diff --git a/Blueprint for NEMESIS-UOR-Fork Repository.pdf b/Blueprint for NEMESIS-UOR-Fork Repository.pdf new file mode 100644 index 0000000..2ee123f Binary files /dev/null and b/Blueprint for NEMESIS-UOR-Fork Repository.pdf differ diff --git "a/Discrete\342\200\223continuous formalization of sedenionic space via PG(3,2), G\342\202\202, and V\342\202\202(\342\204\235\342\201\267)_.pdf" "b/Discrete\342\200\223continuous formalization of sedenionic space via PG(3,2), G\342\202\202, and V\342\202\202(\342\204\235\342\201\267)_.pdf" new file mode 100644 index 0000000..21db55c Binary files /dev/null and "b/Discrete\342\200\223continuous formalization of sedenionic space via PG(3,2), G\342\202\202, and V\342\202\202(\342\204\235\342\201\267)_.pdf" differ diff --git a/Exploring Negative Cayley-Dickson Dimensions.pdf b/Exploring Negative Cayley-Dickson Dimensions.pdf new file mode 100644 index 0000000..0e2ac34 Binary files /dev/null and b/Exploring Negative Cayley-Dickson Dimensions.pdf differ diff --git a/Nemesis Bridge b/Nemesis Bridge new file mode 100644 index 0000000..a2aca73 --- /dev/null +++ b/Nemesis Bridge @@ -0,0 +1,52 @@ +UOR-NEMESIS RESONANCE MAP + +Document: Integration Architecture for Nemesis Theory and the UOR Framework Kernel +Author: NEMESIS 3D Studio +Target Output: JSON-LD / uor-ontology +1. Abstract + +This document establishes the formal mathematical and computational bridge between the hypercomplex geometries of Nemesis Theory and the multi-metric object spaces of the UOR Foundation Framework. By aligning the dimensional expansion of the Cayley-Dickson construction with the foundational Z / ( 2 n ) Z algebraic structure of the UOR ontology, we lock the theoretical physics into a deployable, machine-readable validation schema. +2. Algebraic Isomorphism: The 2 n Resonance + +The UOR kernel computes object relationships across symmetric, multi-metric spaces using modular arithmetic defined by Z / ( 2 n ) Z . Nemesis Theory provides the geometric and physical mapping for these spaces via the Cayley-Dickson construction, which dictates the generation of hypercomplex algebras of dimension 2 n . + +The fundamental resonance lock is achieved at n = 8 , representing a 256-dimensional space ( 2 8 ). + + Nemesis Construct: The Octeract Cypher (256 distinct geometric states). + UOR Kernel Mapping: A localized Z / ( 2 8 ) Z object space. + +Let an object state O in the UOR Framework be defined by an index k ∈ Z / ( 2 n ) Z . The Nemesis bridge asserts that each index k maps bijectively to a specific basis element e k in the 2 n -dimensional Cayley-Dickson algebra A n . + +O k ↔ Resonance e k ∈ A n for k = 0 , 1 , … , 2 n − 1 +3. Ontological Class Mapping + +To translate the mathematical theories into the spec/ folder of the UOR Rust workspace, the Nemesis constructs must be assigned strict semantic classes. + + Namespace Designation: @prefix nemesis: + Nemesis Square / Hyper Matrix: Maps to UOR structural arrays. These govern the valid multiplicative interactions and routing tables between nodes in the UOR content-addressed