Architectural Blueprint: Fractal-Based Associative Memory andPost-Quantum Cryptographic Primitives 1. The Pérez Hourglass: Structural and Mathematical FoundationsThe Pérez Hourglass represents a paradigm shift in computational topology, moving away fromiterative error-correction engineering toward a structure that is natively "born fault-tolerant." Thisbidirectional, self-similar geometric structure emerges from the superposition of Pascal’sTriangle—the additive representation of matter—and its mirrored, subtractive counterpart,defined as "digital antimatter." This architecture provides a fundamental unification of numbertheory and quantum entanglement, resolving the 1769 Lichtenberg Conjecture and mappingthese principles onto a physical qubit lattice.The following table identifies the specificmathematical convergences that dictate the Hourglass topology:Mathematical Convergence in the HourglassMathematical Component,Functional Role in the ArchitecturePascal’s Triangle (Matter),Provides the additive foundation for northern hemisphere bitmaskingand state initialization.Antimatter Mirror (Subtraction),"Forms the southern hemisphere, enabling ""error-draining""through subtractive dynamics."Fibonacci & Lichtenberg Sequences,Emergent sequences determining the qubit layout andexponential scaling of code distance.Ramanujan’s Mock Theta Functions,First discrete analogue using triangular exponents ( $T_k= k(k+1)/2$ ) instead of squares (2025–2026 Discovery).Perfect Parity Entanglement,"Enforces global coherence; even entries always face even, andodd face odd."The transition from abstract geometry to physical advantage occurs at the "waist" of thehourglass. This narrow intersection acts as a self-stabilizing computational environment wherethe topology dictates that local errors—bit-flips or phase-flips—are "drained" and annihilated bythe subtractive dynamics of the southern hemisphere. This passive suppression transformsenvironmental noise into manageable dynamics, creating a self-healing order that provides themathematical substrate for all memory and security functions.
Architectural Blueprint: Fractal-Based Associative Memory andPost-Quantum Cryptographic Primitives 1. The Pérez Hourglass: Structural and Mathematical FoundationsThe Pérez Hourglass represents a paradigm shift in computational topology, moving away fromiterative error-correction engineering toward a structure that is natively "born fault-tolerant." Thisbidirectional, self-similar geometric structure emerges from the superposition of Pascal’sTriangle—the additive representation of matter—and its mirrored, subtractive counterpart,defined as "digital antimatter." This architecture provides a fundamental unification of numbertheory and quantum entanglement, resolving the 1769 Lichtenberg Conjecture and mappingthese principles onto a physical qubit lattice.The following table identifies the specificmathematical convergences that dictate the Hourglass topology:Mathematical Convergence in the HourglassMathematical Component,Functional Role in the ArchitecturePascal’s Triangle (Matter),Provides the additive foundation for northern hemisphere bitmaskingand state initialization.Antimatter Mirror (Subtraction),"Forms the southern hemisphere, enabling ""error-draining""through subtractive dynamics."Fibonacci & Lichtenberg Sequences,Emergent sequences determining the qubit layout andexponential scaling of code distance.Ramanujan’s Mock Theta Functions,First discrete analogue using triangular exponents ( Tk=k(k+1)/2 ) instead of squares (2025–2026 Discovery).Perfect Parity Entanglement,"Enforces global coherence; even entries always face even, andodd face odd."The transition from abstract geometry to physical advantage occurs at the "waist" of thehourglass. This narrow intersection acts as a self-stabilizing computational environment wherethe topology dictates that local errors—bit-flips or phase-flips—are "drained" and annihilated bythe subtractive dynamics of the southern hemisphere. This passive suppression transformsenvironmental noise into manageable dynamics, creating a self-healing order that provides themathematical substrate for all memory and security functions.
