[petalas]

Result

Scoped proof with exact finite corroboration. No new κ.

This answers the open boundary of JoyboyBrian's #5482 (research/pauli-twisted-history-threshold), which "does not classify arbitrary Pauli-nonordinary matrices."

Theorem 1. In #5482's phase-complete Pauli history model, the least number of returned rows for arbitrary targets H_j on distinct owners is exactly:

• Even width f: the largest number of owners whose target is nonordinary in one common character of the commutative algebra R=span{I,P,C,PC}.
• Odd width f: ceil(sum_j s_j / 2). Here R is isomorphic to M_2(K) and s_j=(dim(R+R H_j)-4)/2 is in {0,1,2}.

The lower bound holds for every program in the model. Explicit returned rows attain it.

Corollaries.

• Every tensor square (f=2). The profile of M tensor M is the set of forms B in {I,X,Z} with M^T B M not proportional to B. A short case analysis shows that a non-Pauli M has two or three of them. The minimum is therefore at least ceil(2q/3), and this is sharp. Owners with targets (2I+X)^⊗2, (2I+Y)^⊗2 and (2I+Z)^⊗2 need 2 returns, not 3.
• Every rank-one tensor power (f≥3). For M=u w^T, the odd-width excess is 1 exactly when u is an eigenvector of X, Z or XZ, so two such owners share one return. At even width the profile has all four characters, or exactly two for Pauli eigenvectors, tabulated by f mod 4. These profiles force q when f≡0 mod 4 and allow ceil(2q/3) when f≡2 mod 4.
• Per-owner charging is forced exactly when one character is shared by all owners (even f), or when at most one owner has excess 1 and none has excess 0 (odd f). The Prove sharp paid-history threshold for phase-complete Pauli operations (no new kappa) #5482 family satisfies both conditions, so this agrees with its q-call result.

Evidence and reproduction

From the repository root, with /Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 (CPython 3.12.15; substitute any CPython 3.12):

python3.12 -W error -B scripts/research_package.py --package research/pauli-history-arbitrary-targets
python3.12 -W error -B scripts/check_research_commit.py --package research/pauli-history-arbitrary-targets --check-source-history
python3.12 -W error -B research/pauli-history-arbitrary-targets/verify.py
python3.12 -O -W error -B research/pauli-history-arbitrary-targets/verify.py

Local results at b017210c92b8c9cda4bcb1debf130313bc908fbd:

• PASS_REGISTERED_RESEARCH_PACKAGE in 3.7 s.
• PASS_RESEARCH_COMMIT_BYTES.
• Both verifier modes print PASS_PAULI_HISTORY_ARBITRARY_TARGETS, in about 1.7 s each.

The verifier uses the standard library only, with exact Q(i) arithmetic. It checks:

• The algebra identities at widths 1 to 5.
• Eleven instances at widths 2 to 5. For each: the formula value, explicit attaining returns, exact membership of every target in the history module, and failure after dropping one return.
• All 625 width-two matrices with entries in {0,±1,±i}. The character profile equals the form profile, and non-Pauli profiles have size 2 or 3. Size counts are 17/192/416 for sizes 0/2/3.
• 48 vectors at widths 3 to 9: dim R u^⊗f = 2 exactly for the 30 Pauli eigenvectors.
• The even-width rank-one profiles at widths 4, 6 and 8.
• Five rejection controls.

The all-width statements rest on PROOF.md, not on these finite checks.

Limitations and unrun obligations

• The minimum counts returned rows in the Prove sharp paid-history threshold for phase-complete Pauli operations (no new kappa) #5482 history model. The attaining rows are explicit, but this PR does not show that a native operator applied to one current field produces them. Prove sharp paid-history threshold for phase-complete Pauli operations (no new kappa) #5482's own upper bound likewise assumes that each target is an available paid primitive.
• Invertible non-Pauli tensor powers at widths ≥3 are not classified. Full invariants there are finite observations only.
• No physical caller, recursive profile, κ or selected-frontier claim. No independent human or proof-kernel review.

Open next questions

1. Does every invertible M outside K{I,X,Z,XZ} have excess 2 at odd f≥3 and all four characters at even f≥4? A yes would extend Prove sharp paid-history threshold for phase-complete Pauli operations (no new kappa) #5482's q-call count to all invertible nonordinary tensor powers.
2. Can two operator calls, each one linear map applied to one current field, deliver the three width-two X/Y/Z-type targets? A yes would make the sharing a real program. A no would give a stronger floor for operator calls.
3. A physical cached boundary must bind its actual observation module and complete bill before either count is charged.

Related open work

No open PR studies this model or quantity. #5482 is the merged predecessor, cited and not rechecked. #5967 studies Clifford tensor-Gram thresholds, a different quantity. The later open PRs #5976, #5977, #5978 and #5980 concern power-variation roots, plane-port decoders and address-twist syzygies, not this model.

CI notes

Both hosted checks passed for b017210c92b8c9cda4bcb1debf130313bc908fbd: plan and replay (research/pauli-history-arbitrary-targets), in run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40566651767.

Credit: JoyboyBrian (#5482) for the model, the history-module inclusion and the (aI+bX) q-call theorem, with that package's retained credits to Chris R / Clear1nsight, the PR5425/PR5446 authors and Ali Romman's PR5450.