The Ladder Index
Every place the substrate framework reads the √2 ladder, sorted by the two poles — structures that lock onto a rung and structures that flee to the gap — and broken out by domain
The Two Poles of the Ladder
The same ladder runs to two poles. Structures that must bind sit on its teeth — the octave and √2. Structures that must never overlap flee to its one gap — the golden ratio. Which pole a structure takes is fixed by what it is for.
The two poles. One ladder runs to two poles, and which one a structure takes is fixed by what it is for: structures that must bind sit on the comb’s teeth (the octave and \sqrt2); structures that must never overlap flee to its one gap (\varphi on a circle, blue noise on a plane, a prime in time). \sqrt2 — the tritone — is the hinge between them. This index catalogs where every structure in the paper falls.
What this page is
The substrate ladder is the claim that the recurring, discrete, log-spaced scales across the paper — grid-cell modules, cochlear octaves, vesicle radii, organelle sizes, EEG bands, earthquake magnitudes — are the discrete-scale-invariance tower of a critical superfluid, with the pairing factor \sqrt2 (\xi^2 = 2\,\xi_\text{GP}^2, the lattice’s anti-phase breath) as the fundamental half-octave and the octave 2 = \sqrt2^{\,2} as two of them. The ladder has one ratio and two ways to meet it (the opening of the chapter):
- A structure can lock onto a rung. To bind, nest, resonate, or hand energy back and forth, parts must share a scale, and the cleanest place to share is a whole-number ratio. Structures whose job is to couple climb onto the teeth — the octave and \sqrt2. On a tooth a structure is plugged into the substrate’s lossless channel: it can hold energy and pass it on.
- Or it can refuse every rung. Some systems fail precisely when their parts fall into step — leaves that shadow one another, rhythms that lock into a seizure, a sensor array that aliases. These sit as far from every tooth as a ratio systematically can: at the gap, whose extremal solution is the most-irrational \varphi = 1.618 on a circle, a disordered-hyperuniform blue noise on a plane, and a prime in whole-number time.
That gives the chapter its single most testable claim, with a sign that can be called before the measurement: name what a structure is for and its pole is fixed — bind, nest, resonate, exchange energy \to a tooth; stay independent, avoid overlap, refuse resonance \to the gap. This page is a catalog of every place in the paper where that reading is made, so a reader can see the pattern without walking the whole sidebar. The structural anchors to hold while reading are the teeth and the gaps, the breath is the ladder, and why structures seek the rungs; the modon index is its companion, cataloging the topology the same way this one catalogs the spacing.
Columns in the tables below:
| Column | What it carries |
|---|---|
| Structure | The named structure, linked to the chapter where the framework develops it |
| Where | The domain and characteristic scale |
| Ratio / value | The rung (\sqrt2, the octave) or the gap value (\varphi, blue noise, a prime), with (measured) or (predicted) where the chapter pins it |
| What the ladder reading adds | The framework-specific claim, prediction, or honest caveat |
Two cautions a reader should carry the whole way down, both earned in Honest accounting. First, \sqrt2 as the specific, universal period is so far carried cleanly by one datum — the grid-cell module ratio 1.42; most other entries are either the looser coarse family (clustering firm, the exact ratio open) or framework predictions awaiting raw machine-measured distributions. Second, the anti-lock pole has three genuinely chemistry-free witnesses — phyllotaxis, the retinal cone mosaic, the periodical cicada — and a much larger set of applications of the same sign-rule inside the framework’s own biology (the genetic code and the cell’s address codes). The applications are marked as such; they extend the rule, they do not independently test it.
Part I — The Lock Pole
Structures on the teeth: parts that must bind, nest, or resonate together climb onto a whole-number ratio, the octave 2 = \sqrt2^{\,2} or the bare half-octave \sqrt2. On a tooth a structure is plugged into the lossless channel — the address of the substrate’s anti-phase exchange.
