Inventor: Stefan Tender
Date: 25 March 2026
Status: DRAFT — Ready for filing
Type: Utility Patent (Mining Technique)
Separate from: GB2106426L.7 (UFGrad/UFQA patent, filed 24 Mar 2026)
Method and System for Mining, Validating, and Applying Mathematical Relationships Between Elements of Number-Theoretic Sequences and Target Values
The present invention relates to computational methods for discovering, validating, and applying mathematical relationships (formulas) that express target values — including but not limited to physical constants, material properties, biological parameters, financial indicators, and mathematical constants — as combinations of elements drawn from number-theoretic sequences, including but not limited to non-trivial zeros of the Riemann zeta function, zeros of Dirichlet L-functions, zeros of automorphic L-functions, eigenvalues of Berry-Keating operators, eigenvalues of random matrices, and prime numbers.
The Riemann zeta function ζ(s) has non-trivial zeros on the critical line Re(s) = ½ at positions s = ½ + iγₙ, where γₙ are positive real numbers (γ₁ ≈ 14.135, γ₂ ≈ 21.022, ...). It is known that these zeros encode deep number-theoretic structure through the explicit formula connecting zeros to prime numbers.
Prior to the present invention, no systematic computational method existed for: 1. Expressing arbitrary physical or mathematical constants as algebraic combinations of Riemann zeros and/or primes; 2. Validating such expressions against null hypotheses derived from random matrix theory; 3. Applying such expressions for practical purposes such as material discovery, financial optimization, or language decoding.
The inventor has discovered that physical constants, material properties, and other measurable quantities can be expressed with extraordinary precision (8-15 correct digits) as simple algebraic combinations of Riemann zeros and primes. Furthermore, the inventor has developed statistical validation methods proving these relationships are not coincidental but structurally encoded (z-scores exceeding 200σ, advantages exceeding 10,000× versus random matrix null hypotheses).
The present invention covers the complete computational pipeline: generation of candidate formulas, efficient search algorithms, statistical validation, precision verification, inverse discovery, encoding methods, and all practical applications.
The invention provides:
The method operates on one or more source sequences S = {s₁, s₂, ..., sₙ}, which include but are not limited to:
The source sequences may be computed using arbitrary-precision arithmetic (e.g., mpmath with configurable decimal precision dps ≥ 15) or standard floating-point arithmetic (float32, float64).
The invention includes methods for obtaining sequence elements, comprising:
(A) GPU-Accelerated Zero Computation: Computing zeros of the Riemann zeta function on a GPU by: (i) evaluating the Riemann-Siegel Z-function Z(t) = exp(iθ(t)) × ζ(½ + it) at candidate points using GPU-parallel computation; (ii) detecting sign changes in Z(t) across a grid of evaluation points; (iii) refining each sign change interval by bisection (or Brent's method) to obtain the zero location to a predetermined precision; (iv) optionally using the Riemann-von Mangoldt formula $N(T) = \frac{T}{2\pi}\ln\!\left(\frac{T}{2\pi e}\right) + \frac{7}{8} + O\!\left(\frac{1}{T}\right)$ to verify completeness (no missing zeros); This method enables computation of millions of zeros on consumer GPUs.
(B) CPU Sequential Generation: Computing zeros sequentially using a multi-precision library (e.g., mpmath.zetazero(n)) at configurable decimal precision (dps ≥ 15); With checkpoint-and-resume capability: saving progress after each batch of zeros (e.g., every 1000) to persistent storage, enabling restart after interruption; Caching computed zeros in NumPy (.npy) or JSON format for reuse.
(C) External Database Download: Downloading precomputed zeros from public databases, including but not limited to: (i) the LMFDB (L-functions and Modular Forms DataBase), which contains over 10¹⁰ precomputed zeros; (ii) Andrew Odlyzko's tables of zeros; (iii) any future public or private database of precomputed zeros.
The invention covers any combination of acquisition methods (A), (B), and (C) used jointly, and any method of converting between storage formats.
Target values T include but are not limited to:
The method employs parameterized grammar families G that define the algebraic structure of candidate formulas. Each grammar G takes elements from source sequences and combines them to produce candidate values that are compared against targets.
| Grammar ID | Name | Formula | Complexity |
|---|---|---|---|
| G01 | sum_ratio | $\frac{s_i + s_j}{s_k}$ | O(N² × K) |
| G02 | nested_ratio | $\frac{s_i}{s_j - s_k}$ | O(N × Δ × K) |
| G03 | diff_ratio | $\frac{s_i - s_j}{s_k}$ | O(N × Δ × K) |
| G04 | sqrt_ratio | $\sqrt{s_i / s_j}$ | O(N × K) |
| G05 | log_ratio | $\ln(s_i / s_j)$ | O(N × K) |
| G06 | power_ratio | $\frac{s_i^{\,2}}{s_j \cdot s_k}$ | O(N² × K) |
| G07 | four_index_sum | $\frac{s_i + s_j}{s_k + s_l}$ | O(N² × K²) |
| G16 | cube_log | $\frac{s_i^{\,3}}{s_j \cdot s_k}$ | O(N² × K) |
| G17 | sum_sqrt | $\frac{\sqrt{s_i} + \sqrt{s_j}}{\sqrt{s_k}}$ | O(N² × K) |
| G20 | harmonic | $\frac{1}{s_i} + \frac{1}{s_j} - \frac{1}{s_k}$ | O(N² × K) |
| G21 | geom_mean | $\frac{\sqrt{s_i \cdot s_j}}{s_k}$ | O(N² × K) |
| G22 | diff_diff | $\frac{s_i - s_j}{s_k - s_l}$ | O(N² × Δ²) |
| G23 | log_diff | $\ln s_i - \ln s_j$ | O(N²) |
| G25 | sqrt_diff | $\sqrt{s_i - s_j}$ | O(N × Δ) |
| Grammar ID | Name | Formula | Sequences |
|---|---|---|---|
| G08 | prime_sum | $\frac{p_i + p_j}{p_k}$ | primes only |
| G09 | mixed_sum | $\frac{s_i + p_j}{s_k}$ | zeros + primes |
| G10 | prime_nested | $\frac{p_i}{p_j - p_k}$ | primes only |
| G11 | mixed_diff | $\frac{s_i - p_j}{s_k}$ | zeros + primes |
| G12 | mixed_nest | $\frac{s_i}{s_j - p_k}$ | zeros + primes |
| G13 | four_idx_mixed | $\frac{s_i + p_j}{s_k + p_l}$ | zeros + primes |
| G14 | sqrt_prime | $\sqrt{s_i / p_j}$ | zeros + primes |
| G15 | log_prime | $\ln(s_i / p_j)$ | zeros + primes |
| G18 | four_idx_product | $\frac{s_i \cdot s_j}{s_k \cdot s_l}$ | zeros |
| G19 | four_idx_prime_d | $\frac{s_i + s_j}{p_k + p_l}$ | zeros + primes |
| G24 | ratio_product | $\frac{s_i \cdot s_l}{s_j \cdot s_k}$ | zeros |
| G26 | diff_prime | $\frac{s_i - s_j}{p_k}$ | zeros + primes |
| G28 | exp_neg | $\exp(-s_i / s_j)$ | zeros |
| G29 | pi_deviation | $\pi - s_i / s_j$ | zeros + const |
| G30 | nest_prime_sum | $\frac{s_i}{s_j + p_k}$ | zeros + primes |
| G31 | mixed_double_diff | $\frac{s_i - p_j}{s_k - p_l}$ | zeros + primes |
Grammars where indices are restricted to subsets of the source sequence, including but not limited to: - Pyramid-prime grammars using only primes from a specified set (e.g., {2, 3, 5, 7, 11, 29}); - Grammars restricted to zeros below/above a threshold; - Grammars restricted to elements with specific p-adic properties.
The grammar family is explicitly open-ended: any mathematical function f(s_{i₁}, s_{i₂}, ..., s_{iₘ}, p_{j₁}, ..., p_{jₙ}) that combines m elements from source sequences and n prime numbers using arithmetic operations (+, −, ×, ÷), roots (√, ∛, ...), logarithms (ln, log₂, log₁₀), exponentials (exp, 2^x, 10^x), trigonometric functions (sin, cos, tan, arcsin, arccos, arctan), hyperbolic functions (sinh, cosh, tanh), special functions (Gamma, Bessel, hypergeometric), and compositions thereof, constitutes a valid grammar within the scope of this invention.
Grammars where the functional form is not specified a priori but is learned by a machine learning model, including but not limited to: - Neural networks (feedforward, recurrent, transformer-based) trained to predict target values from sequence elements; - Symbolic regression systems (genetic programming, sparse regression) that discover formula structures; - Reinforcement learning agents that explore formula space; - Large language models prompted or fine-tuned for formula discovery.
In addition to discrete grammar enumeration, the invention encompasses an alternative paradigm where formula discovery is performed by optimizing continuous parameters. In this approach:
This optimization-based approach constitutes a valid method within the scope of this invention, regardless of whether the functional form is specified a priori (as a grammar) or discovered during optimization.
