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arxiv: 2606.05017 · v3 · pith:Y7B3T7UFnew · submitted 2026-06-03 · 💻 cs.AR · cs.MS

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

Dmitrii Vasilev This is my paper

Pith reviewed 2026-06-28 03:43 UTC · model grok-4.3

classification 💻 cs.AR cs.MS
keywords GoldenFloatgolden ratiofloating-point formatsstatic splitexponent width rulemulti-width arithmeticRTL generatorFPGA codec
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The pith

A single golden-ratio rule sets exponent widths for floating-point formats matching nine existing cases exactly.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines GoldenFloat as a family of static-split floating-point formats whose field widths follow one closed rule derived from the golden ratio. For any total width N of at least 4 bits the exponent field receives e equals round of (N minus 1) divided by phi squared bits, with the fraction field taking the remaining N minus 1 minus e bits. This choice reproduces the exponent widths that appear in the nine realized formats GF4 through GF256 without any adjustment. The work supplies an open RTL generator, a Lucas-sequence exact accumulator verified to high precision, and a working FPGA codec to support practical use of the family. A sympathetic reader would care because the rule supplies a parameter-free way to allocate bits between range and precision that stays consistent as total width changes.

Core claim

The paper claims that the exponent width rule e equals round of (N minus 1) divided by phi squared, where phi is the golden ratio, produces a static-split floating-point family that exactly reproduces the realized exponent widths of the nine formats GF4, GF8, GF12, GF16, GF20, GF24, GF32, GF64, GF256 and extends the identical rule without modification to GF128, GF512, and GF1024, accompanied by open generators, an integer-backed accumulator, and an FPGA implementation.

What carries the argument

The closed-form exponent rule e = round((N-1)/phi^2) with phi the golden ratio that fixes the static split between exponent and fraction bits for every total width N.

If this is right

  • The same rule generates consistent formats at any additional width such as 128, 512, or 1024 bits.
  • A single RTL generator can produce correct implementations for the entire family once the rule is fixed.
  • An integer accumulator using Lucas sequences remains exact for every width generated by the rule.
  • The family can be positioned and compared directly with other parameterised width-spanning formats such as posit and takum.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Designers could select floating-point formats for different hardware budgets using one predictable formula rather than separate tables for each width.
  • The open conjecture on toolchain coherence could be settled by running the pre-registered matched-substrate FPGA and software ablation experiments.
  • The Lucas-exact path might extend naturally to other exact integer identities if the formats are adopted at additional widths.

Load-bearing premise

That a formula chosen because it reproduces the exponent widths of nine existing formats supplies a principled closed-form basis for generating all future widths in the family.

What would settle it

Direct computation of e = round((N-1)/phi^2) for each of the nine widths N equals 4, 8, 12, 16, 20, 24, 32, 64, 256 and verification that every resulting e matches the exponent width actually used in that format.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper presents GoldenFloat (GF), a static-split floating-point family for total widths N >= 4 defined by the closed rule e = round((N-1)/phi^2) (phi = (1+sqrt(5))/2) with f = N-1-e. It claims this rule exactly reproduces the realized exponent widths of nine formats (GF4, GF8, GF12, GF16, GF20, GF24, GF32, GF64, GF256; 9/9 match) and extends consistently to GF128/512/1024. Supporting artifacts include an open multi-width RTL generator with CI differential sweep, a Lucas-exact integer accumulator verified at 500-digit precision, and a GF16 FPGA codec passing 35/35 tests at 323 MHz on Artix-7; an RTL erratum is noted in §5.5.

Significance. If the rule's functional form can be independently motivated, the family would supply a parameter-free generator for multi-width floats together with open, machine-checked verification artifacts (RTL generator, Lucas-exact path, FPGA codec) that are strengths relative to other width-spanning proposals such as posit or takum. The Lucas-exact accumulator and continuous-integration oracle are concrete, reproducible contributions.

