A line has dimension 1 and a filled square has dimension 2. Replace an object by N copies of itself, each scaled by a ratio r, and its similarity dimension is log N / log(1/r). Chapter 1½ builds a curve from four pieces at ratio 2−3/2, which gives d = 4/3 exactly. It is not the Koch curve, whose dimension is 1.2619. Page 2 and Page 3 move one step up, to chaotic flows in three variables. Page 4 builds a set of dimension π and says what must be true of the space around it.
The curve was checked in exact integer arithmetic. No two pieces cross at any level tried, but from level 3 on pieces touch: 12, 70, 316 touching pairs at levels 3 to 5, and 5,544,270 at level 12. Touching is compatible with the open-set condition that the dimension formula needs, and that condition has not been proved.
A floating-point version of the crossing test gave false crossings at level 5, exactly at the touch points, and its counts were dropped. Page 5 lists what was withdrawn and what is open.
The Reading Room, Page 6, carries Part 11 of the serial, set from its source.
0.3305. Touching pairs per piece on the tent curve at levels 9 to 12 run 0.3299, 0.3302, 0.3304, 0.3305, rising ever more slowly. That is below 1/3, so an earlier suggestion that the value tends to 1/3 is not supported. The extrapolation assumes the differences shrink geometrically, which is itself unproved.
Figures from Book 4, Chapter 1½: book4/fractals.html, with the verifier fractals-verify.py and the touch counters beside it.
A chaotic flow in three variables contracts volume, so its attractor is thinner than a solid, yet it never settles into a curve. For each system the verifier integrates the equations with a fourth-order Runge–Kutta step, discards a transient, and computes the full Lyapunov spectrum by the Benettin method (QR renormalization, tangent map carried alongside the orbit). The Kaplan–Yorke dimension is then read off the spectrum. It is a conjectured estimate, not a proved value, and any exponent within 0.01 of zero is counted as zero.
Two checks guard the code. For systems whose divergence is constant, the exponents must add up to it exactly. The Burke–Shaw field must be unchanged by the half-turn (x, y, z) → (−x, −y, z). All constant-divergence rows pass the sum check to four decimals, and both symmetry tests pass at 200 random points each. For Lorenz, λ1 = 0.909 and the Kaplan–Yorke dimension is 2.062, consistent with the usual values (about 0.906 and 2.06), which are quoted from memory of the literature and were not checked against a source for this chapter.
| System | Common belief | In this run |
|---|---|---|
| Halvorsen, a = 1.89 | Drawn everywhere as a chaotic attractor. | λ1 = +0.0015, indistinguishable from zero, the others negative: a limit cycle. At a = 1.4 the largest exponent is 0.673 and the system is chaotic. Both are on the plate. |
| Aizawa | One set of parameters, one attractor. | Same parameters, two starts. One gave λ1 = 0.096, chaotic. The other, the one many write-ups use, gave exponents (0, −, −): a limit cycle. A single parameter set can hold more than one attractor. |
| Thomas, b = 0.208186 | Chaotic. | λ1 = +0.0023 here, not distinguishable from zero. At b = 0.18 it is 0.030 and the system is chaotic. It sits at the edge. |
| Sprott’s case A | A strange attractor. | Exponents sum to about zero (−0.0039). It is conservative, so it has no attractor: a chaotic sea among regular orbits that does not contract. On the plate as a contrast. |
Each exponent is a finite-time estimate from one orbit. Where two starts were tried they agree to about two decimals, and no more precision is claimed. This is not a proof of chaos for any system.