The target value T may be expressed as a spectral decomposition using zeros as frequencies:
$$T \approx \sum_n a_n \cdot f(\gamma_n,\, x)$$
where f is a spectral kernel function (e.g., exp(iγₙx), x^{iγₙ}, cos(γₙx), sin(γₙx), J_ν(γₙx)), x is a tunable parameter, and the coefficients aₙ are determined by optimization or analytic methods. This approach is motivated by the explicit formula in prime number theory, where the prime counting function is expressed as a sum over zeros of ζ(s), and represents an alternative to arithmetic formula mining.
Rather than constructing formulas that combine individual sequence elements, this category of analysis computes global statistical, topological, information-theoretic, or spectral properties of the ENTIRE source sequence (or a subsequence thereof) and matches these aggregate properties against target values. Methods include but are not limited to:
(i) Topological Data Analysis (TDA): Computing persistent homology of the sequence or its derived point cloud, extracting Betti numbers β₀, β₁, ..., persistence diagrams, bottleneck distances, and Euler characteristics, and matching these topological invariants against target values or against invariants of known physical systems.
(ii) Information-Theoretic Measures: Computing Shannon entropy, Rényi entropy, Kolmogorov complexity estimates, mutual information, transfer entropy, or Fisher information of the gap distribution, the digit distribution, or the spacing distribution of the source sequence, and comparing these quantities against target values or information-theoretic properties of physical systems.
(iii) Graph-Theoretic Properties: Constructing a graph from the source sequence (e.g., visibility graph, recurrence network, correlation graph with edges between elements within a threshold distance), and computing graph invariants including spectral gap, chromatic number, clustering coefficient, modularity, degree distribution, and matching these against target values.
(iv) Fractal and Multifractal Analysis: Computing box-counting dimension, Hausdorff dimension, Hurst exponent, multifractal spectrum f(α), and detrended fluctuation analysis (DFA) exponents of the gap sequence, and matching against properties of physical or mathematical systems.
(v) Spectral Analysis of Gap Sequences: Computing the power spectral density, autocorrelation function at multiple lags, or wavelet scalogram of the gap sequence, and matching spectral features (peak frequencies, spectral slopes, characteristic scales) against target properties.
These methods capture properties of the sequence that are NOT expressible as algebraic combinations of individual elements, and thus constitute a complementary approach to formula-based mining.
Integer relation algorithms may be employed as a STANDALONE method (i.e., without requiring prior histogram generation, peak detection, or grammar enumeration) to discover exact algebraic relationships between a target value T and elements of the source sequence S. These methods include:
(i) PSLQ Algorithm: Given a target T and a basis set B = {b₁, b₂, ..., b_k}, the PSLQ algorithm searches for integer coefficients m₁, m₂, ..., m_k such that m₁b₁ + m₂b₂ + ... + m_kb_k = 0 to within a specified precision, where the basis set may include T itself, sequence elements γₙ, primes p_j, and mathematical constants (π, e, φ, ln(2), etc.).
(ii) LLL Algorithm (Lenstra-Lenstra-Lovász): A lattice basis reduction algorithm applied to the same integer relation problem, offering different computational tradeoffs.
(iii) HJLS Algorithm (Hastad-Just-Lagarias-Schnorr): An alternative integer relation algorithm with different convergence properties.
(iv) Ferguson-Forcade Algorithm: The predecessor of PSLQ, applicable to the same class of problems.
The basis sets for integer relation detection may include any combination of: (a) the target value T; (b) individual zeros γₙ; (c) prime numbers p_j; (d) products of zeros γᵢγⱼ; (e) mathematical constants; (f) logarithms of zeros ln(γₙ); (g) square roots of zeros √γₙ; (h) powers of zeros γₙᵏ; and up to 40 or more distinct basis elements. The maximum coefficient magnitude is bounded (e.g., |mᵢ| ≤ 50) to ensure discovered relations are non-trivial.
For each grammar G and target T, the method computes a "need" value from T and a subset of sequence elements, then uses binary search (e.g., numpy searchsorted) to find the closest matching element in a sorted array:
need = T × sₖ − sⱼ (for G01)
idx = searchsorted(sorted_s, need)
error = |sorted_s[idx] - need| / |T|
Complexity: O(N² log N) per target for 3-index grammars.
For inverse discovery (finding unknown targets), the method: 1. Evaluates ALL possible formulas for a grammar, producing candidate values; 2. Bins these values into a high-resolution histogram (bin_width ≈ 2×10⁻⁴) using GPU-accelerated binning (e.g., torch.histc on CUDA); 3. Smooths the histogram and computes z-scores per bin; 4. Identifies peaks with z > threshold as candidate constants;
This method evaluates up to 10¹¹+ formulas in seconds on modern GPUs.
For small sequence sizes or when completeness is required, all possible index combinations are enumerated. The invention covers any enumeration strategy including nested loops, generator functions, and iterator-based approaches.
The method includes using trained models to predict which index combinations are most likely to produce matches, thereby pruning the search space. This includes: - Embedding sequence elements in learned vector spaces; - Using attention mechanisms to identify promising index combinations; - Transfer learning from previously discovered formulas; - Active learning strategies that focus computation on the most informative regions.
The method includes distribution of search across: - Multiple GPU devices on a single machine; - Multiple machines in a cluster or cloud environment; - Heterogeneous hardware (CPU + GPU + TPU + FPGA); - Quantum computing devices used to accelerate the search.
The search method includes early termination when a formula is found whose error is below a predetermined threshold (e.g., 10⁻²⁵), avoiding unnecessary computation of remaining index combinations. Additionally, for null hypothesis trials, early termination may occur after a minimum number of trials (e.g., 3) when the advantage ratio exceeds a predetermined level (e.g., 50×), avoiding computation of remaining trials.
The method includes adaptive management of GPU device memory (VRAM), comprising: (i) allocating a configurable fraction of available VRAM (e.g., 75% for standard grammars, 10% for four-index grammars) as a memory budget; (ii) computing the maximum number of sequence elements that can be stored within the budget; (iii) automatically chunking larger sequences into sub-batches that fit within the budget; (iv) applying try/except error handling per grammar to gracefully skip grammars that exceed available memory.
The invention employs random matrix theory to validate discoveries:
Unfolded using Wigner semicircle CDF;
Repeat mining on GUE pseudo-zeros (N_trials ≥ 10);
Permuting the order of sequence elements (gaps or values) and re-running the analysis.
Computing the best result across all grammars for each constant, and taking the combined verdict as the maximum advantage achieved by any single grammar.
When combining results from multiple grammars, the invention normalizes each grammar's contribution to be equal (geometric mean of magnitudes), preventing any single grammar from dominating. This is the application of the UFGrad principle (gradient magnitude equalization) to formula discovery.
Splitting the source sequence into independent halves and verifying that discoveries reproduce across both halves.
For grammar families that combine Riemann zeros with prime numbers (hybrid grammars), the method includes an additional null hypothesis test: (i) replacing the real prime number sequence with a synthetic sequence of uniformly distributed random integers having the same count and approximate range; (ii) repeating the mining procedure with these fake primes; (iii) comparing the error achieved with real primes versus fake primes; A result is classified as REAL only if the advantage of real primes over fake primes exceeds a threshold (e.g., 10×).
The validation framework extends beyond random matrix theory to encompass ANY statistical null hypothesis that provides a baseline for comparison:
Any combination of these methods constitutes a valid statistical validation within the scope of this invention. The method is agnostic to which specific null hypothesis is employed; the essential inventive step is the comparison of mined formula precision against a baseline derived from any reasonable null model.
The method includes verification of discovered formulas at arbitrary precision:
The method includes a mode where no target value is specified a priori:
The method further includes constructing relational equations between pairs of unidentified peaks: (i) For each pair of unidentified peaks (U_a, U_b), computing U_a + U_b, U_a - U_b, U_a × U_b, U_a / U_b, √(U_a/U_b), ln(U_a/U_b); (ii) Matching each computed value against all known constants and against all other unidentified peaks; (iii) Scoring each equation by combining error magnitude with z-score significance; (iv) Building a constraint network linking peaks through equations; (v) Identifying hub peaks (those appearing in the most equations across the most domains).
The method includes using unidentified peaks as predictions for undiscovered physical quantities: (i) Classifying each peak by numerical range into candidate domains (e.g., 200-400 for superconductor critical temperatures in Kelvin, 0.5-5.0 for bandgap energies in eV); (ii) Cross-referencing with element mass ratios to suggest candidate material compositions; (iii) Ranking predictions by z-score and number of supporting Riemann zero formulas.
The method includes resolving disputes between competing measured values of the same physical quantity: (i) Mining each competing value independently using identical grammar families and sequence lengths; (ii) Comparing the number of correct digits achieved for each competing value; (iii) Ranking competing values by precision and number of independent formulas; (iv) Reporting the value that achieves the highest precision as the Riemann-preferred value.
The method includes extracting structured information from gaps between consecutive sequence elements:
Specific instances include but are not limited to N=24 (Egyptian consonants), N=22 (Hebrew letters), N=26 (Latin alphabet), N=28 (Arabic letters), and any other value of N.
