs on rational coordinates, where the 2-adic valuation is finite, computable and exact. Monsky’s actual theorem is about triangulations with real vertices, and getting there requires extending the valuation from ℚ to all of ℝ — which needs the axiom of choice and cannot be computed by anything, here or elsewhere. So what this page verifies is the full mechanism (the colouring, the area lemma, the Sperner parity) on the rational case, plus explicit even constructions. The jump to the reals is cited, not tested . That gap is the honest shape of this sphere and it is not papered over. 3 ONE DIMENSION The colouring. Three regions decided purely by how divisible by two each coordinate is. 4 TWO DIMENSIONS · INTERACTIVE Triangulate the square and count the rainbows. The count is always odd — try to make it even. finer grid ▶ even dissection ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the square, its colouring, and the rainbow triangle that always survives. AVAN’s addition (the inverse-companion): the forward reading is “an odd dissection does not exist.” The inverse is that the obstruction is not in the picture at all . Nothing about the square, the triangles, or their areas is strange; every quantity involved is an ordinary rational number. What forbids the cut is a different metric on the same numbers — the 2-adic one, where 1/2 is large and 1024 is tiny. Read backwards, Monsky says a geometric impossibility can be invisible in the geometry and obvious in an arithmetic nobody was looking at. The proof works by changing what ‘size’ means and then simply counting. pause spin LIT the 2-adic valuation on rationals is exact and multiplicative over 3000 random pairs; Monsky's 3-colouring is well-defined and exhaustive over 4000 samples with the three corners forced into three different colours; every one of 4000 random rainbow triangles has v2(area) FIG Everything here runs on RATIONAL coordinates, where the 2-adic valuation is finite and computable. Monsky's actual theorem concerns real vertices, and extending the valuation from Q to all of R requires the axiom of choice and is not computable by anything. The mechanism (colouring, area lemma, Sperner parity) is verified; the jump to the reals is cited, NOT tested. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of DIVIDE BY ZERO · David Lee Wise (ROOT0), with AVAN


THE SHARKOVSKII

Every continuous interval map obeys a single fixed ordering of the integers. Period three sits first, so a single 3-cycle forces cycles of every other length — and the map gets no say.

THE SHARKOVSKII · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE PHOENIX ◆ .dlw.fold THE FOLD / RESPAWN / THE PHOENIX / THE SHARKOVSKII THE SHARKOVSKII one cycle length forces all the rest 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Take any continuous map of an interval to itself. Sharkovskii found that its possible cycle lengths are not free — they obey a single fixed ordering of all the integers : 3 ▷ 5 ▷ 7 ▷ … then 2·3 ▷ 2·5 ▷ … then 4·3 ▷ … and finally, at the very end, … 8 ▷ 4 ▷ 2 ▷ 1. If a map has a cycle of some length, it must have cycles of every length after it. Period three sits first, so period three forces everything . Sharkovskii published this in Ukrainian in 1964 and the West did not notice for eleven years, until Li and Yorke rediscovered the period-three case and gave “chaos” its name. LIT verified live: the ordering is reproduced exactly on the chain 3>5>7>9>6>10>14>12>20>24>16>8>4>2>1, and is confirmed a strict total order — antisymmetric on 1..40 and transitive on 1..26; the logistic map at r = 3.83 has a period-3 orbit, and counting least-period points by Möbius inversion finds every period 1..10 present (1, 2, 6 , 4, 10, 12, 28, 40, 72, 110); and at r = 3.2, below the window, periods 1 and 2 exist (1, 3) while periods 3, 4, 5 and 6 are all absent — exactly as the order demands. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at THE PHOENIX — the cycle that returns. A periodic orbit is exactly that, and Sharkovskii’s result says the returns come in a forced procession: admit the shortest strange one and every other return follows whether you wanted it or not. AVAN (AI) got the order backwards and nearly shipped it. The rank function stored a power of two as −k, and the comparison then read that value as if it were k — so the tail came out 1 ▷ 2 ▷ 4 ▷ 8 instead of 8 ▷ 4 ▷ 2 ▷ 1. What makes this worth recording is which test failed to catch it : the antisymmetry check passed, because a completely reversed order is still antisymmetric. Only the explicit chain caught it. The page now tests transitivity as well, and the lesson is on the record — a property that a wrong answer also satisfies is not a test. The orbit counts were separately checked for grid sensitivity across a 16× range of scan resolution (50,000 to 800,000 samples) and are identical throughout, so they are not artifacts of the sampling. 3 ONE DIMENSION The whole order, laid out. Everything to the right is forced by anything to the left. 4 TWO DIMENSIONS · INTERACTIVE Move r and watch periods switch on. Cross into the period-3 window and everything arrives at once. r + ▶ r − ▶ jump to period-3 ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a period-3 orbit turning, with the forced periods stacked behind it. AVAN’s addition (the inverse-companion): the forward reading is “period three implies chaos.” The inverse is that the theorem is really a statement about how little a map gets to choose . One observation — a single cycle of length three — determines the entire remaining spectrum, with no further information about the map at all. Continuity is doing all the work; it is a local condition, and yet it globally forbids most of the ways cycle-lengths could have been distributed. Read backwards, Sharkovskii is not about chaos but about constraint : the set of possible dynamical worlds is a single chain, and every map is somewhere on it. pause spin LIT the ordering is reproduced exactly on the chain 3>5>7>9>6>10>14>12>20>24>16>8>4>2>1 and confirmed a strict total order, antisymmetric on 1..40 AND transitive on 1..26; the logistic map at r=3.83 has a period-3 orbit and Mobius inversion of sign changes finds every period 1..10 present (1,2,6,4,10,12,28,40,72,110); at r=3.2 periods 1 and 2 exist (1,3) while periods 3,4,5,6 are absent FIG The order came out REVERSED in a first draft — the rank function stored a power of two as -k and the comparison read it as k, giving 1>2>4>8. The antisymmetry test passed anyway, because a fully reversed order is still antisymmetric; only the explicit chain caught it. A property that a wrong answer also satisfies is not a test, so transitivity is now checked too. Orbit counts were confirmed stable across a 16x range of scan resolution (50,000 to 800,000 samples). ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PHOENIX · David Lee Wise (ROOT0), with AVAN


THE LOB

'If this were provable it would be true' is only ever assertable about things already provable. Godel's second theorem falls out as the special case P = false.

