M47a The Action Sector of the HyperCore - Translation Duality, the Associator as a Commutator, and Free Histories

The Theory of Operations studies not only scalar operations but also the transformations created when operands act repeatedly. For a binary operation star, each element a defines a translation

L_a(x) = a star x.

These translations compose associatively and generate the operation’s action semigroup.

M47a "The Action Sector of the HyperCore" proves that, for a commutative operation, scalar non-associativity becomes non-commutativity of its translations. It classifies two-chart actions through their transition maps, proves that the translations of commutative power Cpow generate free histories, and separates the classical dynamics of one translation from the maximal complexity of the complete family.

The Symmetric Core consists of commutative, associative room operations obtained by transporting addition through iterated-logarithmic charts. In every room,

L_a L_b = L_(a star b),

so a finite history collapses to one translation determined only by the multiset of operands. Order is forgotten and the growth entropy is zero.

The HyperCore’s rungs from rank three onward are commutative but non-associative. Their translations still compose associatively, because maps always do, but they need not commute. The action sector therefore fills the associative, non-commutative quadrant absent from the scalar rungs themselves.

Translation duality

For every commutative magma, the following are equivalent:

• the scalar operation is associative;
• all elementary translations commute;
• the action semigroup is commutative.

When subtraction is available, the associator is exactly a commutator of translations:

As(a,b,c)

= (a star b) star c-a star (b star c)

= [L_c,L_a](b).

Scalar non-associativity is therefore not repaired or discarded when passing to actions. It is converted from dependence on bracketing into dependence on order.

At the scalar level, commutativity erases operand order while bracketing remains significant. At the action level, associativity of composition erases bracketing while word order remains significant. This is the Order-Bracketing Exchange.

Every caterpillar bracketing, in which each internal product has an original operand as one factor, corresponds to an ordered word of translations applied to a terminal operand. Other bracketings require translations by composite products and differ from ordinary words by associators. The passage from scalar products to actions is therefore a recoding of bracketing information, not a loss of it.

The translation centre of a commutative magma equals its left and right nuclei. The middle nucleus may differ. For Cpow on (1,infinity), the centre and all three nuclei are empty. When the absorbing endpoint 1 is included, each contains only 1.

Histories and their growth

For a finite set G of operands, let gamma_G(n) count distinct actions represented by words of length at most n. The growth entropy is

h(G) = lim_(n->infinity) (1/n)*log_2(gamma_G(n))

when the limit exists.

In an associative room, the growth is polynomial and

h(G)=0.

If the translations generate a free semigroup, then

gamma_G(n) = sum_(j=1)^n |G|^j

and

h(G) = log_2(|G|).

This is the largest possible growth for |G| generators. It measures the number of bits of ordered history retained per action step.

The action is therefore generative for associativity: it exists because associativity fails, its growth measures what that failure produces, and it collapses when the operation is replaced by its associative straight factor.

The transition decides the action

A two-chart operation has the form

O_(phi,psi)(a,b)

= psi^(-1)(phi(a)+phi(b)).

Its transition is

Theta = phi o psi^(-1).

In the measuring chart, every translation becomes a shift followed by the same transition:

phi o L_a o phi^(-1)

= Theta o T_(phi(a)).

Two translations commute exactly when the difference of their chart values is a period of the bend displacement

Theta-id.

The transition divides the action into three principal regimes:

1. Abelian: Theta is a translation. The scalar law is associative and its action collapses.
2. Medial: Theta(w)=c*w+d, with c != 1. The action is affine and metabelian; arithmetic relations among words depend on the slope c.
3. Free: Theta' becomes unbounded at one end. Every distinct word defines a distinct transformation.

The three regimes do not exhaust all possibilities. Periodic bend displacements can produce lattices of commuting translations, while affine actions with operand-dependent slopes can carry isolated relations without becoming medial.

The free action of Cpow

Commutative power is

Cpow(a,b)

= exp(sqrt(a*b*ln(a)*ln(b))).

Its measuring chart and transition are

F(x) = (1/2)*ln(x*ln(x)),

Theta_3(w) = (w+exp(w))/2.

The bend displacement has no nonzero period, so no two distinct translations of Cpow commute.

Because

Theta_3'(w) = (1+exp(w))/2

tends to infinity as w -> infinity, every set of Cpow translations generates a free semigroup. Every ordered sequence of encounters is a different map. For finite G,

gamma_G(n) = sum_(j=1)^n |G|^j,

h(G) = log_2(|G|).

The action on (1,infinity) is the free semigroup on the continuum of translations. It has no idempotents, periodic elements, nontrivial units or minimal ideal. Including the absorbing element 1 adds the constant map L_1 as a zero and the unique idempotent.

