M42h The TetraCore Chart and Its Row Shift - A Corrected Record [deprecated]

The Theory of Operations distinguishes the two Cores from the floors and charts built around them. The historical “TetraCore” is not a third Core, but the right-end chart of rank four’s étage line: a partial floor obtained from the superlogarithm and carrying the Symmetric Core’s shelf in transported form.

M42h "The TetraCore Chart and Its Row Shift" preserves the principal result of the original paper, the row shift, and uses it to correct the former two-dimensional room grid, rank labels, period doubling, rank-four Triad and inter-shelf identifications.

Fix a base 1<B<exp(1/e) and let slog_B be the regular superlogarithm, normalised by

slog_B(1)=0.

With F_r=slog_B^[r], define the right-end array

TC[r,k] = F_r^(-1) o room_k o (F_r,F_r),

where r is the nesting and k the Symmetric-Core room. Nesting zero recovers the ordinary shelf, while nesting one gives the historical TetraCore chart.

The central theorem is the row shift. If

h = exp o slog_B,

then

TC[1,k] = h^(-1) o room_(k+1) o (h,h).

Thus the nesting-one row is not a new independent room direction. It is the original shelf shifted upward by one room and read through the single chart h. In particular:

• TC[1,0] is multiplication in the chart h;
• TC[1,1] is Apow in h;
• TC[1,2] is room three in h;
• TC[1,3] is room four in h.

At rank four, the Symmetric Core contributes room three. In the right-end array, this is therefore TC[1,2], not TC[1,1]. This conclusion from the first edition stands, but the row shift supplies its correct reason. The rank-four columns are the HyperCore rung C_4, Symmetric-Core room three and the AscentCore step Asc(Cpow). They are not obtained by transporting the rank-three Triad through one superlogarithmic substitution.

The right-end chart is only partially defined on the real line. Its superlogarithm maps the interval below the attracting fixed point onto a restricted range, so the transported addition law is a local group near its zero. The TetraCore chart is therefore a partial floor and the chiral right endpoint of the étage line. The TetraEtage at the line’s centre remains rank four’s binding floor.

What a chart carries

Conjugation through a chart transports algebraic identities, but an equation involving an external constant must transport that constant as well. Consequently, the Hermit equation of a room does not acquire a doubled period merely because the room is read through slog_B.

The operation called the TetraZeug is the Symmetric Core’s MultPow formula transported through h. Its native Hermit equation is

W^2 = i*pi,

where W=slog_B(u).

This equation already belongs to MultPow; it is not produced by doubling the room-two Hermit period. Room two itself, read in h, retains its quadratic equation with the constant carried through the chart.

The proposed rank formula

R(r,k) = (k+1)*(1-r)+4*r

is likewise corrected. At nesting zero it reproduces the height of room k; at nesting one it gives the rank of the whole right-end floor. Away from those boundaries it is only a label of a place, not an operational rank. In particular, the same label may be attached to positions for which no operation has been constructed.

A half-nesting cell TC[1/2,k] would require a half-iterate of slog_B. No such construction is supplied. The former identification of TC[1/2,2] with the rank-7/2 ISHE is therefore withdrawn. Objects at 7/2, including the HyperCore protocol ISHE-H and the shelf presentation ISHE-S, arise from different constructions.

Corrected scope

The first edition treated the two indices of the array as a genuinely two-dimensional room geometry. The row shift shows instead that nesting and room describe one shelf seen through related charts. It also removes the basis for the claimed faithful lift of the Triad, Hermit-period doubling, abstract rank-four anchor and “conscious” Hypertriad.

The former gravity and holonomy readings are withdrawn. The cascade discrepancy belongs to room three of the Symmetric Core, not to transport between the Symmetric Core and a third Core. The proposed graviton room carries no derived gravitational coupling or field equation.

M42h v1.0 is therefore a correction and provenance record centred on one durable theorem. The TetraCore is a chart of the shelf, not a Core; its nesting-one row is the shelf shifted by one room and read through h; room three appears as TC[1,2]; and transported equations retain their constants. The two-dimensional grid, interpolated-rank interpretation, period doubling, lifted Triad, half-nesting ISHE and gravity programme are withdrawn.

Status

M42h v1.0 is deprecated and retained for provenance and historical audit. Its standing mathematics is carried by M13 "The Symmetric Core", especially the row shift; M36b "Nesting and Room", for the floor array and its labels; and M21a "The TetraEtage and the Étage Line", for the complete right-end geometry.