INFORMATION_BUDGET.md

Resolve the full collar information budget

Maintainer follow-up (#1049): support containment in the relative-modular moment test is now compared in the canonical Gaussian rational field, using a zero residual. Symbolic expression-tree equality falsely rejected a full-rank complex conditional-product state and a nearby non-Markov state. The new regression exercises exact Markov, public CMI, collar CMI and the alignment budget, with both full-rank and singular conditional products; real and complex support escapes must still raise. This changes acceptance of valid inputs, not the moment theorem, frozen evidence or paper claims.

The review summary and inline comment identify this same defect. The eleven direct controls in test_complex_markov_support.py cover both entries' requirements; five fail against the reviewed 10fffe19 implementation. The broader follow-up retains 24 independently constructed complex conditional-product families, with unequal sector weights and complex states at both endpoints. In each case, an explicit within-sector correlation preserves both marginals and destroys conditional independence. Exact Markov, public CMI and collar CMI must accept the product and distinguish the perturbed state. Twenty-three of these 24 controls fail on the reviewed implementation; all 35 controls pass with canonical support equality. The branch includes main 7f99c8cc.

This is a focused evidence repair under #1033, built on main 3d848b7f. It repairs false zero information scores and makes the existing alignment criterion more informative. It builds on the sector-label and Gibbs audits without depending on the open first-law PR #1048. No physical model or new physical hypothesis is selected.

Reproduced failures

The first commit, 568cc679, contains fourteen tests that fail before the implementation changes. The previous shared routines computed information by subtracting binary64 entropies. With ordinary trace and natural logarithms:

Supplied state	Previous result	Independent result
diag(1/4+e,1/4-e,1/4-e,1/4+e), e=2^-30	MI = 0; collar alignment accepted at tolerance e^2	MI = 6.938893903907228e-18, greater than the tolerance
Classical three-bit parity perturbation p=1/8+e*(-1)^(a+b+c), e=2^-30	CMI = 0	CMI = 2.7755575615628914e-17
I_8/8+e X tensor Y tensor X, e=1e-150	CMI = 0	CMI approximately 3.2e-299
I_4/4+1e-200 X tensor X	MI = 0	Positive, below binary64 range; the numerical API must raise
A masked density entry	Hidden payload accepted	Missing evidence must be rejected

The scalar controls use the separate formula [(1+x)log(1+x)+(1-x)log(1-x)]/2, where x=d*e. Further controls use independent 2-by-2 characteristic roots and matrix logarithms. They also retain the fourth-order example rho=I_4/4+e*(X tensor I+I tensor X), whose MI is 128 e^4+O(e^6). Its second-order entropy terms cancel; resolving only the leading state perturbation does not resolve its information.

Four terms, with no additional state-selection premise

Fix the already-declared collar tensor factors A,L,R,D. For every normalized finite state, including singular states, the chain rule gives

I(AL:RD) = I(L:R) + I(A:R|L) + I(L:D|R) + I(A:D|LR).             (1)

To prove this, expand I(AL:RD) first across R,D, then expand each term across L,A:

I(AL:R)   = I(L:R)   + I(A:R|L),
I(AL:D|R) = I(L:D|R) + I(A:D|LR).

Every term is nonnegative by subadditivity or strong subadditivity. These are established finite quantum-information results; see Watrous, chapter 5. Equation (1) therefore gives an exact equivalence: the specified cut is a product precisely when all four terms vanish. It also gives a quantitative budget: the whole defect is at most the sum of any four valid upper bounds. No faithfulness, Gibbs model, small-coupling expansion or continuum limit is required for this identity.

The usual collar CMI is only the last term. Each term can be the sole obstruction: correlate just the corresponding pair of classical bits and leave the other two bits independent. The retained Bell counterexample has zero collar and middle terms but two endpoint terms of 2 log(2) each, giving total alignment defect 4 log(2). Thus its obstruction is located, rather than merely labelled nonaligned.

alignment_information_budget reports all four terms for every positive sector, their weighted values, the existing maximum-sector defect, and the numerical chain-rule residual. A small weight can make the average small while the maximum remains large. Zero-weight sectors have no effect.

The whole budget is a variational distance

Let the retained center and cuts be fixed, with rho=direct_sum_j p_j rho_j and normalized conditional states. Compare with an arbitrary aligned state on this same declared algebra, tau=direct_sum_j q_j (sigma_L,j tensor sigma_R,j). Block logarithms and partial traces give the exact decomposition

D(rho || tau)
 = D(p || q)
   + sum_j p_j [ I(L_j:R_j)_rho_j
                 + D(rho_L,j || sigma_L,j)
                 + D(rho_R,j || sigma_R,j) ].                 (2)

Support escape has its usual infinite-divergence meaning. All added terms are nonnegative. Consequently the minimum is J=sum_j p_j I(L_j:R_j), attained by retaining p_j and replacing each conditional state by its own marginal product. Equality fixes these weights and marginals on positive sectors; choices on absent sectors are irrelevant. This proves the minimizer directly, without an optimizer. For the collar, L_j means A bL,j and R_j means bR,j D, so (1) resolves this same minimum into four contributions.

