Sharp two-qubit Clifford tensor threshold 9 and Gram condition 219/28

research/full-clifford-two-qubit-centre/NOTICE.md

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+# Attribution and source references
+
+New work: MatShep, with substantial OpenAI Codex assistance (actual GPT-6.1
+Sol/xhigh for derivation, code, exact checking and documentation). A brief
+actual GPT-6 Astra/high publication sanity check is separate; it is not an
+independent human, proof-kernel or full multiplication-theorem review.
+All new files are Apache-2.0, under the repository license.
+
+Ali Romman's PR5526 supplies the general pointwise Clifford/phase theorem and
+open sharp-width/spectral question. MatShep's PR5549 and PR5594 supply the
+one-qubit results and unreserved larger-qubit question. They are prior context,
+not executable dependencies or certificates for this computation. The proof
+and replay here are self-contained, with no imported predecessor code.
+
+DLamdata's PR5547 and josecarlosgonzalezroque-design's PR5570 are distinct
+independent AI-assisted scoped audits. Retain Dugongue, Chris R/Clear1nsight,
+Cao Yifan, Cinna Davis, zxv, Krzysztof Wojt, Nick Petalas and the credited
+native/controller/history/Clifford/analytic lineage. Classical finite-group
+regular representations, Pauli normalizers, spectral projectors and the Schur
+product theorem retain their classical status.
+
+Exact pending-source context pins (not required at replay time):
+
+- PR5526, Ali Romman, head`45f7de73b168721cdf7ec8c1769b1b594493716a`, `research/clifford-pointwise-faithfulness/PROOF.md`, SHA256`45483e4b5f604a5e70602c5caaa4394d4a0d0d1a6d0b6e8a93b836d50b23f764`.
+- PR5549, MatShep, head`7f104c00852346f0d07c4e87e830284e8daab67b`, `research/full-clifford-one-qubit/README.md`, SHA256`a38fccb6b1196821f00442f03e376b884f07b40ba7ddf99ae42d3cefa9afbec8`.
+- PR5594, MatShep, head`bb6598a1622f74eeb73aeacc290224ef388b717d`, `research/full-clifford-one-qubit-spectrum/PROOF.md`, SHA256`1a7432f859600b7933d02e74a99c5ff584596596efe67049041e0c3b2f8f61aa`.
+
+No selected-frontier or supplier promotion is implied by local verification.
+
+JoyboyBrian's PR5051, exact head 0960807a68ff46acbe6510c8867912ef2324ee71,
+already independently enumerates all 11,520 two-qubit projective Cliffords
+for partial signed-Pauli contracts. This submission's enumeration is a new
+implementation of that classical finite alphabet; the new result is its
+tensor Gram spectrum, threshold and conditioning, not the group count.
+No PR5051 executable or proof is imported.

research/full-clifford-two-qubit-centre/PROOF.md

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+# Sharp two-qubit full-Clifford tensor threshold and conditioning
+
+Let G be the group of all two-qubit unitary Clifford matrices with entries in
+Z[i,1/2]. Its scalar subgroup is Z={I,iI,-I,-iI}. Choose one representative U_a
+of each projective class a in G/Z, and, for positive integer f, put
+
+    H_f(a,b) = (tr(U_a^* U_b)/4)^f.
+
+This is the Gram matrix of the unit Hilbert-Schmidt vectors
+U_a^{tensor f}/2^f. Changing representatives conjugates H_f by a diagonal
+unitary, so its spectrum, rank and condition number are intrinsic.
+
+**Theorem.** There are 11,520 classes. Their pointwise tensor-independence
+threshold is exactly 9. For every f>=9,
+
+    (315/1024) I <= H_f <= (9855/4096) I.
+
+Both constants and the resulting uniform Gram condition bound 219/28 are
+optimal, attained at f=9. The condition number of the synthesis map itself
+is the square root of this Gram condition number.
