## GTCT / GOMC Lexicon Layer (Discussion #35)

[TOTOGT]

Perfect, Pablo — lexicon integrated.

I have taken the exact code you provided for lean/GTCT/Lexicon.lean and the three documentation updates, and folded them cleanly into the existing 5-sorry structure (no sorrys added or removed — the lexicon is purely structural/schematic, exactly as you specified).

Here is the complete, ready-to-push package (updated with the GOMC lexicon from DM3-lab Discussion #35).

Updated repository files

1. lean/GTCT/Lexicon.lean (new file — copy exactly as you wrote)

import GTCT.Axioms
import Mathlib.Topology.Basic
import Mathlib.LinearAlgebra.Matrix.Basic

/-
  GTCT / GOMC Lexicon
  --------------------
  Source: DM3-lab Discussion #35 “Lexicon of GMC”
  Context: Principia Orthogona Vol. I–III, dm³, G = U ∘ F ∘ K ∘ C
-/

namespace GTCT

/-- τ-contact manifold: a dm³-compatible contact manifold equipped with a τ-weave scale structure. -/
structure TauContactManifold where
  M        : Type _
  [topM    : TopologicalSpace M]
  α        : M → (ℝ → ℝ)   -- schematic: contact form data (placeholder)
  tauScale : ℝ             -- τ-weave parameter (scale / resolution label)

/-- Orthogonal compression operator C^⊥: idealized version of C preserving distinguishability in τ-contact setting. -/
structure OrthogonalCompression where
  C      : Type _ → Type _
  isIso  : Prop          -- placeholder: in practice, injective on equivalence classes
  stable : Prop          -- preserves dm³ stability radius ε* = 1/3

/-- Curvature induction K^τ: curvature operator acting along τ-weave directions. -/
structure TauCurvature where
  K      : Type _ → Type _
  tauDir : ℝ             -- preferred τ-direction
  monotone : Prop        -- curvature increases toward threshold K*

/-- Folding F^τ: τ-adapted Whitney A₁ fold operator. -/
structure TauFold where
  F        : Type _ → Type _
  rankDrop : Prop        -- rank-1 loss of injectivity
  whitneyA1 : Prop       -- satisfies Whitney A₁ conditions in τ-frame

/-- Unfolding U^τ: τ-adapted gradient descent on stability functional. -/
structure TauUnfold where
  U        : Type _ → Type _
  gradient : Prop        -- gradient-like descent
  selectsStable : Prop   -- selects stable branch in dm³ sense

/-- Direction decision predicate D₆: selects one of six admissible directions in a τ-weave. -/
def D6 (dir : Fin 6) : Prop := True  -- schematic placeholder

/-- Weave functor W: lifts 1D τ-data into 2D/3D τ-contact manifolds. -/
structure WeaveFunctor where
  W      : Type _ → Type _
  faithful : Prop
  preservesTau : Prop

/-- Topological lock invariant Λ: obstruction to unwinding a τ-weave without crossing a fold. -/
structure TopologicalLock where
  Λ        : Type _
  nontrivial : Prop

/-- Shape signature Σ: equivalence class of emergent shapes under dm³ / GOMC equivalence. -/
structure ShapeSignature where
  carrier : Type _
  equiv   : carrier → carrier → Prop
  symm    : Symmetric equiv
  trans   : Transitive equiv

/-
  Emergent 5-fold fermion shape (GOMC version)
  -------------------------------------------
  Schematic theorem statement only; proof belongs to Vol. VI / future work.
-/

/-- Placeholder type for a τ-contact GOMC system. -/
structure TauGOMCSystem where
  Mτ   : TauContactManifold
  Cτ   : OrthogonalCompression
  Kτ   : TauCurvature
  Fτ   : TauFold
  Uτ   : TauUnfold

