The MPRC Framework
A Discrete Geometric Theory of Nature.
Five-Paper Dyadic Cubic Series
Five individual v1 preprints by Muhammad Arshad, published on Zenodo under CC BY 4.0. These records are the authoritative citations; the MPRC chapters and collective booklet provide an index to the series.
| Paper | Title | Canonical DOI |
|---|---|---|
| P1 | Quantinion Simplex Cubic Decomposition over 2-Power Rings: Global Identifiability from Binary Contractions and Exact Bit Lifting | 10.5281/zenodo.22850767 |
| P2 | Exact Free Capacity of Symmetric Rank-One Cubic Dictionaries over Dyadic Rings | 10.5281/zenodo.22850810 |
| P3 | An Eight-State Operator and Transpose Geometry over Dyadic Rings: Trace-Zero Modules, Mixed-Radix Addressing, and Tensor-Lift Kernels | 10.5281/zenodo.22850830 |
| P4 | Dimension-Free Identifiability of (n+2) Symmetric Cubics over Dyadic Rings | 10.5281/zenodo.22850834 |
| P5 | Koszul–Sidon Rigidity for (n+3) Cubic Decompositions over Dyadic Rings | 10.5281/zenodo.22850842 |
Paper 5 includes its PDF, Python verifier, and JSON certificate. Citations, verification programs, and release provenance →
MPRC is a discrete geometric framework in which space is a cubic lattice of spherical nodes, each carrying a four-component ternary state vector — Movement, Position, Rotation, Charge. From this single structure emerge gravity, electromagnetism, particle stability, atomic prediction, cryptographic key agreement, and a verified Goldbach verification engine.
The numbered chapters collected here represent the current verified state of the framework. Each chapter is self-contained but builds on a shared geometric foundation. No chapter introduces a free parameter without geometric justification or explicit acknowledgment that derivation is pending.
Research Chapters, One Ring
I · Ring Foundation
Speed & Velocity on Z₂₅₆
Formal definitions for ring trajectory, velocity, speed, orientation bit, and zero net angular momentum. The foundation every MPRC motion chapter relies on. AGC-invariance proved.
Definitions · v1.1Discrete Lattice Mechanics
Space as a cubic lattice of spherical MPRC nodes. The state vector |ψ⟩ = [M,P,R,C]. Gravity as tension. GPS verified. B₄ symmetry, Pati–Salam structure derived.
GPS VerifiedGrand Unified Theory — DLM v2
Continuation targeting seven open gaps. Node geometry formalized with two circles and six pillars. Discrete Laplacian derived. Spin-statistics closed. Dark matter remains open.
Active ResearchII · Atomic Physics
QH4 Atomic Prediction Protocol
Arshad's Sieve on Z₂₅₆. Stability metric S derived from ring geometry. Metals protocol achieves 0.21% MAPE across 12 elements Z=3 to Z=92. Gas phase protocol verified across 17+ species.
0.21% MAPEZ-Cancellation: When Charge Disappears
In the gas phase, atomic number Z cancels exactly from the stability equation. Gas stability is mass and temperature driven — not charge driven. Verified across 14 gas species.
14 Species VerifiedIII · Gravity & Cosmology
Gravity as E·B Field: Artemis III Predictions
MPRC derives gravity as the inward E·B field output of the Z₂₅₆ ring. Newton verified as the geometric projection (ratio 1.00000000 on 11 datasets). Three falsifiable pre-mission predictions stated for Artemis III.
Predictions Stated · Mission 2027Lunar-Nodal Climate Hypothesis
18.61-year lunar nodal cycle as a falsifiable driver of multi-decadal climate variability. MPRC ring geometry applied to tidal forcing. Pre-stated predictions against observational datasets.
Hypothesis · Predictions StatedIV · Mathematics & Conjectures
Arshad's Sieve — Theorem Register
Modular power-sum framework on Z₂₅₆. 12 proven theorems — Type-A, Type-B, Type-AB walls; 16/9 disk growth identity; three-region partition; spiral non-closure. 3 open problems.
12 Proven · 3 OpenQuartic Centroid Conjecture
Centroid–residue decoupling in 4-term arithmetic products. √10 as the fundamental residue unit of the entire arithmetic-4 family. f(n)=3n geometric scaling law. Competition problem solved in 90 seconds.
