Grassmann–Casimir compatibility

# E47 Grassmann–Casimir Compatibility and Spectral Stabilization Theorem

**A first-principles constructive proof with exact Python verification**  
Nicholas Kouns’ E47 construction — validation dated 8 October 2026.

**Execution result: 34/34 exact checks passed (SymPy 1.14.0).** These checks validate the theorem stated below; claim statuses for the supplied narrative are recorded separately.

## Theorem

Let V = V₂ ⊗ V₂ ⊗ V₂ over ℂ, with the standard positive spin inner product, and let C = J²_tot and K₄₇ = (C − 6I)(C − 30I). Then:

1. dim V = 125 and ker K₄₇ = E₆ ⊕ E₃₀ has dimension 47.
2. There exist E ⊂ V, dim E = 90, and a rank-43 linear map A: V → ℂ⁴³ such that dim ker A = 82 and E ∩ ker A = ker K₄₇.
3. For arbitrary E of dimension 90 and A of rank 43, dim(E ∩ ker A) = 47 holds exactly when E + ker A = V, equivalently rank(A|E) = 43. Identifying this intersection with ker K₄₇ further requires ker K₄₇ ⊂ E ∩ ker A.
4. With P₄₇ the orthogonal projector onto ker K₄₇ and Γ = I − K₄₇²/99144,

   ‖Γⁿ − P₄₇‖₂ = (15/17)ⁿ, for every integer n ≥ 0,

   in the physical spin inner product. Thus Γⁿ → P₄₇ and Ωc = rank(P₄₇)/dim V = 47/125 = 0.376 exactly.

The 82-dimensional kernel belongs to A. The Casimir selector K₄₇ has a 47-dimensional kernel. These distinct operators make the two dimensional constructions compatible.

## Lemma 1 — Derivation of the Casimir census

The weights of V₂ are −2, −1, 0, 1, 2. The tensor character is

χV(t) = (t⁻² + t⁻¹ + 1 + t + t²)³.

For nonnegative weights m = 0,…,6, the weight multiplicities are

wₘ = (19, 18, 15, 10, 6, 3, 1).

Each spin-j irreducible contributes one vector to every weight from −j through j. Therefore its multiplicity is aⱼ = wⱼ − wⱼ₊₁, giving

V ≅ V₀ ⊕ 3V₁ ⊕ 5V₂ ⊕ 4V₃ ⊕ 3V₄ ⊕ 2V₅ ⊕ V₆.

Since C acts as j(j+1)I on Vⱼ:

| j | C eigenvalue | Irreducible copies | Eigenspace dimension |
|---|---:|---:|---:|
| 0 | 0 | 1 | 1 |
| 1 | 2 | 3 | 9 |
| 2 | 6 | 5 | 25 |
| 3 | 12 | 4 | 28 |
| 4 | 20 | 3 | 27 |
| 5 | 30 | 2 | 22 |
| 6 | 42 | 1 | 13 |

Consequently K₄₇ vanishes precisely on E₆ ⊕ E₃₀. Its nullity is 25 + 22 = 47; its rank is 78.

## Lemma 2 — Grassmann intersection and constraint independence

Set B = ker A. Rank-nullity gives dim B = 125 − 43 = 82. Grassmann’s identity gives

 dim(E ∩ B) = dim E + dim B − dim(E + B)
             = 172 − dim(E + B) ≥ 47.

Equality holds if and only if E + B = V. Equivalently, by applying rank-nullity to A|E,

 dim(E ∩ ker A) = 90 − rank(A|E).

Thus 43 independent ambient constraints must remain independent on E to obtain exactly 47. Dimension data alone permit intersection dimensions from 47 through 82. Equality of dimensions with E47 alone does not identify the subspaces; containment supplies that identification.

