M43 F10 The Projective Base - Rays, the Bending Identity, the Harmonic Fork and Flat Shelves

The Theory of Operations represents the rooms of the Symmetric Core through iterated-logarithm charts: ln turns multiplication into addition, ln(ln) does the same for associative power Apow, and higher iterates generate the higher rooms.

F10 " The Projective Base" studies operations obtained by mixing these charts. Since proportional charts define the same transported operation, the resulting operations are naturally points of a projective space, called the projective base. The rooms form its coordinate axes, while mixtures of neighbouring charts fill the lines and higher-dimensional projective spaces between them. The monograph develops this geometry through the rank-three projective line, proves that commutative power Cpow is obtained by bending an associative operation, derives the harmonic fork and its Klein symmetry, describes the Lambert gluing between adjacent floors, and identifies where chart transport is flat and where folds or seams can carry nontrivial loop data.

Let

ell_k = ln^[k]

be the iterated logarithms, and let V be their linear span. A nonzero mixture

A_alpha = sum_k c_k*ell_k

defines, on any interval where it is injective, the straight transport

Op_alpha(a,b)

= A_alpha^(-1)(A_alpha(a)+A_alpha(b)).

Every such operation is commutative, associative and cancellative because it is addition transported through a chart. Conversely, every continuous strictly increasing associative operation on an interval has such an additive representation.

Two analytic charts define the same straight operation exactly when they differ by a nonzero scalar. The operations are therefore the rays of V, and their parameter space is

B = P(V).

The rooms of the Symmetric Core occupy its coordinate axes. At rank n, the mixtures of

ell_1,...,ell_(n-1)

form a projective space P^(n-2). This projective count agrees at rank three with the number of operational indices. Above rank three, its identification with the complete rank architecture remains structural rather than proved.

Straight and bent operations

The principal rank-three line is generated by ln and ln(ln). Write

A_s(x) = (ln(ln(x))+s*ln(x))/2.

Its straight operation is

C_hat_s(a,b)

= A_s^(-1)(A_s(a)+A_s(b)).

The corresponding bent operation measures through A_s but decodes through ln(ln):

C_s(a,b)

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

The cases s=1 and s=-1 are

C_1 = Cpow,

C_(-1) = Croot.

F10 proves the bending identity

C_s = g_s o C_hat_s,

where

g_s = ell_2^(-1) o A_s.

Thus the HyperCore’s rank-three rung is an associative operation followed by one non-associative readout. For s=1,

C_hat_+(a,b)

= exp(W(a*b*ln(a)*ln(b))),

while

g_+(u) = exp(sqrt(u*ln(u))).

The straight family is associative. The bent family is not. In its measuring chart, the bent operation becomes the warped sum

u star_s v = Theta_s(u+v),

with

Theta_s(t) = (t+s*exp(t))/2.

This gives the associator explicitly as the difference between two iterates of Theta_s. The non-associativity of Cpow is therefore not unexplained deformation: it is the exact effect of decoding an additive chart sum through a different chart.

Three readings of one chart

The rank-three symmetrisation, its conserved quantity, storage mean and final operation are three readings of the same chart A_+.

The protocol conserves

A_+(a)+A_+(b),

equivalently

K_3(a,b) = sqrt(a*b*ln(a)*ln(b)).

Its storage mean is the quasi-arithmetic mean

M_3(a,b)

= A_+^(-1)((A_+(a)+A_+(b))/2)

= exp(W(K_3(a,b))).

The rank-three operation uses the same sum but decodes it through ln(ln):

Cpow(a,b)

= ell_2^(-1)(A_+(a)+A_+(b)).

The chart therefore supplies three related objects:

• its conserved sum;
• its averaged value, producing the storage mean;
• its bent readout, producing Cpow.

The elevator selects the midpoint ray [ell_2+ell_1]. The conservation law itself follows from the separated rank-three channel and the exchange of its logarithmic components.