The decoding method further comprises: (i) Viterbi multi-POS decoding: assigning each matched word one or more parts of speech (POS) from a set including at minimum Verb, Noun, Adjective, Preposition, Pronoun, Particle, and Numeral; selecting the POS assignment that maximizes a grammar-weighted score according to a transition matrix encoding natural language word-order rules (e.g., VSO order for Egyptian: Verb→Noun = high score, Noun→Noun = penalty); (ii) Gap tolerance: allowing a configurable number of unmatched characters between consecutive words (e.g., 1-2 characters representing missing vowels in abjad scripts), with a penalty factor per gap; (iii) Word scoring: weighting each matched word by len(word)² × N^len(word) × grammar_multiplier, so that longer words contribute exponentially more significance; (iv) Sentence chain extraction: grouping consecutive matched words (within gap tolerance) into sentences; reporting sentence length, position in stream, and cumulative score; (v) Narrative assembly: ordering extracted sentences by position to reconstruct a sequential narrative or set of instructions; (vi) Coverage analysis: reporting what fraction of the total stream is covered by matched words; (vii) Null hypothesis testing: shuffling the stream (preserving character frequencies but randomizing order) and comparing grammar scores, sentence counts, and coverage against the shuffled baseline.
Selecting subsequences of gaps based on p-adic valuation: - v_p(n) = number of times prime p divides the quantized gap integer; - Filtering by depth = v₂(n) + v₃(n) to reveal hidden order; - Using divisibility by specific numbers (12, 36, 72, 432) as filters.
Inscribing regular k-gons on the critical line (viewed as a circle on the Riemann sphere) and analyzing gap properties at vertices, edges, and interiors of chambers: - Coulomb-type radial profiles; - Energy conservation (signed sum = 0); - Negative energy in lunules; - Chain fold operations (iterative half-length transformations).
Classifying consecutive gap sequences by sign patterns (above/below mean) in d-dimensional spaces, analyzing transition matrices, and detecting clustering and anti-clustering patterns.
The method is applied to the following domains (including but not limited to):
Mining candidate values for superconducting transition temperatures, bandgaps, lattice constants, and other material properties. Predicting alloy compositions by matching elemental mass ratios to formula indices.
Using GUE repulsion properties of Riemann zero spacings to diversify financial portfolios. Applying random matrix theory denoising to correlation matrices. Mining financial indicators as Riemann zero combinations.
Encoding personal data (birthdates, numerical identifiers) as target values and mining their Riemann zero representations to generate unique mathematical profiles.
Mining dimensions of ancient structures (pyramids, temples), sacred numbers, and calendar cycles as combinations of Riemann zeros and primes. Decoding potential messages in gap sequences using ancient alphabets.
Mining cosmological constants (Hubble constant, dark matter fraction, CMB parameters) and discovering cross-domain connections (e.g., galaxy rotation velocity = CMB acoustic peak index = 220).
Mining molecular properties (binding energies, partition coefficients, pKa values) as Riemann zero combinations to screen drug candidates.
Using the unpredictability of high-index Riemann zero combinations as a source of cryptographically structured randomness. Key derivation from formula indices.
Applying gap-based encoding and geometric chamber analysis to detect patterns in experimental data streams.
Maintaining and licensing access to databases of discovered formula-constant mappings for use in scientific research, engineering design, and other applications.
Software tools that visualize the landscape of Riemann zero formulas, allowing interactive exploration of connections between mathematics and physics.
Mining for DNA base pair ratios, codon frequencies (64 codons), amino acid counts (20 standard), protein folding constants, enzyme kinetic parameters (Michaelis-Menten K_m, V_max), and nucleotide binding energies.
Mining for orbital mechanics constants (escape velocities, orbital periods, Titius-Bode coefficients), aerodynamic parameters (Reynolds critical numbers, drag coefficients, Mach number thresholds), and planetary physical properties.
Mining for physiological reference ranges (blood gas values, hormonal levels, cardiac intervals), pharmacokinetic parameters (half-lives, clearance rates), and anatomical proportions.
Mining for frequency ratios in tuning systems (equal temperament 2^(1/12), just intonation ratios), harmonic series relationships, resonance frequencies of standard instrument geometries, and psychoacoustic parameters.
Mining for bond energies, molecular orbital energies, lattice constants and crystal structure parameters, electronegativity differences, and reaction rate constants (Arrhenius parameters).
As used herein, the term "computer" or "processor" includes, without limitation: classical digital processors (CPU, GPU, TPU, DSP), quantum processors (gate-based quantum computers, quantum annealers, photonic quantum processors, topological quantum processors), hybrid classical-quantum systems, field-programmable gate arrays (FPGAs), application-specific integrated circuits (ASICs), neuromorphic processors, analog computers, cloud computing services, distributed computing systems, and any combination thereof. A "computer-implemented method" is a method executed on any such computer or processor.
As used herein, the term "database" includes any structured or unstructured collection of data, whether stored electronically, printed in a publication, communicated verbally, or otherwise made available, including published papers, preprints, patents, websites, lookup tables, spreadsheets, and programmatic APIs.
As used herein, "using" a mathematical relationship includes any act of applying, employing, relying upon, incorporating, referencing, or otherwise making practical use of the numerical value produced by evaluating that relationship, whether performed by computer, by hand, by instrument, or by any other means.
Claim 1 (Independent): A computer-implemented method for discovering mathematical relationships between elements of a number-theoretic sequence and a target value, comprising: (a) obtaining a source sequence S comprising N elements computed from a number-theoretic function; (b) selecting a parameterized grammar family G that defines algebraic combinations of elements from S; (c) for at least one target value T, searching the space of formulas defined by G to identify one or more candidate formulas whose evaluated value approximates T within a predetermined, adaptive, learned, or dynamically computed error threshold; (d) validating the identified candidate formulas by comparing their precision against a null hypothesis derived from random matrix theory; (e) outputting the validated formulas with their precision metrics and validation scores.
Claim 2: The method of Claim 1, wherein the source sequence S comprises non-trivial zeros γₙ of the Riemann zeta function ζ(s), where ζ(½ + iγₙ) = 0.
Claim 3: The method of Claim 1, wherein the source sequence S comprises zeros of a Dirichlet L-function L(s, χ) for a Dirichlet character χ.
Claim 4: The method of Claim 1, wherein the source sequence S comprises zeros of an automorphic L-function, including L-functions associated with modular forms, elliptic curves, or Maass forms.
Claim 5: The method of Claim 1, wherein the source sequence S comprises eigenvalues of a Berry-Keating-type Hamiltonian H = xp or its symmetrized variants, computed on a logarithmic spatial grid.
Claim 6: The method of Claim 1, wherein in step (b), the grammar family G further uses prime numbers pₙ in combination with elements of S.
Claim 7: The method of Claim 1, wherein the grammar family G includes at least one of: (i) sum_ratio: $\frac{s_i + s_j}{s_k}$; (ii) nested_ratio: $\frac{s_i}{s_j - s_k}$; (iii) diff_ratio: $\frac{s_i - s_j}{s_k}$; (iv) mixed_sum: $\frac{s_i + p_j}{s_k}$, where pⱼ is a prime number; (v) mixed_diff: $\frac{s_i - p_j}{s_k}$; (vi) mixed_nest: $\frac{s_i}{s_j - p_k}$; wherein i, j, k are indices into the sequences.
Claim 8: The method of Claim 1, wherein the grammar family G includes at least one of: (i) sqrt_ratio: $\sqrt{s_i / s_j}$; (ii) log_ratio: $\ln(s_i / s_j)$; (iii) power_ratio: $\frac{s_i^{\,2}}{s_j \cdot s_k}$; (iv) geom_mean: $\frac{\sqrt{s_i \cdot s_j}}{s_k}$; (v) sum_sqrt: $\frac{\sqrt{s_i} + \sqrt{s_j}}{\sqrt{s_k}}$; (vi) sqrt_prime: $\sqrt{s_i / p_j}$; (vii) log_prime: $\ln(s_i / p_j)$.
Claim 9: The method of Claim 1, wherein the grammar family G includes four-index formulas using four elements from S and/or primes, including at least one of: (i) $\frac{s_i + s_j}{s_k + s_l}$; (ii) $\frac{s_i + p_j}{s_k + p_l}$; (iii) $\frac{s_i \cdot s_j}{s_k \cdot s_l}$; (iv) $\frac{s_i + s_j}{p_k + p_l}$; (v) $\frac{s_i - s_j}{s_k - s_l}$; (vi) $\frac{s_i - p_j}{s_k - p_l}$.
Claim 10: The method of Claim 1, wherein the grammar family G includes any mathematical function f(s_{i₁}, ..., s_{iₘ}, p_{j₁}, ..., p_{jₙ}) combining m sequence elements and n primes using operations selected from the group consisting of: addition, subtraction, multiplication, division, square root, n-th root, natural logarithm, logarithm base b, exponential, power, trigonometric functions, inverse trigonometric functions, hyperbolic functions, and compositions thereof.
Claim 11: The method of Claim 1, wherein the grammar family G is defined by a machine learning model trained to discover formula structures, the model being selected from the group consisting of: neural networks, symbolic regression systems, genetic programming, reinforcement learning agents, and large language models.
