Professor’s Cube
PROFESSOR'S CUBE
Shared-Carrier S3 Symmetry Bridge
Exhibit Dossier - Geometric Computation Borough
Finite representation theory | C5^3 permutation-Laplacian mechanics | Python certificates | finite-dimensional quantum simulation
Curatorial thesis: the Professor's Cube is a concrete coordinate realization of the common 125-state tensor carrier. It is not the E47 kernel. The exact bridge is the residual S3 factor-permutation symmetry shared by the SU(2) Casimir and the C5^3 product Laplacian.
Carrier: V = C^5 tensor C^5 tensor C^5, dim V = 125
E47 branch: C -> K=(C-6I)(C-30I) -> P47, rank 47
Cube branch: L = L_C5 (+) L_C5 (+) L_C5 -> P0, rank 1
Exact bridge: [C,U_sigma]=[L,U_sigma]=[P47,U_sigma]=0 for sigma in S3
Canonical descendant: Pcan = P6 Psym, rank 5
Evidence classes used here: E0 exact finite algebra; E1 executable reconstruction; E2 numerical unitary simulation.
1. Parsimonious-complete exhibit image
One cube, two operators, one exact symmetry bridge, one canonical rank-5 descendant, one existing Python certificate, one new S3/quantum certificate, and one sealed boundary.
Figure 1. Professor's Cube - Shared-Carrier S3 Symmetry Bridge proof plate.
2. Archive wall: work previously filed under Rubik / Base-5
These plates are retained as lineage artifacts and re-routed into the Professor's Cube exhibit. Their visual history is preserved; their mathematical claims are read through the corrected C5^3 boundary.
Figure 2. Archive wall assembled from current-conversation Professor's Cube / Rubik / Base-5 source images.
3. Existing executed Professor's Cube Python
Canonical executable recovered from Google Drive: rubik_unification_exact_proof.py. It constructs the 125-site carrier, all fifteen quarter-turn layer permutations, the C5^3 Laplacian, the eigenmode-impact benchmark, the corrected energy identity, low-band retention, and convergence to the constant-mode projector.
Carrier dimension: 125; Laplacian kernel dimension: 1; spectral gap 1.381966011250; lambda_max 10.854101966250.
15/15 moves are orthogonal 125 x 125 permutation matrices.
15/15 moves have order four and preserve the uniform mode.
All 15 moves have operator defect ||P^T L P - L||_2 = 2 sqrt(2) and Frobenius defect 8 sqrt(3) = 13.856406460551.
The corrected continuity theorem uses E = (1/2) rho^T L rho for dE/dt = -||L rho||^2.
The C5^3 contraction limit is rank one. The E47 ratio 47/125 is not derived by this Laplacian flow.
Recovered script SHA-256: 0011598e5d6d43385886fcbdce77a41eaa800afe8efd8e40368aeb57ae477aea
Recovered benchmark table
Move
Mode01 dE/E %
Mode01 overlap
Mode05 dE/E %
Mode05 overlap
R
53.1
0.817
20.7
0.676
L
53.1
0.817
12.9
0.850
R2
53.1
0.817
9.4
0.929
M
53.1
0.817
19.9
0.693
L2
53.1
0.817
14.2
0.822
U
58.8
0.797
18.8
0.816
D
58.8
0.797
19.5
0.801
U2
58.8
0.797
18.4
0.825
E
58.8
0.797
19.7
0.796
D2
58.8
0.797
18.0
0.834
F
9.9
0.966
25.9
0.433
B
9.9
0.966
2.6
0.953
F2
9.9
0.966
14.1
0.696
S
9.9
0.966
9.9
0.790
B2
9.9
0.966
28.5
0.375
4. New S3 symmetry bridge and canonical projector certificate
A new independent NumPy/SciPy validator constructs the spin-2 SU(2) generators, total Casimir, E47 projector, C5^3 Laplacian, all six tensor-factor permutations of S3, the S3 central idempotents, the canonical rank-5 projector, and a finite-dimensional unitary simulation.
Quantity
Result
Carrier
125
rank(P47)
47
rank ker(L)
1
max ||[C,U_sigma]||_F
9.730e-15
max ||[L,U_sigma]||_F
0
carrier S3 ranks
35 / 10 / 80
E47 S3 ranks
5 / 0 / 42
rank(Pcan)
5
||Pcan^2-Pcan||_F
9.760e-14
||K Pcan||_F
4.088e-11
max S3-fixed residual
2.290e-14
certificate
PASS 20/20
Validator SHA-256: 588a0ba2c095da3b3c83bfb1787522006cbe903bc5858947fc578d55d6b62cd2
Certificate file: professors_cube_s3_bridge_quantum_certificate.json
Representation-theoretic closure
Full carrier: 125 = 35 symmetric + 10 antisymmetric + 80 standard S3 isotypic dimensions.
E47 intersection: 47 = 5 symmetric + 0 antisymmetric + 42 standard dimensions.
The j=2 multiplicity space has S3 content 1 + 2 standard; the trivial line selects the canonical copy of D2.
