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HOMOTOPY SUPER SMOOTH
DIFFERENTIAL QUANTUM
TYPE SYSTEM

HOMOTOPY

  1. UNIVERSE
  2. FORALL
  3. SIGMA
  4. EXISTS
  5. REAL
  6. PATH
  7. GLUE
  8. E-M

SUP-SMTH

  1. GRADED
  2. FLAT
  3. SHARP
  4. BOSE
  5. FERMI
  6. TENSOR
  7. SMTH
  8. SUPSMTH

DIFF-K

  1. SPECTRA
  2. GRPD
  3. FORM
  4. KU
  5. DIFF
  6. DIFFKU

QUANTUM

  1. CONFn
  2. BRAID
  3. QUBIT
  4. LINEAR

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Urs operates as an OCaml framework, manipulating pure ASTs, similar to a Lisp-like environment, pending a dedicated syntax.

$ git clone [email protected]:groupoid/urs $ ocamlfind ocamlc -o urs -package z3 -linkpkg src/urs.ml

SYNTAX

SEMANTICS


Homotopy

A foundational framework where types are ∞-groupoids, and propositions emerge as homotopical structures. Equality is a space of paths, extensible to higher dimensions, weaving homotopy theory into type-theoretic reasoning. With constructs like , , , , , , , and , it blends topological invariance with constructive logic. This system enables synthetic reasoning about continuity, deformation, and equivalence, treating mathematical spaces as dynamic objects rich with paths, homotopies, and higher coherences—a language for capturing the shape of mathematical truth.

Graded Cohesion

A cohesive, graded extension of type theory infused with supergeometry, where types inhabit universes indexed by parity, symmetry, and smoothness. Using , , , , , , , , and , it equips types with tensor structures, group actions, and super-modality operators to model differential cohesion, infinitesimal geometry, and graded symmetries. This system unifies the logic of supermanifolds, derived geometry, and higher smooth stacks, enabling type-theoretic reasoning about spaces enriched by quantum and differential symmetry.

Differential K-Theory

A homotopy-theoretic framework augmented with differential structure, where types encode spectra, groupoids, and refined cohomological data. Leveraging , , , , , , and , it internalizes differential K-theory to provide a synthetic language for characteristic classes, index theory, and topological invariants. By integrating stable homotopy, smooth geometry, and categorical semantics, it articulates the interplay of curvature, bundles, and higher cohomological operations within a unified type system of paths and forms.

Qunatum Linearity

A type-theoretic foundation for quantum systems, internalizing configuration spaces, braids, and higher categorical symmetries within a resource-sensitive linear logic. Built on , , , and , it captures quantum statistics via braided monoidal categories, quantum group actions, and path spaces of indistinguishable particles. This system supports quantum computation and field theory semantics, using linear types to model entanglement, interference, and non-classical information flow, ensuring quantum processes respect spatial and algebraic phase coherence.

Bibliography


TED-K