Research audit: CK Horner reversal raises scalar-grid guard from 3 to 5 bits (no new kappa)

[bitplane]

The bidirectional Horner circuits in #837 preserve their exact endpoint maps and gate savings, but can need a finer intermediate fixed-point grid. This mathematical audit establishes the exact fractional guard requirements of all six pinned scalar programs and supplies a complete-word rounding counterexample. No new κ.

For inputs on 2^-p Z^N, with an unbounded integer part and working grid 2^-(p+g) Z, exact representability of every intermediate for every input is equivalent to g being at least the largest reduced binary denominator exponent of any prefix-map coefficient. The package gives the proof, complete prefix replays, and sharpness witnesses.

Scalar projection	Forward binary/NAF baseline	#837 bidirectional
CK forward	3	5
CK backward	8	8
JP8 forward	4	4
JP8 backward	27	27
SW4 forward	3	3
SW4 backward	19	19

The shared forward/backward guard stays 8 for CK, 27 for JP8, 19 for SW4. Eighteen of the 962 individual coefficients require one or two additional bits in the chosen reversed circuits, occurring in 21 CK, 4 JP8 and 5 SW4 additions per orientation.

For t += (31/8)*s, reverse NAF saves one real primitive but raises local depth from three to five. With s=1, halving rounded toward minus infinity at three fractional bits gives 4 instead of 31/8. In the complete CK forward scalar program, native basis input 16 at three bits leaves errors L43=-1/8, L422=+1/8, L425=+1/8; four bits still fail, and five restore the exact endpoint. The original forward-only program is exact at three bits.

Reproduce and validation

python3 -B -I research/dyadic-fractional-grid/verify.py
python3 -B -I research/dyadic-fractional-grid/crosscheck.py
python3 -B -I research/dyadic-fractional-grid/test_grid.py

Expected main marker:

PASS_DYADIC_FRACTIONAL_GRID programs=6 coefficients=962 extra_grid_coefficients=18 kappa=unchanged

• Integer rows and a separate sparse-Fraction interpreter agree across 12 complete traces and 1,016,110 scalar primitives, including full depth-stream digests and endpoints.
• All 3,842 candidate circuits, 12 independently evaluated sharpness witnesses, 24 insufficient local guard executions and the complete CK rounding witness are checked.
• Arbitrary 716+9 or 728+9 native/private columns, macro boundaries, private erasure, event inventory and paid profiles are retained.
• Python 3.12.3 and 3.14.4 passed; nine tests passed on each. A minimal checkout with spaces in its path also passed in isolated, optimized Python without site packages.
• Six package controls reject changed source/payload bytes, an understated guard, an altered witness and overwriting an existing receipt; changed gzip/tar containers with identical consumed payloads pass.
• make selected-record-check passed. Full selected cold replay was not rerun: this adds only research/dyadic-fractional-grid/ and changes no selected artifact.

Scope, pins and attribution

This resolves a finite precision obligation that #837 explicitly left open. It does not retract its endpoint or count claims. Frame/phase changes and recursive children are not evaluated as physical numerical operators here, so this is not a full compiler precision certificate, an overflow or floating-point stability budget, or evidence that the selected #799 construction fails. Fractional guards are not total register width. All-size realization, hosting, precision/tape and analytic assumptions remain.

Base: f567884aeb8851b6e794af1184119a3bfbe1e631. Frozen #837 head: 0a2977195efc8fb2a918fbfb694ec8d294387f08. Historical #626 and #741 sources, SPECs, manifests and checkpoint records are bound in SOURCES.json; compressed proof inputs are compared by their consumed payloads. Seed 60, alternatives, scope and overlap check are in CLAIM.json; no sprint coordination issue was available.

Credit Sagar Thakrar's #696 binary copy/unit and #723 forward NAF work, Chris Pagolu's #734 bounded Horner prefixes, and the retained #622/#626/#741 construction and verification lineage named in the README. Our #837 lowering and its #696 module are vendored verbatim with attribution. Related #821 magnitude-growth, #830 transposed work and #764/#862 cyclic-inverse precision results address different obligations.

Prepared for Gareth Davidson (@bitplane) with substantial OpenAI Codex assistance. Apache-2.0. Both new arithmetic implementations were written by one AI agent; this is not independent human review or formal verification of the full multiplication theorem.