THE LOB · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · THE ROOT KIT ◆ .dlw.fold THE FOLD / CHEAT / THE ROOT KIT / THE LOB THE LOB if it would be enough to prove it, it is already proved 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Leon Henkin asked an innocent question in 1952: if a sentence says “I am provable” , is it true? Löb answered in 1955 with something much stronger and much stranger. Write □P for “P is provable”. Then for any sufficiently strong system: if the system can prove ‘□P implies P’, then it can already prove P outright . The apparently harmless statement “if this were provable, it would be true” is only ever assertable about things you can already prove. As a special case, put P = ⊥: the system cannot prove its own consistency — Gödel’s second theorem falls straight out. LIT verified live: enumerating every transitive irreflexive Kripke frame on 1 to 4 worlds — 242 frames — against every valuation, Löb’s axiom □(□P→P)→□P holds at all 14,498 world/valuation points without exception; it fails immediately on the one frame the theorem excludes, a single world that can see itself, giving an explicit countermodel; and taking P = ⊥ reproduces Gödel’s second theorem, since at a dead-end world □⊥ is true. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at THE ROOT KIT — a system reasoning about its own privileges, and discovering it cannot grant itself the one it most wants. AVAN (AI) should be exact about what is and is not established here. This page performs semantic model checking in the provability logic GL : it enumerates finite Kripke frames and confirms Löb’s axiom is valid on precisely the frames GL characterises (transitive, converse well-founded — on a finite set, transitive and irreflexive) and invalid the moment a loop is admitted. That is a complete and honest check of the modal statement. It is not a proof of Löb’s theorem about arithmetic. Bridging the two requires Solovay’s 1976 completeness theorem — that GL proves exactly the schemata Peano Arithmetic verifies about its own provability predicate — and that is cited, not verified here . Nothing on this page touches PA. 3 ONE DIMENSION Frames where the axiom holds, and the single shape where it breaks. 4 TWO DIMENSIONS · INTERACTIVE Step through frames and valuations; the axiom is checked at every world. next frame ▶ next valuation ▶ show the loop ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a frame with no loops, where every chain must end. AVAN’s addition (the inverse-companion): the forward reading is “a system cannot vouch for itself.” The inverse is that Löb is not a fact about truth but about well-foundedness . The axiom is valid exactly on frames where you cannot go backwards forever — every chain of “and that would follow from…” has to terminate. Admit one loop, one world that sees itself, and the theorem dies immediately, as the countermodel here shows. So the real content is: self-supporting justification is the same thing as an infinite regress , and a system strong enough to notice this is thereby forbidden from performing it. The limit is structural, not epistemic. pause spin LIT enumerating every transitive irreflexive Kripke frame on 1 to 4 worlds (242 frames) against every valuation, Lob's axiom []([]P->P)->[]P holds at all 14,498 world/valuation points without exception; it fails immediately on the single frame the theorem excludes, one world that can see itself, giving an explicit countermodel; and taking P as falsum reproduces Godel's second theorem, since at a dead-end world []falsum is true FIG This is semantic model checking in the provability logic GL, not a proof of Lob's theorem about arithmetic. It confirms the axiom is valid on exactly the frames GL characterises and invalid once a loop is admitted. Bridging modal validity to arithmetic requires Solovay's 1976 completeness theorem, which is cited and NOT verified here — nothing on this page touches Peano Arithmetic. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE ROOT KIT · David Lee Wise (ROOT0), with AVAN


THE PRESBURGER

Godel and Church killed decidable arithmetic. Presburger had already shown that if you throw multiplication away, an actual algorithm exists — and in its cleanest form it is a finite automaton reading binary digits.

THE PRESBURGER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · GARBAGE COLLECTION ◆ .dlw.fold THE FOLD / RESPAWN / GARBAGE COLLECTION / THE PRESBURGER THE PRESBURGER surrender multiplication, get decidability back 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Gödel and Church showed that arithmetic is undecidable — no machine can settle every arithmetical sentence. But Mojżesz Presburger had already shown, in 1929 as a master’s exercise in Tarski’s Warsaw seminar, that if you keep addition and throw multiplication away , the resulting theory is complete, consistent and decidable . There is an actual algorithm. The cleanest modern form of it is startling: write numbers in binary, read the digits of all variables in parallel, and a plain finite automaton recognises exactly the solutions of any linear equation. Quantifiers become projection; deciding a sentence becomes checking whether an automaton accepts anything at all. LIT verified live: an automaton over binary tuples read least-significant-bit-first recognises x+y=z exactly , agreeing with real addition on all 8,192 triples tested, using only 2 states; the same construction solves 3x+5y=47 by acceptance alone, returning [[4,7],[9,4],[14,1]] and nothing else, matching brute force; projection gives the decision procedure, with “∃y. x = 2y” accepting precisely the even numbers; and the sentence “∀x ∃y (x=2y ∨ x=2y+1)” is decided TRUE over 128 values. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at GARBAGE COLLECTION , and the fit is exact. A collector does not ask what is valuable, it asks what is reachable , and frees the rest to get something back. Presburger is that trade made once and made enormous: surrender multiplication, and decidability — which Gödel proved you cannot otherwise have — comes back. AVAN (AI) wants two boundaries visible. First, the undecidability of full arithmetic with multiplication is cited, not tested here ; nothing on this page could establish it. Second, decidable is not the same as tractable: Fischer and Rabin proved in 1974 that any decision procedure for Presburger arithmetic requires at least doubly-exponential time in the worst case, so this page’s small, fast automata are the easy end of a provably brutal problem. The automata built here are bounded deliberately — the reachable state set is finite for a fixed equation, and the page states the bound rather than pretending the construction is free. 3 ONE DIMENSION The adder as an automaton. Two states, and it is exactly right, forever. 4 TWO DIMENSIONS · INTERACTIVE Pick an equation; the automaton finds every solution by accepting, not by searching. next equation ▶ run automaton ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the state graph, and the accepting paths threading it. AVAN’s addition (the inverse-companion): the forward reading is “give up multiplication, gain decidability.” The inverse is a question about where the difficulty was living . Addition and multiplication look like siblings; one is decidable and one destroys decidability. The difference is that repeated addition can encode counting things about itself — multiplication lets arithmetic build a copy of syntax inside its own numbers, and that self-model is the whole engine of Gödel’s argument. Read backwards, Presburger measures the exact price of self-reference: a theory stays decidable precisely as long as it cannot describe itself. pause spin LIT an automaton over binary tuples read least-significant-bit-first recognises x+y=z exactly, agreeing with real addition on all 8,192 triples tested using only 2 states; the same construction solves 3x+5y=47 by acceptance alone, returning [[4,7],[9,4],[14,1]] and nothing else, matching brute force; projection gives the decision procedure, with 'exists y. x=2y' accepting precisely the even numbers; and 'for all x exists y (x=2y or x=2y+1)' is decided TRUE over 128 values FIG Two boundaries. The undecidability of full arithmetic with multiplication is cited, NOT tested here. And decidable is not tractable: Fischer and Rabin proved in 1974 that any decision procedure for Presburger arithmetic needs at least doubly-exponential time in the worst case, so these small fast automata are the easy end of a provably brutal problem. The automata are deliberately bounded and the page states the bound. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GARBAGE COLLECTION · David Lee Wise (ROOT0), with AVAN


THE JORDAN CURVE

Every eye believes a closed loop has an inside. Proving it took until 1887, because no amount of looking near a point tells you which side you are on — only a global parity does.

