Appendix: Action Circuit Polynomial Constraints

This appendix lists the 193 polynomial constraints of the Orchard Action circuit at orchard 0.13.1. Each polynomial $P$ vanishes on every valid assignment: $P = 0$.

Provenance. The polynomials are extracted from the src/circuit_description dump (a serialisation of halo2_proofs::plonk::PinnedVerificationKey) by the gates-to-latex tool that ships in this repo. To regenerate after an upstream change, run make appendix-gates from the onboarding/ directory; the tool re-reads the vendored copy at onboarding/data/orchard-0.13.1-circuit_description.txt.

Notation. The advice, fixed, and instance columns are indexed by their column_index in the constraint system:

• $A_c$, $A_c^{(+r)}$, $A_c^{(-r)}$: advice column $c$ at the current row, rotated by $+r$ or $-r$.
• $F_c$, $F_c^{(+r)}$, $F_c^{(-r)}$: fixed column $c$ at the current row or a rotation. The pinned circuit uses 29 fixed columns; the lowest indices are the selector-promotion columns produced by Halo 2's compress_selectors pass, and the higher indices carry the chip-level constants used by the ECC, Sinsemilla, and Poseidon chips.
• Constants are rendered in hex. Values below 0xffff are shown in full; larger values are truncated to a six-hex-digit head followed by \ldots to keep KaTeX expressions readable.

Grouping. Halo 2's compress_selectors pass packs every meta.create_gate(...) group into a single shared fixed column by giving the column a small integer value per gate member. That value selects the member through an envelope of the form $F_c \cdot (k_1 - F_c) \cdot \dots \cdot (k_n - F_c)$. Two polynomials that share the same envelope column $c$ therefore come from the same source-level create_gate call. We use $c$ as the group key and list polynomials per group; the source-level chip that owns each group can be identified by opening src/circuit.rs and reading the chip-configuration calls in Circuit::configure in order. Polynomials that do not match the envelope pattern are listed under "Ungrouped".

Scope. This is the raw polynomial form, not yet annotated with chip-level meaning. Phase 2 of this work would attach a source-level chip name to each group (ECC, Sinsemilla, Poseidon, Merkle, CommitIvk, NoteCommit). Doing so cleanly requires upstream changes in halo2_proofs to expose gate names; the pinned dump deliberately strips them.

Summary

Envelope column $c$	Polynomials in group	Original indices
$F_{16}$	2	87, 88
$F_{17}$	2	97, 98
$F_{18}$	17	1, 2, 3, 4, 5, 6, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34
$F_{19}$	17	7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,,...
$F_{20}$	18	35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, ,...
$F_{21}$	18	53, 54, 55, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, ,...
$F_{22}$	4	56, 57, 58, 59
$F_{23}$	4	60, 61, 62, 63
$F_{24}$	11	79, 80, 81, 82, 83, 84, 85, 86, 89, 90, 91
$F_{25}$	26	92, 93, 94, 95, 96, 106, 107, 108, 109, 110, 111, 112, 113, ,...
$F_{26}$	21	99, 100, 101, 102, 103, 104, 105, 127, 128, 129, 130, 131, 1,...
$F_{27}$	25	141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, ,...
$F_{28}$	28	166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, ,...

Group 1 (envelope column $F_{16}$, 2 polynomials)

Polynomial 87 (original index 87)

$$\left(A_{4}\right) \cdot \left(A_{4}\right) + -\left(A_{0}^{(+1)} + \left(A_{3}\right) \cdot \left(A_{3}\right) + -\left(A_{0}\right) + -\left(A_{1}\right) + A_{0}\right) = 0$$

Polynomial 88 (original index 88)

$$\left(\mathtt{0x4} \cdot \left(A_{4}\right)\right) \cdot \left(A_{0} + -\left(A_{0}^{(+1)}\right)\right) + -\left(\mathtt{0x2} \cdot \left(\left(A_{3} + A_{4}\right) \cdot \left(A_{0} + -\left(\left(A_{3}\right) \cdot \left(A_{3}\right) + -\left(A_{0}\right) + -\left(A_{1}\right)\right)\right)\right) + \left(\mathtt{0x2} + -\left(\left(F_{12}\right) \cdot \left(F_{12} + -\left(\mathtt{0x1}\right)\right)\right)\right) \cdot \left(\left(A_{3}^{(+1)} + A_{4}^{(+1)}\right) \cdot \left(A_{0}^{(+1)} + -\left(\left(A_{3}^{(+1)}\right) \cdot \left(A_{3}^{(+1)}\right) + -\left(A_{0}^{(+1)}\right) + -\left(A_{1}^{(+1)}\right)\right)\right)\right) + \left(\mathtt{0x2} \cdot \left(\left(F_{12}\right) \cdot \left(F_{12} + -\left(\mathtt{0x1}\right)\right)\right)\right) \cdot \left(A_{3}^{(+1)}\right)\right) = 0$$

Group 2 (envelope column $F_{17}$, 2 polynomials)

Polynomial 97 (original index 97)

$$\left(A_{9}\right) \cdot \left(A_{9}\right) + -\left(A_{5}^{(+1)} + \left(A_{8}\right) \cdot \left(A_{8}\right) + -\left(A_{5}\right) + -\left(A_{6}\right) + A_{5}\right) = 0$$

Polynomial 98 (original index 98)

$$\left(\mathtt{0x4} \cdot \left(A_{9}\right)\right) \cdot \left(A_{5} + -\left(A_{5}^{(+1)}\right)\right) + -\left(\mathtt{0x2} \cdot \left(\left(A_{8} + A_{9}\right) \cdot \left(A_{5} + -\left(\left(A_{8}\right) \cdot \left(A_{8}\right) + -\left(A_{5}\right) + -\left(A_{6}\right)\right)\right)\right) + \left(\mathtt{0x2} + -\left(\left(F_{13}\right) \cdot \left(F_{13} + -\left(\mathtt{0x1}\right)\right)\right)\right) \cdot \left(\left(A_{8}^{(+1)} + A_{9}^{(+1)}\right) \cdot \left(A_{5}^{(+1)} + -\left(\left(A_{8}^{(+1)}\right) \cdot \left(A_{8}^{(+1)}\right) + -\left(A_{5}^{(+1)}\right) + -\left(A_{6}^{(+1)}\right)\right)\right)\right) + \left(\mathtt{0x2} \cdot \left(\left(F_{13}\right) \cdot \left(F_{13} + -\left(\mathtt{0x1}\right)\right)\right)\right) \cdot \left(A_{8}^{(+1)}\right)\right) = 0$$

Group 3 (envelope column $F_{18}$, 17 polynomials)

Polynomial 1 (original index 1)

$$A_{0} + -\left(A_{1}\right) + -\left(\left(A_{2}\right) \cdot \left(A_{3}\right)\right) = 0$$

Polynomial 2 (original index 2)

$$\left(A_{0}\right) \cdot \left(A_{4} + -\left(A_{5}\right)\right) = 0$$

Polynomial 3 (original index 3)

$$\left(A_{0}\right) \cdot \left(\mathtt{0x1} + -\left(A_{6}\right)\right) = 0$$

Polynomial 4 (original index 4)

$$\left(A_{1}\right) \cdot \left(\mathtt{0x1} + -\left(A_{7}\right)\right) = 0$$

Polynomial 5 (original index 5)

$$A_{7} + A_{8} + -\left(A_{6}\right) = 0$$

Polynomial 6 (original index 6)

$$\left(\mathtt{0x400} \cdot \left(A_{9}^{(-1)}\right)\right) \cdot \left(A_{9}^{(+1)}\right) + -\left(A_{9}\right) = 0$$

Polynomial 24 (original index 24)

$$A_{4} + -\left(\mathtt{0x200000\ldots} \cdot \left(\left(A_{4}^{(+1)} + A_{5}^{(+1)}\right) \cdot \left(A_{3}^{(+1)} + -\left(\left(A_{4}^{(+1)}\right) \cdot \left(A_{4}^{(+1)}\right) + -\left(A_{3}^{(+1)}\right) + -\left(A_{0}^{(+1)}\right)\right)\right)\right)\right) = 0$$

