M44a The Regressing Columns - A Corrected Record [deprecated]

The Theory of Operations represents the rooms of the Symmetric Core as abelian groups obtained by transporting addition through iterated-logarithmic charts. Each room therefore has a natural regressor, formed by replacing addition in its chart with subtraction.

M44a "The Regressing Columns" reorganises subtraction, division and logarithm as successive readings of this single construction. It preserves the forced phases, nest identities, exact quarter-turn cycles and critical base found in the original paper, while correcting their former interpretation as separate “Regressing Holograms.” The three columns are not independent structures: they are regressors of the rooms, with the logarithm reported one room below the associative-power regressor.

If room k has chart ell_k=ln^[k] and operation

x odot_k y = exp^ell_k(x+ell_k(y)),

its regressor is

x ominus_k y = exp^ell_k(x-ell_k(y)).

The first three cases are subtraction, division and associative-power regression:

a ominus_0 b = a-b,

a ominus_1 b = a/b,

a ominus_2 b = exp(ln(a)/ln(b)).

The ordinary logarithm is the room-two regressor read one room lower:

log_b(a) = ln(a ominus_2 b).

This frame offset explains why subtraction and division have rigid reflection dynamics while Logarithmation develops a base-dependent multiplier.

The apparent annihilating invariants of the three columns are one chart identity:

ell_k(b ominus_k a) = -ell_k(a ominus_k b).

Native symmetrisation therefore collapses to the room identity. Applying a quarter-turn to the chart difference instead gives the forced phases

Csub(a,b)=i*(a-b),

Cdiv(a,b)=(a/b)^i,

Clog(a,b)=(log_a(b))^i

with the appropriate orientation and branch conventions. These phases lie on the native imaginary axis of their rooms. Subtraction gives the ordinary imaginary axis, while division and logarithm give unit-circle phases.

At the diagonal, commutativity and differentiability cannot both be retained. A continuous symmetric square root of -(a-b)^2 is necessarily

+/- i*|a-b|

and has a cone along the diagonal. A differentiable choice is instead

+/- i*(a-b),

which is antisymmetric. The former “conical branch obstruction” is therefore not an unavoidable failure of analytic continuation; it is the price of insisting on a commutative branch.

The roads joining a regressor to its reverse are sideways rotations of the chart difference. For division, the road is

exp((ln(a)-ln(b))*exp(i*pi*r)).

At r=1/2, it crosses the unit circle and equals the forced phase. Continuing the dial to

r=1/2+i*s

contracts the chart difference toward the identity by the factor exp(-pi*s). The effect is a complex-flux continuation of the sideways morph, not a separate Wick rotation or physical chirality.

Nests and quarter-turn centres

Freezing one operand produces the reflection nests

z -> B-z,

z -> B/z.

In their native charts, both are the reflection

w -> beta-w

with multiplier -1. Their centres are respectively B/2 and sqrt(B).

Commutativising the nests applies a quarter-turn:

z -> i*(B-z),

z -> (B/z)^i.

In the room chart, both become

w -> i*(beta-w).

The multiplier is -i, so every integer orbit is an exact four-cycle around the centre

(1+i)*beta/2.

In the original coordinates, the centres are

(1+i)*B/2

and

B^((1+i)/2).

These points are neutral centres under integer iteration, not attractors. Attraction appears only after continuing the iteration height in an appropriate imaginary direction. The reflection contracts at rate exp(-pi) per unit of imaginary height; the quarter-turned nest contracts at rate exp(-pi/2) with the orientation determined by the chosen winding.

The fractional heights form an explicit one-parameter group. In the room chart,

T^[h](w) = c+exp(-i*pi*h/2)*(w-c),

where c=(1+i)*beta/2. This gives closed formulas for composition, height inversion and recovery of the base. It answers the corresponding inverse problem for the subtraction and division columns.

Commutative Logarithmation

The logarithm column has frame offset one. Its commutativised nest is

CLogn(z) = (log_B(z))^i.

Its principal fixed point and multiplier are

z* = exp(-i*W(i*ln(B))),

q = -1/W(i*ln(B)).

The fixed point becomes attracting exactly when

B > B_CL,

where

B_CL = exp(exp(sin(D)))

= 7.1086250777584670897...

and D is the Dottie number satisfying

D=cos(D).

Equivalently, writing theta*=pi/2-D, the threshold satisfies

theta*+sin(theta*)=pi/2.

The threshold is the quarter-turn point of the wider critical-base curve for

z -> (log_B(z))^p.

That curve runs from Euler’s power-tower boundary exp(1/e) to the period-doubling threshold exp(e). The critical base is therefore part of one offset-one dynamical family, not a rank-four counterpart of an unrelated progressing bound.

The stability theorem is local. Just above B_CL, the principal attracting fixed point may coexist with an attracting three-cycle and another attracting fixed point. The inequality B>B_CL does not imply that every initial value converges to the principal fixed point.

Corrected scope

The original paper called the three columns separate regressing holograms and searched for a spine joining them. The spine is simply the ladder of Symmetric-Core rooms, and the charts transport one column into the next exactly.

Several other terms are corrected:

• the quarter-turn fixed points are centres, not integer-iteration attractors;
• the agreement between a dial value and a fractional height does not identify chirality with height;
• base dependence begins with frame offset one, not because of rank four itself;
• the critical-base theorem is local rather than a global basin theorem;
• the former inverse-sector analogy with the commutative power wall is withdrawn.

M44a v1.0 preserves a substantial body of exact mathematics inside a deprecated record. Subtraction, division and logarithm become regressors of successive rooms; their forced phases are quarter-turns of one chart difference; their nests are transported reflections; and their commutativised nests have explicit centres, four-cycles, fractional heights and inverses. The logarithm column introduces a Lambert fixed point and a critical base because it is reported at offset one. The former hologram taxonomy, attractor language, chirality-height identification and rank-based explanation are repaired or withdrawn.

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