TFPT scientific source excerpt
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000248 | \section*{Verified Selector --- \texttt{DIMENSION.SELECTOR.4D.01} (why $d=4$, conditionally)}
000249 | \addcontentsline{toc}{section}{Verified Selector: DIMENSION.SELECTOR.4D.01}
000250 | % =====================================================================
000251 | \noindent\emph{Typing.} A \emph{conditional} identity \mC\ (ledger row
000252 | \texttt{DIMENSION.SELECTOR.4D.01}), machine-checked in
000253 | \veri{v975\_dimension\_selector\_4d.py} ($16/16$, sympy-exact).  The $\mu_4$ structure of the
000254 | compiler must \emph{not} be read as a proof of four spacetime dimensions
000255 | (\texttt{DIMENSION.UPLIFT.FIREWALL.01}); what \emph{is} checkable is a selector theorem under a
000256 | named axiom set.  \textbf{Statement.}  Under
000257 | \textbf{A1}~$d>2$ (propagating local gauge fields, $d-2\ge1$ transverse polarizations),
000258 | \textbf{A2}~a dimensionless Yang--Mills coupling,
000259 | \textbf{A3}~real self-dual/anti-self-dual 2-form sectors (the Euclidean Hodge star an involutive
000260 | endomorphism of 2-forms), and
000261 | \textbf{A4}~chiral Weyl representations (even $d$),
000262 | the dimension $d=4$ is \emph{unique and minimal} on the exact scan window $d=2..12$ --- and the
000263 | selector is \emph{overdetermined}: A2 alone pins $\{4\}$ ($[g_{\rm YM}]=(4-d)/2$ forced
000264 | symbolically from dimensionless-action power counting) and A3 alone pins $\{4\}$
000265 | ($*:\Omega^2\to\Omega^{d-2}$ is a self-map iff $d=4$; the explicit $6\times6$ star on
000266 | $\Lambda^2(\mathbb R^4)$ satisfies $*^2=I$ with eigenvalues $\pm1$ at multiplicities $3/3$ ---
000267 | the real $\mathfrak{su}(2)_+\times\mathfrak{su}(2)_-$ split), while A1\,\&\,A4 alone give
000268 | $\{4,6,8,10,12\}$: chirality does \emph{not} select $4$ by itself (honest ablation ledger).
000269 | Weyl chirality is certified by exact Clifford algebras $d=2..7$ (even $d$: $\gamma_*$
000270 | anticommutes with every $\gamma_\mu$, $\gamma_*^2=I$, $\mathrm{tr}\,\gamma_*=0$; odd $d$: the
000271 | full product is central and scalar --- no grading).  Signature is typed honestly: in Lorentzian
000272 | signature $*^2=-1$ on 2-forms in $d=4$, so the \emph{real} split is a Euclidean/OS-side
000273 | statement.  Mutants are caught (the wrong power counting $[g]=(6-d)/2$ selects $\{6\}$; the
000274 | ``A4 alone selects 4'' overclaim is refused).  \textbf{Honest boundary.}  $\mu_4$ and
000275 | $g_{\rm car}$ enter nowhere --- deliberately, because ``$\mu_4\Rightarrow$ 4D'' is exactly the
000276 | forbidden reading.  This is a selector, \emph{not} a construction and \emph{not} a compiler
000277 | derivation: \emph{axiom provenance is the open half} --- if A1--A4 cannot themselves be forced
000278 | from $\{c_3, g_{\rm car}\}$, the selector remains a consistency statement, not a prediction
000279 | (the registered kill criterion).  Status: \texttt{DIMENSION.SELECTOR.4D.01} \mC; every 4D
000280 | construction contract stays \mO.
000281 | 
000282 | % =====================================================================