[aliromman]

This enabling proof gives the exact minimum native Gaussian rank over all symplectic completions of any compatible partial signed Pauli map. A dimension-uniform constructive Lagrangian argument attains the formula; a Pauli correction fixes all compatible signs without changing the rank. The complete kappa calculation is unchanged.

Proof and scope, exact checker, replay declaration, and source provenance are pinned to the reviewed commit. Base: canonical main ff3cc58e370d0f678e529246f8c73e8094e18fcf, priority revision 7.

CPython 3.12.14 registered normal and optimized replays passed: 10,113 complete extension fibers, 386 signed lifts, 848 literal matrix-column identities and five controls. Both print PASS_PARTIAL_PAULI_GAUSSIAN_MINIMUM; the launcher prints PASS_REGISTERED_RESEARCH_PACKAGE. All five package files match their committed bytes, and the base is an ancestor of the package commit.

python3 -B scripts/research_package.py --package research/partial-pauli-gaussian-minimum

Use Python 3.12 and the standard library. No external runtime inputs or third-party Python packages are required. Hosted planning and the registered package replay passed on this exact head in run 39998914288. No predecessor scientific suite, selected cold replay, native replay, compiler, Lean or Docker build was run. The optional full-history preflight is unavailable in this shallow checkout. Finite checks corroborate the written proof; they do not constitute independent human or proof-kernel acceptance.

Implement the quotient construction on one actual contact constraint set, retain its literal phase and charge all monomial passes. Source preparation, native operations, precision, movement, and supplier/outer admission remain separate obligations.

Credit @JoyboyBrian's PR #5051 supplies the signed partial-map contract, compatibility test, fixed-prefix extension and turn convention in pauli.py. @Dugongue's PR #4841 supplies the native Gaussian rank, monomial-factor bill and absolute-phase contract in the compiler proof. @petalas's partial-transvection work is a separate tariff predecessor, as recorded in the released handoff. Symplectic basis extension and the Clifford/Bruhat methods are classical; this result addresses minimization over the prescribed extension fiber. The checker adapts the credited source's literal turn conventions and extension steps, and independently enumerates the full small completion fibers.

Prepared by @aliromman with substantial OpenAI Codex assistance; Apache-2.0.

[jwr404]

Independent review (James Reed, with Claude Code). Head cc0f40e71133a9ddb922bf39c93785c1a43017b9.

Verdict: the formula checks out beyond the PR's n ≤ 2 census, and the argument reads correctly.

Independent n = 3 test

I wrote a separate C program that shares no code with verify.py:

• It enumerates all of Sp(6, 2) by column DFS, giving exactly 1,451,520 elements (the correct group order).
• For each element it precomputes dim(Z + gZ) − 3, with Z the z-type Lagrangian.
• For each random test it takes a random g₀, a random subspace U of uniform random dimension 0–6 and h = g₀|_U. It computes the exact fiber minimum over every g with g|_U = h and compares it with dim B + t − 2c, where A = U ∩ Z, B = h(A), c = dim(B ∩ Z) and t = dim(h(U) ∩ Z ∩ B^⊥).

Over 20,000 random partial maps there are 0 mismatches. The fiber minima histogram is 0: 7,351, 1: 5,414, 2: 4,960, 3: 2,275, so every value is exercised. The run takes about 60 s; I can attach the program if you want it as an optional extra check.

Proof read

• Transverse-extension lemma. The pairings in the displayed Lagrangian vanish mod 2, and the components outside W give disjointness. ✓
• Maximum intersection m − r₀.
  • The upper bound holds since L ∩ R misses D = R ∩ W. ✓
  • For attainment, π: R → D is the identity on D and onto D on N = R ∩ W^⊥ (possible since dim N ≥ r₀). With K = ker π, K ∩ W = 0 because K ∩ D = 0 and K ⊆ R. The image of W ∩ K^⊥ in K^⊥/K then has dimension r₀, half the ambient. I checked this dimension count; it relies on K + N = R, which your construction of π gives.
• Reduction to B^⊥/B. L ⊇ B, L ∩ U′ = B, dim(R ∩ W) = t − c and dim(L ∩ Z) = c + dim((L/B) ∩ R) combine to give n − [c + (n−b) − (t−c)] = b + t − 2c, matching the formula. ✓
• Sign correction. A Pauli correction realizes any linear sign functional, by nondegeneracy, without changing gZ. ✓

The registered replay passes locally (PASS_REGISTERED_RESEARCH_PACKAGE).

Minor: the README could note explicitly that the minimum is n − max dim(Z ∩ gZ). That form is what the quotient argument computes, and it makes the sign-independence remark immediate.