Claim 12: The method of Claim 1, wherein step (c) comprises searching using binary search on a sorted array of sequence elements, with complexity O(N² log N) or better per target per grammar.
Claim 13: The method of Claim 1, wherein step (c) comprises GPU-accelerated computation, including: (i) storing sequence elements as tensors on GPU device memory; (ii) computing candidate formula values using parallel tensor operations; (iii) identifying minimum-error matches using GPU-accelerated sorting or reduction operations.
Claim 14: The method of Claim 1, wherein step (c) comprises batched processing of multiple target values simultaneously on a GPU, with adaptive chunk sizing based on available GPU memory.
Claim 15: The method of Claim 1, wherein step (d) comprises: (i) generating M sets of pseudo-zeros from random matrices of a specified ensemble (GUE, GOE, or GSE) using the Dumitriu-Edelman tridiagonal representation; (ii) performing the same search procedure on each set of pseudo-zeros; (iii) computing an advantage ratio as the median pseudo-zero error divided by the real zero error; (iv) classifying the result as REAL if the advantage ratio exceeds 10×, MARGINAL if the advantage ratio is between 2× and 10×, or STRUCTURAL if the advantage ratio is below 2×.
Claim 16: The method of Claim 15, wherein the random matrix generation uses the Dumitriu-Edelman tridiagonal method comprising: (i) generating diagonal elements from N(0, 1/√2); (ii) generating sub-diagonal elements as χ_{n-k} / √2 for k = 1, ..., N-1; (iii) computing eigenvalues using a tridiagonal eigensolver with O(N) complexity; (iv) unfolding eigenvalues using the Wigner semicircle cumulative distribution function.
Claim 17: The method of Claim 1, wherein step (d) includes early termination of null hypothesis trials when the advantage ratio exceeds a predetermined threshold after a minimum number of trials.
Claim 18: The method of Claim 1, wherein step (c) is performed using multiple grammar families simultaneously, and step (d) includes UFGrad equalization comprising normalizing each grammar's contribution to a combined result such that all grammars contribute equally regardless of their individual match densities.
Claim 19: The method of Claim 1, further comprising precision verification by: (i) computing the sequence elements appearing in the best formula at arbitrary precision using a multi-precision arithmetic library; (ii) evaluating the formula at arbitrary precision; (iii) reporting the number of correct decimal digits of the formula's value compared to the target.
Claim 20: The method of Claim 19, wherein previously computed high-precision sequence element values are cached in persistent storage and reused across multiple mining operations.
Claim 21: The method of Claim 1, wherein the source sequence S contains at least 1,000 elements, preferably at least 10,000 elements, more preferably at least 30,000 elements, still more preferably at least 50,000 elements, yet more preferably at least 1,000,000 elements, and most preferably at least 7,000,000 elements.
Claim 22: The method of Claim 1, wherein the target value T represents a physical constant selected from the group consisting of: the fine structure constant, particle masses and mass ratios, coupling constants, mixing angles, the Hubble constant, dark matter fraction, CMB temperature, Boltzmann constant ratios, and nuclear binding energies.
Claim 23: The method of Claim 1, wherein the target value T represents a material property selected from the group consisting of: superconducting transition temperature, electrical bandgap, lattice constant, elastic modulus, thermal conductivity, refractive index, piezoelectric coefficient, and magnetic susceptibility.
Claim 24: The method of Claim 1, wherein the target value T represents a mathematical constant selected from the group consisting of: π, e, φ (golden ratio), ζ(n) for integer n ≥ 2, Catalan's constant, Feigenbaum constants, Euler-Mascheroni constant, twin prime constant, and Bessel function zeros.
Claim 25: The method of Claim 1, wherein the target value T represents a biological parameter, a financial indicator, a personal data encoding, an archaeological measurement, a cosmological parameter, or a musical frequency.
Claim 26 (Independent): A computer-implemented method for discovering previously unknown mathematical constants encoded in a number-theoretic sequence, comprising: (a) obtaining a source sequence S comprising N elements computed from a number-theoretic function; (b) for each of a plurality of parameterized grammar families, evaluating all possible formulas to generate a set of candidate values; (c) constructing a histogram of candidate values with a predetermined bin width; (d) computing a local z-score for each histogram bin by comparing the bin count against a smoothed local density; (e) identifying peaks where the z-score exceeds a predetermined threshold as candidate unknown constants; (f) cross-referencing identified peaks against a database of known constants; (g) for peaks not matching known constants, attempting closed-form identification.
Claim 27: The method of Claim 26, wherein step (c) is performed using GPU-accelerated histogram computation (torch.histc or equivalent).
Claim 28: The method of Claim 26, wherein step (b) includes UFGrad equalization: for each grammar, the histogram contribution is normalized such that all grammars contribute equally to the combined histogram, by computing the geometric mean of histogram masses across grammars and scaling each grammar's histogram to that mean.
Claim 29: The method of Claim 26, wherein step (f) includes matching against a database comprising at least 500 known physical, mathematical, and material constants from at least 20 distinct scientific domains, preferably at least 2,000 constants from at least 40 domains, more preferably at least 9,000 constants from at least 50 domains, and most preferably at least 20,000 constants; and wherein matching is performed at multiple tiers of stringency: ULTRA (relative error < 10⁻¹⁵), STRICT (relative error < 10⁻⁸), MEDIUM (relative error < 10⁻⁶), and LOOSE (relative error < 10⁻⁴).
Claim 30: The method of Claim 26, wherein step (g) includes at least one of: (i) the PSLQ algorithm for finding integer relations, applied with at least 16 systematically constructed basis sets including: a standard basis {1, π, e, φ, ln2, √2, √3, √5, V}, a squared basis {π², e², φ²}, zeta value bases {ζ(3), ζ(5), ζ(7)}, Gamma function bases {Γ(1/3), Γ(1/4), Γ(1/5)} and their products/ratios, logarithmic bases {ln2, ln3, ln5, ln7}, Bessel zero bases {J_{0,1}, J_{0,2}, J_{1,1}}, and integer/rational bases, with maximum coefficient magnitude bounded (e.g., ≤50); (ii) matching against trigonometric values at rational multiples of π, including sec, tan, csc, cos, sin, and cot evaluated at mπ/n for integers m and n with n ≤ 360; (iii) testing for algebraic numbers (roots of polynomials of degree 2 through 6 with integer coefficients bounded in magnitude); (iv) systematic search over expressions of the form π^a × e^b × (p/q) × φ^c × √n, and separately over expressions of the form 2^d × 3^e × 5^f × 7^g for small integer exponents; (v) testing Gamma function values and their products/ratios: Γ(p/q), Γ(p/q)×Γ(r/s), and Γ(p/q)/Γ(r/s) for positive rational arguments with small denominators.
Claim 31: The method of Claim 26, further comprising mining formulas for each identified peak using the method of Claim 1.
Claim 32: The method of Claim 26, further comprising constructing equations between peaks, including: (i) unknown = known + known; (ii) unknown = known − known; (iii) unknown = known × known; (iv) unknown = known / known; (v) unknown + unknown = known; (vi) unknown × unknown = known; (vii) unknown − unknown = known; (viii) unknown / unknown = known; (ix) unknown / unknown = mathematical constant; (x) unknown = known × unknown; (xi) powers: unknown², √(unknown), 1/unknown, unknown³; and scoring each equation by combining error magnitude with z-score significance, and classifying equations into tiers: GOLD (error < 10⁻⁸), SILVER (error < 10⁻⁶), and BRONZE (error < 10⁻⁴).
Claim 33 (Independent): A computing system for mining mathematical relationships between elements of a number-theoretic sequence and target values, comprising: (a) a processor; (b) a memory storing a source sequence S of N elements computed from a number-theoretic function; (c) a grammar module configured to define parameterized grammar families specifying algebraic combinations of sequence elements; (d) a search engine configured to search the formula space defined by a grammar to find candidate formulas matching a target value within a threshold; (e) a validation module configured to test candidates against a null hypothesis derived from random matrix theory; (f) an output module configured to present validated formulas with precision and significance metrics.
Claim 34: The system of Claim 33, further comprising a GPU accelerator with at least 8 GB of device memory, wherein the search engine executes tensor operations on the GPU accelerator.
Claim 35: The system of Claim 33, further comprising a GPU accelerator with at least 48 GB of device memory, wherein batched processing of multiple targets is performed simultaneously.
Claim 36: The system of Claim 33, further comprising a distributed computing component configured to distribute the search across multiple computing nodes connected by a network.
Claim 37: The system of Claim 33, further comprising a persistent cache of sequence elements computed at high precision (≥ 30 decimal digits).
Claim 38: The system of Claim 33, further comprising an inverse discovery module configured to evaluate all formulas for a grammar, build a histogram, and identify peaks as candidate unknown constants.
Claim 39 (Independent): A non-transitory computer-readable medium storing instructions that, when executed by a processor, cause the processor to perform the method of any one of Claims 1-32.
Claim 40: The computer-readable medium of Claim 39, wherein the instructions are configured for execution on a system comprising at least one GPU accelerator.
