NULL_NET_AUDIT.md

Audit of the finite Gaussian null-net report

This is a focused repair of finite diagnostics promoted to stronger null-net receipts, under #1033. It concerns the supplied anti-periodic, half-filled free-fermion ring in null_net_receipts.py. It does not construct a physical null net or change the axioms. The report now gives full-subspace leakage, an actual bound for the particular expectation it measures, and complete finite momentum-envelope sums. These replace selected-packet, finite-trend and truncated-shape evidence that had been assigned stronger meanings.

Reproductions before the repair

On main 7f5805a2, the four regression tests committed before the repair fail with these results:

• The full N=16 covariance has eigenvalues between approximately -0.140729 and 1.140729, so it cannot be a fermionic covariance. Its projector defect ||C^2-C||_F is approximately 0.527622.
• instrument_null_net((16,)) witnesses half-sided compression although the maximum leakage probability over the B subspace is 0.653606 in both time directions. The selected packet's leakage ratio is 9.5669.
• That same single-stage report witnesses mixed-GNS Cauchy convergence and percent-level modular Lie closure, despite having no refinement difference and no Lie samples.
• lie_closure_receipt(32, rmax=0) returns a NaN residual after a warning. It does not reject the empty signal. NaN itself did not pass the old percent-level comparison; the empty-family all(...) did.

The old docstring also asserted that a nonzero imaginary antisymmetric Hermitian commutator lies approximately in a span of real symmetric generators. Section 4 gives an exact obstruction. Finite residual decrease was incorrectly labeled a certified convergence rate.

1. A valid full covariance, and why finite arcs are faithful

Put k_t=(2t+1) pi/N and fill cos(k_t)>0, with N divisible by four. The normalized discrete Fourier columns form a unitary matrix. If V contains its occupied columns, then C=V V*, so C^2=C, C*=C, and rank C=N/2. In the chosen gauge its entries are real:

[ C_{ij}=\frac1N\sum_{\cos k_t>0}\cos(k_t(i-j)). ]

The anti-periodic seam has C(N-d)=-C(d). Replacing distance d by min(d,N-d), as the old helper did, therefore changes signs. The repair uses the actual distance |i-j|. Independent complex Fourier projectors, including the complementary unoccupied projector, check the whole matrix. Local arc entries of size at most N/2 were unaffected by this seam bug; the old interior bond expectation and arc modular generators do not change for this reason.

There is also a structural proof of finite arc faithfulness, independent of the small computed eigenvalue gaps. For an m-site consecutive arc with m<=N/2, restrict V to those rows. Any m distinct occupied columns give a Vandermonde matrix with distinct nodes exp(i k_t), up to nonzero column factors. It has rank m. Hence C_arc=V_arc V_arc* > 0. The same argument using unoccupied columns gives I-C_arc>0. Every occupation eigenvalue therefore lies strictly between zero and one. The reduced Gaussian density has positive eigenvalues, products of occupation eigenvalues and their complements, and is faithful.

The numerical gap calculation uses its own 120-digit context and rejects an unresolved gap. The Vandermonde proof supplies strict positivity for every mathematical finite stage, not a uniform lower bound or a cyclicity/standardness result on a common limiting GNS space. The finite relative commutant dimension 4^(|A|-|B|) remains valid: these are full matrix factors. In CAR notation, complement generators need the B-parity twist to commute with B; bare odd operators anticommute. Site coverage likewise remains a finite algebra-generation fact.

2. One selected observable has a proved limit

Keep exactly the original mathematical observable, with the seam bond excluded:

[ E_N=\frac1N\sum_{j=0}^{N-2}f(j/N) \langle c_j^\dagger c_{j+1}+c_{j+1}^\dagger c_j\rangle, \qquad f(x)=e^{-200(x-1/4)^2}. ]

The finite geometric sum over occupied momenta gives C_N(1)=1/(N sin(pi/N)). Set S_N=(1/N) sum_{j=0}^{N-2} f(j/N), I=integral_0^1 f, and c_N=2/(N sin(pi/N)). Thus E_N=c_N S_N and the limit is L=(2/pi) I, approximately 0.07978843321.

