feat(phd-kat-rw): L-KAT-RW — KART × KAN × Trinity GF16 Related Work patch

docs/phd/chapters/ch_27.tex

@@ -204,6 +204,189 @@ \section{7. Discussion}\label{discussion}
 
 The TRI27 DSL formalised here is intentionally minimal. The present two theorems establish only determinism and exhaustiveness; a complete verified compiler from TRI27 to FPGA RTL would require additional theorems on type safety, termination, and translation correctness --- all planned for v5 of the dissertation. The most significant limitation is that the current semantics does not handle variable out-of-scope errors gracefully: \texttt{eval} returns \texttt{None}, but there is no formal type-system proof that well-typed programs never produce \texttt{None}. A dependent type approach (à la Agda or Idris) would subsume this. The \texttt{If3} constructor as currently implemented is also a two-branch conditional rather than the intended three-branch form; extending it to \texttt{If3\ e\ e1\ e2\ e3} with a \texttt{trit}-dispatched branch selection is deferred to the next proof sprint. Chapter 28 (FPGA implementation) and App.H (VM specification) build directly on the TRI27 kernel defined here.
 
+\section{8. Related Work: Trinity GF(16) and the Kolmogorov--Arnold Lineage}\label{sec:related-kart}
+
+% Lane L-KAT-RW (salvage) · trios#380 / trios#572 · author Dmitrii Vasilev <raoffonom@icloud.com>
+% ORCID 0009-0008-4294-6159 · Anchor: phi^2 + phi^{-2} = 3 · DOI 10.5281/zenodo.19227877
+% Salvage rewrite per queen ruling https://github.com/gHashTag/trios/issues/572#issuecomment-4407395169
+%   - "isomorphism" / "finite-field analogue" / "dual" -> "structurally analogous"
+%   - primary cite shifts to finite_field_expressivity_2025 (ICLR 2025, Weil conjectures)
+%   - Kolmogorov-Arnold cited only as structural inspiration (secondary)
+%   - finite-group VSA NeurIPS 2022 cited as closest prior art
+%   - KANtize (arXiv:2603.17230) cited as closest KAN+quant prior
+
+The ternary alphabet \(\{-1, 0, +1\}\) on which TRI27 is built admits a deeper
+function-theoretic reading. We close this chapter by situating TRI27 and its
+GF(16)-coded successor (Ch.~\ref{ch:gf16-algebra},
+Ch.~\ref{ch:hardware-bridge}) within three converging lines of prior work:
+the finite-field expressivity programme of
+\cite{finite_field_expressivity_2025}, finite-group vector-symbolic
+architectures \cite{finite_group_vsa_2022}, and post-hoc ternarisation of
+Kolmogorov--Arnold Networks \cite{kantize_2026}. The Kolmogorov--Arnold
+Representation Theorem (KART) \cite{kolmogorov_kar_1957,arnold_kar_1957} is
+cited as classical structural inspiration only --- not as a result Trinity
+proves, ports, or extends. The purpose of this section is descriptive: the
+formal correspondence between GF(16) VSA-binding and a two-level superposition
+in the spirit of KART is stated as
+Theorem~\ref{thm:kart-gf16} in Ch.~\ref{ch:hardware-bridge} as a
+\emph{structural analogy}; here we record the historical and architectural
+lineage only.
+
+\subsection*{8.1. The Kolmogorov--Arnold Representation Theorem (Structural Inspiration)}
+
+Kolmogorov, in resolving Hilbert's 13th problem in the negative for continuous
+functions, proved that every continuous function
+\(f : [0,1]^n \to \mathbb{R}\) admits an exact representation
+\cite{kolmogorov_kar_1957,arnold_kar_1957}
+\[
+  f(x_1, \dots, x_n) \;=\; \sum_{q=0}^{2n} \Phi_q\!\left(
+    \sum_{p=1}^{n} \phi_{q,p}(x_p)
+  \right),
+\]
+where the inner functions \(\phi_{q,p} : [0,1] \to \mathbb{R}\) and the outer
+functions \(\Phi_q : \mathbb{R} \to \mathbb{R}\) are continuous and depend on
+one variable each. The decomposition is exact, not approximate: the
+multivariate complexity of \(f\) is fully absorbed into a finite superposition
+of univariate pieces. This theorem was for sixty years regarded as a curiosity
+of real analysis, since Kolmogorov's inner functions are pathologically
+non-smooth and resisted constructive use.
+
+We stress what KART does \emph{not} say. It does not assert that
+\(\phi_{q,p}\) are smooth, learnable, or computable in finite precision; it
+only asserts the existence of such a decomposition for continuous \(f\) over
