GoldenFloat: A Phi-Derived Static-Split Floating-Point Family from GF4 to GF256 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). 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 (9/9) and extends consistently to GF128, GF512, GF1024. The rule is positioned alongside posit, takum, OCP-MX, and the IEEE P3109 multi-width float draft. We make no per-rung accuracy or superiority claim against any of them. The breadth/toolchain-coherence framing is recorded as an open conjecture with a pre-registered falsification path. A falsification ledger (FL-002) records open questions and the experiments that would settle them. An RTL-correctness erratum dated 2026-05-31 is reported; the fabricated TTSKY26b dies carry the defective multiplier portfolio, and the corrected generator is the regeneration baseline.

• The paper introduces GoldenFloat, a floating-point family (GF4 to GF256) using a phi-derived static partitioning of exponent and fraction bits.
• It details a closed analytic ladder rule and empirical validation against alternative formats like posit and takum.
• It demonstrates hardware verification with an FPGA implementation and a Lucas-exact accumulator, outlining pathways for future research.

GoldenFloat: A Phi-Derived Static-Split Floating-Point Format Family

Introduction and Motivation

GoldenFloat (GF) introduces a static-split floating-point format family spanning widths from 4 to 256 bits, parameterized by a closed analytic rule anchored in the golden ratio $\varphi = (1+\sqrt{5})/2$. Unlike adaptive formats such as posit and takum, GF fixes the exponent-width and significand-width per bit-width $N$ independently of the encoded value. The static split is defined as $e = \mathrm{round}((N-1)/\varphi^2)$ exponent bits and $f = N-1-e$ fraction bits, leveraging the algebraic property $\varphi^2 = \varphi + 1$. This formulation is motivated both by numerical reproducibility of desirable exponent/fraction ratios and by structural properties that facilitate integer-backed exact accumulation through the Lucas recurrence.

GF is positioned in the context of recent proliferations of non-IEEE-754 number formats for ML hardware. It is explicitly compared to posits, takum, OCP-MX, and emerging standards such as IEEE P3109, sharing the goal of covering a broad ladder of bit-widths under a unified rule, but with a distinct choice of static, phi-derived partitioning.

Ladder Rule and Numerical Properties

The ladder rule $e = \mathrm{round}((N-1)/\varphi^2)$, with $\varphi^2 \approx 2.618$, consistently reproduces the exponent-widths for the canonical widths GF4, GF8, GF12, GF16, GF20, GF24, GF32, GF64, and GF256 and extends smoothly to wider formats including GF128, GF512, and GF1024. Empirically, the realized ratio $e/(N-1)$ converges to $1/\varphi^2 \approx 0.38197$ as $N$ increases.

An exhaustive "look-elsewhere" search across the interval $N$0 at $N$1 resolution demonstrates that the matching of exponent-widths by $N$2 is not numerically unique—there are 82 other matching ratios under nine-width matching and 47 under twelve-width extension. Bonferroni-corrected familywise probabilities confirm that the phi-derived choice is not statistically distinguished purely on ladder-reproduction criteria. Therefore, the justification for phi is explicitly algebraic and structural rather than statistical coincidence.

Algebraic and Hardware Anchor: The Golden Ratio Encoder and Lucas Identity

The relevance of $N$3 is anchored in the multiplier-free property of the Golden Ratio Encoder (GRE) [Daubechies et al., 2010], where the recursion at base $N$4 admits addition-only implementation due to $N$5. GF capitalizes on this by transferring the same algebraic leverage to its format family.

A novel contribution of GF is the explicit citation and utilization of the Lucas identity: for all integers $N$6, $N$7, with $N$8 the $N$9-th Lucas number. This property, verified both symbolically in $e = \mathrm{round}((N-1)/\varphi^2)$0 and numerically to 500-digit precision, supports integer-backed accumulation for powers-of-$e = \mathrm{round}((N-1)/\varphi^2)$1 terms, enabling an exact accumulation path without a posit-style quire or explicit Kulisch register. This theoretically supports hardware implementations that carry long accumulations in integer space using Lucas recurrences, though no hardware realization of such an accumulator is presented in the current work.

Hardware Implementation and Verification

Three artefacts are provided:

• An open RTL generator emitting codecs and arithmetic modules for all GF4–GF256 rungs, formally verified against a reference and integrated into CI.
• A Lucas-exact accumulator path, with high-precision validation of the underlying Lucas identity for $e = \mathrm{round}((N-1)/\varphi^2)$2.
• A GF16 FPGA implementation on Artix-7 (Xilinx XC7A35T), passing full codec/multiplier testbenches at 323 MHz.

