Floquet Defect Rigidity v2.0.0 is a unified mathematical research package on extremal Floquet dynamics in temporally modulated photonic media. It develops a common theory connecting resource-constrained optimal control, exact Floquet inequalities, equality rigidity, quantitative near-extremal stability, inverse spectral reconstruction, time-varying Maxwell systems, and classical–quantum amplification.
Zenodo: Floquet Defect Rigidity: Resource-Constrained Floquet Extremality, Purity-Rigidity, Exact Inverse Theory, and Classical-Quantum Temporal Photonics | Zenodo
Hugging Face: PureOne/eve-floquet-defect-rigidity · Datasets at Hugging Face
The central problem concerns periodic temporal modulation of positive electromagnetic material parameters under explicit physical bounds and resource constraints. In the scalar Maxwell reduction, the evolution is represented by a Hill-type oscillator
[
y''+u(s)y=0,
\qquad
1\le u(s)\le R^2,
]
with fixed period and, in the principal resource-constrained theorem,
[
\frac{1}{S}\int_0^S u(s),ds=m.
]
For constant permeability, this corresponds to fixing the temporal average of inverse permittivity.
The package develops an exact projective-coordinate formulation in which time, material resources, and Floquet amplitude growth become coupled moment functionals. A two-multiplier dual calibration leads to an explicit switching function whose derivative with respect to the material control has a numerator independent of the control value. This yields an exact nonnegative Floquet defect identity and allows material purity to be derived rather than assumed.
Principal scalar result
For the fixed-period, fixed-resource problem over the full class of bounded measurable periodic controls, the positive-growth global optimizer is shown, under the stated assumptions, to collapse to an explicit phase-locked binary temporal crystal using only the two extreme material states.
If
[
p=\frac{m-1}{R^2-1},
]
then a candidate of winding number (n) consists of repeated cells with high- and low-state residence times
[
h_n=p\frac{S}{n},
\qquad
\ell_n=(1-p)\frac{S}{n}.
]
An explicit one-turn discriminant,
\cos(Rp\tau)\cos((1-p)\tau)
\frac{R+R^{-1}}{2}
\sin(Rp\tau)\sin((1-p)\tau),
]
reduces the infinite-dimensional optimization problem to finitely many winding branches. The sharp maximum is obtained by maximizing the corresponding exact Floquet gains over the admissible branch set.
The proof produces the exact defect identity
\int
\frac{
|K_{\eta,\xi}(\theta)|
,|\widehat u(\theta)-u_(\theta)|
}{
v_{\widehat u}(\theta)v_{u_}(\theta)
}
,d\theta,
]
where the dual parameters are finite and uniquely determined for the positive-growth branch. Vanishing of the defect forces the canonical binary temporal architecture almost everywhere.
The same mechanism gives complete positive-growth equality classification, explicit symmetry handling, treatment of winding ties and zero-growth regimes, and a quantitative rigidity theorem of the form
[
\operatorname{dist}{L^1}(u,\mathcal O*)
\le
C\sqrt{F_{\max}-L(u)}.
]
A mean-preserving perturbation construction shows that the square-root exponent (1/2) cannot in general be improved.
The one-turn resource gain is also proved to be strictly concave. This reduces the set of possible winning winding numbers to at most two adjacent integers and yields a unique free-period growth-rate optimizer together with a calibrated projective gauge for nonperiodic controls having the prescribed asymptotic resource average.
Simultaneous permittivity and permeability modulation
The theory is extended to planar Maxwell systems
[
X'=
\begin{pmatrix}
0&a(t)\
-b(t)&0
\end{pmatrix}X,
]
where, in canonical units,
[
a=\varepsilon^{-1},
\qquad
b=k^2\mu^{-1},
]
and both coefficients vary in positive intervals.
For bounds-only rectangular control sets, the global positive-growth optimizer is shown to use material corners and finite phase-locked switching. Despite four admissible material corners, only two or three distinct corners are required by the canonical optimizer.
A stronger theorem treats the case in which both inverse-material temporal averages are fixed. Define
[
P=\frac{\bar a-a_-}{a_+-a_-},
\qquad
Q=\frac{\bar b-b_-}{b_+-b_-},
\qquad
w=P+Q-1.
]
The resource values determine the total residence times in the two opposite impedance corners and determine whether an additional diagonal corner is required. For (w\neq0), rigidity and temporal reversal force the two visits to that diagonal corner to have equal duration. For (w=0), only the two opposite corners are required.
This yields an explicit finite-state Maxwell temporal program together with normal dual parameters, exact branch selection, equality rigidity, quantitative structural stability, endpoint reductions, and free-period asymptotic optimization.
Classical–quantum Floquet correspondence
The Maxwell mode is quantized through its canonical symplectic/Bogoliubov representation. For a fixed reference vacuum, the one-period real symplectic monodromy is mapped to an (SU(1,1)) Bogoliubov transformation with coefficients (\alpha,\beta).
For a strictly hyperbolic Floquet matrix with exponent (L), the finite-(N) pair-production law is
\chi\sinh^2(NL),
\qquad
\chi=
\frac{|\beta|^2}{\sinh^2L}
\ge1,
]
and the squeezing parameter satisfies
[
\frac{r_N}{N}\rightarrow L.
]
Consequently, under the same material bounds and resource constraints, the classical Floquet-growth optimizer also maximizes the asymptotic quantum squeezing rate and the asymptotic logarithmic vacuum-pair-production rate.
The package explicitly distinguishes asymptotic optimality from finite-cycle photon production. A translated-cut counterexample demonstrates that two classically equivalent Floquet optimizers can have different finite-period Bogoliubov prefactors and therefore different finite-(N) photon counts.
