class constant

Part V, Dynamics and Limits · defined in Chapter XXIV, A Number Nature Could Refute

The gap of the tower’s matter walk, its smallest eigenvalue in modulus, which the program reads as a mass.

Plate W.30The roots of the quartics

q_2

and

q_3

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

-\tfrac1{10}

times the root of

q_3

nearest zero.

The tower builds coarser observers, composites, from crowds of registers. A composite writes coarse letters, and its coarse depth is the number it keeps. Composites that conserve their number are matter; matter moves by a walk on its possible coarse words, and the smallest energy of that walk is its mass in the composite’s own units. At coarse depth three, with the weights the program has stated, the eigenvalue nearest zero is $-g$ with $g=0.021386958933918933047\ldots$

Its origin is a price paid for blindness. A composite cannot audit which member of its crowd realized a coarse move, so it gives the move one amplitude, the uniform sum over the members. For matter each hop carries two such blind sums, each a scaled eighth-turn of modulus $\sqrt2/2$ , so the hop has modulus $\tfrac12$ ; radiation’s hop pays nothing. With the hop modulus set to one, matter is massless at the natural mobility, its memory and re-anchoring moves balancing on a 48-dimensional kernel; at modulus $\tfrac12$ the balance breaks and $g$ is the residue. The number belongs to the stated weights and to the tower’s flat transports: with the lift’s spinor transport on the hops the converged gap is 0.0289563, and it runs towards zero as memory’s transport changes, so no mass number consistent with the lift exists yet. That the gap is a mass is a reading.

Theorem(The class constant is a quartic) computed

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)=\big[(y+1)(3y+5)\,q_2(y)\,q_3(y)\,p_7(y)\,p_{12}(y)\big]^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 $q_3=q_2+y(y-1)$ .

As mathematics

An algebraic number of degree four: $y=-10g$ is a root of the irreducible quartic $q_3(y)=3y^4+5y^3-16y^2-27y-5$ , an exact factor of the characteristic polynomial of an explicit invariant block of 58 states. Without memory, at coarse depth two, five alternating classes of coarse words carry the constant, with characteristic factor $q_2(y)=3y^4+5y^3-17y^2-26y-5$ , and memory shifts it to $q_3=q_2+y(y-1)$ . That $-g$ is the eigenvalue nearest zero of the whole 648-state matter sector is a numerical fact, computed to thirty-six digits and checked at coarse depths three and four only.

The number moves with every choice the law leaves open: the weight of memory, a private energy the law permits (which takes it from 0.0214 to 0.0038), the generator axiom, and the transports of hops and memory. It is dimensionless, and it becomes a statement about nature only through a physical unit of time, which the observer language provably cannot supply.

Its name in another field	Bridge
a quartic algebraic number	built
a mass, in the composite’s own units	a reading

A second sense

The word sits beside other classes of the volume: the reports are classes of future indistinguishability, an observer’s auditable classes are the orbits it cannot see into, and the fifteen objects are conjugacy classes of subgroups. The class constant is none of these. Seams keeps class for conjugacy and stabilizer classes and calls the number the spectral gap $g$ .

Built from: tower
In the volume: IThe Founding SentenceIIBlind ObserversVIIThe Branchial TreeXIXRulial InvariantsXXIVA Number Nature Could Refute