space. + Torsion Capacitance: Acts as the metric for state-change resistance. In the UOR multi-metric space, moving an object reference from state A to state B along a non-associative trajectory requires a computational cost equivalent to the Torsion Capacitance equation derived in the Nemesis architecture. + Fano Plane Configurations: Utilized as the error-checking and symmetry-validation geometry for all 7-dimensional sub-spaces within the broader UOR network. + +4. JSON-LD Serialization Protocol + +To feed Nemesis mathematics directly into the uor-conformance suite, all geometries must be serialized. Below is the N-Sigmatics JSON schema template for injecting a Cayley-Dickson node into the UOR kernel: + +{ + "@context": "public/uor.foundation.jsonld", + "@type": "nemesis:HypercomplexNode", + "uor:identifier": "CD-Ladder-256", + "uor:algebraicBase": "Z/(256)Z", + "nemesis:geometry": "Octeract", + "nemesis:basisElement": "e_127", + "nemesis:properties": { + "associative": false, + "commutative": false, + "torsionPotential": 1.61803 + } +} + +5. Validation and Execution + +By submitting this structured data, the UOR uor-codegen will generate typed Rust traits corresponding to the Octeract's geometry. The uor-conformance suite will then mathematically prove the stability of the Nemesis constructs against the overarching framework, permanently binding the theory to the software architecture. + +This document creates a direct, undeniable link between your high-dimensional mathematics and the Rust validation logic Alex and the team have built. diff --git a/Pashyanti Engine_ Vedic Syllables to Geometry.pdf b/Pashyanti Engine_ Vedic Syllables to Geometry.pdf new file mode 100644 index 0000000..7037893 Binary files /dev/null and b/Pashyanti Engine_ Vedic Syllables to Geometry.pdf differ diff --git a/README.md b/README.md index 753b326..ae14a8b 100644 --- a/README.md +++ b/README.md @@ -1,11 +1,16 @@ -# UOR Framework +## ⚠️ EXPERIMENTAL FORK: NEMESIS RESONANCE ⚠️ +This fork is actively integrating the hypercomplex scalar geometries of **NEMESIS Theory**. +For the theoretical proofs, operational logs, and equations governing the `spec/nemesis/` ontology, refer to the core archive: +👉 **[NEMESIS-Theory Repository](https://github.com/markrnd87-cmd/NEMESIS-Theory)**# UOR Framework [![DOI](https://zenodo.org/badge/DOI/10.5281/zenodo.19068826.svg)](https://doi.org/10.5281/zenodo.19068826) +![a785ceaa-31fe-45d5-99a5-1a289fb160d2](https://github.com/user-attachments/assets/b9c37c35-87e9-4a05-b0b5-1808decd9f94) The Universal Object Reference (UOR) Framework is a Rust workspace implementing the [UOR Foundation](https://uor.foundation/) ontology — a mathematical framework for content-addressed, symmetric, multi-metric object spaces with algebraic structure based on Z/(2^n)Z. +Gemini_Generated_Image_7pymxc7pymxc7pym ## Ontology @@ -72,3 +77,4 @@ MIT — see [LICENSE](LICENSE). --- *This README is machine-generated by `uor-docs`. Do not edit by hand.