The substrate’s own anchor rungs
The ladder is not abstract — the substrate lays its own ruler out in two places, and these are the marks every other entry is read against.
| Structure | Where | Ratio / value | What the ladder reading adds |
|---|---|---|---|
| In-plane pairing \xi_\text{GP} \to \xi | the vacuum, \sim 70\;\mum \to 100\;\mum | \sqrt2 (exact) | The bare rung itself. The close-packed core lattice is exactly twice as dense as the lattice of fermions, \xi^2 = 2\,\xi_\text{GP}^2, so the two scales it sets stand at \sqrt2 — the factor of two is the pair; its square root is the rung. |
| Vertical sheet spacing d_\text{GJO}/2 \to d_\text{GJO} | the vacuum, 8\;\mum \to 16\;\mum | 2 (the octave) | The counter-rotating boundary to the like-handed sheet spans exactly one octave — the substrate’s own vertical geometry is two half-octaves stacked. Read off polarized light, not biology, in crystal optics. |
Quantum and molecular
| Structure | Where | Ratio / value | What the ladder reading adds |
|---|---|---|---|
| Aromatic ring (benzene) | \sim 1.4 Å | integer-nest (4n+2) | The molecular archetype of the lock pole: every chiral circulation balanced by its counter-rotating twin (the pairing-two), the ring plugged whole into the lossless channel. The diamagnetic ring current is the coin at its smallest closed instance. |
| Cooper pair vortex | \xi_\text{BCS} \sim 38–100 nm | the paired breath | The cleanest paired breath in the framework: two electrons of one chirality breathing anti-phase across a shared vortex — the motif the whole ladder replicates, here gone macroscopic. |
| Type-I/II threshold \kappa = 1/\sqrt2 | Ginzburg–Landau | \sqrt2 (exact, textbook) | The self-dual point is \xi_\text{BCS}^2 = 2\lambda_L^2 — the same pairing-two (length^2 = 2\timeslength^2) as \xi^2 = 2\,\xi_\text{GP}^2. A structural match, not yet derived from substrate parameters. |
| Codon base-stacking | \sim 3.4 Å | integer-nest | Aromatic stacks nesting at the lock pole; the machinery that reads the code locks even as the sequence it carries anti-locks. |
Brain and perception
| Structure | Where | Ratio / value | What the ladder reading adds |
|---|---|---|---|
| Grid-cell modules | entorhinal cortex | 1.42 \approx \sqrt2 (measured) | The single cleanest datum in the chapter — adjacent module scales step by 1.42 against \sqrt2 = 1.414, agreement to 0.4\%, across animals. The bare rung, read directly. |
| Cochlea and the musical octave | basilar membrane | octave 2; tritone \sqrt2 | The one place biology builds an explicit frequency analyzer, it builds a logarithmic ruler whose axis of symmetry is the \sqrt2 tritone — octaves, never overtones. |
| Theta–gamma coupling | cortical column | integer (\sim 2 = \sqrt2^{\,2}) | Binding rides an integer count of gamma cycles on each theta cycle — the cleanest tooth, the octave sub-lattice the column occupies when it binds (Lisman–Idiart). |
| EEG band centers | cortex, 10^5\times size range | geometric, \sim 2 (measured) | Penttonen–Buzsáki find the centers in a geometric progression conserved across mammals — the log ladder itself. The octave structure is robust; the open question is whether the fine step is \sqrt2 or \varphi (the rest-EEG leans \varphi — see Part II). |
Cell and organelle
| Structure | Where | Ratio / value | What the ladder reading adds |
|---|---|---|---|
| Vesicle coat radii | \sim 30–100 nm | \sqrt2-spaced (predicted) | “The tightest test the cell offers.” Clathrin/COP coats nest by powers of \sqrt2; the cleanest comb test the paper proposes, awaiting raw cryo-EM radii. |
| Curvature scaffolds | clathrin pit \sim 100 nm, caveola \sim 70 nm | \sqrt2 (predicted) | A caveola \to clathrin pit is one half-octave (70\times\sqrt2 \approx 99). The membrane curvature catalogue is the vesicle-coat catalogue one stage earlier — one dataset at two stages. |
| Microtubule resonance cascade | 25 nm wall | self-similar tower | Bandyopadhyay’s “fractal, scale-free” cascade is a literal DSI claim — but a coarse one: bands fold between the \sqrt2 teeth (period \ne \sqrt2), the two-families tension with numbers. |
| The single cell’s nested modons | \sim 100 nm \to \xi | \sqrt2 and octaves | Organelle, vesicle, and the 8/16\;\mum boundary-and-sheet spacings sampled inside one coherence cell — five or six rungs in one place, the cleanest fold the paper offers. |
Classical and macroscopic
The same discrete-scale-invariance signature in classical media — exactly where Sornette finds log-periodicity. These are coarse-family by default: the log-periodic structure is firm, the exact \sqrt2 period is the open question.