For completeness, 0<=f<=1 and its total variation on [0,1] is less than 2. On each mesh cell, its left-rectangle quadrature error is bounded by the cell width times its variation on that cell. Summing gives |(1/N) sum_{j=0}^{N-1} f(j/N)-I|<=2/N. Removing the last term gives |S_N-I|<=3/N.

For x=pi/N<=pi/8, x-x^3/6<=sin x<=x. Therefore

[ |E_N-L|\leq\frac{2}{\pi} \frac{3/N+x^2/6}{1-x^2/6} \leq B_N:=\frac23\frac{3/N+a_N}{1-a_N}, \qquad a_N=\frac{(22/(7N))^2}{6}. ]

The second inequality uses 3<pi<22/7 and monotonicity in the nonnegative argument a. B_N is an exact rational, tends to zero, and is less than 3/N for N>=8. The latter inequality is equivalent to a_N(2N/3+3)<1; using a_N<2/N^2 bounds its left side by 4/(3N)+6/N^2, already below one at N=8 and decreasing thereafter. Consequently |E_N-E_M|<=B_N+B_M<3/N+3/M. This controls every allowed pair, not only a sampled decreasing sequence. In particular, stages above 6/epsilon are Cauchy to epsilon for this observable.

The report stores exact rational bounds. Its expectation and limit floats are numerical evaluations, not outward-rounded enclosures. Separate Fourier sums and adaptive quadrature test the values and bounds. This proves convergence of the single supplied observable. It does not prove the mixed inner products, compatible embeddings and observable-family convergence that the mixed-GNS receipt consumes.

3. Why packet asymmetry cannot certify half-sided inclusion

Let P project onto a finite nonempty proper subspace B, Q=I-P, and V(t)=exp(i h_A t). The internal factor W(t)=exp(-i h_B t) preserves B, so

[ \sup_{\psi\in B,\ |\psi|=1}|QV(t)W(t)\psi|^2 =|QV(t)P|^2. ]

Write A=P V(t) P as a square matrix on B. Unitarity gives

[ (QVP)^(QVP)=I_B-A^A, \qquad (PVQ)(PVQ)^=I_B-AA^. ]

The matrices A*A and AA* have the same eigenvalues, including zero multiplicities, since A is finite and square. The two leakage blocks have the same singular values apart from harmless rectangular zeros. Since V(-t)=V(t)*, the maximum leakage probabilities are identical for t and -t. This holds also for complex Hermitian generators and unequal B and complement dimensions; it is stronger than real-matrix time reversal.

Moreover, if a unitary sends finite B into B, its image has the same dimension and equals B. Proper one-sided subspace inclusion is impossible. The same elementary dimension argument excludes proper inclusion of a finite-dimensional algebra into itself by conjugation. This does not exclude half-sided modular inclusions of infinite-dimensional algebras or of real standard subspaces in their proper setting. It explains why this particular finite test cannot serve as their receipt.

An exact three-dimensional counterexample makes the packet problem sharp. Let U cycle e0 -> e1 -> e2 -> e0 and B=span(e0,e1). The packet e0 has zero forward leakage and unit backward leakage, while the worst-case leakage is one in both directions. U has a Hermitian logarithm, so it is a member of a continuous finite unitary flow. Exact symbolic matrix checks and a numerical realization test this example independently.

On the actual supplied rings at model time 0.12:

N	Selected packet leakage ratio	Maximum leakage probability, either sign
16	9.5669	0.653606
32	38.2244	0.983561
64	2941.09	0.999994

The large packet ratio survives as a directional transport diagnostic. It cannot discharge half-sided inclusion. Zero leakage ratios are recorded as undefined where appropriate, rather than divided by an invented floor. The flow's numerical unitarity defect is reported and must be <=1e-10; that tolerance is a numerical resolution guard, never a proof of inclusion.

