[JoyboyBrian]

The width-two standalone paid-history threshold survives enlarging ordinary operations from {I,X} to the phase-complete Pauli group <iI,X,Z>. For targets (aI+bX)^tensor f with ab(a²-b²) != 0, width one is ordinary, while every width at least two needs at least one new single-row paid return per distinct independent target owner. The matching upper bound is conditional on each target being an available paid primitive. No new kappa or physical caller admission is claimed.

PR5446's general projective-group argument only gives width at least seven for this group. PR5450's smaller-alphabet projector does not remain one-dimensional after adding Z. The new proof handles the actual phase twist: an even-width character and odd-width full module separation establish the uniform bound without assuming scalar coefficients for a matrix-algebra component. Arbitrary width-dependent ordinary words, scalar coefficients, independent dirty inputs and paid feedback remain allowed.

The package includes an exact Q(i) Kronecker interpreter, an independently implemented finite-field bit-index oracle and six optimized semantic mutation controls that bypass stored-receipt comparison. It changes only research/pauli-twisted-history-threshold/; no predecessor code, frozen source, selected result or shared tooling is modified.

Validation on CPython 3.12.2 / macOS arm64:

• All five registered commands pass: Gaussian-rational and independent modular engines in normal/optimized modes, plus six semantic mutants.
• 24 literal Q(i) cases, 64 phase-group lifts, 18 full owner/history controls, 366 scalar identities and 13 boundary controls; the separate oracle covers 512 finite-field tensor cases and 3,072 separation identities.
• Committed-byte/source-history checks pass for all nine payloads. Official export and offline integrity checks pass.
• All five commands pass again from the read-only export with Git absent from PATH; ten files including the receipt remain byte-identical.
• Separate Codex proof review found no blocking mathematical issue; this is not independent human or formal-kernel acceptance.

Reproduce:

python3.12 -B scripts/research_package.py --package research/pauli-twisted-history-threshold
python3.12 -B scripts/check_research_commit.py --package research/pauli-twisted-history-threshold --check-source-history

Proof SHA256: f3f3972133a614cf9406cee6cfb78e62f51fad4fd57a1ed367467c75af1c2001.
Publication head: 57b126b32cf027119a4268f512547deebc521e18.
Base: 32890bc5de034ab8fc93ac88ead7279a6a416d04, priority revision seven.
Canonical investigation: #5476.

Credit Chris R / Clear1nsight and retained PR5425/5446 history authors, Ali Romman's PR5450 and larger-group handoff, classical Pauli/character/tensor theory, and retained native/helper/compiler/analytic contributors. JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0. Immutable external proof citations are inspected references, not unmerged Git-object replay dependencies.

Remaining obligations: bind a real caller's versioned initial observations before charging this standalone result; establish physical paid-primitive availability, complete prices, precision, storage and tape costs separately. No full selected-frontier replay, native supplier admission, Lean or hosted CI is claimed.