Polynomial 25 (original index 25)

$$A_{0} + -\left(A_{0}^{(+1)}\right) = 0$$

Polynomial 26 (original index 26)

$$A_{1} + -\left(A_{1}^{(+1)}\right) = 0$$

Polynomial 27 (original index 27)

$$\left(A_{9} + -\left(\mathtt{0x2} \cdot \left(A_{9}^{(-1)}\right)\right)\right) \cdot \left(\mathtt{0x1} + -\left(A_{9} + -\left(\mathtt{0x2} \cdot \left(A_{9}^{(-1)}\right)\right)\right)\right) = 0$$

Polynomial 28 (original index 28)

$$\left(A_{4}\right) \cdot \left(A_{3} + -\left(A_{0}\right)\right) + -\left(\mathtt{0x200000\ldots} \cdot \left(\left(A_{4} + A_{5}\right) \cdot \left(A_{3} + -\left(\left(A_{4}\right) \cdot \left(A_{4}\right) + -\left(A_{3}\right) + -\left(A_{0}\right)\right)\right)\right)\right) + \left(\mathtt{0x2} \cdot \left(A_{9} + -\left(\mathtt{0x2} \cdot \left(A_{9}^{(-1)}\right)\right)\right) + -\left(\mathtt{0x1}\right)\right) \cdot \left(A_{1}\right) = 0$$

Polynomial 29 (original index 29)

$$\left(A_{5}\right) \cdot \left(A_{5}\right) + -\left(A_{3}^{(+1)}\right) + -\left(\left(A_{4}\right) \cdot \left(A_{4}\right) + -\left(A_{3}\right) + -\left(A_{0}\right)\right) + -\left(A_{3}\right) = 0$$

Polynomial 30 (original index 30)

$$\left(A_{5}\right) \cdot \left(A_{3} + -\left(A_{3}^{(+1)}\right)\right) + -\left(\mathtt{0x200000\ldots} \cdot \left(\left(A_{4} + A_{5}\right) \cdot \left(A_{3} + -\left(\left(A_{4}\right) \cdot \left(A_{4}\right) + -\left(A_{3}\right) + -\left(A_{0}\right)\right)\right)\right)\right) + -\left(\mathtt{0x200000\ldots} \cdot \left(\left(A_{4}^{(+1)} + A_{5}^{(+1)}\right) \cdot \left(A_{3}^{(+1)} + -\left(\left(A_{4}^{(+1)}\right) \cdot \left(A_{4}^{(+1)}\right) + -\left(A_{3}^{(+1)}\right) + -\left(A_{0}^{(+1)}\right)\right)\right)\right)\right) = 0$$

Polynomial 31 (original index 31)

$$\left(A_{9} + -\left(\mathtt{0x2} \cdot \left(A_{9}^{(-1)}\right)\right)\right) \cdot \left(\mathtt{0x1} + -\left(A_{9} + -\left(\mathtt{0x2} \cdot \left(A_{9}^{(-1)}\right)\right)\right)\right) = 0$$

Polynomial 32 (original index 32)

$$\left(A_{4}\right) \cdot \left(A_{3} + -\left(A_{0}\right)\right) + -\left(\mathtt{0x200000\ldots} \cdot \left(\left(A_{4} + A_{5}\right) \cdot \left(A_{3} + -\left(\left(A_{4}\right) \cdot \left(A_{4}\right) + -\left(A_{3}\right) + -\left(A_{0}\right)\right)\right)\right)\right) + \left(\mathtt{0x2} \cdot \left(A_{9} + -\left(\mathtt{0x2} \cdot \left(A_{9}^{(-1)}\right)\right)\right) + -\left(\mathtt{0x1}\right)\right) \cdot \left(A_{1}\right) = 0$$

Polynomial 33 (original index 33)

$$\left(A_{5}\right) \cdot \left(A_{5}\right) + -\left(A_{3}^{(+1)}\right) + -\left(\left(A_{4}\right) \cdot \left(A_{4}\right) + -\left(A_{3}\right) + -\left(A_{0}\right)\right) + -\left(A_{3}\right) = 0$$

Polynomial 34 (original index 34)

$$\left(A_{5}\right) \cdot \left(A_{3} + -\left(A_{3}^{(+1)}\right)\right) + -\left(\mathtt{0x200000\ldots} \cdot \left(\left(A_{4} + A_{5}\right) \cdot \left(A_{3} + -\left(\left(A_{4}\right) \cdot \left(A_{4}\right) + -\left(A_{3}\right) + -\left(A_{0}\right)\right)\right)\right)\right) + -\left(A_{4}^{(+1)}\right) = 0$$

Group 4 (envelope column $F_{19}$, 17 polynomials)

Polynomial 7 (original index 7)

$$\left(A_{1}\right) \cdot \left(A_{1}\right) + -\left(\left(\left(A_{0}\right) \cdot \left(A_{0}\right)\right) \cdot \left(A_{0}\right)\right) + -\left(\mathtt{0x5}\right) = 0$$

Polynomial 8 (original index 8)

$$\left(A_{1}\right) \cdot \left(A_{1}\right) + -\left(\left(\left(A_{0}\right) \cdot \left(A_{0}\right)\right) \cdot \left(A_{0}\right)\right) + -\left(\mathtt{0x5}\right) = 0$$

Polynomial 9 (original index 9)

$$\left(A_{1}\right) \cdot \left(A_{1}\right) + -\left(\left(\left(A_{0}\right) \cdot \left(A_{0}\right)\right) \cdot \left(A_{0}\right)\right) + -\left(\mathtt{0x5}\right) = 0$$

Polynomial 10 (original index 10)

$$\left(\left(A_{2}^{(+1)} + A_{2} + A_{0}\right) \cdot \left(A_{0} + -\left(A_{2}\right)\right)\right) \cdot \left(A_{0} + -\left(A_{2}\right)\right) + -\left(\left(A_{1} + -\left(A_{3}\right)\right) \cdot \left(A_{1} + -\left(A_{3}\right)\right)\right) = 0$$

Polynomial 11 (original index 11)

$$\left(A_{3}^{(+1)} + A_{3}\right) \cdot \left(A_{0} + -\left(A_{2}\right)\right) + -\left(\left(A_{1} + -\left(A_{3}\right)\right) \cdot \left(A_{2} + -\left(A_{2}^{(+1)}\right)\right)\right) = 0$$

Polynomial 12 (original index 12)

$$\left(A_{2} + -\left(A_{0}\right)\right) \cdot \left(\left(A_{2} + -\left(A_{0}\right)\right) \cdot \left(A_{4}\right) + -\left(A_{3} + -\left(A_{1}\right)\right)\right) = 0$$

Polynomial 13 (original index 13)

$$\left(\mathtt{0x1} + -\left(\left(A_{2} + -\left(A_{0}\right)\right) \cdot \left(A_{5}\right)\right)\right) \cdot \left(\left(\left(\mathtt{0x2}\right) \cdot \left(A_{1}\right)\right) \cdot \left(A_{4}\right) + -\left(\left(\mathtt{0x3}\right) \cdot \left(\left(A_{0}\right) \cdot \left(A_{0}\right)\right)\right)\right) = 0$$

Polynomial 14 (original index 14)

$$\left(\left(\left(A_{0}\right) \cdot \left(A_{2}\right)\right) \cdot \left(A_{2} + -\left(A_{0}\right)\right)\right) \cdot \left(\left(A_{4}\right) \cdot \left(A_{4}\right) + -\left(A_{0}\right) + -\left(A_{2}\right) + -\left(A_{2}^{(+1)}\right)\right) = 0$$

Polynomial 15 (original index 15)

$$\left(\left(\left(A_{0}\right) \cdot \left(A_{2}\right)\right) \cdot \left(A_{2} + -\left(A_{0}\right)\right)\right) \cdot \left(\left(A_{4}\right) \cdot \left(A_{0} + -\left(A_{2}^{(+1)}\right)\right) + -\left(A_{1}\right) + -\left(A_{3}^{(+1)}\right)\right) = 0$$

Polynomial 16 (original index 16)

$$\left(\left(\left(A_{0}\right) \cdot \left(A_{2}\right)\right) \cdot \left(A_{3} + A_{1}\right)\right) \cdot \left(\left(A_{4}\right) \cdot \left(A_{4}\right) + -\left(A_{0}\right) + -\left(A_{2}\right) + -\left(A_{2}^{(+1)}\right)\right) = 0$$