The chapter does not claim the systems share an invariant. They share a form, three variables and a smooth nonlinear flow, and each has its own computed numbers.
| Item | Source, and its limit |
|---|---|
| Lorenz, Chen and Lü equations and classical parameters | Read from arXiv 1409.8649. The constant divergences, −10 and −19, are the desk’s arithmetic from the paper’s formula, and the verifier reproduces them. |
| Aizawa | From a web write-up that is internally inconsistent about the parameter e: it is listed but missing from the written z-equation. The form with the e term was used, as the write-up’s own code does. |
| Burke–Shaw | Parameters s = 10, v = 4.272 and the half-turn symmetry are from Paul Bourke’s page, which gives the equations only as an image. The equations used are the standard form written from memory. The symmetry check agrees with his statement and does not prove the form is the published one. |
| Sprott’s case A | From a course page. |
| Rössler, Halvorsen, Thomas | Standard forms from memory, not fetched for this chapter. |
| Page | What it is good for, and what to read with care |
|---|---|
| Lorenz · Chaos Theory and dm³ | The equations, fixed points, exponents (+0.906, 0, −14.572) and Kaplan–Yorke dimension 2.06 agree with this chapter’s run (+0.9086 and 2.062). Its claim that the Lorenz flow sits in a contact manifold as a Reeb flow is not supported: a Reeb flow preserves volume, and Lorenz contracts it, with divergence −13.667, the same number the verifier reproduces. |
| μ · The Lyapunov Operator, F · Catastrophe Theory and Disaster Theory | Readable accounts of the exponent and of Thom’s catastrophes, each now carrying a corrections box listing statements that do not hold: fold location, catastrophe labels, the n-bonacci rate, the Reeb divergence. Nothing in them is a computed result about the systems on this chapter’s plate. |
| The Banking Butterfly | A settlement-arithmetic paper that borrows the Lorenz picture as an analogy. It says itself that the mechanism is not chaotic, and it does not use catastrophe theory. |
This chapter’s role is narrower than those pages: it adds the other systems and a computed check for each.
Continued on Page 4 · PiA vector space has a whole-number dimension. A set has other dimensions that measure how it fills space, and they take any real value from 0 up to the dimension of the space it sits in. A set of dimension 3.14159… needs at least four dimensions around it, and four is enough.
| Step | Derivation |
|---|---|
| The Cantor set | Let s = π − 3 = 0.14159… Two pieces, each a copy of the whole scaled by r, one at each end of [0, 1]. The similarity equation 2·rs = 1 gives r = 2−1/s = 0.0074812…, so 1/r = 133.67… Since r < 1/2 the pieces do not touch. |
| Times a solid | Take C × [0,1]3 in R4. At scale ε = rk the count is N(ε) = 2k·(r−k)3 = ε−(3+s) = ε−π, using 2 = r−s. |
| What is quoted | That the Hausdorff dimension of the dust is s (open set condition; Moran, Hutchinson) and that the product with R3 adds the dimensions (Falconer) are standard theorems, quoted, not derived. |
The verifier’s output matches: log 2 / log(1/r) = 0.141592653590 = s, and 2·rs = 1.000000000000. At the level-aligned scales the exact count gives s at every level checked, 1 to 6. A straight fit across ten decades gave 0.158 against the true 0.142, which is what box counting does on a very sparse set, and the reason to prefer the exact level count.
The grid constructions of Chapter 1½ give dimensions ln N / ln m for whole numbers N and m. In the unit 4-cube N can be as large as m4. The verifier found m = 37 with N = 84,460 giving 3.141593, an error near 4×10−8. It does not give π exactly: that would need mπ to be a whole number, and this check does not settle whether it can. The non-lattice ratio, 1/r = 133.67…, sidesteps the question.
Proposition, proved on paper: lattice dimensions are dense in (0, 4], with 0 ≤ d − ln N / ln m ≤ ln 2 / ln m. This is elementary and may well be known; no literature search was made.
| Route | What it gives |
|---|---|
| Certified bracketing | mπ lies strictly between two consecutive integers for every m from 2 to 1,000,000, by interval arithmetic. Nearest miss: 2.9×10−7, at m = 973,492. Nothing beyond the limit. |
| Six exponentials theorem | At most two primes p can have pπ algebraic. Says nothing directly about composite m. |
| Gelfond–Schneider, Baker | Do not apply: π is neither an algebraic exponent nor an algebraic coefficient. |
| Schanuel’s conjecture | Would, as far as the author knows, settle it. Conditional on an unproved conjecture; the implication is not derived. |
| Heuristic | Fractional parts of mπ behave like random numbers. Evidence, not proof. |
Machine-checked, by the author on Lean v4.32.0 with only the standard axioms: the algebra of the construction (PiDimension.lean) and the density statement with the crude bound (LatticeDensity.lean). Neither proves a Hausdorff dimension. The certificates for mπ are computed, not Lean-checked.
