P=NP - Hyperbolic 3-SAT

Continuous relaxation on the Poincaré line. At ratio ≈ 4.267 (phase transition), IEEE 754 loses gradient signal to underflow - GeoNum preserves it.

GeoNum Precision Demo - MLE

Find the peak of L(μ) = Π_i P(x_i|μ,σ) via coordinate scan. IEEE 754 direct product underflows to 0 at large N - making every candidate look identical. GeoNum multiplies in log-space, staying exact at any N. No log-sum trick required.

Geometric Manifold Prediction Geodesic extrapolation · Jacobi field uncertainty

Module Ballistics Phase Diagram 2D FEM Geometric MD Helicity GHD Transport Batch Mode Plasma Medical Cosmos Advanced 3-Body

§ Formal Specification - v2.2

Approximation class. GeoNum is piecewise approximate with O(1/2048) quantization error per zone, drift-tracked. Each encode introduces quantization error ε_q ∈ [0, 1) shade units (systematic truncation). Under zone-normalized drift propagation and outside the cancellation regime, drift satisfies a conditional stability bound: |L̂ − L_true| ≤ (d/SHADES) × W_z + O(ε_mach). Within this regime drift is a certified bound, not merely an indicator. Cancellation is explicitly classified rather than silently degraded.

3.1 Encoding

Given x ∈ ℝ, x ≠ 0:
  σ  = sign(x)
  ℓ  = log₁₀|x|         [IEEE 754 Math.log10]
  z  = min{k ∈ ℤ : B_k > ℓ}   [zone boundaries {B_k}, half-open: B_{z-1} ≤ ℓ < B_z]
  W_z = B_z − B_{z−1}           [zone width in log₁₀ space]
  ι  = (ℓ − B_{z−1}) / W_z ∈ [0, 1)   [normalized intensity]
  s  = ⌊ι × 2047⌋                       [shade  - truncation, NOT round-to-nearest]
  δ₀ = ι × 2047 − s  ∈ [0, 1)          [initial quantization drift]

Zone boundary rule: values with ℓ = B_z exactly → zone z+1 (upper boundary excluded).
Rounding: floor (truncation toward zero). Systematic underestimate of up to 1/2047 shade.
Base case bound: |L̂ − L| ≤ (δ₀/SHADES) × W_z + O(ε_mach)  ✓ proven.

3.2 Addition (same sign, σ_a = σ_b)

ℓ_a = B_{z_a−1} + ι_a × W_{z_a}   [getLogValue()  - uses intensity, not shade]
ℓ_b = B_{z_b−1} + ι_b × W_{z_b}
ℓ_sum = ℓ_max + log₁₀(1 + 10^{ℓ_min − ℓ_max})   [stable log-sum-exp; avoids 10^ℓ overflow]

  Stable form: avoids materializing 10^ℓ_a or 10^ℓ_b directly.
  Safe for |ℓ| > 308. The 10^{ℓ_min − ℓ_max} term is always ≤ 1.

Drift update (zone-normalized  - satisfies conditional stability theorem):
  δ_result = δ₀_result + δ_a × (W_a / W_c) + δ_b × (W_b / W_c)

  W_a, W_b = zone widths of operands; W_c = zone width of result.
  Scaling by W/W_c converts operand drift into result-zone shade units.
  This is the only propagation rule that preserves the log-space error bound
  across zone boundaries. Fixed coefficients (e.g. × 0.01) are not correct.

3.3 Subtraction (σ_a ≠ σ_b) - Closed via log-diff-exp

ℓ_diff = ℓ_larger + log₁₀(1 − 10^{ℓ_smaller − ℓ_larger})   [log-diff-exp]

  Stable form: log₁₀(−expm1(Δ·ln10)) where Δ = ℓ_smaller − ℓ_larger ≤ 0.
  Math.expm1(x) = eˣ − 1, accurate near x ≈ 0 where naive 1 − exp(x) cancels.
  GeoNum NEVER projects to IEEE 754 for subtraction. Algebra is fully closed.

Conditioning: κ = 1 / |1 − 10^Δ|  (condition number of subtraction at this point)
  As ℓ_a → ℓ_b: Δ → 0, κ → ∞  (catastrophic cancellation  - fundamental, not fixable)

Drift amplification (derived from conditioning, not tuned):
  δ_result = (δ_a + δ_b) × κ   where κ = 1 / max(4ε_mach, |1 − exp(Δ·ln10)|)

  Floor 4ε_mach prevents division by zero at exact float equality.
  This is a structural floor derived from float64 representational limits, not a constant.

