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Arithmetic Dynamics as the Seventh World: The Law of Forgetting for z^2+c

Primarily human-written textHuman understanding: all partscs.DS — Data Structures and Algorithmsmath.NT — Number Theory

Contributed by Zeng Chunping ↗

Version 1 / Nov 03, 2026 / CC BY 4.0

Abstract

Among the seven worlds of the central-charge program (number fields, function fields, graphs, finite-type dynamics, knots, tropical geometry, and arithmetic dynamics), the seventh is the only one whose critical exponent is continuously tunable, and the only one whose "cusp layer" is visible point by point over the primes. For phi(z) = z^d + c reduced modulo p, the orbit structure splits into two layers: an algebraic-closure layer (universal, smooth, forgotten: N_m = #Fix(phi^m) = d^m + 1, independent of c) and an F_p-rational layer (arithmetic, fingerprinted, remembered: Per_p(phi), the Frobenius fixed-point count). We prove: the dynamic Lehmer constant L_dyn = h_{x^2+1}(0) = 0.203677261369740... (a complete minimization proof over Q); the three-axis separation theorem for the splitting family c = (1-m^2)/4 (amplitude = 2 is a free fingerprint carried entirely by the n = 1 layer; alpha = 0.5 is a theorem for all m >= 5; the Jensen gap J has no m-dependence and separates the monomial and Chebyshev cases that the scale exponent alpha confounds); the sqrt(2) anisotropy of escape near the cusp, A(phi) = |sin(phi/2)|/sqrt(6) (Joukowski coordinates, 15 directions at two scales digit-perfect); the Sha growth law lim_n log_2(Sha_n)/2^n = 2*alpha - 1 with the Stoll mod-square family classified (7/7 separation); the criterion alpha = 1 if and only if disguised group endomorphism, across polynomials and rational maps including both Lattes quotients; and the number-field PCF criterion h(0) = 0 if and only if c in M intersect O_K with all embeddings in M, verified against Baker-DeMarco with a six-field Lehmer table. The universal far-tail rate is identified with the random-mapping class, Phi = (4-pi)/4 = 0.2146, with a quantitative accounting of the PCC contamination that had drifted earlier measurements. All corrections found during the research (including two withdrawn headline predictions) are recorded in an appendix.

Tools used

Magma
Magma

References

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Version history

  1. v1Submitted by Zeng ChunpingInitial depositCurrentOct 30, 2026