+a real-valued domain. The Trinity claim in Ch.~\ref{ch:hardware-bridge}
+concerns a finite-field two-level decomposition that is \emph{structurally
+analogous} to the KART pattern, not a finite-field re-statement, port, or
+analogue of the classical theorem itself. The classical theorem is cited as
+axiomatic background and as visual scaffolding for our two-level
+\(\Phi \circ \phi\) shape; it is not a Trinity contribution
+(see \admittedbox{Kolmogorov 1957/1961, Arnold 1963}{classical real-analysis
+theorem; cited as structural inspiration only, not proven, ported, or
+extended in this dissertation}). The substantive prior art for Trinity GF(16)
+is the finite-field expressivity programme reviewed in §8.3.
+
+\subsection*{8.2. Kolmogorov--Arnold Networks and Their Quantisation}
+
+Liu \emph{et al.}~\cite{liu_kan_2024,liu_kan2_2025} reified the KART
+\emph{shape} as a learnable architecture by placing the inner functions
+\(\phi_{q,p}\) on the \emph{edges} of a multilayer perceptron rather than
+fixing scalar activations on the \emph{nodes}. Each \(\phi_{q,p}\) is realised
+as a learnable cubic spline over a B-spline basis with eight FP32 control
+points (\(8 \times 32 = 256\) bits per knot vector), and the outer function
+\(\Phi_q\) is realised as a per-node SiLU. Liu \emph{et al.} reported that for
+Besov-class targets KAN attains the parameter-efficiency bound
+\(O(N^{2+1/s})\) compared to the MLP bound \(O(N^{2n/s+1})\), beating the
+curse of dimensionality on smooth targets while remaining computationally
+identical to a sparse MLP at inference time. We emphasise that KAN is a
+real-valued architecture motivated by the KART \emph{shape}; it does not
+constitute a finite-field statement of the theorem and does not establish a
+finite-field expressivity result.
+
+The most directly comparable prior work to Trinity GF(16) on the
+KAN-quantisation axis is KANtize \cite{kantize_2026}, which post-hoc ternarises
+the spline coefficients of a trained KAN to \(\{-1, 0, +1\}\) with reported
+\(<\!1\%\) accuracy regression on standard tabular benchmarks. KANtize remains
+a real-valued architecture --- the underlying spline interpolation, the SiLU
+outer function, and gradient back-propagation are all kept in floating point;
+only the stored coefficients are ternarised. Trinity GF(16) takes a
+structurally different step (§8.3): the inner functions \(\phi_{q,p}\) are
+\emph{native} GF(16) look-up tables rather than ternarised real splines, and
+the outer function is a \texttt{popcount} over GF(2)-encoded outputs rather
+than a real-valued non-linearity. We do not claim that this difference yields
+a quantitative expressivity advantage over KANtize; that comparison is left
+open as Problem~\ref{open:gf16-besov}.
+
+\subsection*{8.3. Trinity GF(16) and Finite-Field Expressivity}
+
+The substantive line of prior work for Trinity GF(16) is the recent
+finite-field expressivity programme of
+\cite{finite_field_expressivity_2025}, which establishes that two-level
+neural decompositions over a finite field \(\mathbb{F}_q\) admit a Weil-style
+counting bound on the cardinality of their neuromanifold and a corresponding
+expressivity statement. Trinity GF(16) instantiates the operative parameters
+of this programme: \(q = 16\); inner alphabet \(\mathrm{GF}(16)\) with
+\(|\mathrm{GF}(16)| = 16\) and width \(n \in \{4, 8\}\) (Ch.~\ref{ch:gf16-algebra});
+outer aggregation by \texttt{popcount} over GF(2) of the
+inner LUT outputs followed by a threshold comparator
+\(\theta = \lceil n \cdot \varphi^{-1} \rceil\) (Ch.~\ref{ch:hardware-bridge},
+\cite{finite_group_vsa_2022}). The \texttt{popcount}-then-threshold pattern
+synthesises to combinational logic on SKY130 with zero DSP slices
+(Ch.~\ref{ch:fpga-implementation}). Theorem~\ref{thm:kart-gf16} establishes a
+\emph{structural analogy} between this two-level decomposition and the KART
+shape; the operative expressivity bound is the Weil bound of
+\cite{finite_field_expressivity_2025}, not the real-analytic statement of
+\cite{kolmogorov_kar_1957,arnold_kar_1957}.
+
+The closest finite-algebra prior art is the finite-group vector-symbolic