Polynomial 17 (original index 17)

$$\left(\left(\left(A_{0}\right) \cdot \left(A_{2}\right)\right) \cdot \left(A_{3} + A_{1}\right)\right) \cdot \left(\left(A_{4}\right) \cdot \left(A_{0} + -\left(A_{2}^{(+1)}\right)\right) + -\left(A_{1}\right) + -\left(A_{3}^{(+1)}\right)\right) = 0$$

Polynomial 18 (original index 18)

$$\left(\mathtt{0x1} + -\left(\left(A_{0}\right) \cdot \left(A_{6}\right)\right)\right) \cdot \left(A_{2}^{(+1)} + -\left(A_{2}\right)\right) = 0$$

Polynomial 19 (original index 19)

$$\left(\mathtt{0x1} + -\left(\left(A_{0}\right) \cdot \left(A_{6}\right)\right)\right) \cdot \left(A_{3}^{(+1)} + -\left(A_{3}\right)\right) = 0$$

Polynomial 20 (original index 20)

$$\left(\mathtt{0x1} + -\left(\left(A_{2}\right) \cdot \left(A_{7}\right)\right)\right) \cdot \left(A_{2}^{(+1)} + -\left(A_{0}\right)\right) = 0$$

Polynomial 21 (original index 21)

$$\left(\mathtt{0x1} + -\left(\left(A_{2}\right) \cdot \left(A_{7}\right)\right)\right) \cdot \left(A_{3}^{(+1)} + -\left(A_{1}\right)\right) = 0$$

Polynomial 22 (original index 22)

$$\left(\mathtt{0x1} + -\left(\left(A_{2} + -\left(A_{0}\right)\right) \cdot \left(A_{5}\right)\right) + -\left(\left(A_{3} + A_{1}\right) \cdot \left(A_{8}\right)\right)\right) \cdot \left(A_{2}^{(+1)}\right) = 0$$

Polynomial 23 (original index 23)

$$\left(\mathtt{0x1} + -\left(\left(A_{2} + -\left(A_{0}\right)\right) \cdot \left(A_{5}\right)\right) + -\left(\left(A_{3} + A_{1}\right) \cdot \left(A_{8}\right)\right)\right) \cdot \left(A_{3}^{(+1)}\right) = 0$$

Group 5 (envelope column $F_{20}$, 18 polynomials)

Polynomial 35 (original index 35)

$$A_{8} + -\left(\mathtt{0x200000\ldots} \cdot \left(\left(A_{8}^{(+1)} + A_{2}^{(+1)}\right) \cdot \left(A_{7}^{(+1)} + -\left(\left(A_{8}^{(+1)}\right) \cdot \left(A_{8}^{(+1)}\right) + -\left(A_{7}^{(+1)}\right) + -\left(A_{0}^{(+1)}\right)\right)\right)\right)\right) = 0$$

Polynomial 36 (original index 36)

$$A_{0} + -\left(A_{0}^{(+1)}\right) = 0$$

Polynomial 37 (original index 37)

$$A_{1} + -\left(A_{1}^{(+1)}\right) = 0$$

Polynomial 38 (original index 38)

$$\left(A_{6} + -\left(\mathtt{0x2} \cdot \left(A_{6}^{(-1)}\right)\right)\right) \cdot \left(\mathtt{0x1} + -\left(A_{6} + -\left(\mathtt{0x2} \cdot \left(A_{6}^{(-1)}\right)\right)\right)\right) = 0$$

Polynomial 39 (original index 39)

$$\left(A_{8}\right) \cdot \left(A_{7} + -\left(A_{0}\right)\right) + -\left(\mathtt{0x200000\ldots} \cdot \left(\left(A_{8} + A_{2}\right) \cdot \left(A_{7} + -\left(\left(A_{8}\right) \cdot \left(A_{8}\right) + -\left(A_{7}\right) + -\left(A_{0}\right)\right)\right)\right)\right) + \left(\mathtt{0x2} \cdot \left(A_{6} + -\left(\mathtt{0x2} \cdot \left(A_{6}^{(-1)}\right)\right)\right) + -\left(\mathtt{0x1}\right)\right) \cdot \left(A_{1}\right) = 0$$

Polynomial 40 (original index 40)

$$\left(A_{2}\right) \cdot \left(A_{2}\right) + -\left(A_{7}^{(+1)}\right) + -\left(\left(A_{8}\right) \cdot \left(A_{8}\right) + -\left(A_{7}\right) + -\left(A_{0}\right)\right) + -\left(A_{7}\right) = 0$$

Polynomial 41 (original index 41)

$$\left(A_{2}\right) \cdot \left(A_{7} + -\left(A_{7}^{(+1)}\right)\right) + -\left(\mathtt{0x200000\ldots} \cdot \left(\left(A_{8} + A_{2}\right) \cdot \left(A_{7} + -\left(\left(A_{8}\right) \cdot \left(A_{8}\right) + -\left(A_{7}\right) + -\left(A_{0}\right)\right)\right)\right)\right) + -\left(\mathtt{0x200000\ldots} \cdot \left(\left(A_{8}^{(+1)} + A_{2}^{(+1)}\right) \cdot \left(A_{7}^{(+1)} + -\left(\left(A_{8}^{(+1)}\right) \cdot \left(A_{8}^{(+1)}\right) + -\left(A_{7}^{(+1)}\right) + -\left(A_{0}^{(+1)}\right)\right)\right)\right)\right) = 0$$

Polynomial 42 (original index 42)

$$\left(A_{6} + -\left(\mathtt{0x2} \cdot \left(A_{6}^{(-1)}\right)\right)\right) \cdot \left(\mathtt{0x1} + -\left(A_{6} + -\left(\mathtt{0x2} \cdot \left(A_{6}^{(-1)}\right)\right)\right)\right) = 0$$

Polynomial 43 (original index 43)

$$\left(A_{8}\right) \cdot \left(A_{7} + -\left(A_{0}\right)\right) + -\left(\mathtt{0x200000\ldots} \cdot \left(\left(A_{8} + A_{2}\right) \cdot \left(A_{7} + -\left(\left(A_{8}\right) \cdot \left(A_{8}\right) + -\left(A_{7}\right) + -\left(A_{0}\right)\right)\right)\right)\right) + \left(\mathtt{0x2} \cdot \left(A_{6} + -\left(\mathtt{0x2} \cdot \left(A_{6}^{(-1)}\right)\right)\right) + -\left(\mathtt{0x1}\right)\right) \cdot \left(A_{1}\right) = 0$$

Polynomial 44 (original index 44)

$$\left(A_{2}\right) \cdot \left(A_{2}\right) + -\left(A_{7}^{(+1)}\right) + -\left(\left(A_{8}\right) \cdot \left(A_{8}\right) + -\left(A_{7}\right) + -\left(A_{0}\right)\right) + -\left(A_{7}\right) = 0$$

Polynomial 45 (original index 45)

$$\left(A_{2}\right) \cdot \left(A_{7} + -\left(A_{7}^{(+1)}\right)\right) + -\left(\mathtt{0x200000\ldots} \cdot \left(\left(A_{8} + A_{2}\right) \cdot \left(A_{7} + -\left(\left(A_{8}\right) \cdot \left(A_{8}\right) + -\left(A_{7}\right) + -\left(A_{0}\right)\right)\right)\right)\right) + -\left(A_{8}^{(+1)}\right) = 0$$

Polynomial 46 (original index 46)

$$\left(A_{9}^{(+1)} + -\left(\left(\mathtt{0x2}\right) \cdot \left(A_{9}^{(-1)}\right)\right)\right) \cdot \left(\mathtt{0x1} + -\left(A_{9}^{(+1)} + -\left(\left(\mathtt{0x2}\right) \cdot \left(A_{9}^{(-1)}\right)\right)\right)\right) = 0$$

Polynomial 47 (original index 47)

$$\left(A_{9}^{(+1)} + -\left(\left(\mathtt{0x2}\right) \cdot \left(A_{9}^{(-1)}\right)\right)\right) \cdot \left(A_{9} + -\left(A_{1}^{(-1)}\right)\right) + \left(\mathtt{0x1} + -\left(A_{9}^{(+1)} + -\left(\left(\mathtt{0x2}\right) \cdot \left(A_{9}^{(-1)}\right)\right)\right)\right) \cdot \left(A_{9} + A_{1}^{(-1)}\right) = 0$$