| Structure | Where | Ratio / value | What the ladder reading adds |
|---|---|---|---|
| Gutenberg–Richter / Sornette log-periodicity | seismicity | log-periodic (predicted comb) | The classical, chemistry-free face of the ladder — a macroscopic comb test on existing seismic catalogs. Fold the log-periodic corrections; if they land on a \sqrt2 comb the reading earns its keep. |
| Lightning branch-point spacing | km-scale bolts | clustering (predicted) | Branch points should cluster at chirality-coherent substrate domain edges — the ladder’s comb test at km scale, LOFAR-resolvable. Coarse family: clustering firm, ratio open. |
| DESI dark-energy bore crests | cosmological | \approx 1.5 (measured-ish) | A chirped, rank-ordered dispersive shock wave — DSI written across the sky, the transcritical crossing M=1 in the role of the critical point. Crest spacing \approx 1.5 is the bore’s own dispersion, not the bare \sqrt2: the coarse family at its largest. |
| Currency denomination ladders | money | \sim 2–5\times | Log-periodic denomination rungs and Sornette crash precursors — the same signature in a medium with no chemistry at all. Coarse family. |
Part II — The Anti-Lock Pole
Structures in the gap: parts that must never resonate, overlap, or be confused sit as far from every tooth as a ratio can. The gap is one principle — maximal incommensurability — in three arithmetics, the variable’s own space picking the geometry (one principle, three faces). Below the three faces is the labelling route: the same job met without geometry, where the symbol space is too cramped to spread.
The circular face — \varphi (reals mod 1, continued fractions)
Elements added one at a time around a centre dodge every rational lock by taking the most-irrational rotation.
| Structure | Where | Ratio / value | What the ladder reading adds |
|---|---|---|---|
| Phyllotaxis | meristem, golden angle 137.5° | \varphi (measured, C=+1.00) | The chemistry-free witness that de-risks the brain. A spatial, neuron-free system landing dead on the \varphi gap — and substrate-level physics, not botany: Levitov 1991 derives the golden angle as the ground state of a layered-superconductor flux lattice (the substrate’s own geometry). |
| Resting-EEG fine ratio | cortex at rest | \varphi (measured, p<10^{-4}) | When the cortex rests it spaces adjacent rhythms at \varphi so they cannot bind; when it binds it locks them at the octave. Seen alone this looked like the substrate’s \sqrt2 losing to the brain’s \varphi — seen beside phyllotaxis it is the ladder’s own complement, the gap the teeth define. |
The planar face — blue noise (the plane, Fourier)
A plane offers no single rotation to be irrational about, so the anti-lock optimum is a disordered hyperuniform packing — as uniform as a lattice on long wavelengths, carrying no Bragg comb.
| Structure | Where | Ratio / value | What the ladder reading adds |
|---|---|---|---|
| Retinal cone mosaic | retina | blue noise (measured, \sigma^2/\langle N\rangle \approx 0.1) | The chemistry-free planar witness. Cones tile blue-noise so aliasing scatters into incoherent noise rather than coherent false pattern (Yellott 1983; Jiao et al. 2014) — the gap’s planar face as \varphi is its circular face. |
| Downstream retinal mosaics | bipolar / amacrine / RGC | blue noise (predicted) | The anti-lock signature should propagate up every image-sampling layer, degree tracking each layer’s aliasing budget. Untested: the literature characterizes these only by a regularity index, never hyperuniformity. |
| Chloroplast array | mesophyll | blue noise (predicted) | The plant-kingdom twin of the cone mosaic, with no shared chemistry: chloroplasts tile the periclinal face to collect light uniformly without self-shadow — an absorptive blue noise where the retina’s is detective. |
| Transformer superposition | residual stream, d_\text{model} sphere | Thomson/Tammes spread (predicted) | Nearly-orthogonal feature directions packed on the hypersphere to store without interference — the spherical twin of blue noise. SAE feature directions should be sub-Poissonian on the sphere. First non-biological planar witness. |
The discrete-temporal face — primes (the integers, gcd)
When the period is counted in whole generations there is no irrational to take; the most a whole number can do is share no factor with the cycles it must dodge.