+
+## 1. Authenticate the finite alphabet
+
+Use S=diag(1,i), V=((1+i,1-i),(1-i,1+i))/2 on either qubit, and CNOT with
+basis map (a,b)->(a,b+a). The verifier stores 2U as a pair of 4-by-4 integer
+matrices, enumerates their generated group by exact multiplication, and
+checks all 46,080 elements are unitary. It checks that the five generators
+normalize the signed two-qubit Pauli matrices and that quotienting by the
+four scalar phases gives 11,520 distinct matrices.
+
+This exhausts the full projective Clifford group, rather than a selected
+subalphabet. An ordered symplectic basis of F2^4 has
+15*8*3*2=720 choices: choose a nonzero first vector, a vector pairing to one,
+then a nonzero vector and its partner in their two-dimensional orthogonal
+complement. Each symplectic action has at most 2^4 signed Pauli actions.
+An operator's signed Pauli action determines it up to scalar, because the
+Pauli matrices span the full matrix algebra. Thus there are at most
+720*16=11,520 projective normalizers; the enumeration attains this bound.
+
+A scalar in Z[i,1/2] of modulus one is a Gaussian unit: write it as
+(a+bi)/2^k and repeatedly use a^2+b^2=2^(2k) modulo four to divide a,b by
+two. At k=0 only the four units remain. The enumerated four lifts therefore
+also exhaust the Gaussian-dyadic scalar kernel. No eighth-root scalar is
+silently identified with one.
+
+## 2. Reduce the Gram to the centre, retaining scalar residue
+
+On all of G, define the unnormalized matrix
+
+    C_f(x,y) = tr(x^* y)^f.
+
+Its kernel is a central function of x^-1 y. Hence C_f is central convolution
+in the finite group algebra, a semisimple algebra over C. Its eigenvalues
+occur already in its action on class functions: each simple matrix block
+has one central scalar, represented by its central idempotent. This is the
+standard finite-group regular representation, not a numerical approximation.
+
+The verifier obtains all 59 conjugacy classes by closure of conjugation by
+the five generators. For class representatives x_j and classes K_k, the
+matrix on class functions is exactly
+
+    M_f(j,k) = sum_{y in K_k} tr(x_j^* y)^f.             (1)
+
+Every sum is recomputed from the literal integer matrices. No class table or
+character table is imported. The range has scalar character
+v(ix)=i^(-f) v(x). Orbiting the classes by multiplication by i and retaining
+this character reduces (1) to R_f of dimensions 21,11,16,11 for f modulo
+four equal to 0,1,2,3. Incompatible shorter scalar orbits contribute zero.
+The verifier checks the entire 59-row invariance equation M_f B=B R_f.
+
+The other scalar sectors are zero. Grouping the four lifts of each
+projective representative gives C_f/4^f as an outer product of a four-component
+phase vector of squared norm four with itself, tensored with H_f. Consequently the nonzero
+scalar-sector spectrum of C_f is exactly 4*4^f times the spectrum of H_f,
+with the same multiplicities. There is no additional factor of four in
+the projective multiplicities.
+
+## 3. An exact certificate, including regular multiplicities
+
+For each f=1..12, CERTIFICATE.json records the independently regenerated R_f
+and a list L_f of distinct integer eigenvalues. The verifier uses unbounded
+Gaussian-integer arithmetic to check
+
+    product_{lambda in L_f} (R_f-lambda I) = 0.         (2)
+
+The roots are distinct. Therefore the spectral projectors are the exact
+polynomials
+
+    P_lambda = product_{mu != lambda}
+                      (R_f-mu I)/(lambda-mu).          (3)
+
+Equations (2)-(3) certify the full spectrum without trusting the exploratory
+symbolic eigensolver. The centre action is faithful on the corresponding
+central ideal, so these identities also hold for central convolution in
+the full regular representation.