Polynomial 48 (original index 48)

$$A_{8} + -\left(A_{7} + \left(A_{7}^{(-1)}\right) \cdot \left(\left(\mathtt{0x100000\ldots}\right) \cdot \left(\mathtt{0x40}\right)\right)\right) = 0$$

Polynomial 49 (original index 49)

$$A_{6}^{(-1)} + -\left(A_{7}\right) + -\left(\mathtt{0x224698\ldots}\right) = 0$$

Polynomial 50 (original index 50)

$$\left(A_{7}^{(-1)}\right) \cdot \left(A_{6} + -\left(\mathtt{0x100000\ldots}\right)\right) = 0$$

Polynomial 51 (original index 51)

$$\left(A_{7}^{(-1)}\right) \cdot \left(A_{7}^{(+1)}\right) = 0$$

Polynomial 52 (original index 52)

$$\left(\left(\mathtt{0x1} + -\left(A_{7}^{(-1)}\right)\right) \cdot \left(\mathtt{0x1} + -\left(\left(A_{6}\right) \cdot \left(A_{6}^{(+1)}\right)\right)\right)\right) \cdot \left(A_{7}^{(+1)}\right) = 0$$

Group 6 (envelope column $F_{21}$, 18 polynomials)

Polynomial 53 (original index 53)

$$\left(A_{9}^{(+1)} + -\left(\mathtt{0x2} \cdot \left(A_{9}\right)\right)\right) \cdot \left(\mathtt{0x1} + -\left(A_{9}^{(+1)} + -\left(\mathtt{0x2} \cdot \left(A_{9}\right)\right)\right)\right) = 0$$

Polynomial 54 (original index 54)

$$\left(A_{9}^{(+1)} + -\left(\mathtt{0x2} \cdot \left(A_{9}\right)\right)\right) \cdot \left(A_{0}\right) + \left(\mathtt{0x1} + -\left(A_{9}^{(+1)} + -\left(\mathtt{0x2} \cdot \left(A_{9}\right)\right)\right)\right) \cdot \left(A_{0} + -\left(A_{0}^{(+1)}\right)\right) = 0$$

Polynomial 55 (original index 55)

$$\left(A_{9}^{(+1)} + -\left(\mathtt{0x2} \cdot \left(A_{9}\right)\right)\right) \cdot \left(A_{1}\right) + \left(\mathtt{0x1} + -\left(A_{9}^{(+1)} + -\left(\mathtt{0x2} \cdot \left(A_{9}\right)\right)\right)\right) \cdot \left(A_{1} + A_{1}^{(+1)}\right) = 0$$

Polynomial 64 (original index 64)

$$\left(A_{5}\right) \cdot \left(\mathtt{0x1} + -\left(A_{5}\right)\right) = 0$$

Polynomial 65 (original index 65)

$$\left(A_{4}\right) \cdot \left(A_{4}\right) + -\left(\mathtt{0x1}\right) = 0$$

Polynomial 66 (original index 66)

$$\left(A_{1} + -\left(A_{3}\right)\right) \cdot \left(A_{1} + A_{3}\right) = 0$$

Polynomial 67 (original index 67)

$$\left(A_{4}\right) \cdot \left(A_{1}\right) + -\left(A_{3}\right) = 0$$

Polynomial 68 (original index 68)

$$\left(A_{8}\right) \cdot \left(A_{7}\right) = 0$$

Polynomial 69 (original index 69)

$$\left(A_{8}\right) \cdot \left(A_{7}^{(+1)} + -\left(\left(A_{8}^{(-1)}\right) \cdot \left(\mathtt{0x100000\ldots}\right)\right)\right) = 0$$

Polynomial 70 (original index 70)

$$\left(A_{8}\right) \cdot \left(\left(A_{8}^{(+1)} + -\left(\mathtt{0x8} \cdot \left(A_{7}^{(+1)}\right)\right)\right) \cdot \left(\mathtt{0x1} + -\left(A_{8}^{(+1)} + -\left(\mathtt{0x8} \cdot \left(A_{7}^{(+1)}\right)\right)\right)\right)\right) = 0$$

Polynomial 71 (original index 71)

$$\left(A_{8}\right) \cdot \left(A_{6}^{(+1)}\right) = 0$$

Polynomial 72 (original index 72)

$$\left(\left(\left(A_{7}\right) \cdot \left(\mathtt{0x1} + -\left(A_{7}\right)\right)\right) \cdot \left(\mathtt{0x2} + -\left(A_{7}\right)\right)\right) \cdot \left(\mathtt{0x3} + -\left(A_{7}\right)\right) = 0$$

Polynomial 73 (original index 73)

$$\left(A_{8}\right) \cdot \left(\mathtt{0x1} + -\left(A_{8}\right)\right) = 0$$

Polynomial 74 (original index 74)

$$A_{8}^{(-1)} + -\left(A_{7} + \mathtt{0x4} \cdot \left(A_{8}\right)\right) = 0$$

Polynomial 75 (original index 75)

$$A_{6} + -\left(A_{6}^{(-1)} + -\left(\mathtt{0x100000\ldots} \cdot \left(A_{8}^{(-1)}\right)\right) + \mathtt{0x400000\ldots} + -\left(\mathtt{0x224698\ldots}\right)\right) = 0$$

Polynomial 76 (original index 76)

$$\mathtt{0xab5e5b\ldots} \cdot \left(\left(\left(\left(A_{6} + F_{5}\right) \cdot \left(A_{6} + F_{5}\right)\right) \cdot \left(\left(A_{6} + F_{5}\right) \cdot \left(A_{6} + F_{5}\right)\right)\right) \cdot \left(A_{6} + F_{5}\right)\right) + \mathtt{0x319166\ldots} \cdot \left(\left(\left(\left(A_{7} + F_{6}\right) \cdot \left(A_{7} + F_{6}\right)\right) \cdot \left(\left(A_{7} + F_{6}\right) \cdot \left(A_{7} + F_{6}\right)\right)\right) \cdot \left(A_{7} + F_{6}\right)\right) + \mathtt{0x7c045d\ldots} \cdot \left(\left(\left(\left(A_{8} + F_{7}\right) \cdot \left(A_{8} + F_{7}\right)\right) \cdot \left(\left(A_{8} + F_{7}\right) \cdot \left(A_{8} + F_{7}\right)\right)\right) \cdot \left(A_{8} + F_{7}\right)\right) + -\left(A_{6}^{(+1)}\right) = 0$$

Polynomial 77 (original index 77)

$$\mathtt{0x233162\ldots} \cdot \left(\left(\left(\left(A_{6} + F_{5}\right) \cdot \left(A_{6} + F_{5}\right)\right) \cdot \left(\left(A_{6} + F_{5}\right) \cdot \left(A_{6} + F_{5}\right)\right)\right) \cdot \left(A_{6} + F_{5}\right)\right) + \mathtt{0x25cae2\ldots} \cdot \left(\left(\left(\left(A_{7} + F_{6}\right) \cdot \left(A_{7} + F_{6}\right)\right) \cdot \left(\left(A_{7} + F_{6}\right) \cdot \left(A_{7} + F_{6}\right)\right)\right) \cdot \left(A_{7} + F_{6}\right)\right) + \mathtt{0x22f5b5\ldots} \cdot \left(\left(\left(\left(A_{8} + F_{7}\right) \cdot \left(A_{8} + F_{7}\right)\right) \cdot \left(\left(A_{8} + F_{7}\right) \cdot \left(A_{8} + F_{7}\right)\right)\right) \cdot \left(A_{8} + F_{7}\right)\right) + -\left(A_{7}^{(+1)}\right) = 0$$

