M43 F15 Pyramidation Rectified

F15 Pyramidation Rectified gives Pyramidation a rectifying coordinate in which its apparently explosive iteration becomes a simple straight-line flow. It is the fifteenth paper of the M43 chain. Its main results are the w-chart normal form, the winding ladder, the Strip Absorption Theorem, the Horizon Theorem, and a precise Collatz bridge at Level Two.

The starting Pyramidation map is the branch-resolved self-power operation Pyr_beta(z) = exp(z Log_beta z).

In the second-logarithm coordinate q = Log Log z, the map becomes q -> q + exp(q). 

Passing to the reciprocal coordinate w = 1 / Log z = exp(-q) gives w -> w exp(-1/w).

F15 identifies this discrete map as the unit-step Euler scheme of the exact continuous flow dw/dt = -1.

Thus the huge value-space explosions are the nonlinear image of uniform translation in the correct native coordinate.

The complex plane is then reorganized into horizontal strips. Collapse strips are S_m: Im(q) in (pi/2 + 2*pi*m, 3*pi/2 + 2*pi*m),

while sufficiently deep strip interiors with negative real part are absorbing. The Strip Absorption Theorem formalizes capture by these regions, and the Manifesto’s reset half-space is reinterpreted as a one-step preimage of a collapse strip rather than as a separate ad hoc mechanism.

The branch structure produces the winding ladder w_k = 1 / (2*pi*i*k), corresponding to z = 1 on branch k, with multipliers lambda_k = 1 + 2*pi*i*k.

This relates Pyramidation to the winding-shell geometry of F11’s regressing Caterpillar sector. F15 records the identification as a strong structural observation but leaves its complete theorem-level proof between the two charts open.

The Horizon Theorem converts an excursion in the rectified chart into visible numerical scale. If P = Re(q) is the peak excursion height, then the subsequent collapse depth is approximately -exp(P), while the visible horizon has approximately exp(P) / ln(10) decimal digits. The enormous apparent scale is therefore generated by ordinary linear motion viewed through repeated exponential coordinates.

F15 also makes its genericity claim relative to a declared ensemble, rather than treating “generic” as an undefined universal statement. The paper correspondingly restructures the earlier RFS picture and explicitly separates what is proved from what remains refused or ensemble-dependent.