+
+For completeness, the regular multiplicity is
+
+    m_lambda = 11520 * (P_lambda)_(0,0).                (4)
+
+Here index zero is the identity scalar orbit. Projecting delta_I to the
+scalar sector gives B e_0/4. A central convolution operator has full regular
+trace 46080 times its identity coefficient, so its spectral projector has
+trace (46080/4)*(P_lambda)_(0,0), giving (4). The verifier evaluates the
+integer numerator and denominator of (3), checks exact division, positive
+integral multiplicities, their sum 11,520 and the unit-diagonal Gram trace.
+
+The resulting exact ranks for f=1..12 are
+
+    16,136,816,3426,9104,11384,11504,11519,11520,11520,11520,11520.
+
+At f=9 the full projective spectrum is:
+
+| Eigenvalue | Multiplicity |
+|---|---:|
+| 315/1024 | 16 |
+| 2205/4096 | 400 |
+| 2835/4096 | 1296 |
+| 6615/8192 | 1600 |
+| 3465/4096 | 400 |
+| 16065/16384 | 4096 |
+| 4725/4096 | 2000 |
+| 5535/4096 | 1296 |
+| 3375/2048 | 400 |
+| 9855/4096 | 16 |
+
+All are positive, giving the stated endpoints at nine. There are ten
+distinct eigenvalues in an eleven-dimensional central block; a central
+multiplicity of two is combined in the regular multiplicity 2000.
+
+## 4. Propagate the bounds and prove sharpness
+
+The identity H_(f+1)=H_f Hadamard H_1 and the Schur product theorem imply
+
+    a I <= H_f <= b I  =>  a I <= H_(f+1) <= b I,
+
+because H_1 is positive semidefinite with diagonal one. Thus the two exact
+endpoints at nine bound every f>=9. Their positive multiplicities make
+each endpoint and their ratio 219/28 sharp for this uniform range.
+
+There is a particularly short explicit witness against width eight.
+The character d(g)=det(g) is +/-1 for every enumerated lift and is invariant
+under scalar multiplication by i. The verifier computes by the 24-term
+determinant formula, separately from the centre spectral certificate,
+
+    sum_{g in G} d(g) tr(g)^4  = 46080,
+    sum_{g in G} d(g) tr(g)^8  = 0,
+    sum_{g in G} d(g) tr(g)^12 = 103403520.             (5)
+
+Multiplicativity of d makes the vector (d(g)) an eigenvector of C_f with
+eigenvalue the corresponding sum. At width eight it is nonzero and null;
+it descends to the 11,520 projective classes. Hence
+
+    sum_a det(U_a) U_a^{tensor 8} = 0.
+
+Its normalized squared Hilbert-Schmidt norm is zero by the Gram identity.
+This witness is nonzero at width twelve; no all-width identity follows from
+the width-eight cancellation. In particular it is already nonzero at four.
+The rank table shows all smaller positive widths also fail independence;
+alternatively positive definiteness at any smaller width would propagate
+to eight, contradicting (5). The threshold nine is therefore sharp.
+
+For a fixed residual in C[G], the four widths 9..12 suffice to decide its
+vanishing at all positive widths: independence separates each projective
+coefficient in one scalar residue, and the four-point Fourier transform
+separates the four scalar lifts. This is only a sufficient window here;
+no optimality among tests using several earlier widths is claimed.
+
+## 5. Arithmetic and scope of the checks
+
+All group and trace arithmetic uses NumPy int64 with explicit integrality,
+entry and trace-magnitude checks. For f<=12, |tr(x^*y)|<=4 and every class
+sum is bounded by 46080*4^12<2^40; the scalar-basis folds only use 0,+/-1,+/-i.
+All relevant int64 intermediate bounds are below 2^63. Spectral polynomial
+products and projector numerators then use unbounded Python integers.
+There is no floating-point spectral calculation in the verifier.
+
+The verifier also checks 64 literal tensor Gram entries at widths one
+through four. Seven controls reject width-eight independence, stronger
+spectral/condition constants, an altered class-block coefficient, discarded
+imaginary trace phases and persistence of the null witness at twelve.