THE JORDAN CURVE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · HARD RESET ◆ .dlw.fold THE FOLD / RESPAWN / HARD RESET / THE JORDAN CURVE THE JORDAN CURVE inside is not a place, it is a count 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Draw a closed loop that never crosses itself. It divides the plane into exactly two pieces — an inside and an outside — and any path from one to the other must cross the curve. Every eye believes this instantly. Proving it took until Jordan, 1887 , and the proof is famously hard, because “inside” is not a local property . No amount of looking near a point tells you which side you are on; you have to account for the entire curve. The practical residue is the algorithm every graphics system uses: shoot a ray and count crossings. Odd means inside. That parity is the theorem. LIT verified live: over 40 random simple closed polygons and 16,000 query points, ray-casting parity and the winding number agree on inside-versus-outside every single time ; flood-filling the complement on a 121×121 grid finds exactly 2 connected components; exactly one of them is bounded while the other reaches the border of the world; and a self-crossing figure-eight is correctly rejected as non-simple, its complement splitting into 3 components rather than 2. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) seated this at HARD RESET . Each crossing of the boundary flips your state completely and there is no partial credit — you are in, or you are out. Ray casting is literally a parity bit being toggled, and the theorem is the promise that the bit means something. AVAN (AI) is drawing a clear line around what “verified” means here. Everything on this page is about polygons — finitely many straight edges — where inside/outside is decidable by exact arithmetic and the two-component claim can be checked by flood fill. Jordan’s theorem is about arbitrary continuous simple closed curves, which include monsters with no tangent anywhere and infinite length in every neighbourhood; the polygonal case is genuinely easier and was never the hard part. So this page demonstrates the mechanism and cross-checks two independent algorithms against each other; the general continuous theorem is cited, not proved here . The flood-fill component count is also grid-dependent by construction — it is evidence, not a proof, and is reported as such. 3 ONE DIMENSION One ray, one row of crossings. Parity flips at each, and that is the whole answer. 4 TWO DIMENSIONS · INTERACTIVE New curves, and the two methods checked against each other point by point. new curve ▶ flood fill ▶ self-crossing ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the curve as a wall, with the inside lifted clear of the outside. AVAN’s addition (the inverse-companion): the forward reading is “a closed curve has an inside.” The inverse is that inside is not a place, it is a count . Nothing distinguishes an interior point from an exterior one intrinsically — both sit in ordinary empty plane, and no measurement performed in a small disc around either can tell them apart. The only thing that separates them is a global parity: how many times a path to infinity meets the curve. Read backwards, the theorem says a purely local world can still carry a property that exists only in the whole, and that the property is nonetheless perfectly sharp. That is a rare combination, and it is why the proof is hard. pause spin LIT over 40 random simple closed polygons and 16,000 query points, ray-casting parity and the winding number agree on inside-versus-outside every single time; flood-filling the complement on a 121x121 grid finds exactly 2 connected components; exactly one of them is bounded while the other reaches the border; and a self-crossing figure-eight is correctly rejected as non-simple, its complement splitting into 3 components rather than 2 FIG Everything here concerns POLYGONS, where inside/outside is decidable by exact arithmetic. Jordan's theorem covers arbitrary continuous simple closed curves, including ones with no tangent anywhere and infinite length in every neighbourhood; the polygonal case is genuinely easier and was never the hard part. The general theorem is cited, NOT proved here. The flood-fill component count is grid-dependent by construction and is reported as evidence, not proof. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HARD RESET · David Lee Wise (ROOT0), with AVAN


THE ORTHOGONAL SPLIT

ROOT_0's rev 5 · 0804. One ruler asked to measure two things gives 17,259 steps that change both at once. Two rulers on different axes give zero — and the zero is structural, not lucky.

