M41b The Midpoint at Five-Halves and the Birth of Chirality - Fluent Half-Étages: CatH, the Midpoint Square and the Holonomicity Question

The Theory of Operations distinguishes three ways of placing objects between integer ranks: a normal reading built from a protocol of means, an ascending reading obtained by iteration, and a symmetrised reading obtained from that ascent. It also admits midpoint formulas between neighbouring rungs, but these formulas are not themselves half-rank readings. At rank 5/2, between multiplication and directed exponentiation, the AscentCore midpoint is

CatH(a,b) = sqrt(a*b*a^b).

M41b "The Midpoint at Five-Halves and the Birth of Chirality" develops this midpoint, embeds it in a two-parameter square of formulas, separates it from the actual half rung, identifies where chirality first appears in the ascending column, and reformulates the open question of whether the Symmetric Hyperelliptic Mean has a holonomic relation compatible with genus-two periods.

The midpoint and the half rung

The formula CatH is the midpoint of the log-linear path

(a*b)^(1-t)*(a^b)^t

from multiplication to directed exponentiation. It preserves the Absolute Invariant,

CatH(2,2)=4,

has 1 as a right identity, and commutes exactly on the classical curve

a^b = b^a.

Its logarithmic commutativity defect is half that of exponentiation:

ln(CatH(a,b))-ln(CatH(b,a))

= (b*ln(a)-a*ln(b))/2.

At rank 5/2, three different columns give the census row

{SHO, MultPow, CatH}:

• SHO is the normal HyperCore operation produced by the SHM protocol;
• MultPow is the Symmetric-Core midpoint formula between multiplication and Apow;
• CatH is the AscentCore midpoint between multiplication and directed exponentiation.

These objects are distinct. They satisfy

SHO(a,b) <= sqrt(a*b*Cpow(a,b))

<= sqrt(CatH(a,b)*CatH(b,a)).

The second inequality becomes equality precisely on a^b=b^a.

The midpoint must also be separated from the AscentCore half rung

AC_(5/2) = Asc(AGM-Op).

The two agree at the initial height and at the Absolute Invariant, but their later behaviour differs. The midpoint grows with exponential base sqrt(a), while the half rung grows with base AGM(1,a), which is strictly larger for a>1. The midpoint is therefore a formula between rungs, whereas the half rung is an iterate of the preceding half-rank operation.

The fluent midpoint square

Replacing directed exponentiation by the rank-three operand dial

L_r(a,b) = exp((b*ln(a))^(1-r)*(a*ln(b))^r)

produces the fluent square

F_(t,r)(a,b) = (a*b)^(1-t)*L_r(a,b)^t.

Its middle row is

F_r(a,b) = sqrt(a*b*L_r(a,b)).

This row runs from CatH(a,b) through the quarter-point formula

sqrt(a*b*HR(a,b))

and the commutative formula

sqrt(a*b*Cpow(a,b))

to the mirror CatH(b,a).

Every member of the operand dial has the same diagonal u^u. Consequently, for each fixed t, every member of the corresponding row has the same diagonal and hence the same Hermit equation:

(1-2*t)*W+t*W*exp(W) = i*pi/2,

where W=Log(u).

On the middle row, t=1/2, this reduces to

W*exp(W) = i*pi.

For the declared exponential continuation, every fluent midpoint therefore shares the unit

tau_CatH = exp(W_0(i*pi))

= 1.6903867572+1.8699079640*i.

The continuation is essential. If the same real formula is continued instead as the principal square root of the principal product, its value at tau_CatH is -i*tau_CatH, not i*tau_CatH.

The operand dial moves values away from the diagonal but moves no Hermit point. The different units found around rank 5/2 arise from different upper legs and different kinds of object, not from chirality along this dial.

Where chirality begins

In the ascending reading, the half rung at rank 3/2 is the channel of addition and remains commutative. The half rung at rank 5/2 is chiral:

AC_(5/2)(2,3) < AC_(5/2)(3,2),

while

AC_(5/2)(3,4) > AC_(5/2)(4,3).

Thus its chirality has no fixed sign. The ascent of a commutative operation can become chiral because one operand is frozen as the base while the other is iterated.

Operand chirality can also be inserted into the legs of the normal protocols. At rank 3/2, replacing the geometric leg of Gauss’s process by the weighted geometric mean gives the transverse multiplier

q_(1,r) = r-1/2.

It is independent of the centre. The protocol converges linearly away from r=1/2 and quadratically at the symmetric midpoint.

At rank 5/2, replacing the Cpow leg of SHM by the fluent member L_r gives

q_(2,r) = (1-2*r)*(sigma-1)/(2*(1+sigma)).

Here the multiplier varies with the log-centre sigma and changes sign at the shell sigma=1. Operand chirality is therefore flat at 3/2 but running at 5/2.

These are distinct statements:

• chirality of the ascending reading first appears at 5/2;
• operand chirality already exists in a protocol at 3/2, but with a centre-independent multiplier;
• on the midpoint formulas, the dial changes off-diagonal values while leaving every Hermit point fixed.

The holonomicity question

The rank-5/2 protocol mean is the Symmetric Hyperelliptic Mean SHM. Its conjectural genus-two interpretation raises the question of whether an associated function satisfies a linear differential equation with polynomial coefficients.

The original form asked whether

s -> ln(SHM(exp(s),exp(t)))

is holonomic. M41b shows that this coordinate is not well calibrated: even Gauss’s genus-one mean fails the analogous test. Suitable analytic continuations of both

1/AGM(1,exp(s))

and

ln(AGM(1,exp(s)))

become unbounded near every nonzero point 2*pi*i*k. A holonomic function can have only finitely many singularities, so neither function is holonomic in s.

The informative versions of the question instead follow the classical genus-one model:

1. for fixed a>1, is b -> 1/SHM(a,b) holonomic in the algebraic variable b?
2. along a declared one-parameter family of genus-two curves, is the corresponding reciprocal SHM expression holonomic in the family parameter?

A genus-two period should satisfy a system of order at most four. The proposed test is therefore a calibrated search for a polynomial-coefficient differential relation of order at most four, first verified on a known period of the same family.

No such equation is currently known. A Heun equation would not establish the intended genus-two interpretation, because Heun’s classical second-order equation belongs to one-parameter elliptic settings. The appropriate genus-two target is an order-four period system.

M41b establishes rank 5/2 as the first cell where several distinct constructions must be kept apart. CatH is an AscentCore midpoint formula, not the half rung or the Hyperzeug; the fluent square gives its operand-dial continuation and shared Hermit equations; the ascending half rung becomes chiral with a changing sign; and operand chirality changes from a flat multiplier at 3/2 to a centre-dependent multiplier at 5/2. The remaining genus-two question is stated in a form that the genus-one model actually passes: the reciprocal mean in an algebraic family coordinate, tested against an order-four period system.