Proven · One Open LinkQH4 Pattern Mining Machine
Λ(ρ,γ,σ,θ) address system for Z₂₅₆. 2 postulates, 3 definitions, 9 theorems — all verified 252/252. O(1) inverse formula. Cross-modal resonance proved: text, image, audio at same position are structurally identical.
252/252 · All 9 TheoremsQH4 Pattern Matching
O(1) structural classification of ring positions by pattern type — the next conjecture to derive. All priors from Ch15 established and locked. Five starting questions stated precisely. Research seed.
Open ConjectureQuantinion Simplex Cubic Decomposition
Global identifiability of an overcomplete n+1-term symmetric cubic over Z/(2^m) when the parity atoms form a binary simplex frame. Recovery uses full-rank binary contractions followed by exact bit lifting. Quantinion specialization: n=8, m=8, r=9.
Proven · Global n+1 RecoveryExact Cubic Dictionary Capacity
Sharp free-dictionary capacity for rank-one symmetric cubics over Z/(2^m): C₃(n)=Σ_{k≤3} C(n,k). In dimension 8 the exact capacity is 92, attained by an explicit dense construction.
Proven · Exact Capacity 92 at n=8Quantinion Operator & Transpose Geometry
Eight-state operator algebra with a 63-dimensional trace-zero module, 21-edge orientation, QH4 mixed-radix address, exact Transpose orbits, and dyadic tensor-lift kernel theorem.
Proven · 63-D Operator SpineTop-KPQ Collision Geometry
(K,P,Q) is a complete invariant of the frozen MPRC collision congruence. The dense Z₂₅₆ diagonal localization collapses to one relative rotor and has exact nilpotence length 576: 𝔭⁵⁷⁵≠0, 𝔭⁵⁷⁶=0.
Top-KPQ Closed · Dense Length 576SILIQ: Goldbach Verification Engine
Z₂₅₆ ring walk for Goldbach pair verification. BPAND gate on biopod geometry. 999,999 even integers tested, zero failures. 8 DERIVED, 0 FAILED — derived from MPRC ring structure.
999,999 Computed — 0 FailuresBell Correlations from Ring Geometry
S = 2√2 derived from Z₂₅₆ without nonlocality. Anti-phase lock at birth. Projected on Hensen et al. 2015 loophole-free data: S = 2.422 ± 0.20. Three predictions Standard QM cannot make.
S=2√2 DerivedV · Cryptography & Security
Lattice Quantum Channel (LQC)
Zero-exchange quantum-resistant communication from MPRC lattice geometry. Key agreement without classical exchange. Conjugate birth invariant d_A + d_B ≡ 0 (mod p). 2²⁵⁴ keyspace.
Derived — Not Yet TestedLQC Three-Layer Safety
Three independent security layers: geometric (L1), physical tension ratchet (L2), temporal Domain Clock Shield (L3). All three derived — exp32–39 pass. Combined entropy budget 280+ bits.
All 3 Layers DerivedVI · Computing
Staging Area — Not Submitted
The following document argues from MPRC geometry that superposition, entanglement, and virtual particles are measurement artifacts — not ontological realities. It is staged here pending sufficient empirical weight. It is not part of the numbered physics spine.
The FSL Observation Thesis
Superposition, entanglement, and virtual particles as artifacts of an observer whose measurement bandwidth falls below the traversal speed of the u-spinor on QH4. QKD as empirical anchor.
Staging · Not SubmittedThe Definition of Time
t = ⟨u|φ⟩ — a projection, not a parameter. Three-observer hierarchy. 16/45 geometric codec exact and invertible. Vacuum nodes verified 4/4. Newton and Einstein recovered as special cases.
Derived · One Open LinkKinematics in Ring Coordinates
Six kinematic formulas — time, displacement, velocity — derived in Z₂₅₆ via the 16/45 codec. Integer mod implementation with no floats. Inverse scaling conjecture ε = 3/128 proven. Spatial codec open.
9 Proven · 3 OpenThe Geometric Core
Every paper in this book derives from a single geometric statement: a 3D+1D spherical node carrying a ternary state vector [M, P, R, C]. The four components are not independent — they are the four faces of the same spherical geometry:
The QH4 ring (Quwa Halaqa — force of the ring) maps this geometry to 256 discrete states: four quadrants of 64, vacuum boundaries at {0,64,128,192}. This is the atom's own resolution — inherited by digital computing, not invented by it.