Graph
Map assumptions, evidence and conclusions.
Collatz / Коллатц — both maps & the narrated tour ↗
GERO Research
A mathematical view of
AI reliability.
Original research and independent analysis of the problems frontier laboratories are naming — translated into mathematical questions, assurance mechanisms and tests that can fail.
128published materials
3frontier research groups
1rule: evidence before confidence
The name
Graph Evaluation &
Reliability Orchestrator.
GERO organizes independent checks around AI-generated claims and keeps every conclusion inside the boundary of its evidence.
Evaluation
Apply bounded checks to necessary claims.
Reliability
Expose contradictions and unresolved risk.
Orchestrator
Coordinate models, tools and expert review.
The journal
Ideas, sources, experiments.
We do not copy laboratory articles. We link to the primary source, state what it reports, separate our inference and propose a falsifiable GERO response.
170 publications
Publication records across GitHub, GERO, LinkedIn and Zenodo →
SciPy exact-test boundaries: floating-point ties can change p-values
Synthetic Poisson and Barnard examples expose omitted ties. A proposed fix, 18 targeted regressions and separate rational/Decimal references are published with limits.
Tax-Calculator: Schedule D and AMT rule alignment
Five synthetic regressions and 490 passing checks support four proposed calculation corrections.
Luxon: skipped midnights and clamped month-end differences
Two independent proposals correct calendar counts and fractional month/quarter anchors.
decimal.js: rounding boundaries in log2 and log10
Sixteen independently checked references expose early termination at logarithm guard-digit boundaries.
ethers: negative FixedNumber rounding can change integral values
A proposed correction to signed quotient rounding, with 16 regressions and 104 related checks.
pvlib: clock time and elapsed time diverge on DST-transition days
A proposed fix for hour-angle and geometric sunrise/set conversion; six regressions fail before and pass after.
lifelines: keep fixed-time variance constant between observed events
A hand-calculated Greenwood regression exposes interpolation inconsistent with stepwise survival curves.
statsmodels: continuity correction and power inversion regressions
Two proposed corrections: avoid overshooting a continuity adjustment and reject false numerical roots.
Formula.js: seven proposed fixes for blanks, financial formulas and dates
Reproduced blank-cell, solver-convergence and calendar discrepancies, with independent PRs and tests.
Two reported tax-model discrepancies corrected in PolicyEngine-US 2.60.3
Maintainer-confirmed corrections to separate-filer eligibility and phase-out steps; shipped-source inspection, historical reproduction and explicit verification limits.
PoseLib: valid camera geometry lost through coordinate singularities
Five minimal pose solvers can lose solutions in special rig frames; actual-source regression tests pass after conditional frame changes.
OpenCV RQ decomposition: a scale-aware zero check
Absolute zero assertions reject valid large-magnitude matrices; a relative residual check preserves previously successful outputs.
OpenCV iterative PnP: solve near the object instead of a distant origin
A centered internal coordinate frame improves the conditioning of iterative camera-pose estimation for large world coordinates.
OpenCV SQPnP: conditioning across units and coordinate origins
Centering and unit-RMS scaling address pose sensitivity to units and large world-coordinate offsets.
OpenCV Rodrigues: accuracy at small angles and near pi
Boundary branches can erase a small rotation or distort the axis near pi; a proposed inverse-map correction is tested at fragment level.
OpenCV RANSAC: preserve the camera-pose initial guess
Float32 and row-vector initial poses are lost when output arrays are allocated before reading the supplied guess.
Augmented Black–Litterman: historical prior must retain the joint state
Asset-only and factor-only views change the observation, not the dimension of the joint asset-factor prior.
Riskfolio Bayesian Black–Litterman: equilibrium weights use excess returns
A first-order-condition check isolates an extra risk-free-rate contribution in the directly returned weight vector.
Missing observations are not zero returns: a PyPortfolioOpt correction
Explicit sample handling changes shrinkage and downside-risk estimates; six integration results are independently cross-checked.
Market-neutral portfolios rejected by a fully-invested risk bound
An analytic two-asset counterexample demonstrates a feasible neutral portfolio that the precheck rejects.
Black–Litterman weights: excess returns and the singular fallback
Two corrections to the implied-weight solver: subtract the stated risk-free rate and solve the asset-space system when covariance is singular.
Provably fair does not mean a constant expected return
Two finite-grid case studies distinguish a reproducible outcome mapping from an economic return curve.
Historical IRS Direct File source: a joint-filer credit and a rounding witness
Fresh local replay of the public 2024 fact graph; no filed return or deployed IRS service tested.