The harmonic fork and the slit

Over the complex numbers, the rank-three projective line becomes a sphere of associative operations. Four distinguished rays occupy

multiplication = infinity,

Apow = 0,

Cpow-ray = 1,

Croot-ray = -1.

Their cross-ratio is -1, so they form a harmonic quadruple. The transformations

s -> -s,

s -> 1/s

generate a Klein four-group preserving this configuration. The same harmonic construction occurs on every line spanned by two adjacent iterated logarithms.

The fluent family from a^b through Cpow to b^a also has a projective reading. At dial position r, its two operands are measured on the reciprocal rays

z_r = r/(1-r),

1/z_r = (1-r)/r.

These rays coincide only at r=1/2, where the fluent family becomes Cpow. Operand exchange acts by projective inversion.

The projective sphere is not globally regular. The derivative of A_s is

A_s'(x) = (1+s*ln(x))/(2*x*ln(x)).

For negative real s, the chart folds at

x* = exp(-1/s).

The negative meridian is therefore a slit. The nonnegative meridian, containing Apow, the Cpow ray and multiplication, remains injective on (1,infinity).

Lambert gluing

Every mixture of two adjacent iterated logarithms is inverted by Lambert’s W. For

A(x) = ell_(k+1)(x)+s*ell_k(x),

one has

A^(-1)(T)

= exp^W(s*exp(T)/s).

Thus a single classical special function, conjugated by the exponential tower, glues adjacent floors. Mixtures separated by larger gaps lead to higher hyper-Lambert kernels. These kernels are defined as solution operators, but their branch structures and arithmetic significance remain open.

Fields, Schwarzians and flatness

A chart A has the translation field

L_A = (A')^(-1)*d/dx.

For the room charts,

L_k = d/d(ell_k).

Adjacent fields satisfy

[L_k,L_(k+1)] = L_k

and therefore generate the affine Lie algebra aff(1). Non-adjacent fields, or any three consecutive fields, generate infinite-dimensional Lie algebras.

The Schwarzian derivative gives a scale-free representation of the projective base. If two chart mixtures have the same Schwarzian on an interval, their charts are proportional and define the same ray. The map

[A] -> Sch(A)

is therefore injective on the base.

For the rank-three family, the Schwarzian has a closed form and develops a double pole precisely at the folds of the slit. This connects three levels of structure:

• rays classify straight operations up to scale;
• logarithmic derivatives classify charts up to affine change;
• Schwarzians classify charts up to Möbius change.

These projective actions must remain distinct. The Möbius group acting on chart values and the Klein group acting on the line of chart mixtures are copies of the same abstract group, but no intertwining map between them is defined.

A loop of charts that remain injective on one common interval is flat: its transitions telescope exactly to the identity. Nontrivial loop data therefore require a named obstruction, such as branching, a fold, loss of a common domain or passage through a seam.

Paths of charts nevertheless transport values. Keeping A_s(f) constant while varying s gives the isonativity flow

df/ds = -f*(ln(f))^2/(1+s*ln(f)).

The flow becomes singular on the slit and is integrated by Lambert’s W. Around each fold, continuation exchanges the branches W_0 and W_(-1), giving local monodromy Z/2.

The proposed identification of the cascade discrepancy with a loop datum at the cascade seam remains an untested conjecture. The paper specifies value and covariance criteria for the test, but the loop has not been constructed or computed. The cascade discrepancy already has an independent definition as the root of its closure equation; the conjecture concerns its geometric placement, not its value.

F10 establishes a projective geometry of chart-generated operations. Straight operations are rays; Cpow is a straight operation followed by one bend; its conserved quantity, storage mean and value are three readings of one chart; and the rank-three line forms a harmonic sphere with a folded negative meridian. Lambert’s function glues adjacent floors, chart fields generate an affine chain, and the Schwarzian embeds the base into quadratic differentials. Loops inside one injective chart family are flat, while folds and seams mark the places where nontrivial loop data may enter. The former “One Connection,” curvature-period and physical interpretations are not required for these results and are withdrawn.

F papers belong to the M43 chain.