[petalas]

Result

Claim type: scoped proof for every target count, with finite corroboration. No new supplier, κ or selected-frontier change.

In the zero-frame dirty-shear model of PR #1306 and PR #2567, take two helpers and T targets with unit-cost single-source XORs and no extra registers. Let a, b, c count the rows (1,0), (0,1), (1,1) of K. For every T ≥ 1 and every K, the restricted and unrestricted minima are equal:

d_P(K) = d_full(K) = a + b + c + min(c, 2).

So target-to-helper feedback never helps for a two-helper shear. This answers the 2×7 part of Open next question 1 in rtyu193-rgb's PR #2567. Through that package's duality it also covers T×2, including 7×2. The 3×6 and 4×5 shapes stay open.

The upper bound is an explicit restricted word: direct reads, plus one shared h0 ^= h1 … h0 ^= h1 block when c ≥ 2. The lower bound counts destinations and allows feedback:

1. every nonzero row needs its target as a destination;
2. if no helper is ever a destination, a target modified once can only receive a pure row;
3. a helper modified once cannot end unchanged.

The full proof is in research/two-helper-zero-frame-feedback/README.md.

Evidence and reproduction

verify.py uses only the standard library and imports no predecessor code. It checks:

• explicit restricted witnesses for every K with T = 1..8 (87,380 matrices), replayed exactly; the gates are linear, so this covers all dirty inputs;
• exact unrestricted and restricted BFS for T = 1..3 (84 matrices) against the formula;
• the formula against PR Finite certificate: no target-feedback saving for any shape with at most eight registers (no new kappa) #2567's pinned 2×4, 2×5 and 2×6 certified minima (5,376 matrices). This calibrates the formula; it is not a replay of that search;
• four rejection controls, including the wrong formula wt(K).

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/two-helper-zero-frame-feedback/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/two-helper-zero-frame-feedback/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/two-helper-zero-frame-feedback
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/two-helper-zero-frame-feedback --check-source-history

All four commands passed locally on CPython 3.12. Each direct run took about 0.45 s and printed PASS_TWO_HELPER_ZERO_FRAME_FEEDBACK witnesses=87380 bfs=84 pr2567=5376 controls=4 feedback_gains=0 no_new_kappa.

Limitations

• Covers H = 2 (and T = 2 by duality), F2, PR Finite certificate: no D-local feedback saving for 3x3 binary shears (no new kappa) #1306's two alphabets, zero frames and no extra registers.
• For T > 8, the result rests on the written proof; the finite checks only corroborate it.
• Independent human and formal proof-kernel review are unclaimed. Lean and Docker were not run.
• Physical precision, hosting and supplier pricing are not addressed.

Open next questions

1. Does feedback help for 3×6, the other nine-register shape besides 4×5? Step 2 of the proof assumes two helpers.
2. Is there a row-type closed form for d_P at every H?
3. Does destination counting extend to twisted two-helper exits, beyond PR Negative result: twisted helper exits have no target-to-helper feedback gain at any fixed 3+3 or five-register exit, plus an every-word contamination-charge lemma (no κ claim) #4071's fixed small shapes?

Related open work

• No open PR studies zero-frame shear feedback minima. Search: open PRs matching "feedback shear", 2026-10-28.
• Classify mixed Q8 call-space shear identities; no pair of shear images separates (no new κ) #6039 (Q8 call-space shear identities) uses "shear" for a different object, identities among Q8 call-space shear images. It does not compute unit-XOR word lengths, so the two PRs share no number.

CI notes

Head d27351424186a70b06cf3dae0eb87e425acd5f2d: both hosted checks passed, plan and replay (research/two-helper-zero-frame-feedback), in run 40608381317.