Claim 41 (Independent): A computer-readable database comprising a plurality of records, each record associating: (a) a target value representing a physical constant, material property, mathematical constant, or other measurable quantity; (b) one or more formulas expressing the target value as an algebraic combination of elements from a number-theoretic sequence; (c) for each formula, the indices into the number-theoretic sequence and/or prime number sequence used; (d) a precision metric indicating the number of correct digits; (e) a validation score indicating the statistical significance versus a random matrix null hypothesis; wherein the database contains at least 100 entries with precision of at least 8 correct digits.
Claim 42: The database of Claim 41, wherein the number-theoretic sequence is the sequence of non-trivial zeros of the Riemann zeta function.
Claim 43: The database of Claim 41, comprising at least 500 entries spanning at least 20 scientific domains.
Claim 44: The database of Claim 41, wherein the validation score for each entry indicates an advantage of at least 10× over a GUE random matrix null hypothesis.
Claim 45: The database of Claim 41, further comprising cross-domain connections linking entries from different scientific domains that share common formula indices.
Claim 46: A method of using the database of any one of Claims 41-45 to screen candidate material compositions for a desired property value, comprising: (a) receiving a desired target property value; (b) querying the database for formulas matching that value; (c) extracting the formula indices; (d) using the indices to suggest related material compositions based on elemental mass ratios encoded in the same formula structure.
Claim 47 (Independent): A method of providing mathematical relationship mining as a service, comprising: (a) receiving, over a network, a request from a client comprising at least one target value and optionally a specification of desired grammar families and validation parameters; (b) executing the method of any one of Claims 1-32 on a server system in response to the request; (c) transmitting, over the network, a response to the client comprising the discovered formulas, their precision, and their validation scores.
Claim 48: The method of Claim 47, wherein the service is provided via a RESTful API, GraphQL API, or gRPC interface.
Claim 49: The method of Claim 47, wherein the client accesses the service through a subscription model, a per-query payment model, or a licensing agreement.
Claim 50: The method of Claim 47, wherein the server system comprises one or more GPU accelerators and performs the mining computation using GPU-accelerated tensor operations.
Claim 51: The method of Claim 47, further comprising caching discovered formulas in the database of Claim 41 for reuse in subsequent requests.
Claim 52 (Independent): A computer-implemented method for extracting structured information from a number-theoretic sequence, comprising: (a) computing gaps gₙ = sₙ₊₁ − sₙ between consecutive elements of the sequence; (b) quantizing each gap to produce an integer using a discretization function D: qₙ = D(gₙ, M) mod N, where D is any discretization function (including but not limited to floor, round, ceiling, truncation, or any monotonic function applied to |gₙ| × M or any other monotonic function of gₙ), M is a scaling factor, and N is a positive integer defining an alphabet size; (c) mapping each integer qₙ to a symbol from an alphabet of N symbols; (d) analyzing the resulting symbol stream for patterns corresponding to a reference pattern set (dictionary, grammar rules, or statistical model).
Claim 53: The method of Claim 52, wherein N = 24 and the alphabet comprises consonants of the ancient Egyptian language.
Claim 54: The method of Claim 52, wherein N = 22 and the alphabet comprises letters of the Hebrew alphabet.
Claim 55: The method of Claim 52, wherein N is any integer between 2 and 256, inclusive, and the alphabet is any mapping from integers to symbols.
Claim 56: The method of Claim 52, wherein step (d) comprises: (i) sliding a window across the symbol stream; (ii) matching window contents against a dictionary of known words; (iii) applying grammatical rules (e.g., Verb-Subject-Object order) to score candidate sentences; (iv) applying a null hypothesis test by shuffling the symbol stream and comparing scores.
Claim 57: The method of Claim 52, wherein the sequence elements are non-trivial zeros of the Riemann zeta function.
Claim 58: A computer-implemented method for analyzing the p-adic structure of a number-theoretic sequence, comprising: (a) computing gaps between consecutive elements of the sequence; (b) quantizing each gap to produce an integer; (c) computing the p-adic valuation v_p(qₙ) for one or more primes p; (d) filtering the gap sequence to select only gaps whose p-adic depth (sum of valuations across selected primes) exceeds a threshold; (e) computing autocorrelation of the filtered subsequence to detect hidden order.
Claim 59: The method of Claim 58, wherein the primes p are 2 and 3, and the p-adic depth is v₂(q) + v₃(q).
Claim 59b: A computer-implemented method for analyzing modular index structure of a number-theoretic sequence, comprising: (a) assigning each element sₙ of the source sequence a residue class rₙ = n mod M, where M is a positive integer; (b) partitioning the gap sequence into M sub-sequences based on said residue classes; (c) computing statistical properties (mean, variance, autocorrelation, transfer entropy) independently per residue class; (d) detecting systematic differences between residue classes, including but not limited to: fixed-point behavior at specific residues (e.g., r = 0 mod M), oscillatory behavior at other residues, and chaotic behavior at remaining residues; (e) interpreting residue classes as topological features of the sequence (vertices, edges, faces of a geometric structure); (f) validating the modular structure against a shuffled null hypothesis.
Claim 59c: The method of Claim 59b, wherein M = 9 and the residue classes are interpreted as: (i) r ∈ {0, 9} as fixed points ("poles"); (ii) r ∈ {3, 6} as oscillatory elements ("edges"); (iii) all other r as dynamic elements ("faces"); and wherein the method detects preferential ordering in the {3, 6, 9} residue classes relative to non-{3, 6, 9} classes.
Claim 60 (Independent): A computer-implemented method for analyzing the geometric structure of a number-theoretic sequence on a manifold, comprising: (a) mapping elements of the source sequence onto a geometric manifold (including but not limited to the Riemann sphere via stereographic projection); (b) inscribing a regular k-gon (for k ≥ 3) on the manifold and partitioning sequence elements into geometric chambers defined by the k-gon; (c) computing statistical properties (mean, variance, autocorrelation) of the sequence elements within each chamber; (d) fitting a radial profile (e.g., Coulomb potential a/(r+b)+c or power law a·r^(-b)+c) to the chamber statistics; (e) validating the geometric structure against a shuffled null hypothesis.
Claim 60b: The method of Claim 60, further comprising temporal or rotational geometric analysis: (i) assigning each k-gon an angular velocity ω_k (constant or time-varying); (ii) computing meeting points where vertices from different rotating k-gons coincide; (iii) analyzing the frequency and pattern of vertex meetings as a function of rotation parameters; (iv) detecting emergent integer structures from meeting geometry (e.g., k = 3 with counter-rotation ω_A = −ω_B producing 9 unique vertex pairs); (v) correlating emergent integers with gap sequence properties at corresponding positions.
Claim 61: The method of Claim 60, further comprising iterative chain fold operations on the gap sequence, including: (i) halving the sequence length by applying an element-wise operation (absolute difference, minimum, maximum, or product) to consecutive pairs; (ii) repeating step (i) until the sequence length is below a threshold; (iii) tracking the evolution of autocorrelation across fold iterations.
Claim 62: The method of Claim 60, further comprising hypercubic octant analysis: (i) classifying consecutive d-tuples of gaps by their sign pattern (above/below a reference level); (ii) computing the frequency of each sign pattern; (iii) analyzing transitions between sign patterns; (iv) detecting dimensional crossover points where all-positive patterns exceed expected frequency.
Claim 63: A method of screening candidate superconducting materials, comprising: (a) obtaining a set of target transition temperatures; (b) for each target temperature, executing the method of Claim 1 to find formulas expressing that temperature as a combination of Riemann zeros and/or primes; (c) ranking candidate temperatures by the number of strict matches (error < 10⁻⁸) and validation score; (d) suggesting material compositions based on elemental mass ratios corresponding to the formula indices.
Claim 64: A method of optimizing a financial portfolio using number-theoretic sequence analysis, comprising: (a) representing asset returns as a correlation matrix; (b) applying random matrix theory denoising using eigenvalue distributions from GUE or Marchenko-Pastur; (c) using GUE repulsion properties to enforce diversification constraints; (d) optionally mining financial ratios as combinations of Riemann zeros using the method of Claim 1.
Claim 65: A method of generating a mathematical profile for a person, comprising: (a) converting personal data (including but not limited to birthdate, name encoding, identifying numbers) into one or more numerical target values; (b) executing the method of Claim 1 for each target value; (c) compiling the discovered formulas into a profile associating the person with specific indices in the number-theoretic sequence; (d) optionally scoring the profile by validation significance.
Claim 66: A method of analyzing archaeological or historical structures, comprising: (a) measuring or obtaining dimensions, proportions, and numerical features of a structure; (b) executing the method of Claim 1 for each numerical feature; (c) analyzing the prime factorization of structural dimensions; (d) comparing structural numbers against formula indices to detect encoded mathematical knowledge.
Claim 67: A method of predicting cosmological parameters, comprising: (a) mining known cosmological parameters using the method of Claim 1; (b) identifying cross-domain connections where different cosmological phenomena share common formula indices; (c) using shared indices to predict relationships between cosmological parameters.
Claim 68: A method of screening drug candidates, comprising: (a) obtaining molecular property values (binding energies, pKa, partition coefficients) for candidate molecules; (b) mining each property value using the method of Claim 1; (c) ranking candidates by validation score and number of strict matches.
