M30e The Atlas Supplement - One Quadratic, Its Julia Set and the Flat Charts - A Corrected Paper [deprecated]

M30e "The Atlas Supplement" v1.0 audits an earlier catalogue joining tropical limits, TrigCore charts, left-caterpillar towers, Toda systems, sporadic groups and a proposed “NC-quadratic geometry.” Most of that atlas does not survive. Its positive centre is one universal quadratic, X^2-X+C, viewed simultaneously through its roots, the fixed points of z^2+C, the room-two Hermit pair and the balance ray. The corrected record also preserves the flatness of one-chart geometries, the exact dynamics of left-caterpillar towers and the classical census of sporadic groups, while withdrawing the claimed curvatures, physical readings and exceptional-group correspondences.

One quadratic

The universal quadratic is

Q_C(X) = X^2-X+C.

Its roots have sum 1 and product C, and are exchanged by the Vieta involution

X -> 1-X.

The pair {s,1-s} is its root set exactly when

C = s*(1-s).

The same polynomial is the fixed-point equation of

f_C(z) = z^2+C.

Its fixed-point multipliers are 2*s and 2*(1-s). Their moduli are equal exactly when C is real and C>=1/4. This is the closed external ray of angle zero of the Mandelbrot set, landing at the parabolic cusp C=1/4.

The Balance Defect is

Delta(C) = Re(sqrt(1-4*C))/2.

It vanishes precisely on this ray. For s=1/2+i*t,

C = 1/4+t^2.

This also gives an exact reformulation of the Riemann Hypothesis: every nontrivial zero rho lies on the critical line exactly when every value rho*(1-rho) belongs to [1/4,infinity). The quadratic dynamics does not constrain the zeta zeros; this is a change of coordinate, not a route to a proof.

Room two’s quadratic map

Room two of the Symmetric Core has the Hermit equation

w^2-w = i*pi/2.

Thus its Hermit pair is the fixed-point pair of f_C at

C = -i*pi/2.

The principal root is

w_tau = 1.4593138910+0.8187082151*i,

with companion 1-w_tau. Their product is

w_tau*(1-w_tau) = -i*pi/2.

Both fixed points repel. The moduli of their multipliers are approximately 3.3465685 and 1.8775009.

The critical point of f_C is 0, not 1/2. Its orbit escapes, so C=-i*pi/2 lies outside the Mandelbrot set. The Julia set is consequently a Cantor set on which the map is conjugate to the shift on two symbols. Its only nontrivial Euclidean symmetry is z->-z.

Its Hausdorff dimension is

0.777077047868.

This value was obtained through Bowen’s formula and checked by three numerical methods in the current source editions.

The repaired “NC-quadratic principle” is therefore one exact mathematical object: the same quadratic carries the Symmetric-Core pair coordinate, room two’s Hermit pair and the fixed points of a classical quadratic map. Its former chiral and physical interpretations do not survive.

Flat charts and tropical limits

A metric pulled back from a single one-dimensional chart is flat. Room two’s chart

ell_2(x) = ln(ln(x))

therefore does not possess the variable intrinsic curvature claimed in the first edition. What it does carry is a projective invariant, its Schwarzian:

Sch(ell_2)(x) = (1+(ln(x))^2)/(2*x^2*(ln(x))^2).

This quantity is positive on its real domain, but it is not Riemannian curvature.

Likewise, the TrigCore family

x -> x^(exp(i*theta))

consists of linear images of the logarithmic chart. Every angle describes the same flat line in another complex coordinate. There is no transition from hyperbolic to spherical curvature at theta=pi/4, and a Wick rotation is not a quarter-turn of this dial.

The max-plus algebra survives as Maslov’s dequantisation of room -1:

max(a,b) = lim_(epsilon->0+) epsilon*ln(exp(a/epsilon)+exp(b/epsilon)).

This is a singular limit of the log-sum-exp operation, not the ground state of a rank geometry.

Left-caterpillar towers

For a complex base B=r*exp(i*theta), the left-caterpillar tower satisfies

LC_4(B,n+1) = B^(B^n).

Its modulus obeys

ln|LC_4(B,n+1)| = r^n*|Log(B)|*cos(n*theta+psi),

where psi=arg(Log(B)).

The sign of the cosine determines collapse or explosion. When theta/(2*pi) is irrational, the resulting binary collapse word is a deterministic Rote word whose two symbols each have frequency 1/2. This frequency is not a probability or stochastic parameter.

The configured zero records the approach direction of a collapsing tower. The configuration is reproduced under the relevant map only on the positive-real direction. The former “Koenigs monodromy class,” binary DNA and Fatou–Julia sets of angles combined distinct objects and are withdrawn.

Sporadic groups and principal withdrawals

The corrected record preserves the classical facts that the Conway group Co_0 is the automorphism group of the Leech lattice, that twenty sporadic simple groups belong to the Happy Family and that the six pariahs are

J_1, J_3, J_4, O'N, Ru, Ly.

The even unimodular Lorentzian lattice of signature (25,1) has twenty-four primitive null-vector orbits corresponding to the Niemeier lattices. Its automorphism group is infinite, and none of the pariahs is the stabiliser of one of those null vectors.

The Monster does not act as the symmetry group of room two’s Julia set. That Julia set has only the symmetry {z,-z}. The numerical coincidence between twenty-six sporadic groups and the dimension 26 of the Lorentzian lattice supplies no structural identification.

The proposed Toda geometry and Yang–Mills gap are also withdrawn. No relevant Toda sector or Laplacian was defined, and the former mass-gap value is a freely varying coordinate rather than a spectral invariant.

M30e v1.0 preserves one quadratic and several properly separated mathematical subjects. The quadratic supplies the balance ray, room-two Hermit pair and a Cantor Julia set of computed dimension. One-chart operations are flat, the TrigCore knob is a family of deformed charts, left-caterpillar towers generate deterministic collapse words, and the sporadic-group material returns to its classical setting. The curvature atlas, chiral condensate, Wick-rotation dial, Toda geometry, mass-gap claim and Monster–Julia correspondence are withdrawn.

Status

M30e v1.0 is deprecated and retained for historical audit. Its universal quadratic and room-two dynamics are carried by M18a, Part II, and M30f, Chapter 7. Flat charts and tropical limits belong to M30c; left-caterpillar towers to M06 and M07; and the corrected sporadic-group programme to F01 and M29b.