Quantum Pinsker gives ||rho-rho_aligned||_1/2 <= sqrt(J/2). The implementation evaluates this inequality numerically, not as an interval certificate. For total mass T the homogeneous version is sqrt(T*J/2), capped at T; the report retains that mass and evaluates the square roots before multiplication to avoid intermediate underflow.

This projection is onto the fixed center and cut. It does not optimize over all possible quantum Markov decompositions. In particular, a quantum CMI alone is not generally the relative-entropy distance to that larger family; see Ibinson, Linden and Winter. The repair does not use that false identification.

Numerical implementation and exact zero decisions

The information evaluator reuses the shared input conversion and dimension checks. It rejects masked, Boolean, nonfinite and lossy-conversion inputs. Accepted Hermitian roundoff is averaged with exact rational arithmetic. Partial traces then sum the supplied entries exactly, including rare-event mass that would disappear when added to a unit diagonal entry.

Positivity is checked algebraically on that Hermitian representative. Its characteristic polynomial supplies the elementary symmetric functions of its real eigenvalues. They are all nonnegative precisely when the matrix is PSD: otherwise det(t I+A) has a positive root, whereas a polynomial with nonnegative coefficients and positive leading coefficient cannot vanish for positive t. Trailing zero coefficients determine exact nullity. A negative eigenvalue admitted by the older roundoff tolerance is therefore rejected instead of clipped into apparently valid evidence.

For a classical joint distribution, define q_abc=p_ab*p_bc/p_b, omitting zero-mass conditioning blocks. This is a nonnegative distribution with the same total mass as p. Its divergence is exactly the CMI. The evaluator sums p log(p/q)-p+q, using exact probability differences and a convergent log1p remainder near equality. These terms cannot cancel a positive correlation against an order-one entropy.

General quantum combinations use private 80-digit arithmetic, escalating to 400 digits when needed, and must agree with a separate evaluation at 40 additional digits. This is a precision diagnostic, not an interval bound on a numerical eigensolve. Exact nullity distinguishes zero eigenvalues from unresolved positive spectral mass. Nonzero output that underflows or suffers excessive subnormal quantization raises explicitly. The global mpmath context is unchanged.

No uncertain entropy cancellation is floored to zero. Exact product checks handle common zero cases without requiring resolved numerical eigenvalues: positivity has already been proved exactly. Dimension-one conditioning factors do not add distinct product splits. Only the nontrivial factors are enumerated, so the number of splits is at most the conditioning-space dimension, regardless of how many trivial factors are declared. An exact isospectral check also recognizes hidden single-product decompositions: compare the power traces of rho_AB tensor rho_BC and rho_B tensor rho_ABC through their common dimension. Newton identities determine their spectra, and the tensor entropy identity then gives zero CMI. This shortcut is sufficient but is not used as a complete Markov criterion.

The final exact test closes that recognition gap. For source r and reference s containing its support, set

M_k(r,s) = Tr(r^k s^(1-k)),             k=1,2,...,

where negative powers act on the exact support of s. Compare the pairs

(rho_ABC, rho_A tensor rho_BC),
(rho_AB,  rho_A tensor rho_B).

Finite zero test. CMI is zero if and only if these moments agree for k=1,...,N, where N=n_ABC^2+n_AB^2.

For sufficiency, the positive relative modular operator Delta=L_r R_(s inverse) has at most n^2 positive spectral points. With vector sqrt(s), its positive spectral measure has moments M_k and its t log(t) integral is D(r||s). Subtract the two measures and combine coincident positive spectral points. There are at most N points. The first N positive moments determine every remaining weight by an invertible Vandermonde system; the points are nonzero, so starting at power one is sufficient. Zero spectral points do not contribute to t log(t). Thus the divergences agree. Their difference is precisely CMI.

For necessity, use the established equality structure rho_ABC=direct_sum_j p_j rho_A,Lj tensor rho_Rj,C, with the label retained in B (Hayden, Jozsa, Petz and Winter). In each sector both moments reduce to p_j Tr[rho_A,Lj^k (rho_A tensor rho_Lj)^(1-k)]; the right factor contributes trace one. This also holds on singular supports. Summing proves equality for every k. A common nonunit source mass multiplies both moment sequences by the same factor at each power.

All these matrix products, inverses on support, traces and comparisons are rational operations for the supplied binary64 entries. is_markov_exact exposes this decision independently of any floating information value; it can reject a non-Markov perturbation even below binary64 output range. The exact moment path is intended for small finite witnesses and can be more expensive than numerical evaluation. It proves a property of the specified matrix, not of an unknown noisy preparation.

