Part V · Dynamics and LimitsChapter XXIV

A Number Nature Could Refute

Plate XXIV.1The matter gap against the hop modulus: zero at 1 and at

\tfrac{\sqrt2}2

g

\tfrac12

, and

\tfrac4{15}c^2

for small

c

XXIV.1
XXIV.2
XXIV.3
XXIV.4
XXIV.5
XXIV.6
XXIV.7

What would count as a prediction, and how close is the program to one?

In 2010 Coldea and collaborators tuned the Ising-chain magnet $\mathrm{CoNb_2O_6}$ to the quantum critical point of its transverse field. Near that point a weak longitudinal field, supplied by the neighbouring chains, binds the chain’s domain walls into a ladder of excitations, and neutron scattering resolved the two lowest; the ratio of their energies came out close to 1.618, the golden ratio. Zamolodchikov had shown in 1989 that the critical Ising field theory perturbed by a magnetic field is integrable, with eight particles whose mass ratios are fixed by $E_8$ , so that $m_2/m_1=2\cos(\pi/5)$ . Nothing in that ratio was fitted: a principle forced it, an exceptional structure appeared because the principle put it there, and any other measured value would have refuted the theory.

This chapter asks how far the program is from a number of that kind. Its principal dimensionless number, exactly computed and with a physical reading attached, is the class constant of matter: the smallest energy of the walk by which matter, the tower’s composites of odd radius, moves on its possible coarse words, its mass in the composite’s own units. The answer to the chapter’s question is plain. The program has no confirmed prediction, and at present none of its numbers yet qualifies as one.

The central result · The class constant is a quartic

In the tower’s matter walk at coarse depth three, with the weights the program has stated, the eigenvalue nearest zero is $-g$ with $g=0.021386958933918933047\ldots$ , and $y=-10g$ is a root of the irreducible quartic $q_3(y)=3y^4+5y^3-16y^2-27y-5$ . The quartic is an exact factor of the characteristic polynomial of an explicit invariant block of 58 states,

\det(yI-10K)=\bigl[(y+1)(3y+5)\,q_2(y)\,q_3(y)\,p_7(y)\,p_{12}(y)\bigr]^2/3^{16},

where $q_2(y)=3y^4+5y^3-17y^2-26y-5$ is the characteristic factor of the five-state block that carries the constant before memory, and $q_3=q_2+y(y-1)$ .

Status

The algebra is exact. That $-g$ is the eigenvalue nearest zero of the whole 648-state matter sector is a numerical fact, computed to thirty-six digits, reproduced independently from the definition to $3\cdot10^{-16}$ , and checked at coarse depths three and four only. The number belongs to the stated weights: change the weight of memory, or add a small private energy that the law permits, and the gap moves while the massless points of radiation stay where they are. And it is dimensionless, a statement about nature only through a physical unit of time, which the observer language provably cannot supply. So the class constant is exact, and it is not a prediction.

The lift sharpens the point: the number is a property of the tower’s flat transports. With the lift’s spinor transport on the hops the converged gap is 0.0289563, and as memory’s transport changes it runs down towards zero, so no mass number consistent with the lift exists yet. The spinor transport carries spin, not matter’s internal labels. Carried instead by transports that keep the octonion product, the gap depends on a discrete colour background, and in the flat background, which is both the vacuum of the added gauge links and the background the matter’s own energy selects, it is the class constant itself, to every digit computed; the spin factor’s holonomy along the hops has not been put into that calculation. The program already produces numbers nature can refute. It does not yet produce one from the law alone with every weight fixed.

