chore(patch): Make public inputs the start of the UH and MH proof #12266

[aakoshh]

Description

Problem*

Upstream test program changes from AztecProtocol/aztec-packages#12266

Summary*

Got some build failures in AztecProtocol/aztec-packages#12592 due to changes made to the honk test programs 2 days ago. I added a patch file in that PR. This PR is bringing the changes in that patch file back to noir.

Additional Context

To create the patch file, I made the fixes based on the changes in 12266 in my local checkout of noir/noir-repo, then did the following:

cd noir/noir-repo
git add . 
git commit -m "patch: Make public inputs the start of the UH and MH proof #12266"
cd ..
scripts/sync.sh make-patch
git add noir-repo.patch
git commit -m "patch file for #12266"

The way to bring it back was something like this:

cd noir/noir-repo
git checkout -b af/patch-12266
# Get rid the fixup and empty patch marker commits
git rebase -i $(cat ../noir-repo-ref)
git remote add upstream git@github.com:noir-lang/noir.git
git push --set-upstream upstream af/patch-12266
cd ..
# Now re-apply the fixup 
scripts/sync.sh update

Documentation*

Check one:

• No documentation needed.
• Documentation included in this PR.
• [For Experimental Features] Documentation to be submitted in a separate PR.

PR Checklist*

• I have tested the changes locally.
• I have formatted the changes with Prettier and/or cargo fmt on default settings.

[github-actions]

Changes to number of Brillig opcodes executed

Generated at commit: ef8b63ff5714eb7c90f98cfbcc3c23d81497a755, compared to commit: c30bcd7e753d835442187bb27a6c14231041ed77

🧾 Summary (10% most significant diffs)

Program	Brillig opcodes (+/-)	%
double_verify_honk_proof_inliner_max	-42 ✅	-0.57%
double_verify_honk_proof_inliner_min	-42 ✅	-0.57%
double_verify_honk_proof_inliner_zero	-42 ✅	-0.57%

---

Full diff report 👇

Program	Brillig opcodes (+/-)	%
verify_rollup_honk_proof_inliner_max	4,799 (-21)	-0.44%
verify_rollup_honk_proof_inliner_min	4,799 (-21)	-0.44%
verify_rollup_honk_proof_inliner_zero	4,799 (-21)	-0.44%
verify_honk_proof_inliner_max	4,169 (-21)	-0.50%
verify_honk_proof_inliner_min	4,169 (-21)	-0.50%
verify_honk_proof_inliner_zero	4,169 (-21)	-0.50%
double_verify_honk_proof_inliner_max	7,379 (-42)	-0.57%
double_verify_honk_proof_inliner_min	7,379 (-42)	-0.57%
double_verify_honk_proof_inliner_zero	7,379 (-42)	-0.57%

[TomAFrench]

Closing as these files have been removed in a different PR.