Near-cancellation (κ → ∞, result ≈ 0):
  Result classified as TRUST.CANCELLED with _cancelledLogMag = max(ℓ_a, ℓ_b).
  getUncertainty() returns the uncertainty at the cancelled scale, not zero.
  isUncertainZero() = true  - value is zero but not certain at this scale.

3.4 Multiply / Divide

ℓ_product  = ℓ_a + ℓ_b   [exact in log space; IEEE 754 double precision]
ℓ_quotient = ℓ_a − ℓ_b

Drift update (zone-normalized  - satisfies conditional stability theorem):
  δ_result = δ₀_result + δ_a × (W_a / W_c) + δ_b × (W_b / W_c)

  Identical rule as addition. Log-space add/subtract of ℓ values is exact to
  IEEE 754 precision  - prior drift does not compound through the operation itself.
  Re-encoding the result introduces δ₀_result ∈ [0,1); zone-normalized operand
  drifts are added on top. This propagation is sufficient for the bound to hold
  inductively across arbitrary multiply/divide chains.

3.5 Zone Partition Contract

Valid zone config iff: B₀ < B₁ < B₂ < …  (strictly increasing)
  and each B_z = sup{ ℓ | ℓ ∈ zone z }   (each boundary is zone supremum)

  Enforced: setZoneConfig() throws RangeError on non-monotone input.
  Consequence: LUT binary search and linear scan produce identical results
  for all valid configs  - equivalence is guaranteed, not coincidental.
  Zone lookup: O(log n) binary search on monotone-filtered boundary table.

3.6 Trust Classification

getTrustLevel() → GeoNum.TRUST.*

  EXACT        drift = 0                   freshly encoded, no accumulated error
  PRECISE      0 < drift < 1              sub-shade; < 1 ULP-equivalent
  VALID        1 ≤ drift < SHADES/10      bounded; conditional stability theorem applies
  DEGRADED     SHADES/10 ≤ drift < SHADES/2   significant accumulated loss
  UNRELIABLE   drift ≥ SHADES/2           error spans half a zone width
  CANCELLED    isUncertainZero()          cancellation  - value meaningless at this scale

compareWeighted(other) → { cmp: -1|0|1, certain: bool }
  certain=false when uncertainty intervals overlap in log-space.
  Solvers should not treat DEGRADED/UNRELIABLE/CANCELLED results as convergence signals.

§ MPFR-256 Benchmark - GeoNum Rust/WASM

Both the JavaScript and Rust/WASM GeoNum implementations are validated independently against pre-computed MPFR-256 reference values (256-bit precision, 77 significant decimal digits). IEEE 754 cannot serve as a reference - each implementation is tested against ground-truth anchors, not against each other. Pass threshold: <0.5% relative error. Rust/WASM is the validated primary implementation.

Domain Coverage

Domain	Zone Config	Benchmark	Status
Theory	Lucas logarithmic	Hawking radiation	✓ 0.27% vs MPFR-256
GHD	sinh-compressed	Drude weight c→0	✓ live below
Quantum	eV-scale linear	-	zone config defined
Fluids	uniform grid	-	zone config defined
Plasma	frequency-aligned	-	zone config defined
Materials	lattice-symmetric	-	zone config defined

Angle for time-series: θ = 0° (fully open) θ = 9.43° θ = 18.87° θ = 28.30° θ = 37.73° θ = 47.17° (closure) Precision: GeoNum GeoNum Adv IEEE 754 Run Analysis

Rigid-Wall Baseline vs FSI Reference - Takizawa et al. 2014

Passage Geometry & Toroidal Integrals per θ

Tessellated Real-Space DFT via Adaptive Precision Zones

GPU-native Density Functional Theory using GeoNum zone tessellation instead of traditional plane-wave basis sets. Each zone is an independent compute unit - embarrassingly parallel on WebGPU compute shaders. Drift tracking provides automatic convergence diagnostics that traditional DFT codes lack.

Single Atom Diatomic Molecule Multi-Electron Atom Ab Initio MD Multi-Atom (Zone DFT) ORCA Input

Run DFT PES Scan GPU Poisson

DFT Engine - Validated Capabilities

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Scope: this is a surrogate model, not a replication or falsification of any quantum-hardware benchmark. It evolves a synthetic structured circuit under a white-noise model - not real device circuit artifacts (QASM/QPY) and not returned hardware counts. It illustrates the scoring mechanism (XEB/HOG, same-layout controls, GeoNum substrate precision). An actual replication requires ingesting the real circuit artifacts and the returned hardware counts and scoring those through the same pipeline.