+architecture programme of \cite{finite_group_vsa_2022}, which establishes that
+permutation-group-coded VSA bindings preserve associativity and admit
+exact-recovery decoding. Trinity GF(16) extends this finite-algebra binding
+to the additive group of \(\mathrm{GF}(16)\) (which is \((\mathbb{Z}/2)^4\))
+and adds a multiplicative-group rotation
+(Ch.~\ref{ch:gf16-algebra}, §3); the binding-decoding pair is
+formalised in Theorem~\ref{thm:kart-gf16}.
+
+Table~\ref{tab:kan-vs-trinity-gf16} records the parameter footprint of the two
+architectures at a matched width \(n = 8\). The ratio reported in the table is
+literal and conservative (a 4\(\times\) reduction in bits per inner function
+and per superposition, derived from the stated bit-budgets of
+\cite{liu_kan_2024} §3.2 and Ch.~\ref{ch:hardware-bridge}); broader claims
+of memory advantage on Besov-class targets, while plausible, would require a
+finite-field formulation of the parameter-efficiency bound and are deferred
+to Problem~\ref{open:gf16-besov} in Ch.~\ref{ch:future-work}. The table is
+not an expressivity claim; it is a footprint comparison at fixed architecture.
+
+\begin{table}[H]
+  \centering
+  \caption{Parameter footprint at matched architecture (\(n = 8\)): KAN versus
+    Trinity GF(16). The KAN row follows the architectural defaults reported in
+    \cite{liu_kan_2024}, §3.2; the Trinity row follows
+    Th.~\ref{thm:kart-gf16}. The rightmost column reports a literal
+    bit-budget ratio at fixed \(n\), not an expressivity claim.}
+  \label{tab:kan-vs-trinity-gf16}
+  \begin{tabular}{lrrr}
+    \toprule
+    Architecture & Inner-function encoding & Bits per \(\phi_{q,p}\) &
+      Bits per super\-position \((n=8)\) \\
+    \midrule
+    KAN \cite{liu_kan_2024} & 8 FP32 spline knots & 256 & 16{,}384 \\
+    Trinity GF(16) (Th.~\ref{thm:kart-gf16}) & 16 GF(16) cells & 64 & 4{,}096 \\
+    Ratio & --- & \(4\times\) & \(4\times\) \\
+    \bottomrule
+  \end{tabular}
+\end{table}
+
+We note one further architectural consequence. In KAN the
+curse-of-dimensionality argument relies on Besov-class smoothness assumptions
+on the target function; the inner splines must be smooth enough to capture
+this regularity. In the GF(16) finite-field setting smoothness is undefined,
+but the domain itself is finite (\(|\mathrm{GF}(16)| = 16\)) and the inner
+LUTs are exact rather than approximate. The two regimes are not directly
+comparable on the same target class; a finite-field reformulation of any
+Besov-style bound is left open as Problem~\ref{open:gf16-besov} in
+Ch.~\ref{ch:future-work}, and the operative expressivity bound used elsewhere
+in this dissertation is the Weil-style finite-field bound of
+\cite{finite_field_expressivity_2025}, not a real-analytic Besov bound.
+
+This section does not introduce any new theorems. Theorem~\ref{thm:kart-gf16}
+(Trinity GF(16) two-level decomposition, structurally analogous to KART) is
+stated and partially proved in Ch.~\ref{ch:hardware-bridge}; the runtime
+witness lives in \filepath{proofs/KAT\_VSA\_Bridge.v} (lemma
+\texttt{finite\_field\_two\_level\_decomposition}, Qed, runtime witness
+documented per queen ruling on
+\href{https://github.com/gHashTag/trios/issues/572\#issuecomment-4407395169}{trios\#572}).
+Theorem~\ref{thm:mru-kart} (Trinity MRU as a two-level superposition,
+structurally analogous to KART) is stated and falsifier-bound in
+Ch.~\ref{ch:mesh-node}, with runtime witness in
+\filepath{proofs/KAT\_VSA\_Bridge.v} (lemma
+\texttt{MRU\_outer\_independence}, Qed). The related lanes L-KAT-12 and
+L-KAT-35 on issue \href{https://github.com/gHashTag/trios/issues/380}{trios\#380}
+track their landing as separate pull requests. We make no claim that Trinity
+GF(16) is a finite-field analogue, isomorphism, dual, or port of KART; the
+relation is one of structural analogy, with substantive prior art rooted in
+\cite{finite_field_expressivity_2025,finite_group_vsa_2022,kantize_2026}.
+
+
 \section{References}\label{references}
 
 {[}1{]} \emph{Golden Sunflowers} dissertation, Ch.3 --- Trinity Identity (\(\varphi^2 + \varphi^{-2} = 3\)).