Where the number comes from

Plate XXIV.1The matter gap against the hop modulus: zero at 1 and at

\tfrac{\sqrt2}2

g

\tfrac12

, and

\tfrac4{15}c^2

for small

c

The tower builds coarser observers, composites, from crowds of registers. Composites of odd radius conserve the occupation number of their fiber and are massive, the program’s matter; those of even radius change it by pairs and are massless, its radiation. The pairs are of occupied modes in one composite’s fiber, not of registers. Within a composite the walk keeps the chirality, whose two values have the same spectrum, matter and antimatter at equal energies, and no energy is paired with its negative, so the composites offer no sea. A composite cannot audit which member of its crowd realized a coarse move, so by the blindness premise it assigns the move one amplitude, the uniform sum over the members that could have made it. For matter the transport between coarse words is $T=\tfrac14\,\sigma_x\gamma_{x_2}\gamma_x\sigma_{x_2}$ , with $\sigma_x=\gamma_x+\gamma_{\bar x}$ ; for radiation the run factors are scalars and the transport is the bare hop.

For radiation memory and re-anchoring balance exactly, so it is massless at the natural mobility $\rho=1$ ; for matter the halving breaks the balance. With the hop modulus set to one, matter at coarse depth three has an exact 48-dimensional kernel; at modulus $\tfrac12$ the kernel is gone and the gap is $g$ . That mass is the cost of blindness is a reading, which has passed one check inside the model: switching the payment off leaves matter as massless as light. Only the second blind sum makes a mass, and at small hop modulus $c$ the gap is $\tfrac4{15}c^2$ ; composition does not stack blind sums, since a composite of composites pays exactly two at every radius. Colour offers no other route on the structures built so far: the gauge links’ coupling runs neither along the tower of scales nor on the crystal inside each meeting, both of which copy loops rather than subdividing them. The arena’s positions do subdivide, but colour’s frames sit with observers; were a colour frame added at every position, the coupling would run with the sign of asymptotic freedom and the strong scale would follow from the stiffness at the finest scale, an addition the program has not made.

Example(The price of one blind sum)

Distinct letters anticommute and each squares to $-1$ , so $\sigma_x^2=\gamma_x^2+\gamma_{\bar x}^2+\{\gamma_x,\gamma_{\bar x}\}=-2$ . The mean $\tfrac12\sigma_x$ of the two unit elements therefore has square $-\tfrac12$ : it is $\tfrac{\sqrt2}2$ times an element of square $-1$ , the average of two unit phases at right angles. Matter’s transport carries two such sums and so has modulus $\tfrac{\sqrt2}2\cdot\tfrac{\sqrt2}2=\tfrac12$ ; radiation’s carries none and has modulus one.

Five alternating classes

Plate XXIV.2The five-state block of matter at coarse depth two: classes as vertices, matrix entries as edges.

Without memory, at coarse depth two, the constant can be written down by hand. The matter sector commutes with the permutations of the axes that fix the vantage’s parity, and in their sign representation five alternating classes of coarse words span an invariant subspace: a run $(y,y)$ on an axis other than the anchor’s, the antipodal word $(y,\bar y)$ , and three phase-paired combinations, each paired with its antipodal partner at the phase $-i$ .

The phases enter only through $\lvert H_{14}\rvert^2$ , $\lvert H_{24}\rvert^2$ , $\lvert H_{45}\rvert^2$ and the one triangle 3,4,5, whose two orientations contribute conjugate terms, so the coefficients are rational. An early search concluded that the constant was not a simple algebraic number, because it looked for integer polynomials with small coefficients; the walk weight $\tfrac1{10}$ hides the true coefficients, which are small only in $y=10x$ , and the negative was withdrawn. A failed search bounds a height, not a degree.

Proposition(The five-state block)

In the basis of the five classes the matter Hamiltonian is $\operatorname{diag}(-\tfrac16,-\tfrac16,0,0,0)$ with $H_{14}=-i\sqrt{30}/60$ , $H_{24}=-\sqrt{30}/60$ , $H_{34}=-\tfrac1{10}$ , $H_{35}=\tfrac1{10}$ , $H_{45}=\tfrac{\sqrt2}{10}e^{3\pi i/4}$ and Hermitian closure. Its characteristic polynomial is

\frac{(6x+1)(6000x^4+1000x^3-340x^2-52x-1)}{36000},

whose quartic factor, written in $y=10x$ , is $q_2(y)$ . The eigenvalue of smallest modulus is $-0.0228391002106770563968\ldots$ , the class constant at coarse depth two.

