Golden Minkowski Model Set
Abstract
The two real embeddings form a golden lattice whose internal window selects model-set points.
Definition 1.1 (Minkowski lattice, window, and labeled model set).
Lean statement: D5/S1/Scale/MinkowskiModelSet.minkowski_model_set_spec
Formalization. D5/S1/Scale/MinkowskiModelSet.minkowski_model_set_spec (✓ std3).
Citation. Michael Baake, Natalie Priebe Frank, and Uwe Grimm (2021). Three variations on a theme by Fibonacci. DOI: 10.1142/S0219493721400013.
Commentary.
The physical and conjugate embeddings give an injective diagonal range. An internal-space window selects physical projections, and the labeled extension pairs selected points with their joint golden coordinates.
Remark 1.2 (Value and code geometries).
Lean statement: D5/S1/Scale/MinkowskiModelSet.minkowski_model_set_spec
Formalization. D5/S1/Scale/MinkowskiModelSet.minkowski_model_set_spec (✓ std3).
Citation. Michael Baake, Natalie Priebe Frank, and Uwe Grimm (2021). Three variations on a theme by Fibonacci. DOI: 10.1142/S0219493721400013.
Commentary.
The same carrier admits a lattice-like value reading and a cut-and-project code reading. The internal window justifies calling the code geometry a model set; it does not by itself provide a Bloch decomposition, a spectral gap theorem, or a periodic classifier for the code layer.
Remark 1.3 (Off-diagonal load and diagonal blindness).
Source. Repository-derived.
Commentary.
The source assigns the two-sided code genuine load only away from the diagonal: its cited off-diagonal results are reported to fail under replacement encodings. The classical zeta diagonal is instead code-blind whenever it is reached only through diagonal decomposition. Whether an off-diagonal invariant can return analytic information to that diagonal remains explicitly open as O-5.
Remark 1.4 (Scaled zero images need an independent engine).
Source. Repository-derived.
Commentary.
In the source cascade, every zeta zero rho produces the image lattice rho/(aphi^2 + bphi^3) whenever the corresponding exponent is nonzero. The leading scale ratio is phi, and the band endpoints interlace pole and critical images, with 1/(2*phi^3) as the stated left endpoint. This self-similar overlay is only a rearrangement of identities built from zeta, so without new input it gives no compressed zero argument. Its positive use is conditional: genuinely independent control of one quasiperiodic band, for example trace-map hyperbolicity, would constrain an entire family of phi-scaled zeta segments. The recursive skeleton is present; the independent engine is still O-5.
Remark 1.5 (The continuation wall is a transported boundary).
Source. Repository-derived.
Commentary.
The cyclotomic Estermann-Kurokawa mechanism relies on explicit control of polynomial-factor zeros. For irrational exponents the source replaces that input by two obligations: scaled independence of zeta zeros on the zero side and Hecke-Mahler zero avoidance on the axis side. Excluding those two failure channels unconditionally is the outstanding N-4 subaccount of O-5. The source then separates three geometries: the proved code spectrum on a circle, the conjectural zeta-zero spectrum on a line, and a conditional continuation wall on an axis whose bricks are transported critical zeros. The no-door reading of that wall is another projection of scaled zero independence. This dictionary rearranges zeta information and supplies no independent zero input; its new content is the claimed boundary of Zqc as an analytic object. A second independent derivation of the full exponent table is also recorded as passing without a new audit exception.
References
- Truth anchor:
D5/S1/Scale/MinkowskiModelSet.minkowski_model_set_spec - Truth anchor:
D5/S1/Scale/MinkowskiModelSet.minkowski_model_set_spec - Dependency: D5/S1/Depth/JointCoordinates