Claim 69: The method of Claim 1, further comprising extracting the specific indices (i, j, k, ...) of the best formula, enabling reconstruction of the exact formula expression.
Claim 70: The method of Claim 1, wherein the search in step (c) uses a delta-limited approach: for nested or difference grammars, restricting the difference between indices (|j - k|) to be at most Δ_max, thereby reducing computation from O(N³) to O(N² × Δ_max).
Claim 71: The method of Claim 1, wherein multiple source sequences are used jointly in a single grammar evaluation, including formulas that mix zeros of the Riemann zeta function with zeros of a different L-function.
Claim 72: The method of Claim 1, further comprising transforming target values before mining, with transformations including: log₁₀(T), 1/T, T², √T, and mining the transformed value.
Claim 73: The method of Claim 1, further comprising NZ-scaling analysis: repeating the mining at increasing source sequence sizes and measuring how the validation advantage changes as a function of sequence length, to distinguish truly encoded constants from numerical coincidences.
Claim 74: The method of Claim 1, wherein the search is distributed across a plurality of computing devices connected via a network, each device processing a subset of grammar families or target values.
Claim 75: The method of Claim 26, further comprising constructing a knowledge graph connecting discovered constants through shared formula indices, identifying hub zeros (sequence elements appearing in many formulas) and cross-domain bridges.
Claim 76: The method of Claim 1, further comprising applying the method iteratively: using discovered formula indices from a first round as constraints or priors for a second round of more targeted mining.
Claim 77: The method of Claim 52, further comprising applying the method to discover messages or patterns in number-theoretic sequences from civilizations or entities not known a priori, by testing multiple alphabets and grammars systematically.
Claim 78: The method of Claim 1, wherein the source sequence elements are computed on a quantum computing device, and the formula search is performed on a classical or quantum computing device.
Claim 79: The method of Claim 33, wherein the system is deployed as a cloud service with auto-scaling compute resources proportional to the number of concurrent mining requests.
Claim 80: The method of Claim 1, wherein the predetermined error threshold is 10⁻⁸ (strict), 10⁻⁶ (extract), or 10⁻⁴ (loose), and the method reports the number of formulas meeting each threshold level.
Claim 81: The method of Claim 1, further comprising: (i) for each discovered formula, computing the sensitivity of the formula value to perturbations in the sequence elements; (ii) ranking formulas by stability (low sensitivity = high stability); (iii) preferring stable formulas for downstream applications.
Claim 82: The method of Claim 26, further comprising temporal monitoring: periodically re-running discovery as additional sequence elements become available and detecting newly resolvable peaks.
Claim 83: The method of Claim 41, further comprising a licensing module that controls access to the database records, requiring authentication and payment or subscription before providing formula details.
Claim 84: The method of Claim 1, wherein the source sequence S comprises elements from eigenvalue spectra of physical systems (e.g., quantum billiards, nuclear energy levels, acoustic resonances) that exhibit GUE, GOE, or GSE statistics.
Claim 85: The method of Claim 1, further comprising a grammar evolution step where: (i) formulas from multiple grammars that match the same target are combined; (ii) the combination is tested as a new grammar; (iii) successful new grammars are added to the grammar library for future mining.
Claim 86: The method of Claim 1, wherein the source sequence S is any sequence of real numbers whose statistical properties (gap distribution, autocorrelation, spectral statistics) match those of eigenvalues of a random matrix ensemble to within a predetermined statistical threshold.
Claim 87: The method of Claim 1, wherein the grammar family G is discovered automatically by an algorithm (including but not limited to genetic programming, neural architecture search, or systematic enumeration of mathematical operations up to a specified complexity) rather than specified manually.
Claim 88: The method of Claim 1, wherein the target value T is not known a priori but is derived in real-time from sensor data, experimental measurements, market data, or other live data sources.
Claim 89: A method of determining whether a given numerical value is encoded in a number-theoretic sequence, comprising executing the method of Claim 1 and reporting the validation score, wherein a score exceeding a threshold indicates encoding.
Claim 90: The method of Claim 1, applied to a transformed version of the source sequence, including but not limited to: taking logarithms of sequence elements, computing pair-wise differences, computing cumulative sums, applying Fourier transforms, or applying wavelet transforms before mining.
Claim 91: The method of Claim 47, wherein the service receives target values in the form of experimental data (spectra, time series, images) from which numerical features are automatically extracted before mining.
Claim 92 (independent): A computer-implemented method for acquiring and managing a set of non-trivial zeros of the Riemann zeta function or zeros of a Dirichlet L-function L(s, χ) for use in constant mining, the method comprising: (a) computing zeros by evaluating the Riemann-Siegel Z-function Z(t) along the critical line t > 0 at a step size Δt, detecting sign changes of Z(t), and refining each sign change by bisection to locate the zero to desired precision; (b) storing computed zeros in a persistent array indexed by ordinal number n, such that γ_n denotes the imaginary part of the n-th non-trivial zero; (c) verifying completeness of the zero set by comparing the count of stored zeros below a height T with the expected count given by the Riemann-von Mangoldt formula $N(T) = \frac{T}{2\pi}\ln\!\left(\frac{T}{2\pi e}\right) + \frac{7}{8} + O\!\left(\frac{1}{T}\right)$; (d) making the stored zero set available for mining operations as described in any preceding claim.
Claim 93: The method of Claim 92, further comprising computing zeros sequentially using an arbitrary-precision library (e.g., mpmath.zetazero(n)) with checkpoint-resume capability, wherein intermediate results are saved after each batch of computed zeros, enabling interruption and continuation of the computation across sessions or machines.
Claim 94: The method of Claim 92, further comprising downloading pre-computed zeros from a public database (such as the L-functions and Modular Forms Database, LMFDB) and integrating said downloaded zeros with locally computed zeros into a unified indexed store.
Claim 95: The method of Claim 92, further comprising maintaining a disk-persistent cache of high-precision zero values, indexed by both ordinal number and precision level (decimal places), wherein: (i) if a zero at ordinal n with precision P is already cached, the cached value is returned without recomputation; (ii) if a zero is cached at lower precision P' < P, it is recomputed at precision P, stored, and the lower-precision entry is retained; (iii) said cache is shared across multiple mining runs and across different grammar types.
Claim 96: The method of Claim 92, wherein the step size Δt for Z-function evaluation is adaptively chosen based on the local density of zeros, using the approximation that the average spacing between consecutive zeros near height t is 2π/ln(t/(2π)).
Claim 97: The method of Claim 92, wherein the zeros are computed using GPU-accelerated evaluation of the Riemann-Siegel Z-function, with multiple intervals evaluated in parallel across GPU threads, and sign changes detected and refined in parallel.
Claim 98 (independent): A computer-implemented method for discovering relationships between unidentified peaks in a number-theoretic histogram, the method comprising: (a) generating peaks by applying the inverse discovery method of Claim 26 to obtain a set of K peaks P = {V₁, V₂, ..., V_K} with associated z-scores; (b) for each ordered pair or triple of peaks, evaluating whether any of the following relationships hold within a threshold ε: (i) V_a + V_b = V_c; (ii) V_a × V_b = V_c; (iii) V_a / V_b = V_c; (iv) V_a − V_b = V_c; (v) V_a + V_b = known constant; (vi) V_a × V_b = known constant; (vii) V_a / V_b = known constant; (viii) V_a − V_b = known constant; (ix) V_a / V_b = mathematical constant (π, e, φ, etc.); (x) V_a = known × V_b; (xi) V_a^n = V_b or known, for n ∈ {2, 3, ½, ⅓, −1}; (c) scoring each discovered relation by combining the error magnitude with a function of the z-scores of the participating peaks (e.g., score = log₁₀(1/error) × √(Z_a + Z_b + Z_c)); (d) classifying relations into tiers: GOLD (error < 10⁻⁸), SILVER (error < 10⁻⁶), BRONZE (error < 10⁻⁴); (e) constructing a constraint network wherein peaks are nodes and relations are edges, enabling identification of hub peaks (peaks participating in many relations).
Claim 99: The method of Claim 98, further comprising a material property prediction step, wherein: (i) each unidentified peak is classified into a candidate physical domain based on its numerical range (e.g., 0.5–15 → ionization energy, 250–400 → superconductor critical temperature, 1–10 → nuclear binding energy); (ii) for peaks classified as candidate superconductor critical temperatures, element mass ratios from a periodic table database are used to suggest candidate elemental compositions; (iii) candidate alloy systems are scored by the number of independent Riemann formula matches and the combined z-score.
Claim 100: The method of Claim 98, further comprising grouping peaks into families, wherein a family is defined as a set of peaks all connected to a common known constant through the relations of step (b), and reporting the size and connectivity of each family.
Claim 101: The method of Claim 98, wherein the search in step (b) is performed using vectorized operations on sorted arrays of peak values with binary search (searchsorted), enabling evaluation of all pairwise and triple-wise relations for K peaks in O(K² log K) time.
Claim 102: The method of Claim 98, further comprising a temporal stability check, wherein the unknown-unknown relations are recomputed using independently generated subsets of the zero sequence (e.g., first half versus second half), and only relations that reproduce in both subsets with consistent error are retained as validated.
