Research · Papers · The quadratic hull and its defects · MF-033
A structural characterization of the c0c2 quadratic defect
On every one of the 8,192 public-input fibres, the nonzero in-hull output flips are `e1`, `u(x)`, and `e1+u(x)`, where `e1=32` flips `next_c1` and `u(x)=16 XOR next_c2(x)*128`.
Published 2026-08-29
For everyone
Plain summary
The c0c2 relation produces output mistakes in a predictable structure. Fixing a public input gives a single public-input fibre. Across all 8,192 fibres, the quadratic hull admits nonzero output changes in only two independent directions. One direction is always label 32, which flips next_c1. The second direction is label 16 when next_c2 is 0 and label 144 when next_c2 is 1. Combining these two yields label 48 or label 176. This accounts for all five observed labels: each fibre contains only two base directions, but the second direction switches with next_c2, giving three independent semantic directions across the entire input space. This result characterizes the shape of the defect; it makes no repair, complexity, or novelty claim, and the register records no prior art.
Result
On every one of the 8,192 public-input fibres, the nonzero in-hull output flips are e1, u(x), and e1+u(x), where e1=32 flips next_c1 and u(x)=16 XOR next_c2(x)*128. The five observed support labels 16,32,48,144,176 therefore span a fibre dimension of two: label 32 is universal, labels 16 and 48 occur precisely when next_c2=0, and labels 144 and 176 occur precisely when next_c2=1. The global semantic span has rank three because u(x) twists with next_c2.
Setting and definitions
Let x denote the public input. A fibre fixes x. An output flip is the difference mask between two outputs, identified by its integer label. The nonzero elements of a two-dimensional Boolean flip space are e1, u(x), and e1+u(x). The global semantic span is the vector space generated by these fibrewise masks across all public inputs.
Method
Exhaustive per-fibre replay established the flip structure across all 8,192 public-input fibres. An independent replay verified the twisted-defect certificate, reproducing the support labels and global rank.
Discussion
The five support labels form a single fibrewise two-plane whose second generator depends on next_c2. This generator twist accounts for the global rank-three defect and reconciles the global span with the five support labels. This is a structural characterization of the c0c2 quadratic defect. The entry records Corrections: NONE, asserts no repair or complexity conclusions, and states no prior-art position; it makes no novelty claim.
For everyone — the takeaway
What this means
Locally, the error pattern is simple. For any fixed public input, every admitted nonzero change comes from two basic steps: the constant 32 change and a second change chosen by next_c2. These combine into three possible nonzero flips per fibre. Across different inputs, that second base change switches form, meaning three directions are needed to capture the full system. This classifies the defect's structure without claiming to repair it.
Register references
Entry: MF-033
Receipts: CONT pro3_c0c2_twisted_defect_replay.json, pro_request3_task1_replay_receipt.json
Prior art: the register does not record this.
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Evidence pack
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Changelog
Last reviewed 2026-08-29
- 2026-08-29Published on this site.