Architectural Specification: Fault-Tolerant Quantum Processors via thePerez Hourglass Lattice1. Executive System SummaryThe strategic transition from Euclidean surface codes to fractal-based architectures is thenecessary pivot to surmount the "decoherence wall" inherent in current hardware scaling.Conventional architectures rely on intensive, active error-correction cycles that consumeprohibitive computational overhead. The Perez Hourglass (PH) processor shifts this paradigmtoward passive topological drainage. By utilizing a bidirectional, self-similar fractal derived fromthe parity filtering of Pascal’s triangle, the PH architecture creates a "digital antimatter"environment. In this regime, bit-flips and phase-flips are not merely detected but aretopologically driven toward a central waist for annihilation, effectively treating noise as atransient state governed by subtractive dynamics.System Logic The processor’s integrity isdefined by two fundamental topological invariants:● Theorem 7 (Local Determinism): Establishes a collision-free mapping where everyface-to-face folded couple in the lattice possesses identical parity and global numericaluniqueness. This ensures an injective coordinate system for state synchronization andHamiltonian stability.● Theorem 8 (Global Integrity): Anchors scaling stability by proving that the exponentialgrowth of the Northern "matter" hemisphere ( 2n ) directly drives the populationgrowth of the Southern "antimatter" hemisphere ( L(n+2)−L(n) ). This maintains aconstant algebraic row sum of +2, enforcing global coherence across the lattice at alldepths.These mathematical pillars provide the formal foundation for a non-standard qubitlattice capable of universal, fault-tolerant operation.
Abstract More than three decades after the discovery of self-organizing neural networks governed by the golden ratio (Perez, 1988, 1991, 1997), a remarkable fractal structure—the “Perez Hourglass”—emerges from Pascal’s triangle modulo 2 through recursive parity filtering (Perez, 2025a). This exact, self-similar hourglass pattern, indexed as OEIS A000975, constitutes the digital incarnation of the Fibonacci sequence and the topological antimatter of Sierpiński’s fractal triangle.We prove that the Perez Hourglass defines a family of sparse, golden-angle qubit lattices that simultaneously enable: A distance-3 Fibonacci-valued CSS quantum error-correcting code with parameters [[F_{2n+1}, 1, F_n]] surpassing the Bravyi–Poulin–Terhal bound, providing exponential suppression of logical errors at constant physical qubit degree; Native implementation of universal golden-phase gates e^{iπ/φ²} and e^{iπ φ²}, yielding quadratic speedup in quantum phase estimation and direct simulation of fractal quantum critical systems; Magic-state distillation factories with O(log log N) overhead—the lowest known asymptotic cost; Topological protection against decoherence via fractal anyon condensation analogous to time-like fractals in quantum gravity; A natural substrate for post-quantum public-key cryptography based on the hardness of decoding random Fibonacci-coded linear systems; A novel class of dense associative memories (Hopfield–Perez golden networks) whose energy landscape is shaped by the Hourglass attractor, achieving storage capacity ~φⁿ (where φ is the golden ratio) and one-shot pattern retrieval. Theorem 7 Enables the First Perfect Associative Memory in History – The Perez Hourglass Associative Memory (PHAM). Keywords fractal quantum computing, golden ratio, topological quantum error correction, post-quantum cryptography, associative memory, Fibonacci coding, CSS codes, magic-state distillation, Ramanujan Mock Theta Function.
More than three decades after the discovery of self-organizing neural networks governedby the golden ratio (Perez, 1988, 1991, 1997), a remarkable fractal structure—the “PerezHourglass”—emerges from Pascal’s triangle modulo 2 through recursive parity filtering(Perez, 2025a). This exact, self-similar hourglass pattern, indexed as OEIS A000975,constitutes the digital incarnation of the Fibonacci sequence and the topological antimatterof Sierpiński’s fractal triangle.We prove that the Perez Hourglass defines a family ofsparse, golden-angle qubit lattices that simultaneously enable:1.A distance-3 Fibonacci-valued CSS quantum error-correcting code withparameters [[F_{2n+1}, 1, F_n]] surpassing the Bravyi–Poulin–Terhal bound,providing exponential suppression of logical errors at constant physical qubitdegree;2.Native implementation of universal golden-phase gates e^{iπ/φ²} and e^{iπ φ²},yielding quadratic speedup in quantum phase estimation and direct simulation offractal quantum critical systems;3.Magic-state distillation factories with O(log log N) overhead—the lowest knownasymptotic cost;4.Topological protection against decoherence via fractal anyon condensationanalogous to time-like fractals in quantum gravity;5.A natural substrate for post-quantum public-key cryptography based on thehardness of decoding random Fibonacci-coded linear systems;6.A novel class of dense associative memories (Hopfield–Perez golden networks)whose energy landscape is shaped by the Hourglass attractor, achieving storagecapacity ~φⁿ (where φ is the golden ratio) and one-shot pattern retrieval. Theorem7 Enables the First Perfect Associative Memory in History – The Perez Hourglass Associative Memory (PHAM) Keywords fractal quantum computing, golden ratio, topological quantum error correction, post-quantum cryptography, associative memory, Fibonacci coding, CSS codes, magic-state distillation.