* +"For 8-bit Biji-Braille character mapping and hypermatrix logic, see our Integration Guide." diff --git a/Technical Report_ Integration of Hypercomplex Geometries as UOR Structure Carrying Substrates.pdf b/Technical Report_ Integration of Hypercomplex Geometries as UOR Structure Carrying Substrates.pdf new file mode 100644 index 0000000..2d706a2 Binary files /dev/null and b/Technical Report_ Integration of Hypercomplex Geometries as UOR Structure Carrying Substrates.pdf differ diff --git a/lake-manifest.json b/lake-manifest.json index ae85904..b704ceb 100644 --- a/lake-manifest.json +++ b/lake-manifest.json @@ -1,5 +1,95 @@ {"version": "1.1.0", "packagesDir": ".lake/packages", - "packages": [], + "packages": + [{"url": "https://github.com/leanprover-community/mathlib4.git", + "type": "git", + "subDir": null, + "scope": "", + "rev": "a6276f4c6097675b1cf5ebd49b1146b735f38c02", + "name": "mathlib", + "manifestFile": "lake-manifest.json", + "inputRev": "v4.16.0", + "inherited": false, + "configFile": "lakefile.lean"}, + {"url": "https://github.com/leanprover-community/plausible", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "1622a8693b31523c8f82db48e01b14c74bc1f155", + "name": "plausible", + "manifestFile": "lake-manifest.json", + "inputRev": "v4.16.0-rc1", + "inherited": true, + "configFile": "lakefile.toml"}, + {"url": "https://github.com/leanprover-community/LeanSearchClient", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "003ff459cdd85de551f4dcf95cdfeefe10f20531", + "name": "LeanSearchClient", + "manifestFile": "lake-manifest.json", + "inputRev": "main", + "inherited": true, + "configFile": "lakefile.toml"}, + {"url": "https://github.com/leanprover-community/import-graph", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "1a6613663c3eb08c401ce0fd1a408412f2c2321e", + "name": "importGraph", + "manifestFile": "lake-manifest.json", + "inputRev": "main", + "inherited": true, + "configFile": "lakefile.toml"}, + {"url": "https://github.com/leanprover-community/ProofWidgets4", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "07f60e90998dfd6592688a14cd67bd4e384b77b2", + "name": "proofwidgets", + "manifestFile": "lake-manifest.json", + "inputRev": "v0.0.50", + "inherited": true, + "configFile": "lakefile.lean"}, + {"url": "https://github.com/leanprover-community/aesop", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "79402ad9ab4be9a2286701a9880697e2351e4955", + "name": "aesop", + "manifestFile": "lake-manifest.json", + "inputRev": "v4.16.0-rc1", + "inherited": true, + "configFile": "lakefile.toml"}, + {"url": "https://github.com/leanprover-community/quote4", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "f0c584bcb14c5adfb53079781eeea75b26ebbd32", + "name": "Qq", + "manifestFile": "lake-manifest.json", + "inputRev": "v4.15.0", + "inherited": true, + "configFile": "lakefile.toml"}, + {"url": "https://github.com/leanprover-community/batteries", + "type": "git", + "subDir": null, + "scope": "leanprover-community", + "rev": "01006c9e86bf9e397c026fef4190478dd1fd897e", + "name": "batteries", + "manifestFile": "lake-manifest.json", + "inputRev": "main", + "inherited": true, + "configFile": "lakefile.toml"}, + {"url": "https://github.com/leanprover/lean4-cli", + "type": "git", + "subDir": null, + "scope": "leanprover", + "rev": "0c8ea32a15a4f74143e4e1e107ba2c412adb90fd", + "name": "Cli", + "manifestFile": "lake-manifest.json", + "inputRev": "main", + "inherited": true, + "configFile": "lakefile.toml"}], "name": "uor", "lakeDir": ".lake"} diff --git a/lakefile.lean b/lakefile.lean index 01972fb..855110a 100644 --- a/lakefile.lean +++ b/lakefile.lean @@ -7,6 +7,9 @@ package uor where ] preferReleaseBuild := true +require mathlib from git + "https://github.com/leanprover-community/mathlib4.git"@"v4.16.0" + @[default_target] lean_lib UOR where srcDir := "lean4" diff --git a/lean-toolchain b/lean-toolchain index 2586f88..8b4f470 100644 --- a/lean-toolchain +++ b/lean-toolchain @@ -1 +1 @@ -leanprover/lean4:v4.16.0 +leanprover/lean4:v4.16.0 \ No newline at end of file diff --git a/lean4/UOR/Algebra/O256/Basic.lean b/lean4/UOR/Algebra/O256/Basic.lean new file mode 100644 index 0000000..ba50e44 --- /dev/null +++ b/lean4/UOR/Algebra/O256/Basic.lean @@ -0,0 +1,123 @@ +import Mathlib.Data.ZMod.Basic + +set_option linter.dupNamespace false +set_option linter.unnecessarySeqFocus false + +namespace UOR.Algebra.O256 + +abbrev Byte := ZMod 256 + +structure Oct256 where + a0 : Byte + a1 : Byte + a2 : Byte + a3 : Byte + a4 : Byte + a5 : Byte + a6 : Byte + a7 : Byte + deriving DecidableEq, Repr + +@[ext] theorem oct256_ext : ∀ {x y : Oct256}, + x.a0 = y.a0 → x.a1 = y.a1 → x.a2 = y.a2 → x.a3 = y.a3 → + x.a4 = y.a4 → x.a5 = y.a5 → x.a6 = y.a6 → x.a7 = y.a7 → x = y := by + intro x y h0 h1 h2 h3 h4 h5 h6 h7 + cases x <;> cases y <;> simp_all + +abbrev O256 := Oct256 + +def zero : O256 := ⟨0, 0, 0, 0, 0, 0, 0, 0⟩ +def one : O256 := ⟨1, 0, 0, 0, 0, 0, 0, 0⟩ + +def add : O256 → O256 → O256 +| ⟨a0, a1, a2, a3, a4, a5, a6, a7⟩, ⟨b0, b1, b2, b3, b4, b5, b6, b7⟩ => + ⟨a0 + b0, a1 + b1, a2 + b2, a3 + b3, a4 + b4, a5 + b5, a6 + b6, a7 + b7⟩ + +def neg : O256 → O256 +| ⟨a0, a1, a2, a3, a4, a5, a6, a7⟩ => + ⟨-a0, -a1, -a2, -a3, -a4, -a5, -a6, -a7⟩ + +def sub : O256 → O256 → O256 +| x, y => add x (neg y) + +def mul (x y : O256) : O256 := + match x, y with + | ⟨a0, a1, a2, a3, a4, a5, a6, a7⟩, ⟨b0, b1, b2, b3, b4, b5, b6, b7⟩ => + ⟨ + a0 * b0 - a1 * b1 - a2 * b2 - a3 * b3 - a4 * b4 - a5 * b5 - a6 * b6 - a7 * b7, + a0 * b1 + a1 * b0 + a2 * b3 - a3 * b2 + a4 * b5 - a5 * b4 - a6 * b7 + a7 * b6, + a0 * b2 - a1 * b3 + a2 * b0 + a3 * b1 + a4 * b6 + a5 * b7 - a6 * b4 - a7 * b5, + a0 * b3 + a1 * b2 - a2 * b1 + a3 * b0 + a4 * b7 - a5 * b6 + a6 * b5 - a7 * b4, + a0 * b4 - a1 * b5 - a2 * b6 - a3 * b7 + a4 * b0 + a5 * b1 + a6 * b2 + a7 * b3, + a0 * b5 + a1 * b4 - a2 * b7 + a3 * b6 - a4 * b1 + a5 * b0 - a6 * b3 + a7 * b2, + a0 * b6 + a1 * b7 + a2 * b4 - a3 * b5 - a4 * b2 + a5 * b3 + a6 * b0 - a7 * b1, + a0 * b7 - a1 * b6 + a2 * b5 + a3 * b4 - a4 * b3 - a5 * b2 + a6 * b1 + a7 * b0 + ⟩ + +instance : Zero O256 := ⟨zero⟩ +instance : One O256 := ⟨one⟩ +instance : Add O256 := ⟨add⟩ +instance : Neg O256 := ⟨neg⟩ +instance : Sub O256 := ⟨sub⟩ +instance : Mul O256 := ⟨mul⟩ + +def conj : O256 → O256 +| ⟨a0, a1, a2, a3, a4, a5, a6, a7⟩ => + ⟨a0, -a1, -a2, -a3, -a4, -a5, -a6, -a7⟩ + +def norm : O256 → Byte +| ⟨a0, a1, a2, a3, a4, a5, a6, a7⟩ => + a0 * a0 + a1 * a1 + a2 * a2 + a3 * a3 + a4 * a4 + a5 * a5 + a6 * a6 + a7 * a7 + +def scalar : Byte → O256 +| a => ⟨a, 0, 0, 0, 0, 0, 0, 0⟩ + +theorem mul_conj_is_scalar (x : O256) : x * conj x = scalar (norm x) := by + cases x with + | mk a0 a1 a2 a3 a4 a5 a6 a7 => + decide + +def e0 : O256 := one +def e1 : O256 := ⟨0, 1, 0, 0, 0, 0, 0, 0⟩ +def e2 : O256 := ⟨0, 0, 1, 0, 0, 0, 0, 0⟩ +def e3 : O256 := ⟨0, 0, 0, 1, 0, 0, 0, 0⟩ +def e4 : O256 := ⟨0, 0, 0, 0, 1, 0, 0, 0⟩ +def e5 : O256 := ⟨0, 0, 0, 0, 0, 1, 0, 0⟩ +def e6 : O256 := ⟨0, 0, 0, 0, 0, 0, 1, 0⟩ +def e7 : O256 := ⟨0, 0, 0, 0, 0, 0, 0, 1⟩ + +@[simp] theorem conj_e0 : conj e0 = e0 := by rfl +@[simp] theorem conj_e1 : conj e1 = -e1 := by rfl +@[simp] theorem conj_e2 : conj e2 = -e2 := by rfl +@[simp] theorem conj_e3 : conj e3 = -e3 := by rfl +@[simp] theorem conj_e4 : conj e4 = -e4 := by rfl +@[simp] theorem conj_e5 : conj e5 = -e5 := by rfl +@[simp] theorem conj_e6 : conj e6 = -e6 := by rfl +@[simp] theorem conj_e7 : conj e7 = -e7 := by rfl + +@[simp] theorem conj_conj (x : O256) : conj (conj x) = x := by + cases x <;> simp [conj] + +@[simp] theorem mul_e1_e1 : e1 * e1 = -e0 := by rfl +@[simp] theorem mul_e2_e2 : e2 * e2 = -e0 := by rfl +@[simp] theorem mul_e3_e3 : e3 * e3 = -e0 := by rfl +@[simp] theorem mul_e4_e4 : e4 * e4 = -e0 := by rfl +@[simp] theorem mul_e5_e5 : e5 * e5 = -e0 := by rfl +@[simp] theorem mul_e6_e6 : e6 * e6 = -e0 := by rfl +@[simp] theorem mul_e7_e7 : e7 * e7 = -e0 := by rfl + +@[simp] theorem mul_e1_e2 : e1 * e2 = e3 := by rfl +@[simp] theorem mul_e2_e3 : e2 * e3 = e1 := by rfl +@[simp] theorem mul_e3_e1 : e3 * e1 = e2 := by rfl + +@[simp] theorem mul_e2_e1 : e2 * e1 = -e3 := by rfl +@[simp] theorem mul_e3_e2 : e3 * e2 = -e1 := by rfl +@[simp] theorem mul_e1_e3 : e1 * e3 = -e2 := by rfl + +example : (e1 * e2) * e4 = e7 := by decide +example : e1 * (e2 * e4) = -e7 := by decide + +theorem not_assoc_example : (e1 * e2) * e4 ≠ e1 * (e2 * e4) := by + decide + +end UOR.Algebra.O256 diff --git a/spec/Nemesis b/spec/Nemesis new file mode 100644 index 0000000..a8caea1 --- /dev/null +++ b/spec/Nemesis @@ -0,0 +1,16 @@ +{ + "@context": { + "uor": "https://uor.foundation/core/", + "nemesis": "https://uor.foundation/nemesis/" + }, + "@id": "nemesis:OcteractCypher", + "@type": "uor:TopologicalSpace", + "uor:name": "Octeract Cypher 256-State Matrix", + "uor:description": "A high-dimensional scalar geometry bridging the 8-bit feature vector to the Cayley-Dickson 256D resonance.", + "uor:algebraicBase": "Z/(256)Z", + "nemesis:properties": { + "dimensions": 256, + "associativity": "Non-associative", + "torsionCapacitanceActive": true + } +} diff --git a/tmpcheck.lean b/tmpcheck.lean new file mode 100644 index 0000000..cc8103d --- /dev/null +++ b/tmpcheck.lean @@ -0,0 +1,6 @@ +import UOR.Algebra.O256.Basic +open UOR.Algebra.O256 +#eval (e2 * e2 : O256) +#eval (e1 * e2 : O256) +#eval ((e1 * e2) * e4 : O256) +#eval (e1 * (e2 * e4) : O256)