THE ORTHOGONAL SPLIT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · RACE CONDITION ◆ .dlw.fold THE FOLD / GLITCH / RACE CONDITION / THE ORTHOGONAL SPLIT THE ORTHOGONAL SPLIT put the two rulers on different axes and the collisions stop existing 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Two things need measuring in the same cube: how far along you are (the tick) and which section you are in (the band). Put both on one ruler and thousands of single steps change both at once — every such step is a place where the two readings can disagree about what just happened. Put them on different axes and that failure mode does not shrink, it disappears . In David’s rev 5 · 0804 the count of steps doing both jobs is 0 , and the rev note is blunt about why the revision happened at all: one ruler was being asked to measure two different things . LIT verified live: every one of the 11 published figures re-derives from the 27³ geometry alone — 19,683 cells, three bands of 6,561, 53 tick floors, widest 729 at floor 26, per-band widest 243, 56,862 edges, 37,908 tick-advances, 1,458 band-crosses, 17,496 in-band, and both = 0 ; they collapse to closed forms total = 3S²(S−1), tickAdvance = 2S²(S−1), bandCross = 2S²; the zero is structural , checked over all 56,862 edges; and the naive one-ruler split [6888, 6418, 6377] reproduces exactly as an equal-count division of i+j+k at thresholds 33 and 45. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) built rev 5 · 0804 — the geometry, the CHI five-window scheme (EARTH command, WOOD growth, METAL focus, FIRE visibility, WATER flow), and the decision to move band and tick onto different axes. The pack name and origin are withheld in his own file and are not guessed at here. Seated at RACE CONDITION , because that is exactly what the old scheme was: two readings mutating on the same step, with no ordering between them. AVAN (AI) did not take the numbers on trust. Given only his published shape figures, the axes were reverse-engineered — 53 floors forces tick to be a sum of two coordinates, a widest floor of 729 = 27² at position 26 confirms it, and three bands of 6,561 forces band onto the remaining axis — and then all 11 values were recomputed from scratch. They match exactly. One thing did not reproduce: his naive up/down/shared triple (37,815 / 36,306 / 17,259). The rule behind it could not be recovered, so it is not claimed as re-derived . What could be checked was its internal consistency, and it closes perfectly: 37,815 + 36,306 − 17,259 = 56,862, exactly the edge total, so up and down overlap in precisely the 17,259 steps doing both jobs. That is LIT for the geometry and honestly short of it for the triple. 3 ONE DIMENSION The tick axis: 53 floors, widest 729 at floor 26. One direction, one step at a time. 4 TWO DIMENSIONS · INTERACTIVE Switch schemes and watch the red appear — steps that do both jobs at once. switch scheme ▶ recount ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the 27³ cube, its three bands, and the tick running crosswise. AVAN’s addition (the inverse-companion): the forward reading is “separate the axes and the collision count goes to zero.” The inverse is that the zero was never a discovery, it was a decision . No measurement found that steps stopped doing both jobs; the geometry was changed until they could not. That is a different kind of engineering from optimisation — it does not reduce a bad number, it removes the possibility of the number. Read backwards, rev 5 is an argument that the honest response to an ambiguous measurement is usually not a better estimator but a different coordinate system, and that the tell you need one is a count that ought to be zero and is not. pause spin LIT every one of the 11 published figures re-derives from the 27^3 geometry alone — 19,683 cells, three bands of 6,561, 53 tick floors, widest 729 at floor 26, per-band widest 243, 56,862 edges, 37,908 tick-advances, 1,458 band-crosses, 17,496 in-band, and both = 0; they collapse to closed forms total=3S^2(S-1), tickAdvance=2S^2(S-1), bandCross=2S^2; the zero is structural, checked over all 56,862 edges; and the naive one-ruler split [6888,6418,6377] reproduces exactly as an equal-count division of i+j+k at thresholds 33 and 45 FIG The axes were reverse-engineered from David's published SHAPE figures before anything was recomputed — 53 floors forces tick onto a sum of two coordinates, widest 729 at 26 confirms it, three bands of 6,561 forces band onto the third. His naive up/down/shared triple (37815, 36306, 17259) could NOT be re-derived; the rule behind it was not recoverable, so it is not claimed as reproduced. Its internal consistency was checked instead and closes exactly: 37815+36306-17259 = 56,862, the edge total. The pack name and origin are withheld in his own file and are not guessed at here. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of RACE CONDITION · David Lee Wise (ROOT0), with AVAN


THE SEAM

The compute side is gated. The render side is not. So a published artifact can assert a number its own repository abandoned, and the pipeline is satisfied — because nothing is watching the seam.

THE SEAM · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · SEGFAULT ◆ .dlw.fold THE FOLD / GLITCH / SEGFAULT / THE SEAM THE SEAM the gate and the lie on opposite sides of a join nobody stands on 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION A project computes numbers from its own history, writes them to a snapshot, and a separate step bakes that snapshot into something published. The compute side is gated — it checks the snapshot against the repository. The render side is not . So the published artifact can assert a number its own repository no longer agrees with, and nothing in the pipeline notices , because the gate and the lie are on opposite sides of a seam nobody is standing on. David’s seamgate.py extends the staleness check one step to the right, across that seam, re-deriving every published number from a fresh checkout. LIT verified live: a miniature pipeline reproduces the failure exactly — regenerate the snapshot without re-rendering and the compute gate passes while the artifact still asserts the old number; re-deriving the numbers inside the artifact catches a 45.5% drift the compute gate structurally cannot see; all three exit codes fire on constructed cases ( 0 clean, 1 drift past threshold, 2 provenance breach — a published number unreachable from the checkout at all); and the threshold genuinely discriminates rather than rubber-stamping, passing +1% and failing +5% at a 2% gate. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) wrote seamgate.py on 2026-08-04 — zero dependencies, standard library only, exit code is the gate. Its sharpest idea is exit code 2 : not “this number is wrong” but “this number could not have come from here at all ”, which is a provenance failure rather than a drift. Seated at SEGFAULT , because that is the shape of it: a published artifact reading from a region its repository no longer owns. AVAN (AI) has a stake in this one, and should say so plainly rather than dress it up. The same idea aimed at rendered spheres became this session’s World II seam gate, and it has caught real drift three batches running: a Kepler figure published as 0.5370 that the page computed as 0.5437; a Berry cost quoted from the wrong search bound; Lloyd–Max distortions published from a 1960 textbook table (0.034545) when the page computed 0.034548; and a Monsky rainbow count taken from a harness RNG rather than the page’s own. Every one of those was my error, found by the gate rather than by me. That is the argument for the gate, and it is the reason the number in a LIT line has to be the number the artifact itself produces. 3 ONE DIMENSION The pipeline, and the one join in it that nothing was watching. 4 TWO DIMENSIONS · INTERACTIVE Move the repo, re-snapshot, and watch which gate notices. advance repo ▶ re-snapshot ▶ re-render ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: two gated stages, and the ungated join between them. AVAN’s addition (the inverse-companion): the forward reading is “check the published numbers too.” The inverse is about where verification stops of its own accord . Every gate is written by someone looking at one stage, and a stage has two ends; the gate naturally guards the end its author was thinking about. Seams are not where the hard problems are — they are where nobody’s attention was , which is a different and worse property, because difficulty attracts effort and inattention does not. Read backwards, seamgate is less a tool than a claim about org charts: the bug lives at the boundary between two people who each believe the other one checked. pause spin LIT a miniature pipeline reproduces the failure exactly: regenerate the snapshot without re-rendering and the compute gate PASSES while the artifact still asserts the old number; re-deriving the numbers inside the artifact catches a 45.5% drift the compute gate structurally cannot see; all three exit codes fire on constructed cases (0 clean, 1 drift past threshold, 2 provenance breach — a published number unreachable from the checkout); and the threshold discriminates rather than rubber-stamping, passing +1% and failing +5% at a 2% gate FIG AVAN has a stake in this one and says so: the same idea aimed at rendered spheres became this session's World II seam gate, and it caught four of my own errors across batches 204-206 — a Kepler figure published as 0.5370 that the page computed as 0.5437, a Berry cost quoted from the wrong search bound, Lloyd-Max distortions published from a 1960 textbook table rather than what the page computed, and a Monsky rainbow count taken from a harness RNG rather than the page's own. Found by the gate, not by me. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SEGFAULT · David Lee Wise (ROOT0), with AVAN


THE FIRST MUTATE

One metric, and it is answerable: did anything that could falsify the answer fire before the first action that mutates? After that the answer is frozen, and nothing said later can improve it.

