Yuris-Identity-Gap-Operator

A Gap-Based Deterministic Operator for Exact Integer Power Sums

A Gap-Based Deterministic Operator for Exact Integer Power Sums

Author: Yuri Freire de Carvalho Espírito Santo
Paper: Read the Full Paper on Zenodo

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📐 Overview

This repository contains the reference implementation and validation logs for Yuri's Identity, a gap-based computational operator formally introduced in the paper "A Gap-Based Deterministic Operator for Exact Integer Power Sums".

Unlike standard IEEE 754 floating-point arithmetic, which suffers from mantissa truncation when handling magnitudes $> 2^{53}$, this operator uses a constructive gap expansion ($\Delta = Q - P$) to propagate power sums as exact integers. This ensures absolute precision for cryptographic auditing, hardware verification, and number-theoretic proofs.

🚀 Key Validation: The "24-Sextillion" Stress Test

As detailed in Section 14.4 of the paper, standard double-precision FPUs fail to preserve the least significant bits of power sums when exponents grow.

We validated the operator with the base pair P=19, Q=31 at iteration a=13.

Method	Result (S)	Precision
Yuri's Gap Operator	24,441,847,155,750,948,002	✅ Exact (Lossless)
Standard IEEE 754	~2.44418 x 10^19	❌ Loss of ~2,048 integers

The algorithm successfully resolves the recurrence $\Sigma_{a+1} = Q\Sigma_a + P^a$ strictly within the integer domain, confirming correctness where floating-point approximation introduces irreversible error.

🛠️ Usage

To reproduce the validation:

python3 proof_validation.py

The core recurrence relation derived in the paper is:

$$P^a + Q^a = 2P^a + (Q-P) \sum_{k=0}^{a-1} Q^{a-1-k}P^k$$

This implementation demonstrates:

1. Gap Propagation: Using $\Delta$ to drive the sum.
2. Modular Integrity: verifying $S \equiv 2P^a \pmod{\Delta}$ at each step.
3. Exactness: Handling arbitrary-precision integers without FPU fallback.

🔗 Citation

If you use this operator or the gap-based formulation in your research, please cite the foundational paper:

@article{EspiritoSanto2026,
  author       = {Espírito Santo, Yuri Freire de Carvalho},
  title        = {A Gap-Based Deterministic Operator for Exact Integer Power Sums},
  year         = {2026},
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.18204583},
  url          = {[https://doi.org/10.5281/zenodo.18204583](https://doi.org/10.5281/zenodo.18204583)}
}

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Verified against IEEE 754 constraints. 2026.