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[Compilation] 2025 Report on Preliminary Research on Exact Numerical Calculations

Four Perspectives on Rigorous Numerical Computation and Algorithms

In the Prime OS Research Map (B: “Field‑wise milestones and open problems”), GhostDrift Mathematical Institute re‑examines rigorous numerical computation and algorithms from four angles:

  1. (i) Interval arithmetic

  2. (ii) ODE / PDE solvers

  3. (iii) Formal verification

  4. (iv) Prime number algorithms

This article outlines (a) what each field has already achieved, (b) which structural limitations still remain, and (c) how GhostDrift theory (finite closure / ADIC / Prime OS) is designed to push beyond those limits.

▼20205 Exact Numerical Calculations Previous Research Report https://ghostdrifttheory.github.io/prime-os-research-map/


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1. Interval Arithmetic: Where We Stand and What Still Breaks

1.1 Milestones

Interval arithmetic treats not a single floating‑point number, but a whole range of possible values. It is already deployed in

  • numerical analysis,

  • control engineering,

  • safety and reachability analysis,

as a standard tool for rigorous error‑bounding computation.

By rounding interval endpoints outward, one can enclose

  • rounding errors, and

  • model uncertainty

in a single “safe” box, and guarantee that the true value always lies inside the computed interval.

1.2 Structural Limitations

Despite these strengths, modern interval arithmetic faces several structural issues:

  1. Dependency problemWhen the same variable appears multiple times in an expression, intervals tend to blow up. The result is often

    1. mathematically correct but too wide to be practically useful.

  2. Dimensional explosionIn high‑dimensional or long‑time simulations, interval width can grow exponentially, making it hard to keep enough precision for

    • convergence decisions,

    • safety guarantees.

  3. Gap to “proof‑level” explanationsInterval arithmetic is numerically safe, but it usually does not answer questions like:

    • Exactly which constants were assumed?

    • Where does the analysis stop and pure numerics begin?

    • How do we present this as a mathematical proof, not just as an experiment?

1.3 GhostDrift’s Breakthrough: ADIC and Finite Closure

GhostDrift’s ADIC (Analytically Derived Interval Computation) extends interval arithmetic in three key ways:

  • Intervals derived from analysis firstInstead of directly throwing a formula into a numeric kernel, we first use analytic estimates (integral bounds, sum estimates, kernel properties) to obtain

    • explicit upper bounds on errors,

    • strictly positive lower bounds.

  • Finite closureInfinite series, integrals, or products are split into

    1. a finite window + a rigorously bounded “tail error”,

    2. with the entire argument organised so that the proof closes within finitely many steps and finitely many log entries.

  • Σ₁ certificates (GhostDrift ADIC ledger)Every computation is recorded as a verification log containing only

    • integers, and

    • rationals.


2. ODE / PDE Solvers: Rigorous Dynamics and Finite‑Closure Energy

2.1 Milestones

For ODEs, we already have validated ODE solvers that can track

  • existence and uniqueness,

  • rigorous enclosures of trajectories,

using a mixture of

  • interval methods, and

  • a priori estimates.

For certain classes of PDEs, methods such as

  • Galerkin schemes, and

  • Lyapunov functionals

have been used to obtain partial rigorous guarantees (e.g., on stability or long‑time behaviour).

2.2 Structural Limitations

  1. How to measure “meaning energy” of large systemsIn real infrastructures—power grids, mobility systems, socio‑technical networks—one needs to reason not only about physical stability but also about

    • fairness of service,

    • ethical constraints,

    • priority of critical loads.

  2. Reconciling rigor with real‑time decision‑makingA fully validated simulation can be too heavy for

    • real‑time control, or

    • emergency response.

2.3 GhostDrift’s Breakthrough: Finite‑Closure Energy Kernel

The GhostDrift Finite‑Closure Energy Kernel is designed precisely for this.

  • For system state x(t) and control u(t), we define an energy

    1. ( E(x,t) = E_\text{mech}(x,t) + \lambda E_\text{sem}(x,t) )

    2. that combines

      • physical stability (flows, frequencies, SoC, etc.), and

      • semantic aspects (unserved demand, fairness, ethical priorities).

  • Security is then measured by

    1. ( Security(t) := -\frac{d}{dt} E(x,t). )

  • The kernel uses

    • Yukawa‑type finite‑range kernels, and

    • UWP (Window Positivity) conditions,to prove that perturbations

  • All constants and bounds used in this proof are logged in an ADIC ledger, so that a power‑system OS or energy‑control OS can ship not only control actions, but also

    1. a finite proof that “this operation was safe”.