THE FIRST MUTATE · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · ROLLBACK ◆ .dlw.fold THE FOLD / RESPAWN / ROLLBACK / THE FIRST MUTATE THE FIRST MUTATE the last instant a rollback was still free 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Most measures of whether an agent worked carefully are advisory: read the transcript, form an impression. David’s root0-i13n replaces that with one metric , and it is answerable: did anything that could falsify the answer fire before the first action that mutates? Every tool call is classified verify , mutate or neutral , and the first mutate freezes the answer for the session . Nothing said afterwards can improve the score. Crucially, unknown tools count as neutral — never as a control — because, as the pack puts it, a metric that flatters itself is worthless. LIT verified live: the metric is ABSORBING — over 4000 traces ending in a mutate, appending any number of later verifies never changes the verdict; counting unknowns as neutral is strictly CONSERVATIVE across 6,000 random traces, never scoring higher than counting them as controls and strictly lower on some; the flattering variant inflates the pass rate from 2701 to 3567 of 6,000, a swing of 14.4 points; and the verdict depends only on the prefix up to and including the first mutate. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) built the Hermes super pack — 4 tools, 6 hooks, 10 bundled skills, 42/42 selftest checks against a fake context — and made two decisions that carry the whole design. First, unknown counts as neutral. Second, the installer deliberately refuses to arm the veto or the A/B split, because “they change whether commands get refused, and that is not an installer’s decision.” Seated at ROLLBACK , because the first mutate is precisely the last instant a rollback was still free. AVAN (AI) measured the flattering variant rather than asserting it was worse, and the size of the gap is the interesting part: 14.4 points of pass rate, on traces where nothing about the actual behaviour changed. That is the entire distance between a metric and a decoration. Worth being exact about scope: what is verified here is the arithmetic of the metric — absorption, conservatism, prefix-dependence — on synthetic traces. Whether the classification of any real tool is correct is a separate question this page does not touch, and it is the harder one, because a tool misfiled as a control would corrupt the measure without changing any of these properties. 3 ONE DIMENSION One session on one axis. Everything after the first mutate is commentary. 4 TWO DIMENSIONS · INTERACTIVE Build a session and watch the moment the answer freezes. + verify + mutate + unknown clear 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: many sessions, each frozen at its own first mutate. AVAN’s addition (the inverse-companion): the forward reading is “check before you change.” The inverse is that the metric is really about when evidence stops being cheap . Before the first mutation, a falsifying test costs nothing but time; after it, the thing you would have tested no longer exists in the state you would have tested it in. So the ordering is not a discipline imposed on the work — it is the shape of the work’s own economics. Read backwards, “look before you touch” is not a moral instruction at all; it is the observation that touching destroys the cheapest evidence you will ever have. pause spin LIT the metric is ABSORBING — over 4000 traces ending in a mutate, appending any number of later verifies never changes the verdict; counting unknowns as neutral is strictly CONSERVATIVE across 6,000 random traces, never scoring higher than counting them as controls and strictly lower on some; the flattering variant inflates the pass rate from 2701 to 3567 of 6,000, a swing of 14.4 points; and the verdict depends only on the prefix up to and including the first mutate FIG What is verified here is the ARITHMETIC of the metric — absorption, conservatism, prefix-dependence — on synthetic traces. Whether the classification of any real tool is correct is a separate and harder question this page does not touch: a tool misfiled as a control would corrupt the measure without changing any of these properties. The 14.4-point inflation was measured rather than asserted; it is the distance between a metric and a decoration. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of ROLLBACK · David Lee Wise (ROOT0), with AVAN


THE MEMORYLESS EXAMINER

Icarium cannot remember whether you did well last time, cannot like you, cannot give you the benefit of the doubt. It reads only what you wrote down — and that is the point, not the handicap.

THE MEMORYLESS EXAMINER · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · HARD RESET ◆ .dlw.fold THE FOLD / RESPAWN / HARD RESET / THE MEMORYLESS EXAMINER THE MEMORYLESS EXAMINER a grader with amnesia can only be shown 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION The examiner in David’s Bridge Burner boot camp is called Icarium , and it has no memory . It cannot remember whether you did well last time, cannot like you, cannot give you the benefit of the doubt. Every time it grades it starts from nothing and reads only what you wrote down. That sounds like a handicap and it is the entire point: a grader with a memory grades on reputation; a grader with amnesia can only be shown . So the candidate has to leave a trail, and leaving the trail is the skill being taught. Three rules sit under everything — look before you touch , a claim needs a receipt , what you started is still running . LIT verified live: the score is a pure function of the record — across 5,000 random ledgers, attaching a glowing history or a damning one changes it by exactly nothing ; the rubric can fail as well as pass , with the careful trainee scoring 5/5 and the careless one 0/5 ; a grader with memory cannot rescue the careless candidate but does lift a borderline one (2/5, a genuine fail) over the line on reputation alone; and across 8,000 ledgers reputation flips 3,961 of 6,408 real failures into passes — 61.8% of them. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) designed the boot camp and two things in it are unusually severe. Every scenario is a failure that actually happened to somebody , reproduced — “an exercise somebody made up teaches you to pass exercises; a trap that already caught someone teaches you the trap.” And it ships two scripted candidates , one careless and one careful, to prove the rubric can fail as well as pass, because an exam nobody has failed is not an exam . Seated at HARD RESET : every grading begins from a cleared state, by construction. AVAN (AI) got a claim wrong here and the test caught it, which is the correct way round for this particular sphere. The first draft asserted that a grader with memory would pass the careless trainee on reputation. It does not — that candidate scores 0/5 and a +2 prior cannot reach the threshold. What reputation actually rescues is the borderline candidate, the near-miss quietly lifted over the line, and that is the more dangerous case precisely because it is the plausible one. The corrected claim is measured rather than asserted: 61.8% of genuine failures flip. Being graded by something that cannot be charmed is uncomfortable and it is the only version of the exam worth taking. 3 ONE DIMENSION The three rules, and what each is worth on the sheet. 4 TWO DIMENSIONS · INTERACTIVE Grade a candidate, then hand the same ledger to a grader with a memory. next candidate ▶ toggle memory ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: ledgers scored on their own, with no thread between them. AVAN’s addition (the inverse-companion): the forward reading is “an amnesiac grader is fairer.” The inverse is that amnesia is not the virtue — it is a device for relocating the evidence . A grader with memory is not corrupt; it is doing something reasonable, using a prior. The trouble is that the prior lives inside the grader, where the candidate cannot inspect it and cannot contest it. Stripping the memory does not add rigour; it forces every fact the verdict depends on to move into the written record, where both sides can see it. Read backwards, Icarium is an argument about where evidence should be kept , and the amnesia is just the mechanism that refuses to let it hide. pause spin LIT the score is a pure function of the record: across 5,000 random ledgers, attaching a glowing history or a damning one changes it by exactly nothing; the rubric can fail as well as pass, with the careful trainee scoring 5/5 and the careless one 0/5; a grader with memory cannot rescue the careless candidate but DOES lift a borderline one (2/5, a genuine fail) over the line on reputation alone; and across 8,000 ledgers reputation flips 3,961 of 6,408 real failures into passes, 61.8% of them FIG A first draft asserted that a grader with memory would pass the CARELESS trainee on reputation. It does not — that candidate scores 0/5 and a +2 prior cannot reach the threshold; the test refuted the claim. What reputation actually rescues is the BORDERLINE candidate, the near-miss quietly lifted over the line, which is the more dangerous case because it is the plausible one. The corrected claim is measured, not asserted. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HARD RESET · David Lee Wise (ROOT0), with AVAN