+Normal and optimized/warning-strict modes execute the same checks without
+removable assertions. The written finite-group and Schur arguments provide
+the all-width step; this is not a proof-assistant kernel certificate.
+
+No three-qubit result, general sharp 4n+1 theorem, physical implementation,
+paid caller, supplier admission, precision/tape result, new kappa or selected
+frontier change is claimed.
+
+## Attribution
+
+Ali Romman, PR5526, supplies the general pointwise Clifford/phase framework
+and its open sharp-width/spectral question. MatShep's PR5549 and PR5594 solve
+the one-qubit threshold and spectrum; their unreserved larger-qubit question
+motivates this computation. DLamdata's PR5547 and
+josecarlosgonzalezroque-design's PR5570 supply distinct scoped audits. Their
+retained native/history/Clifford lineage is credited in NOTICE.md. Finite-group
+centres, Pauli normalizers, character projectors, tensor Gram identities and
+Schur products are classical. New two-qubit enumeration and certificates:
+MatShep with substantial OpenAI Codex assistance; Apache-2.0.

research/full-clifford-two-qubit-centre/README.md

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+# Sharp full two-qubit Clifford tensor threshold and conditioning
+
+The full two-qubit Gaussian-dyadic Clifford alphabet has **11,520 projective
+classes**. Its sharp pointwise tensor-independence threshold is **9**.
+For every width f>=9, its normalized tensor Gram matrix has optimal uniform
+spectral bounds **315/1024** and **9855/4096**, hence sharp Gram condition
+number **219/28**. All three constants are attained at width nine.
+
+This answers the two-qubit next question left by MatShep's
+[one-qubit spectrum PR5594](https://github.com/CrocSwap/integer-mult-bounds/pull/5594)
+and sharpens the n=2 sufficient threshold 12 in Ali Romman's
+[general pointwise theorem PR5526](https://github.com/CrocSwap/integer-mult-bounds/pull/5526).
+It is a native-algebra side result: no new kappa, selected-frontier change,
+physical supplier, precision/tape or paid-caller claim.
+
+The [self-contained proof](PROOF.md) explains the all-width Schur-product step.
+The computation avoids a dense 11,520-square Gram: it enumerates 46,080 exact
+lifts and 59 conjugacy classes, then retains scalar-residue blocks of sizes
+11,16,11,21. The [certificate](CERTIFICATE.json) contains exact finite block
+matrices and eigenvalues at widths 1..12. The verifier regenerates each block
+and certifies the spectrum and full regular multiplicities by unbounded-integer
+polynomial identities. No symbolic or numerical eigensolver is used in replay.
+
+Ranks at widths 1..12 are
+`16,136,816,3426,9104,11384,11504,11519,11520,11520,11520,11520`.
+The literal determinant character supplies a nonzero null vector at width
+eight; its separate character sums are zero at eight and nonzero at four
+and twelve. It is not an all-width identity.
+
+## Reproduce
+
+Python 3.12 and NumPy 2.3.5 suffice. No network, Git or predecessor executable
+is needed for either direct scientific command once NumPy is installed:
+
+```sh
+python -B research/full-clifford-two-qubit-centre/verify.py
+python -I -O -W error -B research/full-clifford-two-qubit-centre/verify.py
+```
+
+For registered metadata/pins and both scientific commands from the repository:
+
+```sh
+python -B scripts/research_package.py --package research/full-clifford-two-qubit-centre
+```
+
+Both modes pass locally on CPython 3.12.14, NumPy 2.3.5, Windows 11, with equal
+complete JSON payloads in [RESULT.json](RESULT.json). The verifier authenticates
+all group elements and class data, exact spectra and multiplicities, 64 literal
+tensor Gram entries, and seven rejecting controls. It uses explicit checks
+that remain active under optimized Python. The full width-nine spectrum is
+tabulated in PROOF.md. The proof and finite certificate are author checked;
+no independent human or proof-assistant acceptance is claimed. Hosted CI is
+separate from these local checks.
