#!/usr/bin/env python3 """ Hypercube → Hyper-Rhomboid Composition: Stack Mapping ====================================================== Maps the hypercube/rhomboid calculus concept onto Research Stack primitives. Key insight: shearing orthogonal tensor axes into a parallelotope is the mathematical dual of PIST n-dimensional encoding, topological state transitions, and Observer-Admissible Cavity manifestation. """ import json, time from pathlib import Path RESEARCH_STACK = Path("/home/allaun/Documents/Research Stack") HYPER_RHOMBOID = { "id": "hypercube-rhomboid-composition", "source": "User conceptual synthesis — hypercube matrix calculus → hyper-rhomboid", "title": "Hypercube → Hyper-Rhomboid Composition: Sheared Tensor Manifolds as Compression Geometry", "date": "2026-05-07", "core_claim": ( "A hypercube of matrix calculus (n-D tensor of partial derivatives) assumes " "orthogonal axes — all variables independent. Composing hypercubes into a " "hyper-rhomboid (parallelotope) applies geometric shear: axes lean into each " "other, modeling entangled dimensions. This is the geometric engine behind " "topological compression, manifold mapping, and information-theoretic gravity." ), "geometric_primitives": { "hypercube": { "definition": "n-dimensional tensor grid with orthogonal (90°) axes", "mathematical_form": "T_{i,j,k,l} ∈ ℝ^{d₁×d₂×d₃×d₄}", "assumption": "All variables statistically independent (Cartesian)", "problem": "Empty geometric space between correlated variables — inefficient packing" }, "hyper_rhomboid": { "definition": "Sheared parallelotope — axes at non-orthogonal angles", "mathematical_form": "S = A·T where A is a shear matrix (non-orthogonal basis)", "property": "Axes lean into correlated dimensions; volume preserved under shear", "gain": "Dense packing, entanglement modeling, manifold approximation" }, "shear_matrix": { "definition": "Linear transform collapsing 90° angles to acute/oblique", "form": "A_{ij} = δ_{ij} + α_{ij} where α encodes correlation strength", "determinant": "det(A) = 1 (volume-preserving shear)" } }, "stack_mappings": { "pist_nd_encoding": { "analogue": "PIST n-dimensional Cartesian → Bundle → Radial encoding", "mechanism": "Cartesian encode = orthogonal hypercube; Bundle encode = sheared rhomboid with fiber dimensions; Radial encode = fully collapsed angular coordinates", "file": "3-Mathematical-Models/pist_biological_polymorphic_shifter_v3_complete.py", "functions": ["pist_nd_cartesian_encode", "pist_nd_bundle_encode", "pist_nd_radial_encode"] }, "topological_state_machine": { "analogue": "State transition = shear operation on state hypercube", "mechanism": "Each transition applies a shear matrix A_t to the state tensor S_t → S_{t+1} = A_t·S_t. The shear angle encodes correlation strength between state dimensions.", "file": "5-Applications/scripts/topological_state_machine.py" }, "ndimensional_gene_hypothesis": { "analogue": "Gene expression = projection of sheared n-D rhomboid onto 3D observer frame", "mechanism": "The gene is an n-D rhomboid (entangled dimensions). The 3D molecular structure is a projection shadow. Epigenetic marks are shear-angle adjustments.", "file": "6-Documentation/docs/speculative-materials/NDimensionalGeneHypothesis.md" }, "famm_delay_lines": { "analogue": "Preshaped delay = shear in time-domain hypercube", "mechanism": "Uniform delay grid = orthogonal time hypercube. Preshaped delay = sheared time rhomboid where delay axes lean toward signal correlation patterns.", "file": "4-Infrastructure/hardware/famm_verilator_bench.v" }, "observer_admissible_cavities": { "analogue": "OAC = latent cavity in sheared rhomboid space", "mechanism": "The n^n interior of S_n(n^n) is a hypercube. Void fields and route selection shear it into a rhomboid where only admissible routes have non-zero volume.", "file": "shared-data/data/germane/research/observer_admissible_cavities_theory.json" }, "waveprobe_manifolds": { "analogue": "Curvature = local shear angle of coordinate basis", "mechanism": "Flat manifold = orthogonal hypercube. Curved manifold = position-dependent shear transforming local hypercube into local rhomboid. Ricci curvature = trace of shear gradient.", "file": "5-Applications/scripts/hdmi_computational_shell.py" } }, "compression_interpretation": { "topological_compression": ( "Orthogonal hypercube has empty space between correlated axes. " "Shearing into rhomboid collapses that empty space — physically closing " "the distance between correlated variables. This is geometric compression: " "same information in less volume." ), "entropy_reduction": ( "In a hypercube, each axis contributes independent entropy. " "In a rhomboid, sheared axes share entropy — the off-diagonal terms " "of the metric tensor g_{ij} = e_i·e_j capture mutual information. " "Compression ratio ≈ det(g)^{-1/2}." ), "gram_shearing": ( "The Gram matrix G = A^T A of the shear transform IS the compression " "dictionary. Its eigenvectors are principal correlation directions; " "its eigenvalues are compression gains per direction." ) }, "information_gravity": { "analogy": ( "Flat orthogonal grid = empty spacetime. " "Sheared rhomboid grid = spacetime with mass. " "The shear angle at each point encodes local information density. " "Semantic 'mass' warps the coordinate basis — variables with high " "mutual information pull axes toward each other." ), "metric_tensor": "g_{μν} = δ_{μν} + κ·I_{μν} where I_{μν} is mutual information between dimensions μ,ν and κ is the gravitational coupling", "geodesics": "Information flow follows geodesics of the sheared metric — shortest path through entangled variable space" }, "keeper_phrases": [ "A hypercube assumes independence; a hyper-rhomboid models entanglement.", "Shearing a tensor is the geometric dual of discovering correlation.", "The Gram matrix of the shear is the compression dictionary.", "Information has mass — it warps the coordinate basis it lives in.", "Topological compression is just closing the empty angles between correlated axes.", "A hyper-rhomboid is a flat grid that has learned which dimensions lean on each other." ], "metadata": { "ingested_at": time.time(), "tags": [ "hypercube", "hyper-rhomboid", "parallelotope", "tensor-calculus", "geometric-shear", "topological-compression", "information-gravity", "manifold-learning", "gram-matrix", "entanglement-geometry" ] } } def ingest(): germane_dir = RESEARCH_STACK / "shared-data/data/germane/research" germane_dir.mkdir(parents=True, exist_ok=True) out_path = germane_dir / "hypercube_rhomboid_composition.json" with open(out_path, 'w') as f: json.dump(HYPER_RHOMBOID, f, indent=2) print(f"✓ Ingested: {out_path}") index_path = germane_dir / "research_ingestion_index.json" index = [] if index_path.exists(): with open(index_path) as f: index = json.load(f) index.append({ "id": HYPER_RHOMBOID["id"], "title": HYPER_RHOMBOID["title"], "date": HYPER_RHOMBOID["date"], "source": HYPER_RHOMBOID["source"], "ingested_at": HYPER_RHOMBOID["metadata"]["ingested_at"], "tags": HYPER_RHOMBOID["metadata"]["tags"], }) with open(index_path, 'w') as f: json.dump(index, f, indent=2) print(f"✓ Index: {len(index)} entries") print(f"\nStack mappings:") for name, mapping in HYPER_RHOMBOID["stack_mappings"].items(): print(f" ↔ {name}: {mapping['analogue'][:80]}...") print(f"\nKeeper phrases:") for p in HYPER_RHOMBOID["keeper_phrases"]: print(f" → {p}") if __name__ == "__main__": ingest()