4. What the commutator envelope can and cannot establish

For real symmetric H and K, D=i[H,K] is imaginary antisymmetric and Hermitian. Every real symmetric S is orthogonal to D in the real Hilbert-Schmidt inner product. Thus

[ |D-S|_F^2=|D|_F^2+|S|_F^2. ]

For nonzero D, the best relative distance to the entire real symmetric space is one. The old stated span of real arc modular generators, number operators and identity is contained in this space and contains zero, so its best distance is also one. An exact Pauli X/Z calculation checks ||i[X,Z]-[[a,b],[b,c]]||_F^2=8+a^2+2b^2+c^2. This obstruction is to that particular claimed span; it does not rule out a larger conformal algebra with the required momentum generators.

The code actually measures the real antisymmetric commutator C through the scalar envelope

[ p(j)=\sum_{r\geq2,\ r\ \mathrm{even}} \frac r2(-1)^{(r-2)/2}C_{j+1-r/2,,j+1+r/2}, ]

retaining terms whose indices fit. This map is not injective even among even-range commutators. For a 6x6 antisymmetric C with C[1,3]=2 and C[0,4]=1 (and their negative transposes), p is identically zero although ||C||_F^2=10. It is itself a commutator: take H=diag(0,...,5) and the symmetric K[i,j]=C[i,j]/(i-j) off diagonal. Multiplying this invisible component by an arbitrary constant leaves every envelope sample fixed. An accurate envelope fit therefore cannot bound full operator error.

The repair sums all fitting even ranges by default. For a supplied cutoff R it also reports the triangle bound

[ |p_{\rm full}(j)-p_R(j)|\leq \sum_{r>R,\ r\ \mathrm{even, fitting}}\frac r2|C_{j+1-r/2,,j+1+r/2}|. ]

This is an exact finite inequality for exact entries. Reported evaluation of that bound is floating-point, not an interval certificate. Independent exact entry-by-entry enumeration checks it, including R beyond the matrix. The historical R=8 fit remains visible:

N	Old R=8 shape residual	Complete finite sum residual
32	0.00165175	0.000903682
64	0.00663635	0.000369533

Removing a fixed-range truncation improves this supplied-family diagnostic. Neither two decreasing residuals nor fitted normalization prove a continuum rate or an operator relation. Empty sample families cannot pass, fewer than two fit coordinates or zero/nonfinite signals are rejected, and no finite trend is promoted to a convergence-rate certificate.

Numerical follow-up audit

An adversarial review of PR head e732046f found two additional failure classes, reproduced by six failing regression cases before their repair:

• Dividing a complex packet by a subnormal scale, such as 1e-320, can overflow NumPy's internal reciprocal and return NaN leakage. The same operation in the Hermitian check produced NaN comparisons, admitting [[0,1e-320],[0,0]] as a complex Hermitian matrix.
• Input promotion converted the unequal integers 2^53 and 2^53+1 into equal doubles, yielding a falsely perfect scalar fit. Squaring the residual of measured=[1,1e-200], target=[1,0] also returned zero although its relative norm is representable near 1e-200.

The repair scales the real and imaginary components separately. It supports packets from the smallest binary64 subnormal through components of magnitude 1.7e308, including complex magnitudes that overflow if formed directly. Unresolvable component dynamic ranges fail explicitly. The Hermitian guard uses a 64*eps tolerance after component scaling and the flow retains its separate numerical unitarity check; neither tolerance proves an exact identity.

Input validation and rational-norm conversion reuse the audited primitives in null_tomography.py. They inspect original sequence elements before promotion, reject booleans, nonfinite values and lossy conversion, and retain representable small results. For the one-scalar shape fit, compute the exact rational least-squares coefficient of the supplied binary samples, round that coefficient once, and replay the returned coefficient against the original samples. Both squared residual and squared measured norm are computed exactly. The diagnostic threshold compares their ratio directly to (1/50)^2; it never compares an underflowed or rounded displayed norm. The displayed norm is still a numerical square root. The exact ratio concerns the given binary samples, not eigensolver, envelope-construction or continuum error. An unrepresentable nonzero coefficient or norm is rejected.