Memory shifts the quartic

Plate XXIV.3The roots of

q_2

and of

q_3

: memory moves each a little, and the class constant is

-\tfrac1{10}

times the root nearest zero.

The shift $q_3-q_2=y(y-1)$ is what the depth-three memory does to the five-state interference when the six copies talk to each other, and memory of depth four changes nothing: at coarse depth four $q_3$ is again a factor and the gap agrees. The roots of $q_3$ are $-0.21387$ , $-1.34662$ , 2.35312 and $-2.45930$ , standing in modulus as $1:6.296:11.003:11.499$ . The exact doubling admits commuting antiunitaries squaring to $-1$ only as a generic pairing, and its origin is open.

The constant has close neighbours that are not low-degree numbers. With memory, matter becomes massless at three mobilities in $(0.3,1.2)$ , among them $\rho=0.61286625\ldots$ and $\rho=0.43198752\ldots$ , and neither satisfies an integer polynomial of degree at most eight with coefficients up to $10^4$ . These carry no small block: a number is low-degree when a small invariant block carries it.

Proposition(The converged block)

At coarse depth three the five-state block survives unchanged as an invariant block, so the roots of $q_2$ remain eigenvalues. Its six copies, one for each first letter, are coupled by the depth-three memory, which links the run words $(x,c,c)$ to $(c,x,x)$ and $(\bar c,\bar x,\bar x)$ in other copies at unit modulus. They close to an invariant subspace of dimension 58 inside the sign component, on which every eigenvalue is exactly doubled and the characteristic polynomial factors as in the central result; the roots of $q_3$ carry the converged constant.

What the number depends on

Plate XXIV.4The matter gap under each choice the law leaves open: weights, private energy, generator, and the transport of hops and memory.

Light’s massless points are balance lines, $\rho=\mu$ and $\rho=2\mu$ , and every weight on them keeps light massless, while the matter gap moves: with $\mu$ ; with a private energy the law permits, to about 0.0038; and with the choice of generator, from 0.022839 to 0.009749 at depth two. The observer language cannot fix a weight, so only a declared measure clause can. One principle has fixed numbers inside the family: requiring light and matter to be massless together pins the rate ratios to the line $(\tfrac{\sqrt2}2,\tfrac{\sqrt2}2,m)$ , leaving the modulus $m$ free, and these are rates, not speeds.

The number depends on transport as well. With the lift’s spinor transport on the hops the converged gap is 0.028956340208; transporting every memory rewrite by its newest record’s promotion gives no stable value, falling through $3.0\times10^{-4}$ , $9.8\times10^{-5}$ and $6.9\times10^{-6}$ ; keeping the tower’s rule of coherence gives $2.3854\times10^{-4}$ . But the spinor transport is spin’s, and it moves the lepton’s line. Carried by transports that keep the product on matter’s internal factor, the choice on each hop is a discrete colour background: over forty-four backgrounds the gap takes ten values from 0.0178 up to 0.021386958934, exactly the class constant in thirty-five of them, while carrying memory’s rewrites the same way sends it towards zero with depth. The flat background, no colour flux, is unique up to gauge, is the gauge links’ vacuum, and is selected independently by the matter’s own filled-state energy, both among the backgrounds sampled and in descents through the whole space of backgrounds; there the gap is the class constant at depths three and four. Spin’s holonomy along the hops, whose bundle over the clocks comes in two forms the program does not yet choose between, has not been put into the calculation.