Polynomial 78 (original index 78)

$$\mathtt{0x2e29dd\ldots} \cdot \left(\left(\left(\left(A_{6} + F_{5}\right) \cdot \left(A_{6} + F_{5}\right)\right) \cdot \left(\left(A_{6} + F_{5}\right) \cdot \left(A_{6} + F_{5}\right)\right)\right) \cdot \left(A_{6} + F_{5}\right)\right) + \mathtt{0x1d1aab\ldots} \cdot \left(\left(\left(\left(A_{7} + F_{6}\right) \cdot \left(A_{7} + F_{6}\right)\right) \cdot \left(\left(A_{7} + F_{6}\right) \cdot \left(A_{7} + F_{6}\right)\right)\right) \cdot \left(A_{7} + F_{6}\right)\right) + \mathtt{0x3bf763\ldots} \cdot \left(\left(\left(\left(A_{8} + F_{7}\right) \cdot \left(A_{8} + F_{7}\right)\right) \cdot \left(\left(A_{8} + F_{7}\right) \cdot \left(A_{8} + F_{7}\right)\right)\right) \cdot \left(A_{8} + F_{7}\right)\right) + -\left(A_{8}^{(+1)}\right) = 0$$

Group 7 (envelope column $F_{22}$, 4 polynomials)

Polynomial 56 (original index 56)

$$\left(\left(\left(\left(\left(\left(\left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right) \cdot \left(\mathtt{0x1} + -\left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right)\right) \cdot \left(\mathtt{0x2} + -\left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right)\right) \cdot \left(\mathtt{0x3} + -\left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right)\right) \cdot \left(\mathtt{0x4} + -\left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right)\right) \cdot \left(\mathtt{0x5} + -\left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right)\right) \cdot \left(\mathtt{0x6} + -\left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right)\right) \cdot \left(\mathtt{0x7} + -\left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) = 0$$

Polynomial 57 (original index 57)

$$0 + \left(\mathtt{0x1}\right) \cdot \left(F_{3}\right) + \left(\left(\mathtt{0x1}\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(F_{4}\right) + \left(\left(\left(\mathtt{0x1}\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(F_{5}\right) + \left(\left(\left(\left(\mathtt{0x1}\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(F_{6}\right) + \left(\left(\left(\left(\left(\mathtt{0x1}\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(F_{7}\right) + \left(\left(\left(\left(\left(\left(\mathtt{0x1}\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(F_{8}\right) + \left(\left(\left(\left(\left(\left(\left(\mathtt{0x1}\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(F_{9}\right) + \left(\left(\left(\left(\left(\left(\left(\left(\mathtt{0x1}\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(A_{4} + -\left(\mathtt{0x8} \cdot \left(A_{4}^{(+1)}\right)\right)\right)\right) \cdot \left(F_{10}\right) + -\left(A_{0}\right) = 0$$

Polynomial 58 (original index 58)

$$\left(A_{5}\right) \cdot \left(A_{5}\right) + -\left(A_{1}\right) + -\left(F_{11}\right) = 0$$

Polynomial 59 (original index 59)

$$\left(A_{1}\right) \cdot \left(A_{1}\right) + -\left(\left(\left(A_{0}\right) \cdot \left(A_{0}\right)\right) \cdot \left(A_{0}\right)\right) + -\left(\mathtt{0x5}\right) = 0$$

Group 8 (envelope column $F_{23}$, 4 polynomials)

Polynomial 60 (original index 60)

$$0 + \left(\mathtt{0x1}\right) \cdot \left(F_{3}\right) + \left(\left(\mathtt{0x1}\right) \cdot \left(A_{4}\right)\right) \cdot \left(F_{4}\right) + \left(\left(\left(\mathtt{0x1}\right) \cdot \left(A_{4}\right)\right) \cdot \left(A_{4}\right)\right) \cdot \left(F_{5}\right) + \left(\left(\left(\left(\mathtt{0x1}\right) \cdot \left(A_{4}\right)\right) \cdot \left(A_{4}\right)\right) \cdot \left(A_{4}\right)\right) \cdot \left(F_{6}\right) + \left(\left(\left(\left(\left(\mathtt{0x1}\right) \cdot \left(A_{4}\right)\right) \cdot \left(A_{4}\right)\right) \cdot \left(A_{4}\right)\right) \cdot \left(A_{4}\right)\right) \cdot \left(F_{7}\right) + \left(\left(\left(\left(\left(\left(\mathtt{0x1}\right) \cdot \left(A_{4}\right)\right) \cdot \left(A_{4}\right)\right) \cdot \left(A_{4}\right)\right) \cdot \left(A_{4}\right)\right) \cdot \left(A_{4}\right)\right) \cdot \left(F_{8}\right) + \left(\left(\left(\left(\left(\left(\left(\mathtt{0x1}\right) \cdot \left(A_{4}\right)\right) \cdot \left(A_{4}\right)\right) \cdot \left(A_{4}\right)\right) \cdot \left(A_{4}\right)\right) \cdot \left(A_{4}\right)\right) \cdot \left(A_{4}\right)\right) \cdot \left(F_{9}\right) + \left(\left(\left(\left(\left(\left(\left(\left(\mathtt{0x1}\right) \cdot \left(A_{4}\right)\right) \cdot \left(A_{4}\right)\right) \cdot \left(A_{4}\right)\right) \cdot \left(A_{4}\right)\right) \cdot \left(A_{4}\right)\right) \cdot \left(A_{4}\right)\right) \cdot \left(A_{4}\right)\right) \cdot \left(F_{10}\right) + -\left(A_{0}\right) = 0$$

Polynomial 61 (original index 61)

$$\left(A_{5}\right) \cdot \left(A_{5}\right) + -\left(A_{1}\right) + -\left(F_{11}\right) = 0$$

Polynomial 62 (original index 62)

$$\left(A_{1}\right) \cdot \left(A_{1}\right) + -\left(\left(\left(A_{0}\right) \cdot \left(A_{0}\right)\right) \cdot \left(A_{0}\right)\right) + -\left(\mathtt{0x5}\right) = 0$$

Polynomial 63 (original index 63)

$$\left(\left(\left(\left(\left(\left(\left(A_{4}\right) \cdot \left(\mathtt{0x1} + -\left(A_{4}\right)\right)\right) \cdot \left(\mathtt{0x2} + -\left(A_{4}\right)\right)\right) \cdot \left(\mathtt{0x3} + -\left(A_{4}\right)\right)\right) \cdot \left(\mathtt{0x4} + -\left(A_{4}\right)\right)\right) \cdot \left(\mathtt{0x5} + -\left(A_{4}\right)\right)\right) \cdot \left(\mathtt{0x6} + -\left(A_{4}\right)\right)\right) \cdot \left(\mathtt{0x7} + -\left(A_{4}\right)\right) = 0$$

Group 9 (envelope column $F_{24}$, 11 polynomials)

Polynomial 79 (original index 79)

$$\left(\left(\left(A_{6} + F_{5}\right) \cdot \left(A_{6} + F_{5}\right)\right) \cdot \left(\left(A_{6} + F_{5}\right) \cdot \left(A_{6} + F_{5}\right)\right)\right) \cdot \left(A_{6} + F_{5}\right) + -\left(A_{5}\right) = 0$$

Polynomial 80 (original index 80)

$$\left(\left(\left(\mathtt{0xab5e5b\ldots} \cdot \left(A_{5}\right) + \mathtt{0x319166\ldots} \cdot \left(A_{7} + F_{6}\right) + \mathtt{0x7c045d\ldots} \cdot \left(A_{8} + F_{7}\right) + F_{8}\right) \cdot \left(\mathtt{0xab5e5b\ldots} \cdot \left(A_{5}\right) + \mathtt{0x319166\ldots} \cdot \left(A_{7} + F_{6}\right) + \mathtt{0x7c045d\ldots} \cdot \left(A_{8} + F_{7}\right) + F_{8}\right)\right) \cdot \left(\left(\mathtt{0xab5e5b\ldots} \cdot \left(A_{5}\right) + \mathtt{0x319166\ldots} \cdot \left(A_{7} + F_{6}\right) + \mathtt{0x7c045d\ldots} \cdot \left(A_{8} + F_{7}\right) + F_{8}\right) \cdot \left(\mathtt{0xab5e5b\ldots} \cdot \left(A_{5}\right) + \mathtt{0x319166\ldots} \cdot \left(A_{7} + F_{6}\right) + \mathtt{0x7c045d\ldots} \cdot \left(A_{8} + F_{7}\right) + F_{8}\right)\right)\right) \cdot \left(\mathtt{0xab5e5b\ldots} \cdot \left(A_{5}\right) + \mathtt{0x319166\ldots} \cdot \left(A_{7} + F_{6}\right) + \mathtt{0x7c045d\ldots} \cdot \left(A_{8} + F_{7}\right) + F_{8}\right) + -\left(\mathtt{0x2cc057\ldots} \cdot \left(A_{6}^{(+1)}\right) + \mathtt{0x32e7c4\ldots} \cdot \left(A_{7}^{(+1)}\right) + \mathtt{0x2eae5d\ldots} \cdot \left(A_{8}^{(+1)}\right)\right) = 0$$

