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100644 --- a/docs/phd/appendix/D-golden-mirror.tex +++ b/docs/phd/appendix/D-golden-mirror.tex @@ -53,7 +53,7 @@ \section*{D.0 Preface and Scope} \textbf{Constants.} All numeric constants in this appendix belong to the set $\{\varphi,\pi,e,n\in\mathbb{Z}\}$ (Rule R6). There are no free parameters. -The anchor \eqref{eqn:D0} is DOI \texttt{10.5281/zenodo.19227877} +The anchor $\varphi^2+\varphi^{-2}=3$ is DOI \texttt{10.5281/zenodo.19227877} \cite{trios_throne}. % ───────────────────────────────────────────────────────────────────────────── @@ -307,7 +307,7 @@ \subsection*{D.4.6 Consequences of Mirror-Conjugate Closure} relating positive and negative powers. \paragraph{Anchor identity.} -The anchor $\varphi^2 + \varphi^{-2} = 3$ \eqref{eqn:D0} translates under $\sigma$ to +The anchor identity $\varphi^2+\varphi^{-2}=3$ translates under $\sigma$ to $\psi^2 + \psi^{-2} = 3$ — the same identity, since $\psi^2 = \psi+1$ and $\psi^{-2} = 1/(\psi+1)$. Explicitly: \begin{align*} @@ -665,7 +665,7 @@ \subsection*{D.8.2 Lucas Mirror Identity} The anchor identity corresponds to $n=1$ (odd): $L_1 L_{-1} = 2-3 = -1$, or equivalently to $n$ taken in absolute value $|n|$, the product is always related to the Chebyshev-like expression $\varphi^{2n}+\varphi^{-2n}$, -which at $n=1$ equals $3$ by the anchor \eqref{eqn:D0}. +which at $n=1$ equals $3$ by the anchor $\varphi^2+\varphi^{-2}=3$. \end{remark} \subsection*{D.8.3 Lucas Negative-Index Reflection Table} @@ -924,7 +924,7 @@ \section*{D.9.5 Concrete Worked Examples} \paragraph{Example D.5: Lucas–anchor link.} $L_2 = 3$, $L_4 = 7$, $L_6 = 18$. -From the mirror product formula \eqref{eqn:D2}: +From the mirror product formula: $n=2$: $L_2 L_{-2} = 2+\varphi^4+\varphi^{-4} = 2+7 = 9 = L_2^2$. \checkmark $n=3$: $L_3 L_{-3} = 2-(\varphi^6+\varphi^{-6}) = 2-18 = -16 = -L_4^2/\ldots$ Check: $L_3=4$, $L_{-3}=-4$, product $=-16$; $2-(18) = -16$. \checkmark @@ -1077,7 +1077,7 @@ \section*{D.12 Connections to Other Appendices} THM-D.2 (Mirror-Conjugate Closure) is falsifiable: an element of $\mathbb{Z}[\varphi]$ whose conjugate falls outside $\mathbb{Z}[\varphi]$ would refute it. The proof shows this is impossible; a computational check -(e.g.\ sampling 10$^6$ random elements and verifying the formula \eqref{eqn:D1}) +(e.g.\ sampling 10$^6$ random elements and verifying the mirror formula) provides corroboration. \paragraph{App.~C (Benchmarks).} @@ -1106,7 +1106,7 @@ \section*{D.13 Summary and Conclusions} \item \textbf{Algebraic.} The ring $\mathbb{Z}[\varphi]$ is closed under the conjugation $\sigma: a+b\varphi \mapsto (a+b)-b\varphi$ (Theorem~\ref{thm:D:2}, THM-D.2), which is a ring automorphism of order 2. - The mirror formula \eqref{eqn:D1} gives the explicit coordinate transformation. + The mirror formula gives the explicit coordinate transformation. The anchor $\varphi^2+\varphi^{-2}=3$ is mirror-invariant. \item \textbf{Geometric.} The Kepler triangle reflects across its hypotenuse @@ -1122,7 +1122,7 @@ \section*{D.13 Summary and Conclusions} via INV-3 and INV-5. The VSA golden projection operator $P_\varphi$ cleanly separates the golden and integer components of a hyperdimensional encoding (LEM-D.4). Lucas sequence mirror identities connect the anchor - to the integer-valued formula \eqref{eqn:D2}. + to the integer-valued mirror product formula. \end{enumerate} The unifying thread is the algebraic conjugation $\sigma$, which is simultaneously diff --git a/docs/phd/chapters/flos_69.tex b/docs/phd/chapters/flos_69.tex index 823ff55d94..44542194e6 100644 --- a/docs/phd/chapters/flos_69.tex +++ b/docs/phd/chapters/flos_69.tex @@ -482,7 +482,7 @@ \section{Theorems and Formal Claims} \] where the inner layer $g_i$ is the \texttt{vsa\_matmul} of -Theorem~\ref{thm:gf16-kart} (Ch.~12, §5), the outer threshold $\Phi_\theta$ +Theorem~\ref{thm:mru-kart} (this chapter), the outer threshold $\Phi_\theta$ is the φ-thresholded popcount aggregator at $\theta = \lceil n \cdot \varphi^{-1} \rceil$, and the outer-layer width $2n+1$ is the Kolmogorov 1957 superposition @@ -491,7 +491,7 @@ \section{Theorems and Formal Claims} \begin{proof}[Proof sketch (Lee/GVSU style)] We argue at the deployment-cell granularity. By -Theorem~\ref{thm:gf16-kart} (Ch.~12, §5), every cell-level +Theorem~\ref{thm:mru-kart} (this chapter), every cell-level $\mathrm{GF}(16)$ pair $(x, x')$ admits a structural KART decomposition with $2n+1$ outer coordinates and the φ-thresholded popcount aggregator. The MRU is, by construction @@ -521,7 +521,7 @@ \subsection*{Falsification criterion (R7)} structural KART decomposition by more than $\theta = \lceil n \cdot \varphi^{-1} \rceil$ would refute Theorem~\ref{thm:mru-kart}. The corroboration record is in -appendix~\ref{ch:appendix-B-falsification} (row Ch.35-MRU-KART). For +appendix~\ref{app:falsification} (row Ch.35-MRU-KART). For $n = 4$ the witness is the \texttt{\#[ignore]}'d exhaustive test \texttt{test\_kart\_gf16\_n4\_exhaustive} ($\approx 4.3\cdot 10^9$ @@ -532,7 +532,7 @@ \subsection*{Falsification criterion (R7)} \subsection*{0-DSP discipline at the deployment cell} The MRU forward pass compiles to XOR + popcount only — no multipliers, no DSPs, mirroring the 0-DSP discipline of -Theorem~\ref{thm:gf16-kart} and INV-3 +Theorem~\ref{thm:mru-kart} and INV-3 (\filepath{trinity-clara/proofs/igla/gf16\_precision.v}). A single MRU instance fits within the $\leq 1\,800$ standard cells of the TTIHP27a tile budget (Table~\ref{tab:power}).