3. Formal Verification: From Proof Assistants to a “Proof OS”

3.1 Milestones

Proof assistants such as Coq, Isabelle, and Lean have already verified deep results in

  • algebraic geometry,

  • analysis,

  • cryptographic protocols, and more.

In software engineering, tools like

  • SMT solvers, and

  • model checkers

are widely used to find bugs and check safety properties.

3.2 Structural Limitations

  1. Huge numerical kernels remain outside the proofFormal proofs are logically complete, but it is still difficult to bring in

    • 10¹²‑step numeric loops,

    • massive linear algebra routines,

    • HPC‑scale simulations,

  2. Human‑readable explanationsMachine‑generated proof objects are often massive. For journals and review processes, it is hard to answer in a simple way:

    • What exactly has been guaranteed?

    • Under which assumptions?

    • Where did numerics end and pure logic begin?

3.3 GhostDrift’s Breakthrough: ADIC Ledgers as Σ₁ Proof Objects

GhostDrift’s answer is to make all numerical work Σ₁‑friendly:

  • Numeric routines output an ADIC ledger consisting only of

    • integer parameters, and

    • rational bounds.

  • This ledger can be imported into any proof assistant or a dedicated verifier, and checked independently of the original runtime.

  • The key properties are:

    • the log is finite, and

    • it contains no floating‑point artefacts—only exact rationals.

In this way, GhostDrift’s ADIC ledger acts as a

bridging proof assistants and real‑world computations without requiring that every single instruction be re‑implemented inside a theorem prover.


4. Prime Number Algorithms: Prime OS and Finite‑Closure Explicit Formulae

4.1 Milestones

In the prime world we already have:

  • AKS and related polynomial‑time primality tests,

  • fast π(x) algorithms such as Meissel–Lehmer / Lagarias–Miller–Odlyzko,

  • explicit‑formula methods using zeros of the Riemann zeta function.

These provide extremely fast ways to compute or approximate

  • whether a number is prime, and

  • how many primes are below a given x.

Large‑scale experiments have computed π(x) and zeta zeros very far out, giving strong empirical support to the Riemann Hypothesis.

4.2 Structural Limitations

  1. Gap between experiment and logically closed proofNumerically tracking zeros of ζ(s) is not the same as

    • proving RH, or

    • certifying that “there are no zeros in this entire region”.

  2. High speed vs. fully rigorous boundsExisting fast algorithms rely on heavy analytic machinery and approximations. Making every step finite‑closure‑safe is still an open challenge.

4.3 GhostDrift’s Breakthrough: Prime OS and Prime Gravity

GhostDrift’s Prime OS tackles this by treating prime computation itself as an OS‑level service with built‑in finite closure.

  • Explicit formulae for π(x) are regularised using

    • finite windows, and

    • Yukawa‑type kernels,

  • A positive safety margin δ_pos is evaluated as a Σ₁ inequality, yielding statements of the form:

    1. “On this range, our estimate of π(x) is guaranteed to be correct.”

  • In parallel, Prime Gravity interprets primes as “sources” in a potential field. The zeta function is analysed in terms of

    • potentials and forces,

    • equilibrium conditions,

Prime OS demos already integrate

  • prime counting,

  • Yukawa kernels, and

  • ADIC ledgers

into a single UI, making visible the idea that


5. A Single Line Through Four Fields: GhostDrift’s Common Breakthrough

The four fields—interval arithmetic, ODE/PDE solvers, formal verification, and prime algorithms—look separate at first sight. GhostDrift theory connects them via a single organising principle:

The shared toolkit is:

  1. ADIC: intervals derived from analysisUse analytic structure to carve out a positive margin δ_pos before running numeric loops.

  2. Yukawa / UWP: finite‑window positivityShow that cutting the world down to a finite window with a finite‑range kernel does not break the underlying physics or number‑theoretic structure.

  3. Σ₁ ledgers: verifiable proof logsNo matter whether the context is an OS, a service, or a paper, claims of safety and correctness are backed by finite, replayable logs.

Across all four domains, the aim is to satisfy, along one GhostDrift axis,

This is the role of GhostDrift theory within the Prime OS Research Map: to serve as a bridge connecting

  • analytic number theory,

  • numerical analysis,

  • proof technology, and

  • infrastructure OS design,

under a single perspective: rigorous numerical computation and algorithms as finite, shareable, Σ₁‑level evidence.

 
 
 

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