Claim 103 (independent): A computer-implemented method for arbitrating between disputed or imprecisely known physical values using Riemann zero mining, the method comprising: (a) receiving two or more candidate values V₁, V₂, ..., V_m for a single physical quantity (e.g., two conflicting measurements of the Hubble constant, H₀ = 67.4 vs H₀ = 73.04); (b) mining each candidate value independently through the source sequence using the method of any of Claims 1–25; (c) comparing the mining results across the candidates, including: (i) the best formula error achieved for each candidate; (ii) the number of strict formulas found for each candidate; (iii) the GUE advantage ratio for each candidate; (d) ranking the candidates by the strength of their Riemann encoding (best error, highest advantage, most formulas); (e) outputting a confidence ranking indicating which candidate value is most strongly encoded in the Riemann zeros.
Claim 104: The method of Claim 103, applied to predict previously unmeasured physical quantities by: (i) generating a range of candidate values spanning a physically plausible interval; (ii) mining each candidate value; (iii) identifying the candidate value with the strongest Riemann encoding; (iv) reporting said value as the predicted measurement with an associated confidence derived from the advantage ratio.
Claim 105: The method of Claim 103, further comprising a cross-grammar consistency check, wherein a candidate value is considered strongly supported only if it achieves REAL classification (advantage > 10×) in at least 3 distinct grammar types.
Claim 106: The method of Claim 103, applied to a set of related physical quantities simultaneously (e.g., all mixing angles and mass splittings of a particle physics model), wherein the mining results are analyzed collectively to identify patterns in the formula indices that reflect the theoretical structure of the model.
Claim 107: The method of Claim 1, wherein the search in step (b) employs early termination, such that the search for a given target value V is terminated before all grammar types are exhausted if any grammar has already found a formula with error below a threshold (e.g., error < 10⁻²⁵), and/or the null hypothesis test is terminated early if the advantage ratio after M' < M trials already exceeds a predetermined threshold (e.g., 50×).
Claim 108: The method of Claim 1, wherein the GPU memory management comprises: (i) computing a memory budget as a fraction (e.g., 75%) of total available GPU VRAM; (ii) automatically determining chunk sizes for batched operations based on the memory budget divided by the per-element memory footprint; (iii) applying distinct, reduced memory budgets (e.g., 10%) to grammar types requiring larger intermediate arrays (such as 4-index grammars); (iv) wrapping each grammar evaluation in error handling that catches out-of-memory conditions and retries with reduced chunk sizes.
Claim 109: The method of any of Claims 26–32, further comprising complexity-scored filtering of identified formulas, wherein the complexity of a proposed closed-form expression is quantified (e.g., by summing the magnitudes and count of all constituent integers, exponents, and function applications), and results exceeding a complexity threshold (e.g., complexity > 30) are rejected as likely spurious.
Claim 110: The method of Claim 33, further comprising narrative extraction from decoded character chains, wherein consecutive words within a maximum gap distance (e.g., 0–2 intervening unmatched characters) are assembled into sentences, sentences are scored by their total weight, and the resulting set of sentences is analyzed for semantic coherence and thematic clustering.
Claim 111: The method of Claim 1, wherein the target values include: (i) zeros of Bessel functions J_ν,k for integer orders ν = 0–5 and roots k = 1–5; (ii) self-referential values consisting of the Riemann zeros themselves γ_n for small n (self-encoding test); (iii) constants from Millennium Prize Problems including the Yang-Mills mass gap, Navier-Stokes critical Reynolds numbers, Birch and Swinnerton-Dyer L-function values and conductors, and Hodge-theoretic invariants (Euler characteristics, Betti numbers).
Claim 112: The method of Claim 1, wherein the search across M_G grammar types for a given target value is prioritized by a grammar ranking derived from prior performance, such that grammars historically producing more REAL results are evaluated first, and lower-ranked grammars are skipped if a sufficiently strong result has already been obtained.
Claim 113: The method of Claim 92, further comprising generating a companion sequence of prime numbers {p₁, p₂, ..., p_{N_P}} using a sieve (e.g., Sieve of Eratosthenes) up to at least the N_P-th prime, where N_P is chosen to cover at least the magnitude of the largest zero element, and storing the primes in a sorted array for use in hybrid grammar evaluation.
Claim 114 (Independent): A computer-implemented method for identifying mathematical relationships between elements of a number-theoretic sequence and a target value, comprising: (a) obtaining a source sequence S comprising N elements derived from or related to a number-theoretic function, including elements computed directly as zeros or special values of the function, elements derived from such zeros (such as gaps, ratios, derivatives, or statistical aggregates), or elements downloaded from an external database of pre-computed values; (b) evaluating one or more candidate expressions, each candidate expression combining at least two elements selected from S and optionally from a companion sequence of prime numbers, using at least one arithmetic or mathematical operation; (c) for at least one target value T, identifying candidate expressions whose evaluated numerical value approximates T within a predetermined, adaptive, learned, or dynamically computed error threshold; (d) outputting the identified candidate expressions together with their precision metrics; (e) applying said identified candidate expressions to produce a technical effect selected from the group consisting of: predicting a measurable physical property of a material or system, screening candidate compositions for a desired property, controlling or adjusting a physical process or device parameter, generating a signal or data structure for use in a downstream computational or physical system, or validating an experimental measurement.
Claim 115: The method of Claim 114, further comprising validating the identified candidate expressions against a statistical null hypothesis, wherein the null hypothesis is generated by any one or more of: (i) random matrix models (GUE, GOE, GSE, CUE, COE, CSE, Wishart, or any other random matrix ensemble); (ii) shuffled or permuted versions of S; (iii) pseudo-random real numbers generated by a PRNG with matching statistical moments; (iv) Monte Carlo simulation of random expression evaluations; (v) bootstrap resampling of expression errors; (vi) Poisson process models; (vii) sequences of pseudo-zeros generated from heuristic models of the number-theoretic function; (viii) cross-validation on independent subsets of S; (ix) comparison against results obtained from a pseudo-random number-theoretic function (e.g., a polynomial approximation to ζ(s) with randomized coefficients); (x) any other statistical baseline suitable for evaluating whether the discovered relationship is non-trivially specific to the actual source sequence.
Claim 116: The method of Claim 114, wherein step (b) comprises optimizing continuous parameters in a parameterized expression involving elements of S, using an optimization method selected from the group consisting of: linear regression, LASSO regression, ridge regression, elastic net, gradient descent (stochastic or batch), Adam or other adaptive gradient methods, evolutionary strategies, covariance matrix adaptation (CMA-ES), simulated annealing, Bayesian optimization, particle swarm optimization, neural network training, Gaussian process regression, and symbolic regression via genetic programming, wherein the optimized parameters include but are not limited to real-valued weights, exponents, coefficients, and index selections.
Claim 117: The method of Claim 114, wherein the source sequence S comprises values derived from the behavior of a number-theoretic function at or near its zeros, including but not limited to: (i) derivatives ζ'(ρₙ) of the Riemann zeta function at its non-trivial zeros ρₙ; (ii) higher derivatives ζ^(k)(ρₙ) at the zeros; (iii) values of the Hardy Z-function Z(t) at specified points; (iv) values of the argument function arg ζ(½ + it) at specified points; (v) residues or Laurent coefficients of ζ(s) near its zeros; (vi) values of the logarithmic derivative ζ'(s)/ζ(s) at specified points; (vii) values of the Riemann-Siegel theta function θ(t) at specified points; (viii) any other numerical quantity computable from the Riemann zeta function or a related L-function evaluated at or near its zeros.
Claim 118: The method of Claim 114, wherein step (b) comprises expressing the target value T as a spectral decomposition using zeros as frequencies: T ≈ Σₙ aₙ · f(γₙ, x) for functions f selected from the group consisting of: f(γ, x) = exp(iγx), f(γ, x) = x^{iγ}, f(γ, x) = cos(γx), f(γ, x) = sin(γx), f(γ, x) = γ^x, and f(γ, x) = J_ν(γx), where x is a parameter and aₙ are real or complex coefficients determined by optimization, least-squares fitting, or exhaustive search over a discretized parameter space.
Claim 119: The method of Claim 114, wherein the source sequence S comprises zeros or eigenvalues of a function selected from the group consisting of: the Riemann zeta function ζ(s), Dirichlet L-functions L(s, χ), automorphic L-functions, Dedekind zeta functions ζ_K(s), Epstein zeta functions, Selberg zeta functions Z_Γ(s), Ihara zeta functions of graphs, Ruelle zeta functions (dynamical zeta functions), Hurwitz zeta functions ζ(s, a), Lerch zeta functions, the derivative ζ'(s), higher derivatives ζ^(k)(s), Artin L-functions, Hasse-Weil L-functions, and any function in the Selberg class S.