## Lemma 3 — Explicit simultaneous realization

Write V = S ⊕ U ⊕ W with S = ker K₄₇, dim U = 43, dim W = 35. Such a decomposition exists by splitting the 78-dimensional complement of S. Define

E = S ⊕ U,
A(s + u + w) = u,

identifying U with ℂ⁴³. Then

ker A = S ⊕ W,
dim E = 90,
dim ker A = 82,
E + ker A = V,
E ∩ ker A = S = E47.

The Python script constructs S from columns of the exact spectral projector, constructs its complement from columns of I − P₄₇, and splits that complement into dimensions 43 and 35. It checks the ranks of the resulting matrices. This is an existence construction; the supplied text does not specify its original E or constraint matrices.

## Lemma 4 — Exact projector and optimal spectral contraction

For c in the Casimir spectrum, define the Lagrange spectral projector

P_c = ∏_{d ≠ c} (C − dI)/(c − d).

Then P₄₇ = P₆ + P₃₀, P₄₇² = P₄₇, K₄₇P₄₇ = 0 and tr(P₄₇) = 47.

The nonzero eigenvalues of K₄₇² are

{11664, 12544, 19600, 32400, 186624}.

Let a = 11664 and b = 186624. The optimal constant positive step for I − εK₄₇² is obtained by balancing the endpoint errors:

1 − εa = −(1 − εb),
ε* = 2/(a + b) = 1/99144,
ρ* = (b − a)/(b + a) = 15/17.

Every complementary eigenvalue lies between the endpoints, and the maximum absolute value of the affine update occurs at an endpoint. The spectral theorem therefore gives

Γⁿ = P₄₇ + ∑_{c ∉ {6,30}} [1 − ((c−6)(c−30))²/99144]ⁿ P_c,
‖Γⁿ − P₄₇‖₂ = (15/17)ⁿ.

In particular, Γ fixes every vector in E47 and exponentially suppresses its orthogonal complement. This proves stability of the invariant subspace. It does not select a unique vector within that subspace: the limit depends on P₄₇x₀. Entropy decrease requires a separately specified entropy functional.

## Exact arithmetic and implementation

The script uses an unnormalized weight basis with

J_z e_m = m e_m,
J_+ e_m = (2−m)e_{m+1},
J_- e_m = (2+m)e_{m−1}.

All matrix entries and calculations are rational. The positive Gram matrix diag(1, 1/4, 1/6, 1/4, 1) makes J_+ adjoint to J_− and establishes equivalence to the usual orthonormal spin basis. On the tensor carrier, its triple tensor product defines the physical norm used in the theorem. Ordinary Euclidean norms on these unnormalized coordinates must not be substituted for that norm.

The script computes characteristic polynomials of the conserved-weight blocks, derives their roots exactly, independently derives the character census, and verifies the square-free annihilating polynomial and spectral projector identities. The finite checks instantiate the algebra above; the spectral argument establishes the all-n limit.

Run with Python 3 and SymPy:

```sh
python3 e47_grassmann_validation.py
```

The JSON result records theorem checks separately from the status of the supplied narrative’s claims.

## Claims requiring additional mathematical inputs

| Supplied claim | Result of this validation | Input needed to extend validation |
|---|---|---|
| dim ker((C−6I)(C−30I)) = 82 | Incompatible with this operator; exact nullity is 47 | Use a distinct rank-43 constraint map A for the 82-dimensional kernel |
| Grassmann intersection = 47 | Proved with transversality; explicit realization verified | Original E and A matrices to validate the particular historical construction |
| Ωc = 47/125 | Exactly derived from selected eigenspace dimensions | None for this finite-dimensional statement |
| Ωc is a root of the cited cubic closure polynomial | Polynomial absent from the supplied text | Cubic coefficients and their derivation |
| A cubic root establishes dynamical stability | Requires an evolution law and its stability analysis | Update map or differential equation; derivative/Jacobian at the root |
| Einstein-to-spectral closure | Not tested by these finite algebra checks | Explicit spacetime metric/action, geometric-to-spectral map, and claimed implication or equivalence with hypotheses |
| Bit-exact repository implementation | Repository code could not be retrieved during this run | Source revision and its arithmetic/rounding specification |
| FPGA/ASIC realization | Not tested | Hardware description, arithmetic model, and implementation evidence |