Proposition(Zeros stay, numbers move)

Write $H_{\mu,\rho}=D^{-1/2}(H_{\mathrm{hop}}+\mu H_{\mathrm{memory}}+\rho H_{\mathrm{reanchor}})D^{-1/2}$ , with $\mu=\rho=1$ in the stated model. At coarse depths two and three, radiation’s anchor-uniform two-forms are eigenvectors with eigenvalue $(\rho-\mu)/5$ , and its traceless symmetric family has eigenvalue $(2\mu-\rho)/10$ , exactly. At $\mu=\tfrac12$ , $\rho=1$ the smallest eigenvalue modulus of the matter sector is below $\tfrac1{50}<g$ , with an exact rational certificate.

The bridge, and a first comparison

Plate XXIV.5The ladder’s one ratio against nature’s successive mass ratios: refuted as a generation map.

The class constant is a mass times a cycle in units of $\hbar$ : every conserving composite obeys $m\,c^2\,\tau=g\,\hbar$ , with $\tau$ one of its own cycles. For the electron the implied cycle is $\tau=g\hbar/(m_ec^2)\approx2.75\times10^{-23}\,\mathrm{s}$ , and nothing known has that period. A bridge needs a clause outside the observer language: an identification of one of the program’s cycles with a physical period, or a ratio of two such constants, one per class, matched by a ratio in nature. Gravity would supply a ratio of the second kind without a second class: its strength measured in ticks fixes the tick’s length in Planck times, a pure number that no rescaling of rates changes. But the lift fixes gravity’s form and not its strength, and each way of fixing the strength that has been tried needs a long-range field felt by every kind of matter, which the record dynamics does not yet have. The constant has no kinetic partner either: it is an internal rest energy, and a relativistic dispersion needs a composite that grows.

A first comparison was made, with the predictions written down before nature’s values were consulted. Successive matter classes seen by one observer stand in one fixed ratio, 36 per two units of memory depth; read as the three generations, this predicts constant ratios. Nature’s are 206.8 and 16.8 for the charged leptons, 589 and 135.6 for the up-type quarks, and 19.9 and 44.7 for the down-type quarks, ranging over a factor of thirty-five, so the ladder is refuted as a generation map. Only bottom over strange lies within thirty percent of 36, and it is recorded so that it is never later taken for a discovery. A near miss between a second construction’s history action, 0.0213469965, and the class constant was refused: the two answer different questions.

Numbers the lift fixes by itself

Plate XXIV.6The first Hecke eigenvalues of light and of the lift’s two first interior modes, at the first seven Eisenstein primes.

Every number so far depends on a weight or a transport that the law leaves free; the lift offers numbers fixed by its arithmetic alone. Its local numbers, light’s share $\tfrac17$ and the Jacobi-sum constant of its scattering, are fixed by the prime seven and say nothing the sky did not. Global numbers come from interior modes, fields on finer covers attached to no light direction, to which the arithmetic attaches a Hecke eigenvalue at each prime. Their use was fixed before they were computed: to check the modes and identify them, not to match physical constants.

Light’s numbers are $\varpi+\bar\varpi$ , the arithmetic of the Eisenstein integers and nothing more. The level-8 mode gives different numbers to the two primes over 13 and over 19, so it is not borrowed from the rationals, and it vanishes at only one prime over 19, so it is not made of the Eisenstein integers’ own numbers: it is new arithmetic. At all seven primes its numbers are those of one elliptic curve over $\Q(\sqrt{-3})$ , with $j$ -invariant $2^{10}(3+\omega)$ , found from four primes and confirmed at three more predicted in advance; that is strong evidence, not a proof. The level- $4\sqrt{-3}$ mode matches no elliptic curve with small invariants. No reading in nature has been declared, and what these numbers mean physically is not yet known. Both modes need depth at the prime 2. So does memory’s counter: what a record holds is read at the lift’s prime $\sqrt{-3}$ , but the tower’s rewrites act within any two axes as a binary counter, so the transport of memory that the mass number needs points to structure at 2. Whether the two meet has not been tested; the lift reads the prime 2 through the field of four elements while the counter is binary, and the coincidence of primes is recorded as a possible link, not a result.