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What this is: paste anyone's published OpenQASM 2/3 circuit (Qiskit, Cirq, IBM, IonQ all export it). The 2-qubit interaction graph is split into independent patches; every patch up to 24 qubits is evolved as an exact independent statevector, so a patch-separable circuit is full-precision validated up to 128 qubits as the tensor product of its patches - never 2^128. If you also paste the author's measured bitstring counts, we run patch-XEB against our independent ideal (the same per-patch cross-entropy used for RCS), with an automatic bit-order probe. A patch wider than 24 qubits is reported BEYOND-REACH with no fabricated number - that genuinely needs a cluster, which is exactly where a published ideal is itself precision-contested. Download a self-describing, replayable bundle so anyone can reproduce the verdict without a cluster.

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What this is: the classical half of quantum-centric supercomputing (the IBM / Cleveland Clinic / RIKEN SQD method), run entirely in your browser with no cluster. Ingest a published active space (FCIDUMP: h_pq, (pq|rs) chemist notation, E_nuc, NORB, NELEC, MS2) plus sampled electronic configurations - either real published quantum samples or a clearly-labelled classical CISD surrogate. We run self-consistent configuration recovery (S-CORE), project the Hamiltonian onto the sampled-determinant subspace by Slater-Condon rules, and find the ground state by Davidson. This reproduces / audits the active-space SQD energy on classical hardware. The molecular environment (the 12,635-atom protein) is not computed here - a supplied classical embedding term is reported separately and never claimed as ours. Subspace dimensions past the browser cap are reported BEYOND-REACH with no fabricated energy: that regime genuinely needs the supercomputer, which is exactly the SQD premise. The Slater-Condon engine is validated every run against closed-form references (Hubbard dimer, non-interacting limit, H2/STO-3G FCI).

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What this is: a 2D extension of Nudged Elastic Band. Instead of a single minimum-energy path, NEM relaxes a 2D membrane of images on the potential energy surface to expose multiple MEPs, transition states, second-order saddles, shared intermediates, and post-TS bifurcations in one pass. Analytic benchmark surfaces (Mueller-Brown, Wolfe-Quapp, Cerjan-Miller, LEPS) run on GPU in seconds with full GeoNum drift tracking per node. DFT-backed surface uses the existing diatomic engine with a per-node 30 s wall cap.

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References: Mueller & Brown, Theor. Chim. Acta 53, 75 (1979). Quapp, J. Theor. Comput. Chem. 2, 385 (2003) (Wolfe-Quapp surface). Cerjan & Miller, J. Chem. Phys. 75, 2800 (1981). Sato, J. Chem. Phys. 23, 592 (1955) (LEPS). Henkelman & Jonsson, J. Chem. Phys. 113, 9978 (2000) (NEB / nudging projection). NEM formalism: Muechler et al. 2025 (motivating paper for this tab).

Solves the Euler-Lagrange boundary-value problem d/dx(dL/dq') - dL/dq = 0 by minimizing the discrete action on a uniform grid. Same regularized-Newton substrate as the NEM tab. Per-node drift comes from Rust GeoNum (wasm geonum_evaluate). Each Lagrangian validates against its closed-form analytic solution (cycloid, sinusoid, great circle).

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References: Bernoulli J., Acta Eruditorum (1696) brachistochrone problem, solution = cycloid. Riemann, Habilitationsschrift (1854) geodesics. Standard harmonic-oscillator BVP per Landau & Lifshitz, Mechanics §5. Discrete-action variational principle per Marsden & West, Acta Numerica 10, 357 (2001).

A single cavity photon mode Zeeman-couples to the collective magnetization of a transverse-field Ising model across its quantum critical point. When the coupling is parallel to the order parameter (n = x) the ferromagnetic and superradiant phases cooperate and the critical coupling g_c vanishes at the QCP; when orthogonal (n = y) they compete, giving a tricritical point and first-order transitions. Reproduces the phase diagram, polariton spectrum, intrinsic squeezing and quantum Fisher information of Sur, Wang, Mahankali, Paschen & Si, Nat. Commun. 17, 4404 (2026). Every reported quantity carries a Rust GeoNum drift / trust tag. The TFIM lattice coordination number z is a geometric property of the chosen mesh (the GMDBS pentagon mesh has z = 7).