Briefing on the Perez Hourglass Theoretical Framework Summary The Perez Hourglass is a novel, bidirectional fractal structure introduced by Jean-Claude Perezthat provides a unifying framework for several disparate fields of mathematics and computing.Derived from Pascal's triangle and a mirrored "antimatter" counterpart, the Hourglass resolvesthe 256-year-old Lichtenberg Conjecture by revealing the geometric origin of the Lichtenbergsequence.This mathematical breakthrough serves as a blueprint for a new architecturalparadigm in quantum computing. The Hourglass structure, when mapped to qubit lattices,induces a rare form of fractal criticality, enabling highly efficient, fault-tolerant quantum errorcorrection codes that surpass existing theoretical bounds. Simulations on IBM Quantumhardware have validated key predictions, including logarithmic violations of the entanglemententropy area law, a hallmark of advanced topological systems.Beyond quantum mechanics, theHourglass's most profound implications stem from a property known as Theorem 7—the globaluniqueness of symmetrically folded numerical pairs. This theorem is the foundation for thePerez Hourglass Associative Memory (PHAM), a proposed class of associative memory withtheoretically infinite capacity, deterministic constant-time (O(1)) retrieval, and unprecedentednoise resilience, reportedly capable of perfect recall from cues with over 99% corruption.Furthermore, these unique numerical properties are leveraged to create a suite of post-quantumcryptographic primitives believed to be resistant to attacks from even unbounded quantumadversaries. The Perez Hourglass thus represents a significant theoretical convergence ofnumber theory, quantum information science, and artificial intelligence
"Perez Hourglass quantum computing fractal theory" : Towards a New Generation of Quantum Computers "Where there is matter, there is geometry.” Johannes Kepler “Through the looking glass, we do not escape reality—we complete it.” Jean Claude PerezPhD in Mathematics and Computer Science, Bordeaux UniversityLuc Montagnier Foundationjeanclaudeperez2@gmail.com (mailto:jeanclaudeperez2@gmail.com) https://creationwiki.org/Jean-claude_Perez ABSTRACT The Pérez Fractal Hourglass emerges as a paradigm-shifting framework at the intersection of number theory, fractal geometry, and quantum information science, resolving the 256-year Lichtenberg conjecture while unlocking scalable quantum computing architectures. By superimposing an extended Fibonacci sequence with a mirrored Pascal triangle—yielding universal evenness theorems, twin symmetries, and a 5D modular oscillator—this model derives the Lichtenberg sequence (OEIS A000975: 1, 2, 5, 10, 21, ...) through precise counting of non-border positive interior elements in the southern hemisphere's difference triangle, governed by the recurrence L_n = 2L_{n-1} + L_{n-3} . Integrating insights from Pérez's seminal 2025 works, this synthesis embeds fractal symmetry-protected topological (FSPT) phases into qubit lattices, inducing logarithmic entanglement entropy violations ( S \sim \log L ) in gapped 1D/2D chains, validated via Qiskit simulations on 21-qubit IBM backends showing S \approx 2.3 bits for L=10 . Dual-rail encoding via hourglass hemispheres bolsters fault tolerance, slashing logical error rates by 30% and enabling measurement-based quantum computation on noisy intermediate-scale quantum (NISQ) devices. This heralds fractal quantum neural networks and holographic error correction, propelling quantum supremacy toward practical, million-qubit eras.
Quantum error-correcting codes are a cornerstone of quantum computing, with broad and profound connections to physics and mathematics. In this work, we introduce the notion of intrinsic locality dimension of stabilizer codes, which is independent of the underlying geometry of quantum codes and naturally extends to non-integer values. Drawing on mathematical tools from fractal geometry and geometric measure theory, the intrinsic locality dimension accommodates flexible architectures and provides a quantitative measure of code connectivity, encompassing both topological codes and algebraic constructions such as bivariate-bicycle-type codes. We show how the intrinsic dimension serves as a fundamental organizing parameter that unifies code properties. In particular, we prove general limitations on code parameters and compatible fault-tolerant logical gates induced by the intrinsic dimension, generalizing the Bravyi--Poulin--Terhal and Bravyi--König bounds for regular topological codes, respectively. Furthermore, we consider implications on thermal properties: toward fully characterizing the geometry requirement for self-correcting quantum memories (SCQMs), we present a conditional no-go result for SCQMs in dimension 3−ε and take stock of existing results on low-dimensional SCQMs. Our theory provides a unifying mathematical framework for understanding the fundamental capabilities and geometric implementations of quantum error correction and fault tolerance.