Proposition(the first Hecke eigenvalues) computed

At the first seven primes $\varpi$ of the Eisenstein integers, each written by the generator congruent to 1 modulo the sky’s prime $\mathfrak p$ , the lift’s modes carry these numbers:

\begin{array}{lcccc}\varpi & \text{norm} & \text{light} & \text{level }8 & \text{level }4\sqrt{-3}\\\hline 4+\omega & 13 & 7 & -5 & 2\\ -3+\omega & 13 & -7 & -3 & -2\\ -5-2\omega & 19 & -8 & 0 & 6\\ 2+5\omega & 19 & -1 & 5 & 0\\ -5\omega & 25 & 5 & -3 & -2\\ 5+6\omega & 31 & 4 & -4 & 4\\ 5-\omega & 31 & 11 & -3 & 0\end{array}

What would count as a prediction

Plate XXIV.7The sieve: only the magnet’s ratio passes all four tests.

Coldea’s ratio meets all four conditions. The class constant fails the second and third and depends on transport as well as weights; the joint-critical ratio depends on the choice of generator and has no speed to attach to; a second construction’s four matter levels $1:1.132723935:4.494142126:4.626866062$ , invariant under all report permutations with no fitted weight, lack a unit of time; and the unequal cone speeds of an added walk are already excluded by nature’s single speed. Two rows reached a comparison and lost it. The ladder was read in nature and refuted, and so was the single Yukawa coefficient of the Cayley point’s cubic norm, which couples the family to the Higgs field with one coefficient for all four types: with three bare copies it makes the down-type quark masses equal to the charged-lepton masses at the matching scale and the quark mixing matrix the identity, so $(m_s/m_b)/(m_\mu/m_\tau)$ , which running barely changes, would be one, where nature gives about one third, and the mixing angles are not zero. Light’s share $\tfrac17$ is fixed by the lift and has no reading in nature, and the numbers of the first interior modes are computed with no reading declared.

The program already produces numbers nature can refute. It does not yet produce one from the law alone with every weight fixed.

Definition(Prediction)

A number is a prediction of the program when (a) it is dimensionless, or its units come from a period already identified in nature, and it is computed from the law without fitting; (b) every weight the law leaves free either is fixed by a stated principle or is shown not to change it; (c) its operational reading, a named measurement in nature, is fixed before the comparison; and (d) some outcome of that measurement would contradict it.

The routes toward a number nature could refute are four, and each is a question the volume has already posed: a rulial invariant across the unselected weights, of which the ones found so far are zeros; an identified period, which would let the class constant pass the third test; a ratio of two class constants, one per class of composite; and a number fixed by the lift’s arithmetic, which passes the second test by construction. The first global numbers of the lift are computed, one mode’s being on all the evidence those of an elliptic curve over the lift’s own number field, and no reading in nature has been declared for them. The joint critical line points at the most physical test of all: a propagation instrument built on it would make the ratio of the speeds of light and matter a dimensionless number that nature fixes at one.

The class constant is not intrinsic to matter: reading the shared letter at a contact audits one of matter’s two blind sums, and a bound pair at full read is lighter than its two parts by up to about sixteen percent, so a rest energy the environment can lower is not yet a constant of nature. The chapter’s question stays open: is there a quantity of the tower’s matter, computed exactly at the joint critical line, with the lift’s transport carrying spin and a transport that keeps the octonion product carrying matter’s internal factor, that is unchanged under every reweighting the observer language leaves free, and what reading in nature, declared in advance, could the numbers of the lift’s first interior modes have? The epilogue sets this against the older question of when an exceptional structure counts as physics.

Words defined here: class constant
Concepts: statusrefutedbridgeabsence
Objects: the sky