no_nontrivial_dimensionless_monomial
plain-language theorem explainer
No nontrivial integer powers of $c$, $\hbar$, and $G$ yield a dimensionless quantity: the only solution of the $(L,T,M)$ balance is $a=b=d=0$. Anyone arguing that a pure-number theory cannot fix absolute SI values of $\hbar$ or $G$ without an external anchor cites this. The proof is a short integer linear-algebra check: unfold the exponent map and discharge the three Diophantine equations with omega.
Claim. Let $a,b,d\in\mathbb{Z}$. Write the length-time-mass exponent triple of the monomial $c^a\hbar^b G^d$ as $(a+2b+3d,\,-a-b-2d,\,b-d)$. If that triple equals $(0,0,0)$, then $a=0$, $b=0$, and $d=0$.
background
The module draws the honest line between first-principles native constants and SI calibration. Recognition Science can force dimensionless native identities such as $\hbar_{\mathrm{RS}}=\varphi^{-5}$ and $G_{\mathrm{RS}}\hbar_{\mathrm{RS}}=1/\pi$, but pure dimensionless data cannot output the absolute SI value of $\hbar$ or $G$.
The local tool is the exponent map sending integers $(a,b,d)$ to the $(L,T,M)$ dimensions of $c^a\hbar^b G^d$. By standard dimensional analysis, $c$ has dimensions $L T^{-1}$, $\hbar$ has $M L^2 T^{-1}$, and $G$ has $L^3 M^{-1} T^{-2}$, so the map is $(a+2b+3d,,-a-b-2d,,b-d)$. The three columns $(1,-1,0)$, $(2,-1,1)$, $(3,-2,-1)$ are the classical dimension vectors of $c$, $\hbar$, and $G$.
Upstream native definitions of $G$ and $\hbar$ supply the physical meaning of those symbols in RS units; CODATA copies are the SI targets the bridge later matches. This lemma is the negative half of the boundary: an anchor is required.
proof idea
Term-mode proof by direct linear algebra over $\mathbb{Z}$. Unfold the dimension map so the hypothesis becomes equality of three integer components. Split the product equality into the three equations
$a+2b+3d=0$, $-a-b-2d=0$, $b-d=0$.
The tactic omega solves this integer system and concludes $a=b=d=0$. No external lemmas beyond product injectivity and integer arithmetic are required.
why it matters
This is the kernel statement behind the dimensional boundary of Recognition Science. It is re-exported almost verbatim as dimensionless_theory_needs_anchor ("a pure-number theory can fix only dimensionless/native relations among $c$, $\hbar$, and $G$; an absolute SI value needs a dimensional anchor") and is packaged into the certificate native_dimensional_boundary_cert together with the nonzero determinant of the dimension matrix and the claim that the SI bridge is calibration, not prediction.
In the broader framework it underwrites why T0–T8 plus native identities still need exactly one dimensional anchor (e.g. $\tau_0$ in seconds) before SI masses and couplings appear, matching the positive half already proved in SIBridgeClosure and the single-anchor calibration path. It is ordinary dimensional analysis, stated so the formal development cannot quietly smuggle an absolute SI value out of pure numbers.
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