Additional audit findings

Commit 4fe2afd1 reproduces four failures in the first version of this PR. An exactly independent state and a conditional Markov state were rejected because their positive eigenvalues could not be resolved at 400 digits. Both use the six-dimensional integer Gram matrix G=(I+2^26 S)^T (I+2^26 S), where S is the unit superdiagonal, then multiply it by the smallest positive binary64 value. Every input entry is representable, although a positive eigenvalue is below 1e-400. Requiring a numerical entropy before honoring an algebraic zero was unnecessary. Exact product zeros now return after exact PSD validation; other unresolved spectral cases reach the exact zero test before refusal. A nearby state with an additional classical conditional correlation still fails explicitly, so this fallback cannot simply declare unresolved input Markov.

The other two failures concern the numerical and exact APIs with 24 dimension-one conditioning factors. They previously attempted 2^24 copies of the same product check on a four-dimensional state. The shared split helper now tests it once. Regression controls also retain nontrivial conditioning partitions and exercise complex singular quantum supports through the full modular-moment test, bypassing its shortcuts.

Compatibility and downstream impact

• Existing MI/CMI function signatures and nats remain unchanged. CMI keeps its four-entropy definition. MI now uses the same formula with an empty conditioning system, whose entropy is -T log T. Consequently MI is T I(rho/T) for accepted trace-roundoff inputs; this removes an artificial trace-offset contribution. No matrix is silently renormalized.
• The collar wrapper preserves entries until exact Hermitian averaging. Its weighted CMI is accumulated as an exact sum of products before final conversion. It cannot turn an unrepresentable nonzero total into zero.
• The live null-net standardness witness inherits the corrected CMI. Its Markov comparison remains below its stated tolerance, and its interacting Gibbs comparison remains non-Markov. The separate Gaussian null-net receipt producer does not call this information evaluator.
• MaxEnt and Einstein receipts continue to use their existing entropy and relative-entropy interfaces. No frozen registration, pinned receipt, claim payload, paper, book or release artifact changes. Exact normalized formulas and physical qualification in those surfaces are unchanged. The public distinctions between a declared finite collar and a physical source remain in place; this work does not select #1025's model or satisfy #1026's start gate.
• The existing finite-quantum-information workflow collects the new tests on Linux and Windows. The byte-frozen mandatory runner is unchanged.

Reproduction and independent checks

PYTHONPATH=code python -m pytest -q code/quantum_information code/collar_alignment code/geometry code/maxent -W error

Controls include independent scalar spectra through fourth order, separate matrix logarithms, noncommuting nonuniform Markov perturbations, exact tensor permutations, rare-event probabilities, zero and singular supports, missing data, negative states, underflow and weighted-sector distinctions. The exact modular-moment verifier is checked against all 255 nonempty classical three-bit supports, using integer conditional-independence identities as the independent answer. Quantum controls include a basis change hiding a product collar and a multi-sector state with different left/right decompositions inside the conditioning space.

The variational test evaluates an independently chosen trial family with wrong marginal states and wrong sector weights, retaining every extra term in (2). Four separate witnesses isolate each term of (1). The proof concerns all finite states in its stated algebra; these controls check the executable diagnostic and its input boundary rather than replacing the proof.

Initial local validation before maintainer review: 859 tests passed on Linux; 853 passed on Windows with six existing extended-precision skips, with warnings treated as errors. The initial implementation added 77 cases, including fourteen original pre-fix failures and four audit regressions committed before their repair. The null-net Gibbs witness has CMI 0.006842238011420933; its floating Markov construction has residual CMI 1.9828816605061793e-31. The latter is below the existing numerical threshold, not an exact-zero certificate for the rounded constructor output.

After the maintainer correction, the additional 35 support controls and the merge of main 7f99c8cc, the same complete local suite passes 989 tests on Linux and 983 on Windows, with six expected Windows extended-precision skips and warnings treated as errors. The PR now adds 112 cases in total. Replaying the reviewed implementation fails five of the eleven direct controls and 23 of the 24 broader complex-state controls; the canonical-arithmetic correction passes all 35. Claim registry, axiom inventory, reader style and whitespace checks also pass.

Sixteen isolated mutations were tested against 75 of the new controls (the full 255-support enumeration and slower multi-sector replay were separately included in the complete platform suites). Every copy contained its own source and tests outside the repository import path:

Deliberate defect	Failing controls
Erase quantum coherences	30
Round conditional reference probabilities first	1
Omit total-mass correction	2
Accept nonzero output underflow	1
Skip exact positivity	1
Check only the first modular moment	7
Reduce quantum precision	9
Suppress one endpoint obstruction	6
Discard later weighted sectors	7
Trust a product without comparing it	29
Return bits through a nats interface	21
Transpose the relative-modular reference	2
Require resolved eigenvalues for an exact product	1
Repeat splits over dimension-one factors	2
Reject unresolved spectra before exact Markov verification	1
Declare unresolved spectra Markov without verification	1

The transposed-reference mutation initially survived the fast subset. The retained complex BC-state control closes that coverage gap; its B marginal is real, so the wrong reference cannot cancel between both sides. The initial fast-test selector also unintentionally excluded four small_classical cases; the final replay uses complete function names to exclude only the two expensive tests. All cases run in the platform suites. The numerical output is still a checked approximation. The PSD, product, support and finite-moment decisions use exact rational arithmetic.