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Affine Cycles: Exact Cut Defects and Minimal Retained State
Affine Cycles: Exact Cut Defects and Minimal Retained State
lmtlssss🜐 · 3 November 2026 · version 1.0.0
Result
Let an ordered word carry affine state maps and affine local costs. Evaluate each cost before its state transition. Compare every cyclic rotation at the state reached by the original word at that cut. If the full state map is F(x)=Mx+b and the rows of R are the accumulated prefix-cost coefficients, the complete defect vector is
D(x)=R((M-I)x+b).
The exact minimum number of retained linear coordinates is rank(R(M-I)). For all transported cut totals, it is the rank obtained by adjoining the full-cost row to R(M-I). The decoder may be arbitrary; the summary is linear and recovery holds for every real initial state. The affine system R(M-I)x=-Rb gives every cut-invariant state.
The principal rational example has three state coordinates, one required defect coordinate, two required total-cost coordinates, and a whole plane of cut-invariant states. Its cycle has no fixed point. The polynomial extension gives a finite derivative-coefficient rank for the same linear-summary contract.
Release contents
Mathematical manuscript with full proofs; exact verification source and certificates; a native Blender scene; a source-derived 3840 × 2705 PNG in the established visual series.
Author: lmtlssss🜐. Version 1.0.0.
Source commit: 0641a4b80c83b870be360657620fe146c84ea4cd.
Robust Amplitude Elimination and Exact Lattice-Phase Selection
Robust Amplitude Elimination and Exact Lattice-Phase Selection
lmtlssss🜐 · 2 November 2026 · Version 1.0.0
A complete exact selection law for a translating common lattice reduces finite robust designs to residue cells and affine envelopes. It gives rational selection probabilities, expected optima and boundary decisions. A four-design example separates the continuous winner from the phase-selection probabilities. A second theorem proves persistent reversal under arbitrarily fine lattice refinement.
Principal result
The principal contributions are the common-lattice phase-selection law, its exact probability and expectation formula, and the persistent-refinement reversal theorem. Scalar monotonicity, interval elimination, homogeneity and line-envelope geometry are established foundations, not separate originality claims.
DOI: https://doi.org/10.5281/zenodo.23897131
The release contains a six-page mathematical preprint, exact verifier and certificates, a native Blender scene, and its source-derived 3840 × 2705 figure.
Canonical mathematical source: 03750c1cce8098fe1ececc246fdd9339d5a2fed3.
Sharp Observation Bounds and Minimal Peak Horizons for Damped Recurrences
Sharp Observation Bounds and Minimal Peak Horizons for Damped Recurrences
lmtlssss🜐 · 2 November 2026 · Version 1.0.1
Exact coefficient inequalities classify the stable conjugate-root recurrences with minimal uniform peak horizon two or three. A symmetric observation hexagon supplies necessity and sufficiency. An explicit rational boundary example has shortest uniform horizon three. Sharp observation constants, rational adaptive certificates and exact Stein sums accompany the new coefficient-region result.
Principal result
The principal result is the closed necessary-and-sufficient coefficient classification for minimal horizons two and three, including all boundary equalities. General finite determination, polar invariant sets and Stein identities are classical. The paper gives the exact second-order regions rather than presenting those general tools as new.
Correction
Version 1.0.1 corrects the row-action matrix in Lemma 6. The theorem statements and exact verification results remain unchanged. The established native Blender scene and 3840 × 2705 mathematical figure retain their original figure DOI and bytes.
DOI: https://doi.org/10.5281/zenodo.23900237
The release includes the six-page manuscript, exact source and certificates, native scene and 4K figure.
Canonical source: e60fefd
Sharp Observation Bounds and Minimal Peak Horizons for Damped Recurrences
Current corrected version: Version 1.0.1 corrects the row-action matrix in Lemma 6. The coefficient-region theorem and exact certificates are unchanged.
Sharp Observation Bounds and Minimal Peak Horizons for Damped Recurrences
lmtlssss🜐 · 2 November 2026 · Version 1.0.0
Exact coefficient inequalities classify the stable conjugate-root recurrences with minimal uniform peak horizon two or three. A symmetric observation hexagon supplies necessity and sufficiency. An explicit rational boundary example has shortest uniform horizon three. Sharp observation constants, rational adaptive certificates and exact Stein sums accompany the new coefficient-region result.
Principal result
The principal result is the closed necessary-and-sufficient coefficient classification for minimal horizons two and three, including all boundary equalities. General finite determination, polar invariant sets and Stein identities are classical. The paper gives the exact second-order regions rather than presenting those general tools as new.
DOI: https://doi.org/10.5281/zenodo.23897133
The release contains a six-page mathematical preprint, exact verifier and certificates, a native Blender scene, and its source-derived 3840 × 2705 figure.
Canonical mathematical source: 03750c1cce8098fe1ececc246fdd9339d5a2fed3.
Cyclic-Shift Energies: Minimal State and Cyclotomic Gaps
Cyclic-Shift Energies: Minimal State and Cyclotomic Gaps
lmtlssss🜐 · 2 November 2026 · Version 1.0.0
A cyclotomic divisibility criterion decides equality in an integer-occupancy spectral bound. At prime periods, algebraic norms yield a quantitative positive gap for nonflat spectra. The bound is attained at exactly five occupancies in a complete 210-state example. Minimal linear state and retained-potential identities supply the representation framework.
Principal result
The principal result is the quantitative cyclotomic spectral gap, its equality obstruction and the attained five-cycle certificate. The Fourier diagonalization, positive-semidefinite rank argument, cyclotomic irreducibility and field-norm integrality are established ingredients and receive attribution.
DOI: https://doi.org/10.5281/zenodo.23897127
The release contains a six-page mathematical preprint, exact verifier and certificates, a native Blender scene, and its source-derived 3840 × 2705 figure.
Canonical mathematical source: 03750c1cce8098fe1ececc246fdd9339d5a2fed3.
A Three-Bit Boundary for Cyclic Boolean–Modular Inversion
lmtlssss🜐 · Version 1.0.0. Complete inverse, sharp eight-preimage bound, periodic extremal family, and exact verifier. DOI: 10.5281/zenodo.23680529. Mathematics / Discrete mathematics.
Twenty Signed Products for a Twelve-State Composition Tensor
lmtlssss🜐 · Version 1.0.0. Explicit twenty-product integer decomposition; real rank between 18 and 20; exact nonnegative tensor-power rank 35^q and signed upper bound 20^q. Paper, exact certificate and verifier, native Blender figure and editable scene. DOI: https://doi.org/10.5281/zenodo.23695066
Exact Counting for a Boolean–Modular Equation with Bounded Tap Span
lmtlssss🜐 · Version 1.0.0. Exact count and witness from a bounded nonlinear frontier, complete proof and executable verification. DOI: 10.5281/zenodo.23697063.
Ordered Carry Frontiers for Parallel Addition
lmtlssss🜐 · Version 1.0.0. Exact carry-state count, residual minimality, stable transitions and an exact masked counter. DOI: 10.5281/zenodo.23702847.
Exact Counts of Sum-and-Carry Functions for Cascaded Addition
lmtlssss🜐 · Version 1.0.0. General exact count, constructive minimal code, proofs and source. DOI: 10.5281/zenodo.23700193.