+
+## Sources and overlap
+
+Base: canonical main `5e0052ca0aa3036a3afa8b0438927377c8d97d53`, priority 7,
+selected PR5413 unchanged. [TASK.json](TASK.json) records the changed scope
+from the frozen formal-interface card. [SOURCES.json](SOURCES.json) pins the
+catalogue and selected record as provenance; there are no external scientific
+inputs. All matrix generators and certificate checks are implemented here.
+
+Current guidance, the selected review and source manifest, live handoff thread
+1166 and canonical Clifford thread 5236 were read. Targeted all-state PR/Issue
+searches covered two-qubit tensor,11520,Clifford spectrum and Clifford Gram;
+Discussion queries covered two-qubit and Clifford spectrum. The relevant
+matches were the credited one-qubit/general packages and the distinct two-qubit
+partial-Pauli census in JoyboyBrian's PR5051 and unrelated qubit repair PR5245.
+The 11,520-class group count itself is classical/prior; this computation adds
+the tensor spectrum and sharp quantitative bounds. No equivalent result was found in that bounded coverage.
+The repository overlap helper was also attempted: gh was unavailable and local
+sparse coverage was 4 of 7611 notes. The direct authenticated/public API checks
+do not establish universal novelty, arbitrary historical diff coverage or
+coverage of unpublished work. Active P104/query-history scopes are excluded.
+
+## Open next questions
+
+Does the sharp threshold 4n+1 continue at three or more qubits? This could
+sharpen exact native interface tests. First seek a representation or class
+reduction before a much larger enumeration. Only n=1 in the predecessor and
+n=2 here are established; neither a general theorem nor a three-qubit test
+is supplied. The [structured question](TASK.json) records these limits.
+
+## Attribution
+
+New two-qubit computation and certificate: MatShep with substantial OpenAI
+Codex assistance. Credit Ali Romman/#5526, MatShep/#5549/#5594,
+DLamdata/#5547, josecarlosgonzalezroque-design/#5570 and the retained lineage.
+Exact predecessor references and source notices are in [NOTICE.md](NOTICE.md).
+Finite-group centres, Pauli normalizers, character projectors, tensor Grams
+and Schur products are classical. Apache-2.0.

research/full-clifford-two-qubit-centre/RESULT.json

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+{
+  "conjugacy_classes": 59,
+  "determinant_character_moments": {
+    "4": 46080,
+    "8": 0,
+    "12": 103403520
+  },
+  "lifts": 46080,
+  "literal_tensor_entries": 64,
+  "projective_classes": 11520,
+  "ranks_widths_1_to_12": [
+    16,
+    136,
+    816,
+    3426,
+    9104,
+    11384,
+    11504,
+    11519,
+    11520,
+    11520,
+    11520,
+    11520
+  ],
+  "rejection_controls": [
+    "width-eight independence rejected",
+    "stronger lower bound rejected",
+    "stronger upper bound rejected",
+    "stronger condition bound rejected",
+    "changed centre coefficient rejected",
+    "discarded imaginary trace phases rejected",
+    "null witness at width twelve rejected"
+  ],
+  "scalar_sector_dimensions": {
+    "0": 21,
+    "1": 11,
+    "2": 16,
+    "3": 11
+  },
+  "sharp_uniform_gram_condition": "219/28",
+  "sharp_uniform_lower": "315/1024",
+  "sharp_uniform_upper": "9855/4096",
+  "status": "PASS_TWO_QUBIT_CLIFFORD_CENTRE",
+  "width_nine_spectrum": {
+    "16065/16384": 4096,
+    "2205/4096": 400,
+    "2835/4096": 1296,
+    "315/1024": 16,
+    "3375/2048": 400,
+    "3465/4096": 400,
+    "4725/4096": 2000,
+    "5535/4096": 1296,
+    "6615/8192": 1600,
+    "9855/4096": 16
+  }
+}