[petalas]

Result

Scoped proof with finite corroboration. No new κ. For every n >= 2, the projective tensor Gram matrix of the full n-qubit Gaussian-dyadic Clifford group is singular at every width f <= 2n. Its pointwise tensor-independence threshold therefore satisfies

2n + 1 <= T(n) <= 4n + 4        (upper bound: #5526)
7 <= T(3) <= 16

Before this, the only exact values were T(1)=5 (#5549) and T(2)=9 (#5675), and there was no lower bound for general n. With cited code lemmas the bound improves to T(n) >= 2n+3 for n = 1 mod 4, n >= 5 (#5635), and to T(n) >= 2n+2 for n = 2 mod 4 (#5902).

This partially answers MatShep's Open next question in #5675 ("does the sharp threshold 4n+1 continue at three or more qubits?"). It proves that T(n) grows linearly. It leaves the exact three-qubit value open.

Argument

• The Hadamard-free affine/quadratic monomial native group (Prove a complete four-width criterion for affine-quadratic native identities #5369/Lower native monomial cutoff and certify the three-axis window 7–10 #5529) is a Clifford subgroup with the same four scalars. Its Gram matrix is therefore a principal submatrix of the Clifford Gram. A null vector of a principal submatrix of a PSD matrix is a null vector of the whole matrix.
• Ali Romman's cuspidal residual from Prove the sharp nine-axis native window and a general cutoff family #5635, projected to zero phase, is a fixed group-algebra element. It annihilates every tuple with proper affine span (Green's theorem, cited through Prove the sharp nine-axis native window and a general cutoff family #5635).
• New all-n code step. A zero-phase tuple of length at most 2n cannot have full affine span. Pad it to length 2n. The row code C plus the all-one word would then be self-orthogonal of dimension n+1 > n. Prove the sharp nine-axis native window and a general cutoff family #5635 states its sharper code step only for n = 1 mod 4.
• The residual is nonzero, because the explicit full-span zero-phase tuple (0, then four copies of each e_j; length 4n+1) sees the diagonal entry Ψ(1) > 0.

Evidence

verify.py uses only the standard library and exact integer and Gaussian-integer arithmetic. It runs in about 3 s, in normal and -O modes.

• Code step, checked exhaustively at n = 2, 3, 4: 7, 34 and 458 zero-phase multisets of size at most 2n, none with full affine span. The explicit length-(4n+1) tuple is checked for n = 2..64.
• Cuspidal coset identity on GL(2,2) and GL(3,2) at 1,194 (g, hyperplane) pairs.
• Literal two-qubit transfer. All 3,072 monomial matrices normalize the Pauli group and form 768 projective classes. At each width f = 1..4 the residual's class vector is nonzero and is annihilated by the exact class Gram (tr(U_a^* U_b))^f. At widths 9..12 its diagonal entry is 128 = |D|·Ψ(1).
• Three rejection controls each fail H_4 v = 0: the trivial character, the sign with its order-three values negated, and one perturbed coefficient.

Reproduce from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B scripts/research_package.py --package research/clifford-threshold-linear-lower
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B scripts/check_research_commit.py --package research/clifford-threshold-linear-lower --check-source-history
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/clifford-threshold-linear-lower/verify.py

All three passed locally on CPython 3.12 (macOS) at commit b6c450dc9ae33f167dc06163a89367c513389f27, and the two repository checks passed again at the current head 49404527570c90f4a19708004c81533398b5e7b5. The markers are PASS_REGISTERED_RESEARCH_PACKAGE, PASS_RESEARCH_COMMIT_BYTES and PASS_CLIFFORD_THRESHOLD_LINEAR_LOWER. Each registered check takes about 3 s.

Limitations

• Green's character theorem enters through Prove the sharp nine-axis native window and a general cutoff family #5635's cuspidal lemma. It is cited, not proof-kernel checked. The n = 2, 3 coset checks only corroborate it.
• 2n+1 is not sharp at n = 2 (T = 9). The theorem neither proves nor refutes T(3) = 13.
• Spectral constants, physical callers, invoices, compiler, precision, tape, stock and κ are out of scope.
• Independent human and formal proof-kernel review are unclaimed. The selected frontier and the five framework assumptions are unchanged.

Open next questions

• Exact T(3), now in [7,16]. This would decide whether 4n+1 continues. A first step is to compute the centre action of the three-qubit Clifford group on class functions at widths 7..13, as Sharp two-qubit Clifford tensor threshold 9 and Gram condition 219/28 #5675 did at two qubits. The blocker is that the group has order 4 × 92,897,280 and no class table is in the repository.
• A full-group residual. The residual here lies in the Hadamard-free subgroup, whose own threshold is only about 2n+3. Moving the all-n bound toward 4n+1 needs a null vector that uses Hadamard-type elements. A first step is an all-n analogue of Sharp two-qubit Clifford tensor threshold 9 and Gram condition 219/28 #5675's width-8 determinant witness; the determinant itself is trivial for n >= 3.

Related open work

• Prove sharp-order Clifford row-sum convergence (no new κ) #5939 (row-sum convergence of the full-projective Clifford Gram at width 4n+4) studies the same Gram matrices from the sufficient/spectral side. This PR computes no row sums, so there is nothing to compare.
• Refreshed after CI: none of this account's other PRs opened since the claim (Prove exact eight-call tariff for four inverse C3 targets (no κ) #5963–Construct optimal theta sharing in encoded-width time (no new κ) #5966, Save one boundary XOR at four calls with both output updates nonzero #5968–Classify P104 lattices invariant under local Clifford gates (no κ) #5971) studies the Clifford tensor Gram or its threshold.

Credit

MatShep posed the question in #5675 and proved T(1), T(2) (#5549, #5675). Ali Romman proved the 4n+4 upper bound (#5526), the native grammar and Fourier framework (#5369, #5529) and the cuspidal residual (#5635). Joseph Anglada introduced the coset method (#5542). The n = 2 mod 4 refinement uses #5902. Advance claim: #1166 (comment)

CI notes

All hosted checks passed at head 49404527570c90f4a19708004c81533398b5e7b5: plan and replay (research/clifford-threshold-linear-lower), run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40540527148. This head changes only the README credit sentence relative to b6c450dc9.