| Structure | Where | Ratio / value | What the ladder reading adds |
|---|---|---|---|
| Periodical cicada | Magicicada broods | primes 13, 17 (measured) | The chemistry-free temporal witness — the datum is a pair of integers. Both prime; a prime period co-emerges with a threat cycle only every \text{lcm}, minimizing predation overlap and hybridization. Substrate-level by the same move as \varphi: Goles 2001 shows primes emerge as the attractor of a generic, cicada-free avoid-resonance dynamic. |
The labelling route — applications inside the framework’s own biology
Where the symbol space is too cramped for a clean geometric spread, a structure meets the same avoid-confusion job not by packing but by labelling — routing the unavoidable confusions onto harmless equivalents. These are applications of the sign-rule, not independent chemistry-free witnesses: the rule reaching into the framework’s own biology, and outliving the geometry that first revealed it.
| Structure | Where | The move | What the ladder reading adds |
|---|---|---|---|
| The genetic code | 64 codons | route confusions onto synonyms | The cleanest labelling case: too cramped to separate all codons, so it makes the unavoidable confusions harmless (synonyms) and keeps the distinguishable swaps to carry meaning. Pays out a number — silent transversions carry \sim 3.6\times the codon-stamp difference of silent transitions, so among “silent” mutations it is the transversions that quietly reshape a fold (ClinVar-testable, blind to the translation-speed account). |
| Olfactory receptor repertoire | \sim 400 ORs | spread the family in feature space | The aromatic cage locks one binding event; the OR family anti-locks so the combinatorial code stays distinguishable — same job as the cone mosaic, by labelling. Predicts the \Phi_\text{pocket} cloud is disordered-hyperuniform in feature space. |
| Vesicle & glycan routing codes | SNARE / Rab / TGN | spread the address space | The machinery locks (the four-helix SNARE bundle rings shut); the address code anti-locks so destinations stay distinguishable. Mis-fusion should fall on the least-distinguishable pairs. |
| Nuclear & mitochondrial import codes | NLS / MTS | spread the targeting addresses | The two double-wrapped organelles join the cell-wide family of address codes: machinery locks to commit an import, the address space spreads so destinations stay distinct. |
| Dentate-gyrus pattern separation | hippocampus | decorrelate into orthogonal codes | The anti-lock partner of CA3 pattern completion: DG expands and sparsifies inputs into near-orthogonal codes so memories stay separable — the labelling route in representation space. |
| Symplast callose gating | plant tissue | gate a continuous medium | Callose closes pores of an otherwise-continuous symplast to keep tissue domains distinguishable — the gating route to the gap, where the cells cannot be spread. |
Part III — Both Poles at Once
A handful of systems do not take a fixed station on the lock-to-refuse line — they occupy both poles, and the framework’s richest finding is that each does so by a different mechanism. The brain alone realizes the poles three ways; the set now reaches from cortex to commerce.
| System | Both poles realized by… | Binds (lock pole) | Stays separable (anti-lock pole) |
|---|---|---|---|
| Cortex | state (slides in time) | octave nesting when awake / binding | \varphi-spaced rhythms when resting |
| Prediction engine | operation | binding hypotheses on the teeth | separating them so bindings don’t collide |
| Eye | subsystem (both at once in space) | the disc stack, a lattice tuned to absorb the photon | the cone mosaic, blue noise that refuses the image |
| Bilateral hemispheres | architecture (two-modon hardware) | callosal coupling pulling the pair onto a shared tooth | resting \varphi-detuning that keeps the pair two |
| Cortical resonator | transition law | the SNIC lock at K_c = |\Delta\omega| | the \varphi-detuning that sets the threshold |
| Hippocampus | circuit stage (in series) | CA3 completion (attractor) | DG separation (decorrelation) feeding it |
| Body modon (HRV) | autonomic regulation | RSA / baroreflex / cardiac coherence locking on rungs | broadband 1/f “refusing to lock,” staying adaptable |
| Market | behaviour | one clearing price, settlement, entrainment | diversification — uncorrelated holdings (the crash is the seizure) |
| Transformer | representational geometry | attention routing, induction, shared subspaces | superposition — near-orthogonal features that don’t interfere |
| Music | tension and release (the slide made into art) | consonance on the teeth — the cadence, the groove, unison | dissonance in the gap — the suspension, the tritone, the unresolved |
| Chloroplast | subsystem + state | the grana disc stack catching light | the chloroplast array tiling without self-shadow |
| Calvin system | discrimination + separation | the 3-turn loop’s integer conservation | RuBisCO’s CO₂/O₂ commit gate (the hardest anti-lock — no synonym, no spread) |
| Vascular system | organ position | the trunk locked for transport efficiency | distal organs’ spread cavitation thresholds (fail first, protect the trunk) |
| Meristem | developmental sequence | canalisation locking each organ into the vein | golden-angle placement separating it from its neighbours |
| Forest network | network topology | the mycelial web binding plant modons into one cell | modularity that won’t let a pathogen percolate |
The reading runs from the cortex sliding by state down to a galaxy that chooses nothing: a sign-rule holding from the meristem to a market to the main asteroid belt is fixed by what a structure is for, not by what it is made of. The first non-living anti-lock demonstrations are physical — Levitov’s flux lattice (\varphi in a superconductor) and the Kirkwood gaps, where survivors flee Jupiter’s integer commensurabilities exactly as the cicada flees its threat cycles.