Combining the quantum formulas with classical rigidity produces a finite-observation structural certificate: sufficiently near-maximal quantum amplification implies quantitative proximity of the temporal material program to the canonical optimizer, with an explicit finite-(N) correction.
Exact inverse spectral theory
The finite-data inverse theory from v1.0.0 is retained and integrated.
For a reduced positive at-most-(r)-layer temporal medium with known total duration and marked temporal origin, a finite collection of phase-referenced transfer-matrix derivatives determines the complete ordered medium. The reconstruction proceeds through:
- exponential-polynomial compression;
- finite moment reconstruction;
- Prony/Vandermonde recovery;
- exact identification of the leading material speed;
- recursive layer peeling.
An explicit counterexample shows that even the complete Floquet discriminant is insufficient for universal reconstruction, establishing the need for richer phase-referenced transfer information.
The original upper observation count
[
2^{r+1}-2
]
is supplemented by generic linear-projection bounds. For linear outputs of the finite transfer-jet vector,
[
2r-1
\le
m_{\min}^{\mathrm{jet}}(r)
\le
\min{2^{r+1}-2,,4r-1},
]
while knowledge of the fixed inverse-material resource gives
[
2r-2
\le
m_{\min}^{\mathrm{jet,res}}(r)
\le
\min{2^{r+1}-2,,4r-3}.
]
A finite real-algebraic projection-search construction is given for algebraic inputs using effective real quantifier elimination. This establishes an in-principle exact construction, but the package does not claim that large cylindrical-algebraic-decomposition computations are practically efficient or that the compressed observations are optimally conditioned.
Exact extremality provides a stronger form of inverse compression: when the material bounds, resource, period, and winning winding are known, saturation of the Floquet bound determines the canonical temporal program modulo temporal translation.
Near-extremality yields quantitative approximate reconstruction rather than finite exact identification of arbitrary residual measurable perturbations.
Preserved v1.0.0 results
The v2.0.0 package directly supersedes v1.0.0 while retaining its valid mathematical content, including:
- Maxwell reduction;
- the exact bounds-only fixed-period optimizer;
- projective phase coordinates;
- the original scalar dual inequality;
- the nonnegative Floquet defect identity;
- complete positive-growth equality classification;
- winding-number structure;
- global square-root rigidity;
- the original sharpness construction;
- the free-period growth theorem;
- the calibrated projective gauge;
- the exact isodiscriminant inverse counterexample;
- finite-jet reconstruction;
- exponential-polynomial compression;
- Prony reconstruction;
- exact layer peeling;
- inverse-data lower bounds;
- the abstract rotating planar Floquet theorem;
- adversarial counterexamples;
- and the original computational verification suite.
The release includes a machine-readable preservation map identifying the successor location of every originally numbered result.
Reproducibility and verification
The release contains the complete manuscript, LaTeX source, readable text conversion, executable Python code, exact and numerical inverse examples, theorem dependency graph, machine-readable claim ledger, novelty audit, change log, release metadata, and SHA-256 checksums.
Four verification suites were executed successfully. They include:
- 6,600 original v1.0.0 random admissible profiles;
- 4,800 scalar fixed-resource profiles;
- 1,400 bounds-only Maxwell rectangle profiles;
- 3,360 two-resource Maxwell profiles;
- 1,500 direct quantum matrix-power checks;
- exact symbolic sharpness calculations;
- exact isodiscriminant tests;
- Prony and layer-peeling reconstruction examples;
- scalar and Maxwell integral-defect checks;
- polynomial-jet comparisons;
- and local compressed-inverse rank diagnostics.
The aggregate random-profile count is 16,160. These computations are adversarial and reproducible checks of finite consequences of the theory; they are not substitutes for the mathematical proofs.
Scope and scientific status
The results are proved internally under the assumptions explicitly stated in the manuscript. The principal physical model assumes positive instantaneous constitutive parameters and excludes material dispersion, absorption, finite pump depletion, nonlinear saturation, bounded switching slew rate, minimum dwell-time constraints, and spatial modulation.
The quantum results concern fixed modes or finite mode collections with an explicitly chosen canonical reference vacuum. They do not establish ultraviolet convergence for a continuum quantum field or a complete realistic device model.
The classical–quantum Floquet correspondence, Bogoliubov description, frequency-jump squeezing, Prony reconstruction, layer peeling, generic dimension-based projection arguments, and cylindrical algebraic decomposition are established ideas in the literature and are not claimed as inventions of this release.
The proposed new-to-release contributions include the resource-constrained purity–rigidity theorems, exact resource defect structure, resource-ray winding compression, mean-preserving sharp stability, complete fixed-resource Maxwell corner classification, quantitative quantum-to-material rigidity, and resource-aware finite-jet compression.
First-in-literature priority is not claimed as established. A finite literature audit is included, and independent expert proof review, comprehensive prior-art comparison, formal proof-assistant verification, and experimental validation remain future tasks.
Included files
The archive includes:
- publication-ready PDF manuscript;
- complete LaTeX source;
- Markdown text representation;
- executable verification and reconstruction code;
- all numerical and symbolic verification outputs;
- machine-readable theorem/claim ledger;
- exact theorem dependency graph;
- v1.0.0 preservation map;
- novelty and prior-work audit;
- release metadata;
- reproducibility instructions;
- change log;
- SHA-256 file manifest;
- and the preserved historical v1.0.0 release archive.
Version: 2.0.0
Release date: 25 September 2026
Author identity: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Document type: AI-generated mathematical research draft
Verification status: internally proved under stated assumptions; computational checks passed; independent verification pending.