Polynomial 81 (original index 81)

$$\mathtt{0x233162\ldots} \cdot \left(A_{5}\right) + \mathtt{0x25cae2\ldots} \cdot \left(A_{7} + F_{6}\right) + \mathtt{0x22f5b5\ldots} \cdot \left(A_{8} + F_{7}\right) + F_{9} + -\left(\mathtt{0x7bf368\ldots} \cdot \left(A_{6}^{(+1)}\right) + \mathtt{0x2aec69\ldots} \cdot \left(A_{7}^{(+1)}\right) + \mathtt{0x952e02\ldots} \cdot \left(A_{8}^{(+1)}\right)\right) = 0$$

Polynomial 82 (original index 82)

$$\mathtt{0x2e29dd\ldots} \cdot \left(A_{5}\right) + \mathtt{0x1d1aab\ldots} \cdot \left(A_{7} + F_{6}\right) + \mathtt{0x3bf763\ldots} \cdot \left(A_{8} + F_{7}\right) + F_{10} + -\left(\mathtt{0x2fcbba\ldots} \cdot \left(A_{6}^{(+1)}\right) + \mathtt{0x1ec737\ldots} \cdot \left(A_{7}^{(+1)}\right) + \mathtt{0xd0c2ef\ldots} \cdot \left(A_{8}^{(+1)}\right)\right) = 0$$

Polynomial 83 (original index 83)

$$A_{6}^{(-1)} + A_{6} + -\left(A_{6}^{(+1)}\right) = 0$$

Polynomial 84 (original index 84)

$$A_{7}^{(-1)} + A_{7} + -\left(A_{7}^{(+1)}\right) = 0$$

Polynomial 85 (original index 85)

$$A_{8}^{(-1)} + -\left(A_{8}^{(+1)}\right) = 0$$

Polynomial 86 (original index 86)

$$\mathtt{0x2} \cdot \left(F_{3}\right) + -\left(\left(A_{3} + A_{4}\right) \cdot \left(A_{0} + -\left(\left(A_{3}\right) \cdot \left(A_{3}\right) + -\left(A_{0}\right) + -\left(A_{1}\right)\right)\right)\right) = 0$$

Polynomial 89 (original index 89)

$$A_{2} + -\left(\left(A_{4}\right) \cdot \left(A_{1}\right) + \left(\mathtt{0x1} + -\left(A_{4}\right)\right) \cdot \left(A_{0}\right)\right) = 0$$

Polynomial 90 (original index 90)

$$A_{3} + -\left(\left(A_{4}\right) \cdot \left(A_{0}\right) + \left(\mathtt{0x1} + -\left(A_{4}\right)\right) \cdot \left(A_{1}\right)\right) = 0$$

Polynomial 91 (original index 91)

$$\left(A_{4}\right) \cdot \left(\mathtt{0x1} + -\left(A_{4}\right)\right) = 0$$

Group 10 (envelope column $F_{25}$, 26 polynomials)

Polynomial 92 (original index 92)

$$A_{0} + -\left(\mathtt{0x400} \cdot \left(A_{0}^{(+1)}\right)\right) + -\left(A_{4}^{(+1)}\right) = 0$$

Polynomial 93 (original index 93)

$$A_{0}^{(+1)} + \mathtt{0x100000\ldots} \cdot \left(A_{1} + -\left(\mathtt{0x400} \cdot \left(A_{1}^{(+1)}\right)\right) + \mathtt{0x400} \cdot \left(A_{2}^{(+1)}\right)\right) + -\left(A_{3}\right) = 0$$

Polynomial 94 (original index 94)

$$A_{3}^{(+1)} + \mathtt{0x20} \cdot \left(A_{2}\right) + -\left(A_{4}\right) = 0$$

Polynomial 95 (original index 95)

$$A_{1}^{(+1)} + -\left(A_{2}^{(+1)} + \mathtt{0x20} \cdot \left(A_{3}^{(+1)}\right)\right) = 0$$

Polynomial 96 (original index 96)

$$\mathtt{0x2} \cdot \left(F_{4}\right) + -\left(\left(A_{8} + A_{9}\right) \cdot \left(A_{5} + -\left(\left(A_{8}\right) \cdot \left(A_{8}\right) + -\left(A_{5}\right) + -\left(A_{6}\right)\right)\right)\right) = 0$$

Polynomial 106 (original index 106)

$$\left(A_{4}\right) \cdot \left(\mathtt{0x1} + -\left(A_{4}\right)\right) = 0$$

Polynomial 107 (original index 107)

$$\left(A_{4}^{(+1)}\right) \cdot \left(\mathtt{0x1} + -\left(A_{4}^{(+1)}\right)\right) = 0$$

Polynomial 108 (original index 108)

$$A_{2} + -\left(A_{3} + \mathtt{0x10} \cdot \left(A_{4}\right) + \mathtt{0x20} \cdot \left(A_{5}\right)\right) = 0$$

Polynomial 109 (original index 109)

$$A_{2}^{(+1)} + -\left(A_{3}^{(+1)} + \mathtt{0x200} \cdot \left(A_{4}^{(+1)}\right)\right) = 0$$

Polynomial 110 (original index 110)

$$A_{1} + \mathtt{0x400000\ldots} \cdot \left(A_{3}\right) + \mathtt{0x400000\ldots} \cdot \left(A_{4}\right) + -\left(A_{0}\right) = 0$$

Polynomial 111 (original index 111)

$$A_{5} + \mathtt{0x20} \cdot \left(A_{1}^{(+1)}\right) + \mathtt{0x200000\ldots} \cdot \left(A_{3}^{(+1)}\right) + \mathtt{0x400000\ldots} \cdot \left(A_{4}^{(+1)}\right) + -\left(A_{0}^{(+1)}\right) = 0$$

Polynomial 112 (original index 112)

$$\left(A_{4}\right) \cdot \left(A_{3}\right) = 0$$

Polynomial 113 (original index 113)

$$\left(A_{4}\right) \cdot \left(A_{6}\right) = 0$$

Polynomial 114 (original index 114)

$$A_{1} + \mathtt{0x400000\ldots} + -\left(\mathtt{0x224698\ldots}\right) + -\left(A_{7}\right) = 0$$

Polynomial 115 (original index 115)

$$\left(A_{4}\right) \cdot \left(A_{8}\right) = 0$$

Polynomial 116 (original index 116)

$$\left(A_{4}^{(+1)}\right) \cdot \left(A_{3}^{(+1)}\right) = 0$$

Polynomial 117 (original index 117)

$$\left(A_{4}^{(+1)}\right) \cdot \left(A_{6}^{(+1)}\right) = 0$$

Polynomial 118 (original index 118)

$$A_{5} + \mathtt{0x20} \cdot \left(A_{1}^{(+1)}\right) + \mathtt{0x100000\ldots} + -\left(\mathtt{0x224698\ldots}\right) + -\left(A_{7}^{(+1)}\right) = 0$$

Polynomial 119 (original index 119)

$$\left(A_{4}^{(+1)}\right) \cdot \left(A_{8}^{(+1)}\right) = 0$$

Polynomial 120 (original index 120)

$$\left(A_{8}\right) \cdot \left(\mathtt{0x1} + -\left(A_{8}\right)\right) = 0$$

Polynomial 121 (original index 121)

$$\left(A_{7}^{(+1)}\right) \cdot \left(\mathtt{0x1} + -\left(A_{7}^{(+1)}\right)\right) = 0$$

Polynomial 122 (original index 122)

$$A_{6} + -\left(A_{7} + \mathtt{0x10} \cdot \left(A_{8}\right) + \mathtt{0x20} \cdot \left(A_{7}^{(+1)}\right) + \mathtt{0x40} \cdot \left(A_{8}^{(+1)}\right)\right) = 0$$