Independent 400-digit projection controls cover twenty scaled data sets; analytic Pauli evolution checks the subnormal complex flow. The rounded subnormal-coefficient regression separately checks that representation error is included in the returned residual. The default ring conclusions and reported precision-level agreement remain unchanged after these repairs.

Contract, custody and downstream impact

The public function names and report path remain for existing callers, but the report is schema 2. Legacy lie_closure_* verdicts refer explicitly to scalar envelopes; the corresponding stronger receipts_witnessed flag is false. Mixed-GNS and HSM witness flags are also false. The finite algebra facts and the new single-observable limit are positive results. compressing_sign becomes lower_packet_leakage_sign; packet ratio and direction verdicts are named as packet diagnostics. This is an intentional meaning/schema correction. No in-repository consumer depends on the removed keys beyond the updated tests.

Ring families must be nonempty, valid and strictly increasing. Full report rings are divisible by eight; the standalone covariance and bond helpers also accept multiples of four. Lie fits require N>=32 and divisibility by eight. Invalid or unresolved numerical inputs fail; JSON rejects NaN and infinity. Matrix, packet, time and fit inputs must be exactly representable as finite real or complex binary64 components. Modular generators retain the supplied model and 120-digit eigendecomposition, rounded to binary64. No physical clock is attached.

claims/claim_registry.yaml cites this report under OPH-GR-E2E-BRANCH-ENTRY, already classified as a conditional composition with open nonemptiness. Its null-net claim explicitly says finite one-particle diagnostics do not instantiate the scaling-limit receipts. The spacetime paper (including Theorem 4.3d in the shared technical fragment), flagship and book use common-GNS standardness, cofinal relative-commutant cyclicity, actual half-sided inclusions and modular-intersection premises. This repair supplies none of those stronger premises and changes none of their theorem statements. Current public surfaces already retain the physical/nonemptiness boundary; no paper/PDF, claim classification, observation rung, premise, Lean theorem or axiom is promoted or rewritten.

The generated report at its existing registered path is deliberately replaced because its witness flags were wrong. The prior report remains recoverable from main 7f5805a2; the corrected report retains the packet numbers and historical R=8 values for comparison. No frozen registration, source routing tape or hash-pinned evidence is changed. The registry points to a mutable generated report, not a pinned hash. There is no M1 or physical-model completion claim, and standing issue #1033 remains open.

Reproduction and independent controls

OPENBLAS_NUM_THREADS=1 python code/geometry/null_net_receipts.py
OPENBLAS_NUM_THREADS=1 python -m pytest -q code/geometry/test_null_net_receipts.py -W error
OPENBLAS_NUM_THREADS=1 PYTHONPATH=code python -m pytest -q code/quantum_information code/collar_alignment code/geometry code/maxent -W error

The existing Linux/Windows finite-geometry and quantum-information workflows run these tests. They include complex Fourier projectors, independent spectral evolution, exact permutation and Pauli controls, an invisible commutator, exact tail sums, separate quadrature, adversarial empty/zero/ nonfinite inputs, and full schema/numerical replay of the generated report. The proofs above supply the general statements; finite tests support their implementation and do not substitute for those proofs.

Local validation after the follow-up audit: 710 Linux tests passed; 705 Windows tests passed with five expected extended-exponent platform skips, with warnings treated as errors. The null-net module has 93 tests. Claim-registry, axiom-consistency and reader-style checks pass. The unchanged Lean trust-inventory checker passes on Linux; on Windows it reports a path separator mismatch (Screen\\... versus Screen/...), not a changed proof inventory. No Lean source or inventory checker is edited here.