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Reference: Sur, Wang, Mahankali, Paschen & Si, "Amplified response of cavity-coupled quantum-critical systems", Nature Communications 17, 4404 (2026), doi:10.1038/s41467-026-73112-1. Phase boundary Eq. 8; squeezing effective Hamiltonian Supplementary Note 3 (Eq. S31); QFI Eq. 3. Polariton spectrum derived from the two-mode Hopfield diagonalization, normalized to reproduce the paper's Eq. 8 phase boundary and Dicke limit. Squeezing reported from the soft-polariton ratio (perfect intrinsic squeezing as E_- -> 0).

A fundamental quasinormal ringdown (l = m = 2, n = 0) for a Kerr black hole is laid down on an extraction sphere as a Weyl scalar psi4(theta, phi, t), decomposed onto spin-weighted spherical harmonics -2Y_lm (Wigner-d form), converted to strain h(t) by fixed-frequency double integration, and fit for the complex quasinormal frequency Mω, which is compared to the published Kerr value. Each extracted mode carries a GeoNum drift / trust tag: the same operation that computes the amplitude reports whether it is physical signal or numerical cancellation. Scope: this validates the extraction and analysis chain on a reference psi4; wiring it to a live BSSN-evolved Weyl tetrad is the next step.

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References: Goldberg et al., J. Math. Phys. 8, 2155 (1967) - spin-weighted harmonics; Leaver, Proc. R. Soc. Lond. A 402, 285 (1985) - Schwarzschild QNM; Berti, Cardoso & Will, Phys. Rev. D 73, 064030 (2006) - Kerr QNM table; Reisswig & Pollney, Class. Quantum Grav. 28, 195015 (2011) - fixed-frequency psi4 -> h integration. Quasinormal frequencies in geometric units Mω (G = c = M = 1).

A Fibonacci-chain quasicrystal (off-diagonal tight-binding, two hoppings t_A, t_B in the Fibonacci sequence A→AB, B→A) is solved by the linear-scaling Kernel Polynomial Method: Chebyshev moments built from one sparse matrix-vector each, Jackson-damped, reconstructed into the density of states - no diagonalization, O(N) so it scales to millions of sites. The spectrum is a Cantor set whose gaps carry integer topological labels m via the gap-labeling theorem, IDOS = {mτ} mod 1 (τ the golden ratio). Same problem as the Aalto tensor-network result (Antao, Sun, Fumega & Lado, Phys. Rev. Lett. 136, 2026); here with a transparent method and a GeoNum drift band on the Chebyshev reconstruction. Scope: 1D Fibonacci chain on the CPU; GPU sparse mat-vec for the million-site regime and 2D Penrose / Ammann-Beenker tilings are the next steps.

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References: Weisse, Wellein, Alvermann & Fehske, Rev. Mod. Phys. 78, 275 (2006) - Kernel Polynomial Method; Kohmoto, Kadanoff & Tang, Phys. Rev. Lett. 50, 1870 (1983) and Bellissard et al. (1989) - Fibonacci gap labeling; motivated by Antao, Sun, Fumega & Lado, Phys. Rev. Lett. 136 (2026), Aalto University. Exact integrated DOS computed by Sturm sequence (negative-pivot count); gap labels by best-fit integer m in {mτ} mod 1.

A 2D discrete nonlinear Schrodinger lattice (light in a nonlinear waveguide array): i ∂_tc = κ∑_neighborsc + χ|c|²c + Uδ_n,mc. From a near-uniform input, modulational instability drives the field toward self-trapped (localized) states; a refractive-index detuning U on the diagonal (a Hubbard-like interaction) opens intermediate breathing and chaotic-like regimes. The conserved norm carries a GeoNum reliability band - the substrate's drift horizon for long nonlinear integration. Reproduces Bezerra & Dias, Nonlinear Dynamics 114:731 (2026).

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Anchors: CW dispersion ω = 2κ(cos Q_x+cos Q_y)+χ|A|²; modulational-instability gain g(Q)=√(-f(f+2χ|A|²)), f=2κ(cos Q_x+cos Q_y-2); norm + Hamiltonian conservation; exact two-site self-trapping threshold χN/κ = 4 (Kenkre-Campbell / Eilbeck-Lomdahl-Scott). Reproduces Bezerra & Dias, Nonlinear Dynamics 114:731 (2026).