Part IV — Reading the Ladder Honestly: What Is and Isn’t a Rung
A concept is understood partly by its boundaries. Four patterns recur across the paper that look like the \sqrt2 ladder and are deliberately not placed on it — keeping them separate is what keeps the ladder falsifiable. This part is the catalog of those distinctions, drawn straight from Honest accounting and Harmonic is the wrong template.
Harmonic cavities — a length, so overtones and integer counts (the keyboard-vs-string cut)
A structure with a length of its own rings at integer multiples or selects whole-number symmetries — the string, not the lengthless keyboard. These are the complement of the \sqrt2 tower, not rungs of it.
| Pattern | Where | Why it is not a \sqrt2 rung |
|---|---|---|
| Microtubule N=13 protofilaments | 25 nm wall | A closed wall has a circumference, so it selects 13 the way a pinned string selects its overtones — an integer closure, not a scale rung. |
| Axoneme 9{+}2 | cilium / flagellum | The nine-fold symmetry is an azimuthal count of a bounded ring — same class as N=13 and the nuclear pore’s eight-fold. |
| Golgi cisternal count / nuclear-pore 8-fold | organelle | Preferred integers (the lock pole’s count of nesting sheets); the gap between cisternae is the ladder length, the count is not. |
| Photonic band gap | photonic crystal | Locks via the harmonic condition because the crystal has a period — the optical twin of the aromatic ring’s integer-k series. |
| p-mode comb / coronal-loop QPP | Sun | The Sun is a bounded acoustic cavity: equal-\Delta\nu overtones, a length-bearing comb. The complement to the substrate’s lengthless ratio. |
| Recurrence-interval ladders | volcanism, slow slip, Wilson cycle | T_0, 2T_0, 3T_0 — a system with its own clock ringing in overtones (a string), explicitly not the geometric \sqrt2 tower. “Stromboli rings like a string.” |
The spatial template — hexagonal/triangular packing (the lock pole’s spatial face)
The substrate has two templates that must not be conflated: the in-plane sheet packing (a geometry) and the \sqrt2 scale-ratio ladder (a tower of sizes). The packing recurs across orders of magnitude, but what recurs is the geometry, not a spacing.
| Pattern | Where | Why it is not a \sqrt2 rung |
|---|---|---|
| Ice I_h, snowflakes | Å → km | Hexagonal symmetry is the lock pole’s spatial face, carried from a benzene ring up to a km-wide crystal — the same template, not a rung on the scale tower. |
| Silicate sheets → patterned ground | Å → decameter | The hexagonal template spanning \sim 11 orders within earth alone; only the symmetry is substrate-set, the polygon size is frost-mechanics-set. |
| Hexagonal cooling joints | basalt columns | The spatial template in cooling lava; column diameter is \kappa-\tau_\text{cool}-set, coarse. |
| Tkachenko vortex lattice / ecliptic plane | atmosphere → solar system | The triangular/planar packing, sibling of the scale ladder — the lattice fixes the arrangement, R_\text{cross} or dynamics fix the size. |
The coarse family — mechanism-set nesting, \times 6–12 steps
Real ladders whose ratio is not a clean power of \sqrt2: clustering is firm, but the spacing is set by domain physics, and whether it is the same tower sampled every several rungs or a second log-period is open.