A format-conformance oracle, Corona, provides an on-die ROM-based format checker for 80 numeric formats, including all GF ladder points. Corona achieves GDS/TAPEOUT readiness and is intended for silicon verification; it is not a compute accelerator but a reference module.

A notable hardware-level erratum is disclosed: a two-bit-narrow multiplier register affected all but GF4 and required correction post-shuttle submission. The corrected RTL passes exhaustive tests for all rungs; however, some submitted silicon carries the bug, which does not impact the theoretical format specification.

GF16 design is the only rung with measured FPGA synthesis and simulation results. No physical silicon has yet been returned; thus, claims regarding area, energy, or comparative throughput remain projections rather than empirical findings.

Empirical Evaluation and Limitations

On a real corpus (tiny_shakespeare) and in a substrate-confounded pipeline, a GF ladder arm did not display a significant bits-per-byte (BPB) advantage or disadvantage when compared to a heterogeneous numeric zoo. Statistical evidence is insufficient, with mean BPB overlap and Bayesian credibility below the required threshold for a per-rung claim. Substrate bias against GF16 (non-native arithmetic pathways) is acknowledged as a confound, and matched-silicon or matched-FPGA results are pre-registered but not yet available.

Comparison to Prior Art

GoldenFloat is situated among parameterized-width float families:

• Posit: Adaptive regime exponents; two distinct $e = \mathrm{round}((N-1)/\varphi^2)$3 schedules; regime adapts dynamic range per value.
• Takum: Single analytic tapered-precision rule; dynamic range/fraction width varies per value; only published peer-reviewed multi-width hardware.
• OCP-MX: Block-scaled microformats with shared scaling fields.
• IEEE P3109: In-development ML-accelerator float standard spanning multiple widths via parameterization.
• LNS, Zeckendorf arithmetic, Fibbinary quantization: Related nonstandard encodings leveraging special bases (logarithmic, Fibonacci).

Distinctly, GF provides a static, value-independent exponent/fraction split across the ladder, with a closed analytic rule and an integer-backed accumulator rooted in the Lucas identity. It makes no claim of per-rung numerical superiority or uniqueness of the chosen base based on the ladder rule alone. Breadth and toolchain coherence, not per-rung accuracy, are proposed as the principal potential advantage—explicitly marked as an open conjecture pending matched-budget hardware comparisons.

Open Questions and Falsification Criteria

Multiple research hypotheses are pre-registered:

• Breadth-as-moat: Toolchain coherence is an advantage if no alternative (e.g., posit, takum) matches GF at equal area and bit budgets—subject to falsification via matched substrate experiments.
• Ladder-rule uniqueness: Uniqueness of the phi-derived exponent/fraction split is unproven.
• GF256 bias formula: Existence of a closed-form expression matching observed GF256 bias awaits proof.
• GF16 silicon parity: Awaiting returned die and measurement parity with fp16-native references.

All claims are structured to allow clear falsification and negative results are acknowledged as first-class outcomes.

Implications and Future Directions

GF introduces a formally elegant and hardware-friendly static-split float family, leveraging golden ratio properties. The clear, value-independent partitioning aids in format regularity and potentially in toolchain simplicity. The integer-backed Lucas accumulator allows theoretical elimination of explicit wide-register accumulators for sums of powers-of-$e = \mathrm{round}((N-1)/\varphi^2)$4, which may have practical implications in hardware resource allocation and rounding error tracking.

However, demonstrations of competitive practical advantage, either in BPB efficiency or numerical accuracy, require substrate-matched hardware and direct head-to-head comparison against alternatives. The breadth-as-moat conjecture remains untested in silicon, and numerical uniqueness of the phi-derived split is not established.

Further developments may include concrete hardware accumulators implementing Lucas recurrences, empirical comparison against takum and posit formats in end-to-end ML workload scenarios, and exploration of unbiased GF256 bias construction.

Conclusion

GoldenFloat provides a statically partitioned floating-point format family generated by a single analytic, phi-derived rule, with verified ladder reproducibility and a theoretically justified integer-backed accumulator. The framework is rigorously documented, with reproducibility-oriented software, hardware artefacts, and clearly articulated open questions. It makes precise, non-sensational claims and embraces falsification, situating itself as a basis for future empirical comparison rather than asserting inherent superiority over existing format families.