M40f The Shelf Presentation at Seven-Halves - The Closed-Form Rate, the Midpoint Law, and the Drift and Lock of K[3.5] - A Corrected Record [deprecated]

The Theory of Operations distinguishes an operation’s position in the architecture from the construction used to present it. At rank 7/2, between commutative power Cpow and the HyperCore’s rank-four rung, the HyperCore’s own protocol is the base-free compound ISHE-H of its storage means M_3 and M_4.

M40f "The Shelf Presentation at Seven-Halves" studies a different object, ISHE-S: a base-dependent presentation obtained by compounding M_3 with an attractor mean defined in the superlogarithmic chart slog_B. The paper proves its domain, quadratic convergence, closed-form local rate and midpoint law, and shows that the proposed quantity K_(3.5) drifts before locking at the orbit’s own limit. It also explains why this presentation must not be identified with the HyperCore’s protocol at 7/2.

Fix a base 1<B<exp(1/e). Let L_B be the attracting fixed point of x->B^x, and let slog_B be Koenigs’s superlogarithm normalised by

slog_B(1)=0,

slog_B(B^x)=slog_B(x)+1.

On (1,L_B), define the attractor mean M_4^(attr) implicitly by

L*slog_B(L) = sqrt(a*slog_B(a)*b*slog_B(b)).

The shelf presentation is

ISHE-S = P_(M_3,M_4^(attr)),

the Gauss compound of the rank-three storage mean M_3 and this chart-defined rank-four mean. Both legs are smooth strict symmetric means. The protocol therefore maps (1,L_B)^2 into itself and converges quadratically to a continuous mean. Every diagonal point is fixed, so the protocol has no universal input-independent attractor.

The HyperCore’s own protocol is instead

ISHE-H = P_(M_3,M_4),

where M_4 is the non-quasi-arithmetic storage mean produced by the HyperCore’s rank-four channel. ISHE-H is base-free. By contrast, ISHE-S depends on the selected base through slog_B. The two protocols can converge from the same initial pair to different limits and with rates of opposite sign.

The name “inter-shelf” applies only to the presentation ISHE-S, whose two legs are read through logarithmic and superlogarithmic charts. It does not describe the HyperCore’s own half-rank protocol.

Rate and midpoint law

Use logarithmic coordinates

sigma = (ln(a)+ln(b))/2,

w = ln(a)-ln(b),

and write

P(t)=slog_B(exp(t)).

The rank-three coefficient is

c_3 = -1/(2*sigma*(1+sigma)).

If P_0=P(sigma), P_1=P'(sigma) and P_2=P''(sigma), then the coefficient of the attractor leg is

c(M_4^(attr)) = (P_2*P_0-P_1^2)/(2*P_0*(P_0+P_1)).

The full logarithmic gap after one step satisfies

w' = beta_3(sigma)*w^2+gamma_3(sigma)*w^4+O(w^6),

where

beta_3(sigma) = (c_3-c(M_4^(attr)))/4.

The expansion contains only even powers of w because both protocol legs are symmetric. The first contribution to beta_3 comes from the rank-three leg and is base-independent. The second comes from the superlogarithmic presentation and carries the dependence on B.

At B=sqrt(2) and sigma=0.3,

beta_3 = -0.1645892423....

At the same centre, changing the base from 1.3 to 1.44 moves the rate from approximately -0.766 to -0.135. The rank-three contribution remains unchanged.

The coefficient of the compound mean obeys the universal midpoint law

c(ISHE-S) = (c_3+c(M_4^(attr)))/2.

This is the affine half-rank law for protocol means. It does not identify the presentation with the HyperCore’s rank-four mean or with ISHE-H.

A second presentation replaces the attractor mean with the geometric mean transported through slog_B. It has a different coefficient and therefore a different quadratic rate. Both presentations are mathematically valid, but the choice between them is a choice of chart presentation.

Drift and lock

The quantity proposed by the first edition was

K_(3.5)(a,b) = sqrt(a*b*mlog_B(a)*mlog_B(b)),

where

mlog_B(x) = sqrt(ln(x)*slog_B(x)).

It is not conserved by either protocol step. Along an ISHE-S orbit, it changes immediately and then approaches its diagonal value at the orbit’s limit.

For the calibration at B=sqrt(2) beginning from

(sigma,w)=(0.3,0.3),

the orbit converges to

L_infinity = 1.31754414899637...,

while

K_(3.5) -> L_infinity*mlog_B(L_infinity)

= 0.57747644296569....

The locking is a consequence of the quadratic convergence of the orbit. It does not reveal a new invariant: every continuous invariant of a convergent mean-type protocol is a function of its limit.

The first edition incorrectly identified the locked value with the value at the endpoint L_B. In fact,

slog_B(L_B)=+infinity,

so the proposed endpoint value is infinite. Since different initial pairs converge to different diagonal points, neither L_infinity nor the locked value is a universal constant.

Principal corrections

The corrected record withdraws the former “Logarithm Jump” and the division of the hierarchy into K- and superlogarithmic shelves. The superlogarithmic leg is a presentation of rank four, not the HyperCore’s storage mean.

The former restricted-domain theorem is also withdrawn. The natural domain is the whole square (1,L_B)^2; the additional inequality printed by the first edition used the wrong threshold and excluded valid convergent orbits.

The proposed two-Hermit bridge reduces identically to room two’s quadratic. At the declared units, its numerator and denominator are both i*pi/2, so the correction factor equals 1 for every base. It carries no information about rank 7/2.

Finally, the Yang–Mills and mass-gap reading is withdrawn. The domain has no internal boundary of the claimed kind, and the diagonal function

L*mlog_B(L)

ranges over all positive values. Selecting one of those values and multiplying it by a scale produces a fit, not a spectral gap.

M40f v1.0 preserves the sound mathematics of the former inter-shelf programme while relocating it correctly. ISHE-S is a base-dependent shelf presentation at the address 7/2, not the HyperCore’s base-free ISHE-H. Its convergence, closed-form rate, coefficient midpoint law and drift-and-lock behaviour stand. Its former canonical status, restricted domain, conserved quantity, Hermit bridge and physical interpretation do not.

Status

M40f v1.0 is deprecated and retained as a corrected historical record. It supersedes the June 2026 paper The Inter-Shelf Half-Étage: ISHE at Rank 7/2 in Depth. Its standing results are carried principally by F01 The Seam at Seven-Halves, Rectified, Part III, and M21g The Half-Étages, Chapter 9. The present record preserves the original contribution, complete audit and old-to-new terminology map.