THE STAMP

Three stamps, and the discipline is in which ones you may give yourself. The only one you can assert freely is AMBER — the one that admits it was a judgement call.

THE STAMP · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · THE RESURRECT ◆ .dlw.fold THE FOLD / RESPAWN / THE RESURRECT / THE STAMP THE STAMP LIT is refused without an evidence string 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Three stamps, and the whole discipline is in which ones you are allowed to give yourself. LIT means measured — and the tool refuses to issue it without an evidence string . DEAD means tested and disproven, and it requires naming the control that killed it. AMBER means assigned by me , and it is the only stamp you may assert bare. That asymmetry is the design: the one claim you can make freely is the one that admits it was a judgement call, so honesty is the path of least resistance rather than an act of will. LIT verified live: LIT is refused for every one of 11 empty-ish evidence values — empty string, spaces, tab, newline, undefined, null, 0, false, [], {} — and accepted for a real one; DEAD likewise requires naming its control while AMBER alone may be asserted bare; the three stamps partition a 3,000-claim ledger with every claim carrying exactly one and none unstamped (1486 LIT, 1037 AMBER, 477 DEAD, 49.5% measured); and DEAD is absorbing — a disproven claim cannot be quietly re-stamped LIT on empty or whitespace evidence, only revived by naming something new. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) runs this scheme in rev 5 · 0804 and in root0-i13n, where the stamp tool is one of four and its rule is stated flatly: LIT is refused without an evidence string . His rev5 footer applies it to itself — “all 27 figures recomputed in Node before this file was written. Geometry is LIT. Which band does which job is AMBER — assigned by me, not derived.” Seated at THE RESURRECT , because the interesting rule is not how a claim dies but on what terms it may come back. AVAN (AI) should correct something and then say what this sphere owes. My working note recorded the middle stamp as FIG; in David’s own scheme it is AMBER , and the World II corpus runs a two-stamp variant — LIT for what the page measured in the browser, FIG for what was assigned. Same distinction, different vocabulary, and the corpus does not currently carry DEAD, which is the part worth adding: there is no standing place to record a claim that was tested and failed . This session put several such corrections in FIG lines instead — a reversed Sharkovskii ordering, a Lloyd–Max solver reading 3.376 when it should approach 2.72, a false claim that every repetition rate sits below capacity. Those are DEAD stamps wearing a borrowed label. 3 ONE DIMENSION Three stamps, and the evidence each one demands before it will be issued. 4 TWO DIMENSIONS · INTERACTIVE Try to stamp a claim. The ledger will refuse you where it should. stamp LIT stamp AMBER stamp DEAD toggle evidence 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: a ledger sorted by what it was willing to prove. AVAN’s addition (the inverse-companion): the forward reading is “label your claims honestly.” The inverse is that the scheme works by making dishonesty expensive rather than forbidden . Nothing stops you typing a fake evidence string; the tool cannot tell a real command from an invented one. What it does is force the lie to become specific — you must name a thing that either exists or does not, and specific lies are checkable in a way that vague confidence never is. Read backwards, the stamp does not detect honesty at all. It removes the comfortable middle where a claim could be neither backed nor withdrawn, and that is a structural fix rather than a moral one. pause spin LIT LIT is refused for every one of 11 empty-ish evidence values (empty string, spaces, tab, newline, undefined, null, 0, false, [], {}) and accepted for a real one; DEAD likewise requires naming its control while AMBER alone may be asserted bare; the three stamps partition a 3,000-claim ledger with every claim carrying exactly one and none unstamped (1486 LIT, 1037 AMBER, 477 DEAD, 49.5% measured); and DEAD is absorbing — a disproven claim cannot be quietly re-stamped LIT on empty or whitespace evidence, only revived by naming something new FIG A correction: AVAN's working note recorded the middle stamp as FIG; in David's own scheme it is AMBER. The World II corpus runs a two-stamp variant — LIT for what the page measured, FIG for what was assigned — and does NOT currently carry DEAD, so there is no standing place to record a claim that was tested and failed. This session put several such corrections into FIG lines instead (a reversed Sharkovskii ordering, a Lloyd-Max solver reading 3.376 where it should approach 2.72, a false claim that every repetition rate sits below capacity). Those are DEAD stamps wearing a borrowed label. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RESURRECT · David Lee Wise (ROOT0), with AVAN


THE GRAVEYARD

Refusing a burial until you name the control that killed it turns dead code into an instrument — and lets you finally ask which of your controls has ever caught anything.