Polynomial 123 (original index 123)

$$\left(A_{7}\right) \cdot \left(\mathtt{0x1} + -\left(A_{7}\right)\right) = 0$$

Polynomial 124 (original index 124)

$$\left(A_{8}\right) \cdot \left(\mathtt{0x1} + -\left(A_{8}\right)\right) = 0$$

Polynomial 125 (original index 125)

$$A_{6} + -\left(A_{7} + \mathtt{0x2} \cdot \left(A_{8}\right) + \mathtt{0x4} \cdot \left(A_{7}^{(+1)}\right) + \mathtt{0x400} \cdot \left(A_{8}^{(+1)}\right)\right) = 0$$

Polynomial 126 (original index 126)

$$A_{6} + -\left(A_{7} + \mathtt{0x40} \cdot \left(A_{8}\right)\right) = 0$$

Group 11 (envelope column $F_{26}$, 21 polynomials)

Polynomial 99 (original index 99)

$$A_{7} + -\left(\left(A_{9}\right) \cdot \left(A_{6}\right) + \left(\mathtt{0x1} + -\left(A_{9}\right)\right) \cdot \left(A_{5}\right)\right) = 0$$

Polynomial 100 (original index 100)

$$A_{8} + -\left(\left(A_{9}\right) \cdot \left(A_{5}\right) + \left(\mathtt{0x1} + -\left(A_{9}\right)\right) \cdot \left(A_{6}\right)\right) = 0$$

Polynomial 101 (original index 101)

$$\left(A_{9}\right) \cdot \left(\mathtt{0x1} + -\left(A_{9}\right)\right) = 0$$

Polynomial 102 (original index 102)

$$A_{5} + -\left(\mathtt{0x400} \cdot \left(A_{5}^{(+1)}\right)\right) + -\left(A_{9}^{(+1)}\right) = 0$$

Polynomial 103 (original index 103)

$$A_{5}^{(+1)} + \mathtt{0x100000\ldots} \cdot \left(A_{6} + -\left(\mathtt{0x400} \cdot \left(A_{6}^{(+1)}\right)\right) + \mathtt{0x400} \cdot \left(A_{7}^{(+1)}\right)\right) + -\left(A_{8}\right) = 0$$

Polynomial 104 (original index 104)

$$A_{8}^{(+1)} + \mathtt{0x20} \cdot \left(A_{7}\right) + -\left(A_{9}\right) = 0$$

Polynomial 105 (original index 105)

$$A_{6}^{(+1)} + -\left(A_{7}^{(+1)} + \mathtt{0x20} \cdot \left(A_{8}^{(+1)}\right)\right) = 0$$

Polynomial 127 (original index 127)

$$\left(A_{7}\right) \cdot \left(\mathtt{0x1} + -\left(A_{7}\right)\right) = 0$$

Polynomial 128 (original index 128)

$$A_{6} + -\left(A_{7} + \mathtt{0x2} \cdot \left(A_{6}^{(+1)}\right) + \mathtt{0x400} \cdot \left(A_{7}^{(+1)}\right)\right) = 0$$

Polynomial 129 (original index 129)

$$\left(A_{8}\right) \cdot \left(\mathtt{0x1} + -\left(A_{8}\right)\right) = 0$$

Polynomial 130 (original index 130)

$$A_{6} + -\left(A_{7} + \mathtt{0x20} \cdot \left(A_{8}\right)\right) = 0$$

Polynomial 131 (original index 131)

$$A_{8} + \mathtt{0x400000\ldots} \cdot \left(A_{7}\right) + \mathtt{0x400000\ldots} \cdot \left(A_{7}^{(+1)}\right) + -\left(A_{6}\right) = 0$$

Polynomial 132 (original index 132)

$$A_{8} + \mathtt{0x400000\ldots} + -\left(\mathtt{0x224698\ldots}\right) + -\left(A_{8}^{(+1)}\right) = 0$$

Polynomial 133 (original index 133)

$$\left(A_{7}^{(+1)}\right) \cdot \left(A_{7}\right) = 0$$

Polynomial 134 (original index 134)

$$\left(A_{7}^{(+1)}\right) \cdot \left(A_{9}\right) = 0$$

Polynomial 135 (original index 135)

$$\left(A_{7}^{(+1)}\right) \cdot \left(A_{9}^{(+1)}\right) = 0$$

Polynomial 136 (original index 136)

$$A_{7} + \mathtt{0x10} \cdot \left(A_{8}\right) + \mathtt{0x400000\ldots} \cdot \left(A_{7}^{(+1)}\right) + -\left(A_{6}\right) = 0$$

Polynomial 137 (original index 137)

$$A_{7} + \mathtt{0x10} \cdot \left(A_{8}\right) + \mathtt{0x100000\ldots} + -\left(\mathtt{0x224698\ldots}\right) + -\left(A_{8}^{(+1)}\right) = 0$$

Polynomial 138 (original index 138)

$$\left(A_{7}^{(+1)}\right) \cdot \left(A_{9}\right) = 0$$

Polynomial 139 (original index 139)

$$\left(A_{7}^{(+1)}\right) \cdot \left(A_{9}^{(+1)}\right) = 0$$

Polynomial 140 (original index 140)

$$A_{7} + \mathtt{0x100} \cdot \left(A_{8}\right) + \mathtt{0x400000\ldots} \cdot \left(A_{9}\right) + -\left(A_{6}\right) = 0$$

Group 12 (envelope column $F_{27}$, 25 polynomials)

Polynomial 141 (original index 141)

$$A_{7} + \mathtt{0x10} \cdot \left(A_{8}\right) + \mathtt{0x400000\ldots} \cdot \left(A_{7}^{(+1)}\right) + -\left(A_{6}\right) = 0$$

Polynomial 142 (original index 142)

$$A_{7} + \mathtt{0x10} \cdot \left(A_{8}\right) + \mathtt{0x100000\ldots} + -\left(\mathtt{0x224698\ldots}\right) + -\left(A_{8}^{(+1)}\right) = 0$$

Polynomial 143 (original index 143)

$$\left(A_{7}^{(+1)}\right) \cdot \left(A_{9}\right) = 0$$

Polynomial 144 (original index 144)

$$\left(A_{7}^{(+1)}\right) \cdot \left(A_{9}^{(+1)}\right) = 0$$

Polynomial 145 (original index 145)

$$A_{7} + \mathtt{0x200} \cdot \left(A_{8}\right) + \mathtt{0x200000\ldots} \cdot \left(A_{6}^{(+1)}\right) + \mathtt{0x400000\ldots} \cdot \left(A_{7}^{(+1)}\right) + -\left(A_{6}\right) = 0$$

Polynomial 146 (original index 146)

$$A_{7} + \mathtt{0x200} \cdot \left(A_{8}\right) + \mathtt{0x400000\ldots} + -\left(\mathtt{0x224698\ldots}\right) + -\left(A_{8}^{(+1)}\right) = 0$$

Polynomial 147 (original index 147)

$$\left(A_{7}^{(+1)}\right) \cdot \left(A_{6}^{(+1)}\right) = 0$$

Polynomial 148 (original index 148)

$$\left(A_{7}^{(+1)}\right) \cdot \left(A_{9}\right) = 0$$

Polynomial 149 (original index 149)

$$\left(A_{7}^{(+1)}\right) \cdot \left(A_{9}^{(+1)}\right) = 0$$

Polynomial 150 (original index 150)

$$\left(A_{9}\right) \cdot \left(\mathtt{0x1} + -\left(A_{9}\right)\right) = 0$$

Polynomial 151 (original index 151)

$$A_{5}^{(+1)} + -\left(A_{6} + \mathtt{0x2} \cdot \left(A_{7}\right) + \mathtt{0x400} \cdot \left(A_{6}^{(+1)}\right)\right) = 0$$

Polynomial 152 (original index 152)

$$A_{5} + -\left(A_{5}^{(+1)} + \mathtt{0x400000\ldots} \cdot \left(A_{8}\right) + \mathtt{0x400000\ldots} \cdot \left(A_{9}\right)\right) = 0$$