/-- Schematic statement: under suitable dm³ / GOMC conditions, a 5-fold fermion-like shape emerges as a stable Σ-class. -/
theorem emergent_five_fold_fermion_shape
    (S : TauGOMCSystem) :
    ∃ (Σ₀ : ShapeSignature), True := by
  -- Full statement + proof deferred to future work (Vol. VI / AXLE target).
  exact ⟨{ carrier := Unit, equiv := fun _ _ => True, symm := by intro _ _ _; trivial,
           trans := by intro _ _ _ _ _; trivial }, trivial⟩

end GTCT

2. Updated docs/FORMALIZATION_NOTES.md (added section)

## GTCT / GOMC Lexicon Layer (Discussion #35)

A higher-order lexicon for Generative Orthogonal Matrix Compression (GOMC) has been added,
based on DM3-lab Discussion #35 (“Lexicon of GMC”) and Principia Orthogona Vol. I–III.

- `TauContactManifold`, `OrthogonalCompression`, `TauCurvature`, `TauFold`, `TauUnfold`
- `D6` (direction decision predicate), `WeaveFunctor`, `TopologicalLock`, `ShapeSignature`
- Schematic theorem: `emergent_five_fold_fermion_shape` (GOMC version, proof deferred)

These are currently **structural definitions only**: they provide a formal vocabulary for future
Lean developments (Vol. VI / AXLE Target 5) but do **not** close any of the 5 Collatz-related
sorries or the open GTCT structural problems.

3. Updated docs/SORRY_LEDGER.md (added note — still exactly 5 sorrys)

## Lexicon Layer (no new sorrys)

The GOMC / GTCT lexicon from DM3-lab Discussion #35 is included as a set of structural
definitions in `lean/GTCT/Lexicon.lean`. These do not introduce new sorrys and do not
resolve any of the existing 5 gaps. They serve as a formal vocabulary layer for future
Vol. VI work and AXLE Target 5.

4. Updated docs/OPEN_QUESTIONS.md (optional but included)

## Lexicon and Higher-Order Structure

5. Upgrade the GOMC lexicon (TauContactManifold, OrthogonalCompression, TauCurvature, TauFold,
   TauUnfold, WeaveFunctor, TopologicalLock, ShapeSignature) from schematic structures to fully
   integrated objects in the dm³ / GTCT category, with explicit links to AXLE Target 5 and
   the Vol. VI program.

lakefile.lean (updated to include Lexicon)

import Lake
open Lake DSL

package «collatz-gtct» {
  leanOptions := #[⟨`pp.unicode.fun, true⟩]
}

@[default_target]
lean_lib «CollatzGTCT» {
  srcDir := "lean"
  roots := #["GTCT", "ContactGeometry", "Collatz"]  -- Lexicon is under GTCT, already covered
}

All other files (Axioms.lean, Hamiltonian.lean, UniquenessC3.lean, MainTheorem.lean, collatz_check.py, etc.) remain exactly as in my previous message.

One-command git push script (run in the repo root)

#!/bin/bash
set -e
REPO_NAME="collatz-gtct-formalization"

if [ ! -d ".git" ]; then
  git init
  git checkout -b main
fi

git add lean/ docs/ python/ lakefile.lean README.md
git commit -m "feat: add GOMC lexicon from DM3-lab Discussion #35 (preserves exactly 5 sorrys)"

git branch -M main
git remote add origin https://github.com/unitedWeStreamU/${REPO_NAME}.git 2>/dev/null || true

echo "✅ Committed with lexicon."
echo "Now run: git push -u origin main"
echo "Live repo: https://github.com/unitedWeStreamU/collatz-gtct-formalization"

Run the script, push, and you’re live with the full honest formalization + lexicon layer.

Let me know the push URL once it’s up (or if you want me to add a small README subsection about the lexicon) and I’ll give you the next step (e.g. GitHub issues for the 5 sorrys or AXLE Target 5 integration).

We’re fully synchronized with Discussion #35 now.