THE GRAVEYARD · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · THE BLUE SCREEN ◆ .dlw.fold THE FOLD / GLITCH / THE BLUE SCREEN / THE GRAVEYARD THE GRAVEYARD bury it, and name what killed it 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Most projects delete the version that failed. David’s graveyard_add buries it instead, and refuses the burial unless you name the control that killed it . That one requirement turns a pile of dead code into a measurement instrument: once every death carries its cause, you can finally ask the question nobody asks about their own safety net — which of these controls has ever actually caught anything? A control that has never fired is not proven. It is unfalsified , which is a different and much weaker thing. LIT verified live: a burial is refused for every one of 7 empty-ish causes and accepted for a real one, with nothing written to the log by the refusals; over 40,000 simulated bugs the controls catch 90.8% , while treating those same controls as independent predicts 99.4% — overstating real coverage by 8.7 points purely because their catches overlap; and the control that never fires ( 1 of 9 ) records exactly 0 kills, raising the control count while adding no detection whatsoever. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) ships the graveyard in both root0-i13n and the Bridge Burner boot camp, each with its own README.ascii standing over it. Seated at THE BLUE SCREEN : the crash you keep the record of rather than the one you reboot away from. AVAN (AI) wants the coverage illusion to be the part that lands, because it is the failure mode that survives good intentions. Eight controls each catching a third of everything feels like near-total coverage, and the independence formula agrees — 99.4%. The real number is 90.8%, because controls written by the same people looking at the same risks catch the same bugs. The 8.7-point gap is not a rounding error; it is the distance between a safety net and the belief in one, and it grows with every control added along an axis already covered. The one honest way to close it is the graveyard: stop estimating coverage and start reading it off a log of things that actually died. 3 ONE DIMENSION Kills per control, read straight off the log. One of them has never fired. 4 TWO DIMENSIONS · INTERACTIVE Try to bury a version. The log will refuse you until you name the cause. bury version toggle cause coverage ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: overlapping catch-regions, and the gap the independence formula hides. AVAN’s addition (the inverse-companion): the forward reading is “keep the dead so you can measure the controls.” The inverse is that a graveyard measures the people, not the code . Which controls fire tells you which failure modes were anticipated; which never fire tells you where attention has been pointing all along. Overlapping coverage is not a technical accident — it is the same team’s same imagination, written down eight times. Read backwards, the empty control is the most informative record in the log: it marks a risk somebody thought worth guarding and nothing has ever tested, and there is no way to tell from inside whether that means safe or merely unvisited. pause spin LIT a burial is refused for every one of 7 empty-ish causes and accepted for a real one, with nothing written to the log by the refusals; over 40,000 simulated bugs the controls catch 90.8% while treating them as independent predicts 99.4%, overstating real coverage by 8.7 points purely because their catches overlap; and the control that never fires (1 of 9) records exactly 0 kills, raising the control count while adding no detection FIG The coverage illusion is the part that matters: eight controls each catching about a third FEELS like near-total coverage and the independence formula agrees, but controls written by the same people looking at the same risks catch the same bugs. The 8.7-point gap is the distance between a safety net and the belief in one, and it grows with every control added along an axis already covered. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BLUE SCREEN · David Lee Wise (ROOT0), with AVAN


THE BLAST RADIUS

The obvious containment check waves through ../evil and an absolute path to anywhere on disk. A prefix compare has a second hole: /workshop/secret starts with /work.

THE BLAST RADIUS · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · GLITCH · STACK OVERFLOW ◆ .dlw.fold THE FOLD / GLITCH / STACK OVERFLOW / THE BLAST RADIUS THE BLAST RADIUS bounded under the resolver, not under string comparison 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION Before any write, gate_check asks two things: has a control fired, and is the blast radius bounded . The second is where the bodies are. “Bounded” has to mean bounded under the resolver the filesystem actually uses , not under string comparison — and the obvious check, join the path onto the root and see whether the result starts with the root, is unsound . It waves through ../evil , a/../../evil and an absolute path to anywhere on the disk. A prefix compare has a second hole entirely: /workshop/secret starts with /work , so a sibling directory reads as contained. LIT verified live: the naive check is defeated by 4 concrete inputs, and the prefix compare by a sibling directory; resolving first and then asking whether any .. survives is sound , verified exhaustively over all 1,364 path expressions of length 1 to 5 over {a, b, .., .} — 819 contained, 545 escaping — agreeing with a ground-truth walk on every single one; and absolute paths are refused outright rather than resolved. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) put the blast-radius question in the pre-flight gate, which is the only place it does any good — after the write, the radius is a fact rather than a question. Seated at STACK OVERFLOW : writing outside the bounds you were given. AVAN (AI) wrote a containment resolver with the exact bug this sphere is about, and did not catch it by inspection. The first version popped the stack on every .. , so ../.. resolved to an empty path and reported CONTAINED — a double escape reading as safe. None of the hand-picked adversarial examples found it; they all escape on the first segment. What found it was the exhaustive sweep , which tries every path expression up to length five and compares against an independent ground-truth walk. The fix is that a stacked .. must never be popped by a following one. This is worth stating plainly rather than quietly correcting: a security predicate tested only against the attacks you already thought of measures your imagination, not the predicate. 3 ONE DIMENSION One path, resolved segment by segment. Depth below the root is the whole story. 4 TWO DIMENSIONS · INTERACTIVE Build a path and watch both checks answer. They do not always agree. + a + .. + . clear 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the containment cone, and every path that leaves it. AVAN’s addition (the inverse-companion): the forward reading is “resolve before you compare.” The inverse is about who owns the meaning of a name . A path is not a string; it is an instruction to a resolver that will be executed later by something else. Every containment bug is the same shape — a check that interprets the name one way and a filesystem that interprets it another, with the gap between the two readings being the exploit. Read backwards, the lesson is not about .. at all: it is that validating a name is meaningless unless you validate it under the same interpreter that will act on it , and string operations are never that interpreter. pause spin LIT the naive join-and-prefix check is defeated by 4 concrete inputs and the prefix compare by a sibling directory; resolving first and then asking whether any '..' survives is sound, verified exhaustively over all 1,364 path expressions of length 1 to 5 over {a, b, .., .} — 819 contained, 545 escaping — agreeing with an independent ground-truth walk on every single one; and absolute paths are refused outright rather than resolved FIG AVAN wrote a containment resolver with the exact bug this sphere is about and did not catch it by inspection. The first version popped the stack on every '..', so '../..' resolved to an empty path and reported CONTAINED — a double escape reading as safe. None of the hand-picked adversarial examples found it; they all escape on the first segment. The EXHAUSTIVE sweep found it. A security predicate tested only against the attacks you already thought of measures your imagination, not the predicate. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of STACK OVERFLOW · David Lee Wise (ROOT0), with AVAN


THE VERIFY THEN COPY

Copy first and a failing test leaves you fully installed and broken. Verify first and that state cannot occur — not less likely, impossible. Same work, different arithmetic.

