The Science

基于 x² = x + 1 的纯数学研究。每条定理经 Lean 4 机器验证。

Pure mathematics from x² = x + 1. Every theorem machine-verified in Lean 4.

Generation 2 — Pure Mathematics from x² = x + 1

Nine papers submitted to international journals, all derived from the golden-mean shift. Every theorem machine-verified in Lean 4.

Shell Geometry from Stationary Detector Thermality in Static KMS Spacetimes

Mathematical physics / QFT in curved spacetime

gen2

mathematical

submitted

Full click-record statistics of an Unruh-DeWitt detector define two codimension-one shells in static KMS spacetimes. In Schwarzschild the shell pair determines the mass parameter via a self-calibratin…

A Sharp Three-Window Threshold and Finite-Memory Conjugacy in Fibonacci Stabilization

Symbolic dynamics and ergodic theory

gen2

symbolic

submitted

A sharp threshold at window size m=3: for m>=3 the stabilized-window map is injective with image topologically conjugate to the full two-shift. The conjugacy transports the full thermodynamic formalis…

Finite-Window Rigidity in Fibonacci Numeration

Symbolic dynamics and formal language theory

gen2

symbolic

submitted

The Zeckendorf fold map is the normal-form map of a finite terminating confluent rewrite system, with a sharp block-bijection threshold at m=3. The right Fischer cover is identified as a suffix graph …

Folded Histograms of Irrational Rotations: Sampling Certificates and Structural Mismatch to the Parry Measure

Applied dynamical systems

gen2

applied

submitted

A two-stage audit framework for coarse-grained orbit data: certifying empirical folded histograms via star-discrepancy, TV, and relative-entropy bounds, then testing compatibility with the Parry equil…

Zeckendorf Folds, Sturmian Rigidity, and Parry Divergence on the Golden-Mean Shift

Ergodic theory and symbolic dynamics

gen2

ergodic

submitted

The canonical Zeckendorf fold on the Sturmian slice is bijective and realizes the higher-block presentation, giving an exact KL formula and sharp support-constrained Renyi minima in closed form….

Canonical Zeckendorf Normalization and the Minimal Berstel Adder

Formal language theory and automata

gen2

formal

submitted

The canonical low-to-high Zeckendorf normalization is non-subsequential with exact prefix-destruction index Delta(n)=n. The Berstel transducer is proved minimal with exact state complexity 10 via kern…

A Quartic Cover of 37a1 and Its Regular S4-Closure

Arithmetic geometry

gen2

arithmetic

submitted

For the conductor-37 elliptic curve, we carry out a complete explicit programme: constructing the regular S4-closure of a degree-4 map, decomposing the Jacobian via rational idempotents into a genus-2…

Upper Fibers and Witness Covers for the Fibonacci Order-of-Apparition Map

Combinatorial number theory

gen2

combinatorial

submitted

We introduce witness covers for the fiber B_n, proving B_n is an upper set in the divisor lattice. Minimal generators are classified via a primitive-vs-ladder dichotomy for atomic prime-power birth mo…

Tilt Dynamics of Cylinder Information and the Parry Measure on the Golden-Mean Shift

Ergodic theory and probability

gen2

ergodic

submitted

Exponential tilts of cylinder information close within the one-step Markov family, making tilt dynamics globally linearizable. The Parry measure is characterized as the unique zero-jitter (constant Re…