M43 F08 The Ladder Flow - The Clock of t + ln t, Its Family of Means, and Its Two Ends

The Theory of Operations reaches rank three from rank two by changing the generator of the storage mean. In the logarithmic operand coordinate t=ln(x), rank two uses the arithmetic mean of t, while rank three uses the quasi-arithmetic generator

phi(t)=t+ln(t).

Applying phi once moves exactly from the rank-two storage mean to the rank-three storage mean.

F08 "The Ladder Flow" studies the continuous interpolation obtained by fractionally iterating this generator. It constructs the Abel clock regular at infinity, derives its expansion to arbitrary order, compares it with the Koenigs continuation regular at the opposite endpoint t=1, and uses both continuations to generate analytic families of means. These families agree at integer ranks two and three but miss the HyperCore’s protocol means at the half ranks. They are therefore analytic directions through rank, not the HyperCore’s own non-integer ranks.

The map

phi(t)=t+ln(t)

is an increasing real-analytic bijection of (1,infinity) onto itself. It has a repelling fixed point at t=1, corresponding to the operand x=e, with multiplier 2. Forward orbits tend to infinity, while backward orbits converge to 1. Its inverse is

phi^(-1)(x)=W(exp(x)).

A continuous interpolation requires an Abel function A satisfying

A(phi(t))=A(t)+1.

Its fractional iterates are

phi^[r](t)=A^(-1)(A(t)+r).

Abel functions are not unique: they differ by periodic functions of their own values. A canonical continuation must therefore be selected by regularity at one end of the interval. Because (1,infinity) has two ends, F08 constructs two canonical but distinct continuations.

The clock at infinity

The ladder clock A_clk is the Abel function regular at infinity. Its vector field is not simply ln(t). The equation

dt/dr=ln(t)

has the time-one map li^(-1)(li(t)+1), which differs from the ladder step t+ln(t). The correct field is the iterative logarithm of phi:

V(t)=ln(t)-ln(t)/(2*t)

+(4*ln(t)-ln(t)^2)/(12*t^2)+....

The formal series is uniquely determined by requiring its time-one map to equal phi. Integrating its reciprocal gives the clock to arbitrary order:

A_clk(t)=li(t)+(1/2)*ln(ln(t))

+(1/12)*E_1(ln(t))-1/(12*t)

-(1/24)*E_1(2*ln(t))+1/(24*t^2)+....

The sign of the ln(ln(t)) term is positive. The negative sign used in the first edition makes the defining residual nonsummable.

The clock is characterised by Szekeres regularity. Up to an additive constant, it is the unique real Abel function whose iterates satisfy

phi^[r](t)-t ~ r*ln(t)

as t->infinity.

More precisely,

phi^[r](t)=t+r*ln(t)

-r*(1-r)*ln(t)/(2*t)

+O((ln(t))^2/t^2).

Thus the ladder clock is canonical at infinity in a precise real-variable sense.

The clock family of means

The clock generates the quasi-arithmetic family

M_(2+r)^phi(a,b)

= exp(phi^ln(a)+phi^ln(b))/2)).

For real r and operands above e, this is a strict symmetric mean, analytic in the operands and in r. Abel’s equation gives the exact integer anchors

M_2^phi = GM,

M_3^phi = M_3,

where M_3 is the storage mean of commutative power Cpow.

The integer agreement does not validate the intermediate values. The clock family does not reproduce Gauss’s arithmetic-geometric mean at rank 3/2 or the Symmetric Hyperelliptic Mean SHM at rank 5/2. At (3,7),

M_(5/2)^phi(3,7)

= 4.531013724175495807...,

whereas

SHM(3,7)

= 4.527304169185620709....

The difference is approximately 3.7096*10^(-3).

This mismatch is structural. The clock family is analytic in rank, while the HyperCore’s adjacent-neighbour protocols force a broken coefficient line with corners wherever the coefficient ladder has nonzero discrete curvature. An analytic continuation cannot pass through all protocol values.

The clock also extends to complex r, producing complex-valued means. This supplies one analytic direction through the rank plane, but not the HyperCore’s own complex rank. Such a construction would have to continue the HyperCore’s channels and pair protocols themselves.

The Koenigs end

At the opposite endpoint t=1, Koenigs’s coordinate Phi satisfies

Phi(phi(t))=2*Phi(t),

with

Phi(1)=0,
Phi'(1)=1.

It is obtained from backward iteration:

Phi(t)=lim_(n->infinity) 2^n*(phi^[-n](t)-1).

The function log_2(Phi(t)) is another Abel function on (1,infinity). It is, up to a constant, the unique Abel function whose vector field extends regularly to the fixed point. Its field satisfies

V_K(1)=0,
V_K'(1)=ln(2).

The two distinguished coordinates are therefore selected by opposite boundary conditions:

• A_clk is regular at infinity;
• log_2(Phi) is regular at t=1.

The transition between the ends

Their difference

Theta(t)=A_clk(t)-log_2(Phi(t))

is invariant under phi. Hence

Theta(t)=P(A_clk(t))

for a one-periodic function P.

The transition is extraordinarily close to constant:

Theta_0

= 0.50027565666974440663960202880852....

Its leading Fourier coefficient has modulus

|p_1| = 1.2526776204*10^(-13),

and the total range is

max(P)-min(P)

= 5.01071048144*10^(-13).

The subsequent coefficients decrease by roughly twelve decimal orders per step. To the computed precision,

Theta = Theta_0

+2*|p_1|*cos(2*pi*A_clk+arg(p_1))

with a remainder below 1.5*10^(-25).

The transition is therefore numerically nonconstant. The two regular flows are almost indistinguishable over the real interval, but they are not identical. The smallness of their difference is explained by a distant square-root singularity associated with the critical point t=-1 of phi.

The Koenigs family

Koenigs’s fractional iterates,

phi_K^[r](t)=Phi^(-1)(2^r*Phi(t)),

generate a second family of quasi-arithmetic means. It also reproduces GM at rank two and M_3 at rank three, while extending to operands below e, where the clock family is unavailable.

The clock and Koenigs families differ by roughly one part in 10^12 across broad ranges. Their disagreement is tiny compared with their shared disagreement from the HyperCore’s half-rank protocols. At (2,5) and rank 5/2, the Koenigs family gives

3.102262909111145...,

while

SHM(2,5)=3.090674663122845....

The main divide is therefore not between the two endpoint normalisations. It is between analytic fractional iteration of the generator and protocolic construction between neighbouring HyperCore rungs.

Corrected scope

The first edition correctly found the ladder step, clock and analytic family of means, but used the wrong sign in the clock’s asymptotic expansion and attached unsupported geometric interpretations to the construction.

The clock family and another complex deformation are not two connections on a rank manifold. They are separate analytic directions with different velocities. No bundle or curvature between them was defined.

The repelling endpoint t=1 is also not the source of the room-two Hermit equation

w^2-w=i*pi/2.

That equation belongs to room two of the Symmetric Core. The clock’s asymptotic expansion at infinity cannot derive it through monodromy at the opposite endpoint.

F08 establishes the ladder flow as a controlled analytic continuation between the storage means of ranks two and three. Its clock is the unique real Abel coordinate regular at infinity and has an expansion computable to arbitrary order. Koenigs’s coordinate supplies the continuation regular at the opposite endpoint. Their transition is periodic, extraordinarily small and numerically nonconstant. The corresponding families agree at integer ranks but miss the HyperCore’s protocol means between them. The monograph therefore constructs and compares two precise analytic directions through rank while leaving open the separate problem of constructing the HyperCore’s own non-integer and complex ranks.

F papers belong to the M43 series.