major comments (2)
  1. [Abstract] Abstract: the central claim that the rule supplies a 'single closed rule' for the entire GF family rests on the 9/9 reproduction of existing exponent widths, yet no derivation is supplied showing why the divisor must be phi^2 or why the round operation follows from floating-point design principles; the perfect match therefore functions as an empirical fit rather than an independent generator.
  2. [§5.5] §5.5: the reported RTL-correctness erratum (dated 2026-05-31) states that the fabricated dies and the presented CI sweep use a defective multiplier portfolio; although a corrected generator is identified as the regeneration baseline, this directly affects the soundness of the verification artifacts that underwrite the 9/9 match and the Lucas-exact claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and for recognizing the value of the open artifacts. We respond point-by-point to the major comments below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the rule supplies a 'single closed rule' for the entire GF family rests on the 9/9 reproduction of existing exponent widths, yet no derivation is supplied showing why the divisor must be phi^2 or why the round operation follows from floating-point design principles; the perfect match therefore functions as an empirical fit rather than an independent generator.

    Authors: We agree that the manuscript presents the rule primarily through its empirical success in reproducing 9/9 realized exponent widths without supplying an a priori derivation from floating-point design principles for the specific choice of phi^2 or the round function. The golden-ratio connection is motivated by the Lucas-exact accumulator identity, but this link is not developed into a derivation of the split rule itself. In revision we will update the abstract to describe the rule as a closed empirical generator and add a short subsection (new §2.3) that explicitly states the empirical discovery process and the Lucas-sequence grounding, while removing any implication of independent derivation. revision: yes

  2. Referee: [§5.5] §5.5: the reported RTL-correctness erratum (dated 2026-05-31) states that the fabricated dies and the presented CI sweep use a defective multiplier portfolio; although a corrected generator is identified as the regeneration baseline, this directly affects the soundness of the verification artifacts that underwrite the 9/9 match and the Lucas-exact claims.

    Authors: Section 5.5 already reports the erratum and identifies the corrected generator as the regeneration baseline. The 9/9 match, CI differential sweep, and Lucas-exact accumulator verifications were performed exclusively with the corrected generator; only the fabricated TTSKY26b dies used the defective multiplier portfolio. To remove ambiguity we will expand §5.5 with a short table that explicitly maps each reported artifact (CI sweep, Lucas-exact path, FPGA codec, 9/9 match) to the generator version used, thereby confirming that the soundness claims rest on the corrected baseline. revision: yes

Circularity Check

1 steps flagged

Exponent rule e = round((N-1)/phi^2) selected to reproduce nine listed formats rather than derived from first principles

specific steps
  1. fitted input called prediction [Abstract / 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 functional form (phi^2 divisor plus round) is justified only by its exact reproduction of the nine target widths; the paper supplies no separate derivation showing why this expression must arise from floating-point arithmetic or hardware constraints. The match is therefore the input that defines the rule, after which the rule is presented as generating the family.

full rationale

The central generator rule is introduced solely by its ability to match the realized exponent widths of the nine concrete formats GF4 through GF256 (explicitly stated as 9/9). No independent derivation from floating-point design constraints, hardware considerations, or mathematical necessity is supplied for the specific divisor phi^2 or the round operation; the form is presented because it fits the target widths and then retroactively labeled a 'closed rule' that generates the family. This matches the fitted-input-called-prediction pattern exactly. No other circular steps (self-citation chains, ansatz smuggling, or self-definitional identities) appear in the provided text. The Lucas-exact accumulator and RTL artefacts are downstream and do not affect the rule's justification.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the mathematical definition of the exponent-width rule using the golden ratio and the empirical observation that it matches nine existing formats; no free parameters beyond the standard golden-ratio constant are introduced, and no new physical entities are postulated.

axioms (1)
  • domain assumption The exponent width for total width N is defined as round((N-1)/phi^2) where phi = (1 + sqrt(5))/2
    This is the single closed rule stated in the abstract as generating the entire family.

pith-pipeline@v0.9.1-grok · 5951 in / 1615 out tokens · 37814 ms · 2026-06-28T03:43:56.770990+00:00 · methodology

discussion (0)

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. An 83-Format Numeric Catalog with Bit-Exact Conformance Vectors: A Vendor-Neutral Reference for FP8, BF16, MXFP4, and Microscaling Formats

    cs.AR 2026-06 unverdicted novelty 2.0

    An 83-format numeric catalog with bit-exact conformance vectors and IEEE P3109 cross-walk serving as a vendor-neutral reference for FP8, BF16, MXFP4, and microscaling formats.

Reference graph

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