This is a construction, not a natural object. π is not special: the same recipe gives e, 3.2 or 3.9. No four-variable flow was computed, so the chapter makes no claim about any natural set near π.
Continued on Page 5 · Corrections and the Open Desk| Where | What changed |
|---|---|
| Ch 1½ · floating-point crossing scan | Superseded by an exact integer scan at nine rational-tangent angles. The floating-point test gave nonzero crossings at level 5, exactly at the touch points where rounding flips the answer. Its counts are not to be quoted. |
| Ch 1½ · touching pairs per piece → 1/3 | Not supported. Levels 9 to 12 give 0.3299 to 0.3305, below 1/3. |
| Ch 1½ · lattice alignment makes 45° special | Tested and dropped. Every rational tangent gives lattice vertices and those angles still cross. What differs is that at 45° the segment directions close up into a finite set. |
| Lorenz · Chaos Theory and dm³ | The claim that the Lorenz flow is a Reeb flow is not supported, because a Reeb flow preserves volume and Lorenz does not. |
| μ, F, Catastrophe and Disaster pages | Corrections boxes added on 29 October 2026 listing the statements that do not hold. |
| Ch 2½ · Halvorsen at 1.89, Sprott’s case A | Shown as a limit cycle and as a conservative system respectively, not as strange attractors. |
| Item | State |
|---|---|
| Open-set condition for the 4/3 tent curve | Checked numerically to level 5 and by touch counts to level 12. Not proved, so d = 4/3 is the similarity dimension and not a proved Hausdorff dimension. open |
| Touch counts beyond level 8 | The fast vertex counter agrees with the slow one at levels 3 to 8 and is assumed to agree beyond. open |
| Odd-level half of the grid lemma | FractalGridLemma.lean covers the even levels only. The tent argument for odd levels, and the identification of the two-step generator with two refinements, are by hand. open |
| Hausdorff dimension of the π set | Rests on two quoted theorems. The sharp lattice bound is on paper only. open |
| mπ a whole number? | Open as far as the author knows. Proved only for m up to 106. open |
| Attractor equations from memory | Rössler, Halvorsen, Thomas and the Burke–Shaw form are not checked against a source. open |
| A four-variable flow with dimension between 3 and 4 | None computed. open |
book5/ExistenceBanach.lean | Written this week; not yet compiled. Nothing is claimed for it until it has been run and its output read. open |
You are invited to join a math study group that meets online, in the library. Everyone is welcome.
To join, or to ask when it meets, write to g6llc@proton.me.
The Reading Room follows on Page 6.
Continued on Page 6 · The Reading RoomThe first sign of other people was smoke — a thin grey thread rising above the canopy, ordinary and domestic in a way that struck Fermat, after five months of storms and reefs and jaguars weighing their scent, as almost unbearably moving. Someone, somewhere ahead of them, was cooking fish over a fire, entirely unaware that two ruined Europeans and a very large, very calm water buffalo were about to walk into the middle of their afternoon.
It was children who found them first — three of them, barefoot, fishing spears forgotten in their hands, staring up at the buffalo and its passengers with an expression that held far more curiosity than fear. One of them said something in a language Fermat didn't recognize, quick and bright, and ran ahead through the trees before he could think of a way to answer.
"They've gone to tell someone," Marie said.
"I imagine they've gone to tell everyone," Fermat said. "I would, in their position. I cannot think of a single thing in my own life half as interesting as what they just saw."