For a supplied cubic f, the exact next check is f(47/125) = 0; no polynomial is invented here. For a supplied geometric map, the next step is to compute the induced metric and Einstein tensor, then verify the stated field equations and the precise direction of the closure theorem. For a supplied hardware implementation, the next step is to compare its arithmetic semantics against the exact reference operations.

## Source handling

The supplied LinkedIn Grassmann post was retrievable and repeats the dimension claim 125, 90, 82, 47 and M = 43, but its retrieved text does not provide E, A, or the cubic polynomial:

https://www.linkedin.com/posts/advancedillness_e47-grassmannidentity-mathematicalphysics-activity-7496006606170562561-FCIV

The supplied repository and the linked Einstein/Invariant Grammar posts were not retrievable through the attempted web access. No repository inspection, hardware run, quantum-device run, or general-relativity equivalence verification is claimed. This proof is independently reconstructed from the spin-2 definitions in the user’s text.

PASS | spin commutators | 
PASS | single-spin Casimir | 
PASS | physical inner product | 
PASS | carrier dimension | 
PASS | total weight conserved | 
PASS | weight -6 polynomial splits | 
PASS | weight -5 polynomial splits | 
PASS | weight -4 polynomial splits | 
PASS | weight -3 polynomial splits | 
PASS | weight -2 polynomial splits | 
PASS | weight -1 polynomial splits | 
PASS | weight +0 polynomial splits | 
PASS | weight +1 polynomial splits | 
PASS | weight +2 polynomial splits | 
PASS | weight +3 polynomial splits | 
PASS | weight +4 polynomial splits | 
PASS | weight +5 polynomial splits | 
PASS | weight +6 polynomial splits | 
PASS | derived Casimir census | {0: 1, 2: 9, 6: 25, 12: 28, 20: 27, 30: 22, 42: 13}
PASS | weight-character decomposition | {0: 1, 1: 3, 2: 5, 3: 4, 4: 3, 5: 2, 6: 1}
PASS | square-free Casimir annihilator | 
PASS | Casimir selector nullity | nullity=47; rank=78; claimed 82 is incompatible with this operator
PASS | exact projector | 
PASS | orthogonal projector in physical metric | 
PASS | optimal positive-step contraction | K^2 spectrum off kernel=[11664, 12544, 19600, 32400, 186624]; epsilon=1/99144; rho=15/17
PASS | kernel fixed by propagator | 
PASS | complement strictly contracts | 
PASS | coherence ratio | 47/125
PASS | spectral complementary basis | 
PASS | constructed Grassmann dimensions | 
PASS | constructed intersection | 
PASS | constraint rank and restricted rank | 
PASS | constraint kernel dimension | 
PASS | Grassmann transversality arithmetic | 

34/34 exact checks passed.
{
  "Casimir kernel equals 82": "FALSE for K=(C-6I)(C-30I); computed nullity 47",
  "Constraint kernel equals 82": "TRUE for a distinct rank-43 map A: V -> C^43",
  "Grassmann intersection equals 47": "TRUE when E+ker(A)=V; explicitly realized here",
  "Original E and 43 constraints": "Not supplied as matrices; construction here is an existence witness",
  "Cubic closure": "Polynomial not supplied; root and dynamical stability untested",
  "Einstein-to-spectral closure": "Metric/action/map and equivalence theorem not supplied; untested",
  "Repository bit-exact implementation": "Repository source unavailable through attempted retrieval; untested",
  "FPGA/ASIC realization": "Hardware description and execution evidence not supplied; untested",
  "Entropy decrease": "No entropy functional specified; spectral error contraction proved"
}