E47 Research Note · Updated 2026-09-29

E47 in Two Pages

K², V₂, recursive invariant selection, induced operator algebra, and the fixed-point quantum channel.

Notion record ↗Supabase mirror ↗Note source ↗Runtime source ↗Algebra/channel validator ↗

Page 1 · What the object is

125 → K → 47

Why V₂ has dimension 5

The starting object is the spin-2 irreducible representation of SU(2), denoted V₂. A spin-j irreducible representation has dimension 2j+1. For j=2, dim V₂=5.

V = V₂⊗³ dim V = 5³ = 125

The Casimir selector

Let C=J_tot². Its eigenvalues on this carrier are {0,2,6,12,20,30,42}, with state multiplicities {1,9,25,28,27,22,13}. Define K=(C−6I)(C−30I).

E₄₇ = ker K = E₆ ⊕ E₃₀ dim E₄₇ = 47

What K² is and why it matters

K²=K†K is positive semidefinite and preserves the same kernel.

Γε = I − εK² ε* = 1/99144 ρ* = 15/17

Ωc = 47/125 = 0.376

Page 2 · Recursive execution

Invariant selection becomes an executable cycle

Σ → Ψ → Γⁿ → Ω → Λ → Σ′ → α → Σ

RESIDUAL-GATED RUNTIME.

Ω is evaluated on the contracted pre-projection state. Only after ||KM_pre||≤τ is satisfied is Λ applied. Reversing those operations would make KΛM≈0 by construction and render the gate tautological.

K=(C−6I)(C−30I)
Γ=I−K²/99144
Λ=P₆+P₃₀
Ω: ||KM_pre|| ≤ 10⁻¹⁰

certificate PASS
rank Λ = 47
ρ⊥ = 15/17
252 contractions in the seeded smoke test
pre-projection residual = 9.353585941533422e−11
post-projection residual = 4.833244591885603e−12

SHA-256: 8ec779cbb5247f9dc733cd2f4e341c8378d5ec6acfc21d56d9d3da33b322e716

INDUCED ALGEBRA + CHANNEL INVARIANTS.

Φ(A)=P₄₇AP₄₇ Φ²=Φ
P₄₇M₁₂₅(ℂ)P₄₇ ≅ M₄₇(ℂ)
dim_ℂ = 47² = 2209

For I=V_c† and M=V_c, MI=P₄₇ and P₄₇Γⁿ=P₄₇, so MIP₄₇Γⁿ=P₄₇ for every n≥0. Γⁿ alone converges asymptotically; the explicit project/reconstruct recursion stabilizes after one cycle.

spec(Γ)|_{C={0,2,6,12,20,30,42}}
{103/153, 10825/12393, 1, 15/17, 9943/12393, 1, −15/17}
ρ(Γ|E₄₇⊥)=15/17

E₆: 5 symmetric + 20 mixed + 0 sign
E₃₀: 22 mixed + 0 symmetric + 0 sign
E₄₇ total: 5 symmetric + 42 mixed + 0 sign
End_SU(2)(E₄₇) ≅ M₅(ℂ) ⊕ M₂(ℂ), dimension 29
End_SU(2)×S₃(E₄₇) ≅ ℂ ⊕ M₂(ℂ) ⊕ ℂ, dimension 6

𝓔(X)=P₄₇XP₄₇ + Tr((I−P₄₇)X)P₄₇/47
𝓔²=𝓔
Fix(𝓔)=P₄₇M₁₂₅(ℂ)P₄₇ ≅ M₄₇(ℂ)

Open the validator source ↗

Claim boundary

47 is not asserted to be the universal dimensionality of nature.
47/125 is the exact rank fraction of this construction, not a universal measured constant.
The finite spectral theorem does not by itself establish downstream physical domains.
Downstream E47 applications require their own explicit maps and evidence.

CORE EXACT CLAIM. V₂⊗³ is a 125-dimensional carrier. K=(C−6I)(C−30I) has a 47-dimensional kernel. K² generates a stable contraction whose powers converge to P₄₇ under the stated step-size bound.

EXECUTABLE INTERPRETATION. Preserve what satisfies the kernel condition, contract what does not, evaluate certainty on the unprojected residual, then project and pass the invariant forward.