The j=5 multiplicity space is the standard S3 representation and has no trivial line.
5. Finite-dimensional quantum simulation
The simulation is deliberately representation-theoretic. It does not claim a physical Professor's Cube quantum device. A dimensionless S3-invariant Hamiltonian is used to test sector conservation under unitary evolution:
Hq = C/42 + 0.173 Psym - 0.071 Panti
Diagnostic
Residual / result
Hamiltonian hermiticity
0
[H,P47]
3.525e-14
[H,Pcan]
2.323e-14
norm drift
4.441e-16
P47 occupancy drift
1.166e-15
Pcan occupancy drift
7.078e-16
canonical-state leakage
4.163e-14
canonical-state S3 displacement
8.778e-15
standard E47 S3 displacement
1.732051
standard E47 leakage
8.931e-15
Simulation conclusion: a canonical Pcan state remains in the rank-5 sector and is fixed by S3 to machine precision; a generic standard-isotypic E47 state transforms nontrivially under S3 but remains in E47.
Runtime status: NumPy/SciPy execution is complete. A historical QuTiP runtime route is not counted as executed here; where older archive material names QuTiP, that execution remains pending unless separately reproduced in a pinned QuTiP environment.
6. Canonical source and simulation inventoryRole
Exact Professor's Cube proof
Google Doc
1oRhmSI3XwjgAnFI74dfjVRJ8QRuhEnaAWiB52Rogo9c
Exact 15-move C5^3 proof and corrections
Original executable
Python
117Bfy7PfMOSMCZFPeAEXgHsdgsmAeHC3
rubik_unification_exact_proof.py
Original certificate
Drive file
1bBuxEB3bcTCZc9FGyZosm_mWfIVkxirO
Machine certificate
Canonical monograph
Google Doc
1RAlm80wiJh_YnzfJmFFgTWloUaqib8fC0Yc7AeN1oKw
RUB5-C01 through RUB5-C12
Publication PDF
1q4HEqBIPyGfJJDD1rk9Ai9rgfTpl6ysx
Canonical 2026-07-30 release
Deterministic proof plate
PNG
1ryU2TcWxGf8ya8asE2P773qZHom-8cjB
Prior Rubik/Professor's Cube proof plate
Carrier-equivalence dossier
Google Doc
1jlwNRb3XPI5bLeq2GXsBRyjxJ1KMdS3BZXG_QyAdGPA
Professor's Cube / E47 boundary map
Cross-plate validator registry
Supabase
PY-E47-PLATE-SPINE
Canonical Python corpus entry
New S3 + quantum validator
Python
588a0ba2c095da3b...
PASS 20/20; local executable prepared for archival
Archive simulation lineage
Earlier Base-5 Plate 5 materials describe eigenmode volumes, face-net images, operator matrices, and a folding/contraction animation route. Those are retained as historical simulation artifacts; claims of convergence to 47/125 are superseded by the corrected rank-one C5^3 Laplacian boundary unless 47/125 is explicitly imported as an E47 scalar.
The current machine-validated bridge is not a folding animation. It is the exact S3 commuting symmetry plus a new unitary sector-preservation simulation.
Supabase already carries a canonical E47/Base-5/Rubik cross-plate validator registry entry whose evidence boundary requires the K^2 -> P47 and L -> P0 contraction targets to remain distinct.
7. Exhibit routing and sealed evidence boundary
SEALED: ker(L_C5^3) has dimension 1, while E47 has dimension 47. The six S3 factor permutations are not the fifteen Professor's Cube layer turns. Shared carrier and shared factor symmetry are exact; physical identity, quantum hardware realization, and cross-domain interpretation require separate models and evidence.
Recommended Green District exhibit spine
Entrance: one 5 x 5 x 5 carrier with address map pi(x,y,z)=25x+5y+z.
Left gallery: E47 Casimir -> K -> P47, rank 47.
Right gallery: C5^3 Laplacian -> P0, rank 1, with the fifteen layer-turn audits.
Bridge hall: the six S3 factor permutations commuting with both C and L.
Inner chamber: Pcan = P6 Psym, rank 5, with 125 -> 47 -> 5 symmetry-resolved anatomy.
Machine room: original Professor's Cube Python plus the new PASS 20/20 S3/quantum certificate.
Archive wall: prior Rubik/Base-5 imagery, preserved as lineage rather than silently rewritten.
Boundary plaque: exact finite mathematics separated from physical or universal claims.
Contact survey
Google Contacts keyword searches for CERN, physics, and quantum returned no matching saved contacts. No outreach was initiated.
Platform role
Notion: exhibit, routes, evidence boundaries, visual archive, and source links.
Google Drive / Documents: canonical dossier, authored proof lineage, images, Python provenance, and certificates.
Supabase: structured artifact registry, machine certificate, quantum-simulation formalism, murmuration sweep, transit events, and agent-run audit.
Agent network: existing agents remain bounded by the human-gate policy; the exhibit is routed as a coordinated research object rather than a permission to change unrelated canonical claims.
Prepared 12 August 2026 - Mathematical City / Green Geometric Computation Borough
