GoldenFloat: A Phi-Derived Static-Split Floating-Point Family from GF4 to GF1024 with a Lucas-Exact Integer Identity

Abstract

We present a hardware-oriented description of GoldenFloat (GF), a static-split floating-point family generated by a single closed rule, and three concrete artefacts: (i) an open multi-width RTL generator covering GF4-GF256 with a continuous-integration differential sweep against a correctly-rounded reference; (ii) an integer-backed Lucas-exact accumulator path verified at 500-digit precision for n = 1, ..., 256; and (iii) a GF16 FPGA codec passing a 35-of-35 testbench at 323 MHz on Artix-7 (Xilinx XC7A35T). A format-conformance oracle (Corona) ships in the same repository and is used as the blackbox check in our continuous-integration audit. The rule and its scope. For each total width N >= 4, the exponent width is e = round((N-1)/phi^2) with fraction f = N-1-e and phi = (1+sqrt(5))/2. The rule reproduces the realised exponent widths of nine formats GF4, GF8, GF12, GF16, GF20, GF24, GF32, GF64, GF256 (9/9) and extends consistently to GF128, GF512, GF1024. The rule is positioned alongside posit (2022 Posit Standard), takum (Hunhold 2024, 2025), OCP-MX (Rouhani et al. 2023), and the IEEE P3109 multi-width float draft, all of which are width-spanning families under a parameterised rule. We make no per-rung accuracy or superiority claim against any of them. What is open. The breadth/toolchain-coherence framing is recorded as an open conjecture with a pre-registered falsification path: a matched-substrate FPGA experiment and a matched-budget software ablation. A falsification ledger (FL-002) records the open questions and the experiments that would settle them. An RTL-correctness erratum dated 2026-05-31 is reported in Section 5.5; the fabricated TTSKY26b dies carry the defective multiplier portfolio, and the corrected generator is the regeneration baseline.