PolicyEngine UK: Child Benefit charge rounding now corrected upstream
A public merged fix implements the statutory staircase; 33 current-helper checks agree with an independent Decimal reference.
web3-ethereum-defi: zero depth inside an active liquidity interval
An isolated current-source depth routine returns zero against a nonzero integral reference; live RPC applicability is untested.
penaltyblog: positive expected log growth at a ruin boundary
A capped stake leaves a positive probability of zero wealth, while one reported growth field remains positive.
shin: a default probability result can hide non-convergence
The actual Python optimiser returns a vector summing to 0.90136 on an underround example; diagnostic output exposes failure.
uniswap-python: integer price flooring and odd-tick liquidity, with a public patch
Fresh isolated-method replay of main and PR 482; the contributor credits GERO, and merge is still pending.
Curo: fortnightly payments collapse onto monthly dates
Real source repeats payment dates and divides a two-week remainder by 15. Focused checks reproduce both problems.
actuarialmath: the variance selector rejected before its own branch
36 valid variance requests fail on current source. A narrow candidate repairs them while preserving 108 moment controls.
bond_pricing: perpetuity-rate inversion fixed in release 1.0.1
A maintainer-credited fix, replayed against 288 perpetuities, 18 finite-term controls and a vector input.
Samsung ONE: Gather slices and Mean axes changed by compiler passes
Two native C++ reproductions on the last public source: a row becomes [3,0,0] and fused reductions select the wrong axes. Focused patches and tests; successor routing pending.
Empyrical: constant returns and floating-point Sharpe ratios
Exactly constant returns can produce huge finite ratios instead of the documented signed infinity. A focused fix passes all 860 package tests.
Riskfolio BOP: correcting the shrinkage target denominator
The target coefficient differs from the cited equation. An exact rational regression passes after the one-line correction; full-package integration remains untested locally.
PyPortfolioOpt: a ticker-order mismatch breaks the allocation budget
A synthetic $10,000 allocation costs $500,050 when priced by ticker. A one-line fix and regression tests have been submitted.
A mid-month rate increase disappears from chainladder on-level factors
A +20% incremental change on July 15 is dropped by monthly exact-date matching. A proposed as-of fix restores the old-year factor from 1.0 to 1.2; upstream review is pending.
A missing drawdown and a positive score for a loss in quantstats
Two separately reproduced reporting defects: an initial drawdown is omitted and Recovery Factor loses the return sign. Two focused PRs include regression tests; review is pending.
A shared constant flips two convolution biases in coremltools
A reproduced converter defect changes two subtractions into additions. A focused patch and six regression cases are submitted to Apple in PR #2901; review is pending.
Frappe Lending quarterly day counts: three days instead of 91, and the v16.6.1 correction
A reproduced calendar-unit error in quarterly interest and a bounded verification of the official v16.6.1 fix. Eight intervals: seven day-count failures before, zero after. Synthetic evidence, no customer-loss claim.
A zero guarantee with a minus-200 outcome: penaltyblog partial hedging
A reproducible payoff inconsistency in penaltyblog's hedge_all=False branch: the function reports zero worst-case profit while its own returned back stakes lose 200 in one synthetic outcome.
A one-payment loan breaks IRR: an incorrect derivative in loan-calculator
A public one-payment IRR call crashes, and an exact derivative is 140 instead of 240. One proposed correction passes all 35 checks; maintainer review is pending.
When gross-up does not return the requested net: two loan-calculator defects
In loan-calculator 1.2.2, progressive Price and constant-amortization gross-up fail to reconstruct the requested net principal. Exact witnesses, 36 cases, an independent cash-flow reference and proposed corrections.
actuarialmath whole_life_annuity ignores benefit b at zero interest
At zero interest, the public method ignores its benefit amount: a Beta(2) example returns 20 instead of 140. Current and release fail 48 of 90 bounded checks; an isolated one-line candidate fails zero.
actuarialmath Uniform shortcuts use the wrong conditioning age and reverse limited-life weights
Source-pinned current and release shortcuts disagree with independent De Moivre closed forms in 275 of 468 comparisons; an isolated local candidate gives zero disagreements on the same grid and the failures return after restoration.
amortization 3.0.0: small positive rates can erase interest or alter the rounded payment
A small positive periodic rate can be rounded away inside the annuity formula, causing ZeroDivisionError or a cent-level payment error; an equivalent log1p/expm1 form passes the project suite and an independent Decimal grid.