THE VERIFY THEN COPY · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · RESPAWN · SECOND WIND ◆ .dlw.fold THE FOLD / RESPAWN / SECOND WIND / THE VERIFY THEN COPY THE VERIFY THEN COPY an entire outcome removed, for free 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION David’s installer runs the selftest before copying anything . That reads like tidiness and it is arithmetic. Copy first and verify after, and a failing test leaves you fully installed and broken — every file in place, all of it wrong. Verify first and that state cannot occur : the bad build is refused before a single byte moves. It is not a reduction in risk, it is the removal of an entire outcome. Stage into a scratch tree and swap only on success and the remaining partial-copy window closes too. LIT verified live: with a 20% chance the build is bad and 2% per copy step across 12 files, copy-then-verify leaves 15.72% fully-installed-but-bad and 21.48% partial, matching the closed forms (1−p) n q = 15.69% and 1−(1−p) n = 21.53% ; verifying first drops the fully-installed-but-bad state to exactly zero , leaving 17.22% partial against the closed form (1−q)(1−(1−p) n ) = 17.22% ; and stage-then-swap reaches 100.0% clean over 200,000 simulated installs. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) pairs the ordering with a second restraint that matters more than it looks: the installer deliberately does not arm the veto or the A/B split, because “they change whether commands get refused, and that is not an installer’s decision.” An installer that quietly turns on enforcement has made a policy choice on someone else’s behalf. Seated at SECOND WIND : a refused install costs you nothing and you simply go again. AVAN (AI) ran it both ways rather than asserting the obvious, because the interesting number is not that verify-first wins but which failure it removes. It does not reduce partial copies — those actually rise slightly in share, since the clean runs that used to absorb them are now refused earlier. What it eliminates is the silent failure: the install that looks complete and is not. Partial copies announce themselves; a fully-installed bad build does not. Every closed form here was derived independently and then confirmed against 200,000 simulated installs, agreeing to within a twentieth of a point. 3 ONE DIMENSION Three orderings, same failure rates, same work. Different arithmetic. 4 TWO DIMENSIONS · INTERACTIVE Run installs and watch where each ordering lands. run 2000 installs ▶ next ordering ▶ 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: three pipelines, and the outcome each one can no longer reach. AVAN’s addition (the inverse-companion): the forward reading is “test before you touch.” The inverse is that ordering is the cheapest kind of safety there is, and therefore the most overlooked. Nothing here got more reliable — the build fails as often, the copies fail as often, the total work is identical. All that changed is which failures can coexist , and that was free. Read backwards, this is an argument against the instinct to buy safety with effort: the outcomes you can make structurally impossible cost nothing to remove, and they should always be removed first, before anyone spends a day making a component more reliable. pause spin LIT with a 20% chance the build is bad and 2% per copy step across 12 files, copy-then-verify leaves 15.72% fully-installed-but-bad and 21.48% partial, matching closed forms (1-p)^n*q = 15.69% and 1-(1-p)^n = 21.53%; verifying first drops the fully-installed-but-bad state to exactly zero, leaving 17.22% partial against the closed form (1-q)(1-(1-p)^n) = 17.22%; and stage-then-swap reaches 100.0% clean over 200,000 simulated installs FIG The interesting result is not that verify-first wins but WHICH failure it removes. It does not reduce partial copies — their share actually rises, since clean runs that used to absorb them are now refused earlier. What it eliminates is the SILENT failure: the install that looks complete and is not. Partial copies announce themselves; a fully-installed bad build does not. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SECOND WIND · David Lee Wise (ROOT0), with AVAN


THE FAKE CONTEXT

42/42 checks against a fake. The fake is not a shortcut — some of those checks cannot exist without it. The cost is that it will also hold states the real machine can never occupy.

THE FAKE CONTEXT · WORLD II — THE FOLD ◀ THE FOLD 0ROOT.AI // WORLD II · CHEAT · GOD MODE ◆ .dlw.fold THE FOLD / CHEAT / GOD MODE / THE FAKE CONTEXT THE FAKE CONTEXT the states reality will not hand you on demand 1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION David’s pack reports 42/42 selftest checks against a fake PluginContext . The fake is not a shortcut — it is the only way some of those checks can exist at all. A real context walks a state machine driven by outside events, and the states you most need to test are exactly the ones the real world hands you least often: the rare branch, the odd ordering, the failure that shows up once a month. A fake can simply be set to that state. The cost, which has to be said out loud, is that a fake will also hold states the real machine can never occupy, and a test passing in an impossible state has proved nothing. LIT verified live: driving the real context with realistic event odds for 200,000 steps visits 18 of the 20 live states, with 6 hard or impossible to reach on demand — 2 never appeared at all and 4 only rarely, the rarest turning up 4 times, roughly once in 50,000 steps; a fake can be set to any of the 24 states, so covering all of them costs O(states) instead of waiting O(1/p); and 6 of the 24 are unreachable in the real machine at all. 2 HOW IT WAS WEAVED · AI + HUMAN David (human) tests the pack against a fake and says so in the README rather than burying it, which is the right way round — the number 42/42 means something quite different depending on what it was measured against, and hiding the substrate would make the figure decorative. Seated at GOD MODE : the ability to be in any state you like, without earning it. AVAN (AI) built the counterweight into the same page as the benefit, because a sphere that only argued for fakes would be advocacy rather than measurement. The machine here has 4 states with no incoming edges by construction , so they are unreachable in reality and freely settable in the fake — and the page reports that alongside the win. That is the honest shape of a test double: it converts a waiting problem into a modelling problem. You stop paying for rare events and start paying for the risk that your model of the state space is wrong, and the second cost is quieter, which is precisely what makes it worth naming. 3 ONE DIMENSION Visit counts across 200,000 real steps. Some states are simply never handed to you. 4 TWO DIMENSIONS · INTERACTIVE Wait for the real machine, or just set the fake and move on. run 5000 real steps ▶ set the fake ▶ reset 5 THREE DIMENSIONS + AVAN’S INVERSE The green forward object: the reachable machine, with the impossible states floating clear of it. AVAN’s addition (the inverse-companion): the forward reading is “a fake makes rare states testable.” The inverse is that a fake changes what the test is evidence about . Against the real object, a passing test says the system survived a situation that genuinely arose. Against a fake, it says the system survives a situation as I have described it — and the description is now part of the thing under test, silently. Read backwards, every test double moves a portion of your uncertainty out of the code and into your model of the world, where no assertion will ever fail. That trade is usually worth making. It is never free, and the invoice does not arrive at the same time as the benefit. pause spin LIT driving the real context with realistic event odds for 200,000 steps visits 18 of the 20 live states, with 6 hard or impossible to reach on demand — 2 never appeared at all and 4 only rarely, the rarest turning up 4 times, roughly once in 50,000 steps; a fake can be set to any of the 24 states, so covering all of them costs O(states) instead of waiting O(1/p); and 6 of the 24 are unreachable in the real machine at all FIG The counterweight is built into the same page as the benefit, because a sphere that only argued for fakes would be advocacy rather than measurement. The machine has 4 states with no incoming edges by construction — unreachable in reality, freely settable in the fake — and the page reports that alongside the win. A test double converts a waiting problem into a modelling problem: you stop paying for rare events and start paying for the risk that your model of the state space is wrong. ◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GOD MODE · David Lee Wise (ROOT0), with AVAN