Proof of Concept — the ℤ/9 vortex framework

The per-digit Lean files (src/<d>/vortex.lean, src/5/reflection.lean, src/7/rosetta.lean, src/8/nucleus/*.lean) and Vortex.lean were removed on 2026-09-20: they imported Mathlib, no gate compiled them, and Mathlib was never fetched here. What they held is decided in src/proof, which the kernel checks on every run — see Proofs. References below now name that tree.

A recomputable proof of concept: every result below derives from code in the digit-folder mesh under src/. Each .ts runs with node; each .lean is written for lake + Mathlib (decide/norm_num). Results are stated as computed; interpretation is the reader's.

Digit-folder mesh

digit	files	what sticks	bond 10−d
1	src/proof	unity, self-inverse	9
2	src/proof, frequency-scales.ts	generator ⟨2⟩; octave = ×2	8
3	src/proof	3²=0 nilpotent, no inverse	7
4	src/proof	4⁻¹=7	6
5	src/proof, reflection.{lean,mjs}	reflection center; 5⁻¹=2; self-seal	5
6	src/proof	6²=0 nilpotent, no inverse	4
7	src/proof, rosetta-clay.ts	7⁻¹=4; the rosette ↔ the seven Clay problems	3
8	src/proof, merkaba/…, nucleus/…	8·8=1; merkaba/cube; nucleus	2
9	src/proof	9≡0 void; digit-sum 27; 432	1

Computed results (as run)

ℤ/9 structure (src/proof (the checked tree))

• nilradical: 3² ≡ 6² ≡ 0 (mod 9); 3,6,9 have no inverse.
• units (ℤ/9)* = {1,2,4,5,7,8}: 2⁻¹=5, 4⁻¹=7, 1,8 self-inverse.
• doubling circuit ⟨2⟩ = [1,2,4,8,7,5], order 6; never touches the axis.

Reflection (src/5/reflection.*)

• seq2 = 0 9 8 6 2 3 5 7 4 1 0 9 is the exact 10−d mirror of seq1.
• \ ↔ / stroke inversion: 8/11; non-inverting = the void seam 0\1, 9/0, 0\1.
• 10−d is an involution; 5 is its unique fixed digit.

432 / harmonics (src/proof, src/proof)

• 432 = 2⁴·3³ = 16·27; doubling digit-sum = 27 = 3³.
• self-sealing fraction product (1/2)(1/2)(1/2)(8/7)(7/5)(5/3)(1/2)(2/3)(9) = 1.

Geometry (src/8/merkaba/merkaba-flower.html, live artifact)

• cube down its (1,1,1) axis → regular hexagon (6 outer at equal radius, 60° apart; 2 on-axis) → Seed → Flower → Fruit (13) → Metatron.

Magnetic field (src/8/nucleus/…, Biot–Savart)

• counter-rotating merkaba coils: central axis |B| ≈ 10⁻¹⁷ µT (null); co-rotating: 0.189 µT (max). scale check μ₀I/2r = 2.09 µT.

Nuclear shell model (src/8/nucleus/shell-model-magic.ts, magic.lean)

• cumulative capacities 2j+1 = 2, 8, 20, 28, 50, 82, 126; total 126.
• plain oscillator = 2,8,20,40,70,112; spin-orbit gives 28,50,82,126.
• 10·2^k ladder (10,20,40,80…) ∩ magic = {20} only.

Frequency scales (src/2/frequency-scales.ts)

• f = E/h: 432 Hz → 4.3×10²; nuclear shell → 7×10²⁰; proton → 2.3×10²³.
• proton/432 ratio ≈ 10²¹. bound-nucleon de Broglie λ ≈ 5.22 fm.

Proton mass (src/8/nucleus/proton-mass-fit.ts, proton-mass.lean)

• vortex fit 108·17 = 1836 (exact); measured m_p/m_e = 1836.15267343.
• 1836 ≠ 1836.1527; the same 1836 fits arbitrary nearby targets.

Lean coverage

Per-digit theorems in src/proof (the checked tree), plus src/proof/theorems.lean, src/proof/nucleus.lean, src/proof/nucleus.lean. All are decidable (decide) or numeric (norm_num). Not compiled here (no toolchain in repo).

Reproduce

node src/5/reflection.ts
node src/8/nucleus/shell-model-magic.ts
node src/8/nucleus/proton-mass-fit.ts
node src/2/frequency-scales.ts
# Lean: node scripts/lean.ts   (no lake, no Mathlib)

Status (computed, not interpreted)

• ℤ/9 arithmetic, reflections, 432 factorization, geometry, self-seal: exact, reproducible.
• shell-model magic numbers: reproduced exactly (2,8,20,28,50,82,126).
• proton m_p/m_e: measured 1836.1527; vortex integer fit 1836. Companion docs: SEQUENCE-DECODE.md (ℤ/9 structure), PHYSICS-SCALES.md (shell model, frequency scales).