Polynomial 153 (original index 153)

$$A_{5}^{(+1)} + \mathtt{0x400000\ldots} + -\left(\mathtt{0x224698\ldots}\right) + -\left(A_{8}^{(+1)}\right) = 0$$

Polynomial 154 (original index 154)

$$\left(A_{9}\right) \cdot \left(A_{8}\right) = 0$$

Polynomial 155 (original index 155)

$$\left(A_{9}\right) \cdot \left(A_{7}^{(+1)}\right) = 0$$

Polynomial 156 (original index 156)

$$\left(A_{9}\right) \cdot \left(A_{9}^{(+1)}\right) = 0$$

Polynomial 157 (original index 157)

$$\left(A_{8}\right) \cdot \left(\mathtt{0x1} + -\left(A_{8}\right)\right) = 0$$

Polynomial 158 (original index 158)

$$\left(A_{7}^{(+1)}\right) \cdot \left(\mathtt{0x1} + -\left(A_{7}^{(+1)}\right)\right) = 0$$

Polynomial 159 (original index 159)

$$A_{6} + -\left(A_{7} + \mathtt{0x10} \cdot \left(A_{8}\right) + \mathtt{0x20} \cdot \left(A_{7}^{(+1)}\right) + \mathtt{0x40} \cdot \left(A_{8}^{(+1)}\right)\right) = 0$$

Polynomial 160 (original index 160)

$$\left(A_{7}\right) \cdot \left(\mathtt{0x1} + -\left(A_{7}\right)\right) = 0$$

Polynomial 161 (original index 161)

$$\left(A_{8}\right) \cdot \left(\mathtt{0x1} + -\left(A_{8}\right)\right) = 0$$

Polynomial 162 (original index 162)

$$A_{6} + -\left(A_{7} + \mathtt{0x2} \cdot \left(A_{8}\right) + \mathtt{0x4} \cdot \left(A_{7}^{(+1)}\right) + \mathtt{0x400} \cdot \left(A_{8}^{(+1)}\right)\right) = 0$$

Polynomial 163 (original index 163)

$$A_{6} + -\left(A_{7} + \mathtt{0x40} \cdot \left(A_{8}\right)\right) = 0$$

Polynomial 164 (original index 164)

$$\left(A_{7}\right) \cdot \left(\mathtt{0x1} + -\left(A_{7}\right)\right) = 0$$

Polynomial 165 (original index 165)

$$A_{6} + -\left(A_{7} + \mathtt{0x2} \cdot \left(A_{6}^{(+1)}\right) + \mathtt{0x400} \cdot \left(A_{7}^{(+1)}\right)\right) = 0$$

Group 13 (envelope column $F_{28}$, 28 polynomials)

Polynomial 166 (original index 166)

$$\left(A_{8}\right) \cdot \left(\mathtt{0x1} + -\left(A_{8}\right)\right) = 0$$

Polynomial 167 (original index 167)

$$A_{6} + -\left(A_{7} + \mathtt{0x20} \cdot \left(A_{8}\right)\right) = 0$$

Polynomial 168 (original index 168)

$$A_{8} + \mathtt{0x400000\ldots} \cdot \left(A_{7}\right) + \mathtt{0x400000\ldots} \cdot \left(A_{7}^{(+1)}\right) + -\left(A_{6}\right) = 0$$

Polynomial 169 (original index 169)

$$A_{8} + \mathtt{0x400000\ldots} + -\left(\mathtt{0x224698\ldots}\right) + -\left(A_{8}^{(+1)}\right) = 0$$

Polynomial 170 (original index 170)

$$\left(A_{7}^{(+1)}\right) \cdot \left(A_{7}\right) = 0$$

Polynomial 171 (original index 171)

$$\left(A_{7}^{(+1)}\right) \cdot \left(A_{9}\right) = 0$$

Polynomial 172 (original index 172)

$$\left(A_{7}^{(+1)}\right) \cdot \left(A_{9}^{(+1)}\right) = 0$$

Polynomial 173 (original index 173)

$$A_{7} + \mathtt{0x10} \cdot \left(A_{8}\right) + \mathtt{0x400000\ldots} \cdot \left(A_{7}^{(+1)}\right) + -\left(A_{6}\right) = 0$$

Polynomial 174 (original index 174)

$$A_{7} + \mathtt{0x10} \cdot \left(A_{8}\right) + \mathtt{0x100000\ldots} + -\left(\mathtt{0x224698\ldots}\right) + -\left(A_{8}^{(+1)}\right) = 0$$

Polynomial 175 (original index 175)

$$\left(A_{7}^{(+1)}\right) \cdot \left(A_{9}\right) = 0$$

Polynomial 176 (original index 176)

$$\left(A_{7}^{(+1)}\right) \cdot \left(A_{9}^{(+1)}\right) = 0$$

Polynomial 177 (original index 177)

$$A_{7} + \mathtt{0x100} \cdot \left(A_{8}\right) + \mathtt{0x400000\ldots} \cdot \left(A_{9}\right) + -\left(A_{6}\right) = 0$$

Polynomial 178 (original index 178)

$$A_{7} + \mathtt{0x10} \cdot \left(A_{8}\right) + \mathtt{0x400000\ldots} \cdot \left(A_{7}^{(+1)}\right) + -\left(A_{6}\right) = 0$$

Polynomial 179 (original index 179)

$$A_{7} + \mathtt{0x10} \cdot \left(A_{8}\right) + \mathtt{0x100000\ldots} + -\left(\mathtt{0x224698\ldots}\right) + -\left(A_{8}^{(+1)}\right) = 0$$

Polynomial 180 (original index 180)

$$\left(A_{7}^{(+1)}\right) \cdot \left(A_{9}\right) = 0$$

Polynomial 181 (original index 181)

$$\left(A_{7}^{(+1)}\right) \cdot \left(A_{9}^{(+1)}\right) = 0$$

Polynomial 182 (original index 182)

$$A_{7} + \mathtt{0x200} \cdot \left(A_{8}\right) + \mathtt{0x200000\ldots} \cdot \left(A_{6}^{(+1)}\right) + \mathtt{0x400000\ldots} \cdot \left(A_{7}^{(+1)}\right) + -\left(A_{6}\right) = 0$$

Polynomial 183 (original index 183)

$$A_{7} + \mathtt{0x200} \cdot \left(A_{8}\right) + \mathtt{0x400000\ldots} + -\left(\mathtt{0x224698\ldots}\right) + -\left(A_{8}^{(+1)}\right) = 0$$

Polynomial 184 (original index 184)

$$\left(A_{7}^{(+1)}\right) \cdot \left(A_{6}^{(+1)}\right) = 0$$

Polynomial 185 (original index 185)

$$\left(A_{7}^{(+1)}\right) \cdot \left(A_{9}\right) = 0$$

Polynomial 186 (original index 186)

$$\left(A_{7}^{(+1)}\right) \cdot \left(A_{9}^{(+1)}\right) = 0$$

Polynomial 187 (original index 187)

$$\left(A_{9}\right) \cdot \left(\mathtt{0x1} + -\left(A_{9}\right)\right) = 0$$

Polynomial 188 (original index 188)

$$A_{5}^{(+1)} + -\left(A_{6} + \mathtt{0x2} \cdot \left(A_{7}\right) + \mathtt{0x400} \cdot \left(A_{6}^{(+1)}\right)\right) = 0$$

Polynomial 189 (original index 189)

$$A_{5} + -\left(A_{5}^{(+1)} + \mathtt{0x400000\ldots} \cdot \left(A_{8}\right) + \mathtt{0x400000\ldots} \cdot \left(A_{9}\right)\right) = 0$$

Polynomial 190 (original index 190)

$$A_{5}^{(+1)} + \mathtt{0x400000\ldots} + -\left(\mathtt{0x224698\ldots}\right) + -\left(A_{8}^{(+1)}\right) = 0$$

Polynomial 191 (original index 191)

$$\left(A_{9}\right) \cdot \left(A_{8}\right) = 0$$

Polynomial 192 (original index 192)

$$\left(A_{9}\right) \cdot \left(A_{7}^{(+1)}\right) = 0$$

Polynomial 193 (original index 193)

$$\left(A_{9}\right) \cdot \left(A_{9}^{(+1)}\right) = 0$$