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Is Invariant Grammar a new Chomsky class?
Not yet. The displayed BNF is context-free in production form, while operator typing and semantic admissibility constitute an additional calculus. A Type-0 classification requires an explicit string-rewriting encoding and equivalence proof. The complete answer, including lexical attractors and image-derived formalism, is in the Invariant Grammar monograph.
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Does $E_{47}$ prove physical Skyrmions or propulsion?
No. The exact kernel supplies a two-sector algebra and supports a conditional target-orbit construction. Skyrmions records the missing orbit selection, coupling normalization, anisotropic PDE, physical scale, experimental realization, and propulsion evidence. Magnetic Skyrmion experiments validate their own materials systems, not the RI identity or zero-inertia extensions.
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Where are the code and numerical results?
The complete answer is now maintained on Python and Numerical Validation. In brief: the canonical finite-dimensional spectral-kernel chain is machine-verified; QuTiP, SciPy, topology, swarm, protein, GEM, propulsion, fluid, and constants programs validate only their explicitly coded models unless an independent experiment or theorem supplies the domain bridge.
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Frequently Asked Questions
Concise answers governed by the current proof-status grammar.
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Let $V_2$ be the five-dimensional spin-2 irreducible representation of $SU(2)$. Define
$V=V_2^{\otimes3},\qquad \dim V=125,$$ $K=(C-6I)(C-30I),$$
where $C$ is the total quadratic Casimir. Then
$`ker K=W_2oplus W_5cong5V_2oplus2V_5,$$
so
$\dim\ker K=25+22=47.$ Therefore $\Omega_c=47/125=0.376.$ This finite-dimensional result is theorem-level.
There is no continuous fitted parameter after the discrete choice to select the $j=2$ and $j=5$ sectors. The choice of those sectors is part of the model definition and must be stated explicitly.
Not from the algebra alone. It is an exact dimension ratio in the canonical representation. A physical application must prove that its state space and selector map to this construction.
The orthogonal projector onto the protected sector is $P_{47}(C)=\frac{C^6-107C^5+4088C^4-66940C^3+430848C^2-624960C}{1814400}.$ It satisfies $P_{47}^2=P_{47}=P_{47}^*$ and has rank $47$.
Because $K^2\ge0$, $e^{-tK^2}\to P_{47}$ as $t\to\infty$. All transverse modes decay while the kernel is preserved. The smallest nonzero eigenvalue of $K^2$ is $11664.$
No. The reference package proves exact software identities for an abstract $47\oplus78$ projector model:
$\operatorname{Tr}P=47,\quad P^2=P,\quad P^\dagger=P,\quad QP=0,$
and a normalized contraction that preserves the selected sector while attenuating the complement.
Its current ambient representation is a synthetic direct sum, not the canonical $V_2^{\otimes3}$ carrier, and its object named K is the complement projector $Q$ rather than $(C-6I)(C-30I)$. It computes no force, momentum transfer, spacetime curvature, GEM field strength, or thrust. The software harness is therefore structural/computational; propulsion remains a conditional physical program.