| Pattern | Where | Why it is not (yet) a clean \sqrt2 rung |
|---|---|---|
| Spatial cell nesting | lipid → complex → raft → cell | Steps of \times 6–12; chemistry occupies roughly every sixth rung, if it is the same tower at all. |
| Closed-conduit family | desmotubule \to MT \to axoneme \to hypha \to vascular | A median rolled into a tube at five scales — self-similar motif, host-set spacings, coarse. |
| Mechanoreceptor frequency rungs | skin, 0.5/5/30/200 Hz | Half-decade steps — the coarse family in the somatosensory domain. |
| Conduction-velocity / temperature / solar-layer ladders | nerve, heat, the Sun | “One mechanism, many scales” — mechanism-set rungs (opacity, ionization, fibre diameter), not a \sqrt2 comb. |
Cutoffs — a boundary, not a rung
\xi and the dissipation scales mark where the tower ends, not a step on it. A resonance against a cutoff is a boundary effect.
| Pattern | Where | Why it is not a \sqrt2 rung |
|---|---|---|
| THz resonance at \xi | crystal optics | \xi is the long-end cutoff (the \xi/d_\text{GJO} \approx 6.9 scale set by different physics), a resonance against a substrate boundary. |
| Blackbody IR cutoff E_\text{min}=hc/\xi | thermal radiation | The tower’s energy-domain floor; below it you fall off the ladder into continuous Tkachenko “weather.” |
| Flame-thickness floor / tree-height ceiling | combustion, hydraulics | \delta \gtrsim \xi and the cohesion-limited height are ends of a channel, not rungs. |
Part V — The Coin: the Ladder as an Energy Economy
The rungs are not only sizes; a rung is an address of the substrate’s lossless channel (why structures seek the rungs). The substrate fixes the price list (the rungs and the \sqrt2 between them); chemistry sets the boundary conditions (which rung a structure pins to); and the breath is the coin — the complete, lossless anti-phase exchange a structure on a rung can spend and receive. Every place energy moves through the substrate is the coin changing hands.
| Where energy moves | What changes hands |
|---|---|
| Three channels of heat | Radiation ejects the coin as a modon; conduction passes it lattice-cell to cell; convection carries it below the Landau speed. Heat is the coin in motion; temperature is how much the surface holds. |
| Lightning gamma | An electron driven past its own circulation speed (0.776c) can no longer hand the coin off losslessly inside the light cone, so it sheds it as a gamma modon — the breath made visible. |
| Fusion in the Sun | Boundary merger mints the coin, then spends it through the three channels from core to photosphere. |
| Photon → ATP | Charge separation un-pairs the anti-phase breath and holds the halves apart — which is why it banks energy (a paired breath is lossless but cannot store; to bank, you must break the pair). |
| Membrane potential | The cell holding the coin across its wrap; the action potential is spending it as a controlled wave. |
The coherence threshold v_L — the same crossing at three scales
A distinct but related axis: the velocity past which the paired breath can no longer complete its handoff and the lossless channel switches off. This is the lock pole engaged-vs-disengaged — not the anti-lock gap.
| Crossing | Where | What happens at v_L |
|---|---|---|
| Fast solar wind saturation | corona, \sim 0.0025c | Below v_L the flow rides the lossless channel and pays nothing; at v_L the breath can’t complete and is spent to vortex modes. |
| MOND → Newton | galaxy, \sim 750 km/s | MOND is the breath intact (paired, coherent); Newtonian gravity is the breath un-paired by the Hubble phase bias. The external-field effect and the Newtonian limit are one parity-breaking. |
| Bullet Cluster | merger, > v_L | Above v_L the pairing decoheres and the breath switches off → bare gravitating mass with no phonon force = the collisionless dark component. Same critical crossing as lightning; density decides whether the coin is shed as light or silenced into dark mass. |
This index records where every structure falls on the ladder. Why the rungs are spaced the way they are — discrete, and a fixed factor \sqrt2 apart — is the argument of The Substrate Ladder; what would settle it is the comb test of What would falsify it: reanalyze the raw clustering datasets the paper already leans on and the peaks should fall on a \sqrt2 comb. The single prediction that ties the whole wide net together is that the inter-rung ratios are powers of \sqrt2 — octaves and half-octaves — and not arbitrary. The most universal pattern in the paper, pinned to its most load-bearing constant.