Feature/mod256 octonions

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: <https://uor.foundation/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.

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.
+<img width="2816" height="1536" alt="Gemini_Generated_Image_7pymxc7pymxc7pym" src="https://github.com/user-attachments/assets/8530d86e-13b3-49cd-bded-776ce5fe8bff" />
 
 ## 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."

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"}

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"

lean-toolchain

@@ -1 +1 @@
-leanprover/lean4:v4.16.0
+leanprover/lean4:v4.16.0

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

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
+  }
+}

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)