The children's voices carried ahead of them through the trees, rising and multiplying, and somewhere in that relayed excitement Fermat understood, with the particular relief of a man finally permitted to stop calculating, that the worst of the journey was behind them. There was nowhere left to fall. Whatever came next — soldiers, questions, the long slow work of inventing a life — could not possibly be worse than the reef, or the crew, or the open sea.
It was exactly then, with the danger finally behind them and nothing left to keep her upright but will, that Marie's grip on him loosened, and she slid sideways off the buffalo's broad back before he could do more than lurch toward her.
He caught her before the mud did — his own arms shaking with an exhaustion he'd been refusing to acknowledge for hours, his legs barely steadier than hers — and hauled her, with a groan he made no effort to disguise, back across his own lap on the buffalo's back, her head falling against his soaked and filthy shirt.
"You are not permitted to die of relief," he told her, quite sternly, though she couldn't hear him. "I did not carry us through a reef and a curse and an army of blue-clawed demons to lose you to a fainting spell."
She did not answer, and the buffalo did not stop walking, and Fermat — dead tired in every part of himself, sixty-four years old and further from home than he had ever imagined being — found the only strength he had left was in his voice, and used it, murmuring to her in a low, steady mixture of Latin and old French that had nothing to do with what he was saying and everything to do with simply not stopping.
"We have hit the ceiling," he told her, over and over, counting it out against the rhythm of the buffalo's slow, unhurried steps. "There is nowhere left to fall. Do you hear me, Marie? The descent has a floor, and we have found it, and now there is only the climbing back up."
The gates of the fort, when they finally reached them, were already crowded with soldiers who had clearly been told to expect something, though nothing in their faces suggested they had believed it until they saw it for themselves: an enormous, mud-slicked, unbothered beast walking out of the mangroves at the exact center of the afternoon, carrying on its back an old man who looked like he had not slept in a year, cradling an unconscious young woman with a tenderness that belonged to no invading army anyone had ever been trained to fear.
A young officer took one look, crossed himself, and turned to shout something back into the fort that Fermat didn't understand but recognized instantly all the same, because it was the sound every group of frightened, curious men made in every language he had ever learned: come and see this.
He straightened his back as best his exhausted spine allowed, arranged his face into the calm, unhurried authority of thirty years spent commanding a courtroom's attention, and began, in careful, formal Portuguese, to explain that he was a shipwrecked merchant, that the lady was his charge, and that he would be very grateful indeed for a place to let her rest.
It was, he thought, the first entirely true thing he had said aloud in five months, if one did not count the merchant part, or the shipwreck, or very nearly everything else. But the gratitude, at least, was real. He held onto that, and let the rest be handled, for now, by the last reserves of a mind that had gotten them this far on arithmetic alone.
Part 11 of 18 · Fiction, drawn from the documented world; the ledger below says which is which. Set from its source file, as written.
Rights notice. Pierre et Mademoiselle is not covered by this issue’s CC BY 4.0 licence. The serial and the work it belongs to — a limited series in development — are © 2026 Pablo Nogueira Grossi / G6 LLC, all rights reserved. No licence to reproduce, adapt, translate or perform it is granted here.
Documented: nothing in the series ledger (FACT_AND_FICTION.md) bears on this part. It has no entry for the smoke, the children, the officer or the gates, and this part names no place, so it asserts nothing documented about any.
Invented: the three children and their running ahead; Marie’s collapse and Fermat’s counting aloud to her; the soldiers at the gates, the officer who crosses himself, and the account of himself as a shipwrecked merchant. Marie is the series ledger’s item 2 and the voyage its item 5; both are invented, and the buffalo is the relocation recorded for No. 8.
Each issue, when it goes out. No list bought, harvested, or lifted from a conference roster. Unsubscribe by replying with the word stop.
docs/rh-criteria-check.py — pre-registered tests of the Lagarias and Robin inequalities, zeros on the line to height 300, and Re(ξ′/ξ) near the zeros; four controls, all fired. Standard library, numpy and mpmath.tools/corpus_count.py · tools/self_reference.py — entity-aware counting, and the vicious-circle test. Both standard library only.sorry, standard axioms only.