Claim 120 (Independent): A computer-implemented method for identifying mathematical relationships between a number-theoretic sequence and one or more target values, comprising: (a) obtaining a source sequence S comprising N ≥ 100 ordered numerical elements derived from or related to a number-theoretic function; (b) computing one or more global functional properties of S or of a derived sequence (such as the gap sequence), the global functional properties being selected from the group consisting of: persistent homology and Betti numbers via topological data analysis, Shannon entropy, Rényi entropy, approximate Kolmogorov complexity, visibility-graph invariants (spectral gap, chromatic number, clustering coefficient, degree distribution), fractal dimension (Hausdorff, box-counting, correlation), Hurst exponent, detrended fluctuation analysis exponent, multifractal spectrum, power spectral density, autocorrelation function, and wavelet coefficients; (c) comparing the computed global functional properties against a database of target values to identify matches within a predetermined, adaptive, learned, or dynamically computed error threshold; (d) validating identified matches using any statistical method to determine that the match is non-random; (e) outputting the validated matches together with their precision and statistical significance metrics.
Claim 121: A method of using a pre-established mathematical relationship between elements of a number-theoretic sequence and a target value, the relationship having been discovered by the method of any one of Claims 1, 114, or 120, comprising: (a) obtaining, by any means including retrieving from a database, reading from a publication, receiving from a communication, computing from stored indices, or downloading from an electronic resource, a mathematical expression E(sᵢ, sⱼ, ...) involving specific elements sᵢ, sⱼ, ... of the sequence, said expression E having been previously validated as approximating a target value T within a specified precision; (b) evaluating said expression E to obtain a numerical value V, whether by computer, by calculator, by manual computation, or by any other means; (c) using V in at least one downstream application selected from the group consisting of: setting a physical parameter of a manufacturing process, computing a property prediction for a candidate material or chemical compound, calibrating an instrument or measurement system, generating a control signal for a device, validating or arbitrating between conflicting experimental measurements, screening or ranking candidate compositions, computing an input parameter for a simulation, encoding or decoding information, generating a structured dataset for machine learning, and designing or optimizing a physical device or system.
Claim 122: The method of any one of Claims 1, 114, or 120, wherein the source sequence S comprises an ordered numerical sequence of any origin, including sequences generated by a generative adversarial network, variational autoencoder, genetic algorithm, or other machine learning or optimization procedure, sequences obtained from experimental measurements (such as energy levels, resonance frequencies, or spectral lines), sequences derived from graph spectra, sequences of eigenvalues of random or structured matrices, or any numerically specified ordered sequence, regardless of whether the sequence has a known number-theoretic interpretation.
Claim 123: The method of any one of Claims 1 or 114, wherein step (b) is performed using an integer relation algorithm selected from the group consisting of PSLQ, LLL, HJLS, and Ferguson-Forcade, applied to a basis vector comprising the target value T and a plurality of sequence elements, products of sequence elements, logarithms of sequence elements, roots of sequence elements, powers of sequence elements, and optionally mathematical constants, with integer coefficients bounded by a maximum absolute value, to identify an integer relation mᵢ such that Σ mᵢ bᵢ ≈ 0 where bᵢ are basis elements.
Claim 124 (Independent): A computer-implemented method for mining mathematical formulas from ordered numerical sequences, comprising: (a) obtaining a source sequence S; (b) receiving or selecting at least one target value T; (c) systematically evaluating candidate expressions, each combining at least two elements of S using at least one arithmetic or mathematical operation, and identifying those candidate expressions whose value approximates T within a specified error threshold; (d) validating identified candidate expressions against at least one statistical baseline to assess non-randomness; (e) for each validated expression, computing the expression value at extended numerical precision using at least 30 significant digits; (f) outputting the validated expressions, their precision, and their statistical significance; wherein the method is applied to produce a technical effect selected from: predicting a physical property, screening candidate materials, calibrating a measurement, generating a control signal, or encoding or decoding information.
Claim 125: A material composition, device component, or process parameter whose value, composition, or configuration was discovered, predicted, or optimized using the method of any one of Claims 1, 63, 114, 120, or 124.
Claim 126: The method of any one of Claims 1, 114, 120, or 124, wherein the method is executed in whole or in part on a quantum processor, including a gate-based quantum computer, quantum annealer, photonic quantum processor, topological quantum processor, or hybrid classical-quantum system, and wherein at least one step selected from obtaining the source sequence, evaluating candidate expressions, computing global properties, or validating results is performed using quantum computation.
Claim 127: The method of Claim 121, wherein the pre-established mathematical relationship was discovered using a quantum processor, and the downstream application comprises any of: quantum error correction parameter setting, quantum gate calibration, quantum sensor calibration, quantum key distribution parameter optimization, or any other quantum information processing application.
Claim 128 (Independent): A computer-implemented method for measuring signal saturation in a number-theoretic sequence, the method comprising: (a) selecting a target value V and a grammar family G; (b) repeating mining at a plurality of source sequence sizes NZ_1 < NZ_2 < ... < NZ_K, wherein each NZ_i comprises at least 1,000 elements; (c) at each size NZ_i, computing a validation advantage ratio R_i = (median baseline error) / (real sequence error), where baseline error is obtained from random matrix surrogate sequences; (d) determining a saturation curve R(NZ) from the plurality of measurements; (e) identifying a saturation onset point NZ_sat where the derivative dR/dNZ drops below a threshold; and (f) reporting the saturation point and the asymptotic advantage ratio, thereby characterizing the finite information content of the source sequence.
Claim 129: The method of Claim 128, wherein the validation advantage ratio R is observed to increase monotonically from NZ=1,000 to NZ=NZ_sat, and to plateau or decrease for NZ > NZ_sat, indicating that the source sequence encodes a finite amount of information about the target value.
Claim 130: The method of Claim 128, wherein the saturation onset NZ_sat is approximately 1,000 to 5,000 elements for the non-trivial zeros of the Riemann zeta function, and the peak advantage ratio R_max ranges from 4x to 10x depending on the grammar family.
Claim 131 (Independent): A computer-implemented method for mining mathematical relationships between elements of a number-theoretic sequence and target values representing sacred, historical, architectural, or symbolic numbers drawn from one or more traditions including but not limited to: Freemasonry (degrees, symbols, historical dates), ancient Egyptian mathematics (royal cubit = pi/6, Eye of Horus fractions, pyramid dimensions), Kabbalah (sefirot, gematria values, Tree of Life paths), Hermetic tradition (Emerald Tablet constants, 42 Books of Thoth), and sacred geometry (golden angle, Vesica Piscis, Flower of Life), the method comprising: (a) obtaining a source sequence S of at least 10,000 elements; (b) defining target values V from at least one tradition; (c) mining formulas using at least two grammar families; (d) validating each match against a GUE random matrix null hypothesis with at least 10 trials; (e) classifying each target as REAL (advantage > 10x), MARGINAL (2-10x), or STRUCTURAL (< 2x); and (f) reporting the percentage of REAL targets per tradition category.
Claim 132: The method of Claim 131, wherein the target values include pi/6 (royal cubit in meters), 1/64 (missing fraction of Eye of Horus), 63/64 (complete Eye of Horus), Masonic degrees (3, 7, 9, 13, 14, 32, 33), and historical dates (1312, 1717, 1801), and wherein at least 80% of target values achieve REAL classification.
Claim 133 (Independent): A computer-implemented method for discovering an optimal architectural configuration of a physical system using Riemann zero mining, the method comprising: (a) defining a domain D comprising at least 20 configurable parameters of the physical system (including material choices, operating temperatures, frequencies, and geometric dimensions); (b) for each parameter, defining a target value V representing a candidate optimal setting; (c) mining each target value against a source sequence of at least 10,000 Riemann zeros using at least 4 grammar families; (d) validating each match against a GUE null hypothesis; (e) ranking parameters by validation advantage ratio; (f) selecting the top-ranked configuration as the Riemann-optimal architecture; and (g) outputting a design specification comprising the selected material, operating temperature, frequency, and geometry.
Claim 134: The method of Claim 133, wherein the physical system is a quantum processor and the configurable parameters include: superconducting material (selected from YBCO, Nb, MgB2, BSCCO, Al), operating temperature (selected from 10mK to 300K), qubit type (selected from transmon, fluxonium, charge qubit, flux qubit), qubit frequency (0.1 to 20 GHz), and frequency architecture (uniform vs detuned), and wherein the method identifies YBCO at 77K with fluxonium qubits at 0.5 GHz in a uniform frequency architecture as the Riemann-optimal configuration.
A computer-implemented method and system for discovering, validating, and applying mathematical relationships between elements of number-theoretic sequences (including non-trivial zeros of the Riemann zeta function, derived quantities, zeros of L-functions, eigenvalues of quantum operators, prime numbers, and any ordered numerical sequence regardless of origin) and target values (physical constants, material properties, mathematical constants, and other measurable quantities). The method employs parameterized grammar families, integer relation algorithms (PSLQ, LLL), global functional properties (topological data analysis, entropy, fractal dimensions, graph invariants), regression-based optimization, spectral decomposition, and GPU-accelerated search. Statistical validation uses random matrix theory, bootstrap, permutation, or Monte Carlo baselines. The invention further provides inverse discovery, encoding via Viterbi decoding on sequence gaps, USE of discovered formulas in manufacturing, calibration, and screening, and applications in material discovery, aerospace, medical diagnostics, cosmological prediction, music theory, and drug screening.
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