Near the absorber, the transition approaches (1/2)*id. A two-letter history of length n is then stored through binary weights down to 2^(-n), so recovering it requires approximately n bits of precision. In the far field, the exponential part dominates and separates distinct histories without bound. The same action is therefore medial in its absorbing germ and free globally.

Intermediate regimes

At the golden medial slope

c = 1/phi,

the shortest relation is

L_a L_b L_b = L_b L_a L_a.

M47a proves that the number of distinct actions of length n is

N_n = F_(n+3)-1.

The cumulative count and entropy are

gamma(n) = F_(n+5)-n-5,

h = log_2(phi).

The proof uses base-phi normal forms. Every collision of histories is generated by the carry

phi^i+phi^(i+1)=phi^(i+2).

A corresponding k-bonacci law is verified through length 22 for the first further cases but remains conjectural.

The Halbzeug MultPow lies in another intermediate regime. Its translations are affine maps with operand-dependent slopes. No two distinct translations commute, yet selected pairs satisfy a relation of length four when

(1+ln(a))*(1+ln(b))=4.

Generic algebraically independent slope pairs generate free actions. Thus complete non-commutation does not by itself imply freeness.

Multiplication group and firewall

Every translation of a two-chart operation is a bijection, so the operation is a quasigroup. In the measuring chart, its multiplication group is

Mlt(O) = <T_R,Theta>,

generated by all chart shifts and the transition.

For Cpow, the transition itself is the translation by

1/Omega = exp(W(1))

= 1.7632228344....

The action semigroup lies inside this multiplication group but contains no inverse of any nonempty word. Its translations are bijections of the operand line, not automorphisms of the scalar operation.

The infinitesimal shifts of the two Cpow charts generate an infinite-dimensional Lie algebra of vector fields. Consequently, no finite-dimensional Lie group acting smoothly on the operand line can contain both a Cpow translation and the continuous chart-shift group.

This scalar-action firewall does not prohibit abstract free subgroups inside compact groups. It says that no finite-dimensional Lie algebra, including no su(N), is generated directly by the scalar action on the operand line. Any gauge group must be introduced through a separately declared linear representation.

One translation and Euler’s tower

A transformation T acts linearly on observables through pullback:

U_T f = f o T.

Pullback reverses composition, and on linear observables the commutator of two lifted translations evaluates the scalar associator.

A single Cpow translation has classical fixed-point dynamics. Fix a>1 and put

b = a^a.

The fixed points of

x -> Cpow(a,x)

are exactly the fixed points of Euler’s exponential map

x -> b^x.

They exist precisely when

a*ln(a) <= 1/e,

equivalently

a <= exp(W(1/e))

and

a^a <= exp(1/e).

The commutative wall is therefore Euler’s tower boundary read at the effective base a^a.

Below the wall there is one attracting and one repelling fixed point. The attracting multiplier is

q_a = (1-W(-a*ln(a)))/2.

The commutative Cpow tower and Euler’s tower at base a^a converge to the same limit.

Below the wall, Koenigs linearisation gives the analytic eigenvalues

q_a^m, for m>=0,

of the pullback operator on analytic germs. Above the wall, an Abel coordinate gives every point of the unit circle as an eigenvalue on bounded continuous functions. At the parabolic wall, only constant analytic eigenfunctions remain.

These statements concern one translation and depend on the chosen function space. The full family remains a free non-commutative semigroup and has no common Koenigs coordinate or immediate joint spectral decomposition.

Realised histories

Rounding, projection or quantisation may identify distinct action histories. The resulting realised equivalence is coarser than equality of transformations. It is preserved when the same earlier history is appended, but generally not when the same later action is applied, because later expansion can reveal differences hidden by the quantiser.

This separates two sources of order:

• intrinsic order, created by scalar non-associativity and retained in translation history;
• protocolic order, introduced by observation, projection, feedback or changing domains.

One completed construction reads repeated application of a single Cpow translation through integer floors and selects prime itineraries. The analogous problem for genuinely free histories involving two or more operands remains open.

M47a establishes that the HyperCore’s non-associative scalar rungs generate associative, non-commutative actions one level higher. The associator becomes a commutator, scalar bracketings become ordered histories and associative rooms separate sharply from generative HyperCore actions. At rank three, every word of Cpow translations is distinct, while one translation is governed by Euler’s tower boundary and classical Koenigs or Abel linearisation. The monograph thereby separates the tractable dynamics of one action, the free growth of the complete family and the additional order introduced by discrete realisation.