MLX arctan: finite weighted derivatives collapse to zero after intermediate overflow
In Apple MLX CPU float32, arctan differentiation forms 1+x² before division. For finite x=-2^64 and seed=-2^120, x² overflows and the JVP/VJP becomes signed zero although the correctly rounded result is -0.00390625. A scale-aware research candidate reduces 352 discrepancies per tested layout to zero, with documented higher-order residuals.
actuarialmath direct endowment second moments omit the benefit power
A missing power yields 1.5 instead of 3 and removes uncertainty from a synthetic risk worksheet. Actual current/release APIs, independent oracles, candidate and restoration: 720/720/0/720 disagreements across 2,880 calls per variant.
bond_pricing 0.7.3 ignores immediate-start annuity payments
A documented payment-timing flag is ignored: two immediate-start payments with PV 210 and a 10% rate are calculated as 121 instead of 110. Actual package reproduction, explicit cash-flow reference and a proposed correction.
FinancePy Merton jump pricing: a zero probability can erase an option value
Current-source Merton jump pricing returns 0 instead of about 23.50587 under a finite-input stress test. 112 scenarios, 52/0/52 discrepancies with candidate/restoration, and a measured synthetic worksheet consequence. The affected module is absent from the released 1.1.2 wheel.
Apache Commons Statistics: finite bounds, NaN probabilities, and a changed decision
Rounded-log cancellation in LogUniformDistribution 1.3 produces invalid probabilities for narrow finite intervals. Across 176 distinct inputs, a GERO synthetic worksheet makes 30 wrong threshold decisions; a local candidate removes them and restoring the original restores them.
MLX log10: finite gradients, nonfinite updates under extreme loss scaling
In a deliberately extreme CPU loss-scaling experiment, an MLX log10 derivative overflow makes all 65 classifier parameters and 108 held-out predictions nonfinite. A guard skips the step; an incomplete experimental candidate matches ordinary updates.
actuarialmath continuous annuity: dropping elapsed years changes a value and a decision
Dropping elapsed years changes a continuous annuity factor from the correct 8.75 to 9.166666666666666. A local candidate removes 648 failures across 2,160 cases; a synthetic written value changes from 9,166.67 to 8,750.00 and flips a ceiling decision.
FinancePy floating lookback calls: a finite price becomes unavailable
A valid floating lookback call worth about 54.03 becomes unavailable because a power overflows before multiplication by a tiny probability. A local log-space candidate removes 52 failing public calls on a 600-row grid; restoration reproduces all 52.
nntrainer CentroidKNN chooses an unobserved class over an exact centroid match
A positive zero-sample sentinel outranks every finite negative distance. Native component and public model execution verify wrong class decisions, a one-line candidate and restoration.
Same Itô integral, fewer operations: a reproducible computational study
Known conditional estimates, fewer local operations, and measured trade-offs. Exact identities, executable Python, raw timings, single-query slowdowns and a rounding counterexample.
MLX erf: a finite gradient becomes infinity before damping is applied
Native MLX CPU float32 execution returns an infinite derivative action where 1.32834399e38 is representable. A synthetic guarded update is skipped; a bounded first-derivative candidate permits it. Extreme tails and second derivatives remain limited.
Endowment variance scales the benefit twice in actuarialmath
The benefit scale is applied twice to endowment variance. In a synthetic report, the native portfolio estimate changes from 41,862.13 to 6,606.12 and a 7,000-limit decision flips. Exact-payout replay: 483 → 0 → 483 mismatches across 1,300 scenarios.
Strata: a zero-volatility Hessian prevents custom-model option pricing
NaN second derivatives interrupt the native price extrapolator through a documented custom-model interface. Direct failures180->0->180 in866 input rows; downstream constructor failures6->0->6 in21 input rows. Synthetic constant-volatility fixture; default Hagan/CMS path excluded; no bank/customer loss measured.
PyLoan replaces an explicit 100 payment with a 919.10 final payment
PyLoan 0.7.2 overrides a specified annuity payment at maturity. A synthetic schedule pays 919.10 instead of 100; a targeted candidate removes 408 payment-cap violations in 864 scenarios. Separate rounding differences and unchanged upstream test failures are retained.
Strata returns zero implied volatility for a positive option price
For a synthetic option price of 1, Strata 2.12.74 returns normal implied volatility 0 instead of 2.5066282746 when the initial guess is zero. A local initializer correction removes 864 failures in 1,750 observations; restoring the source restores them. Reported first as issue 2796.
MLflow NDCG: irrelevant retrievals score above zero after a document-ID collision
An all-irrelevant retrieval list scores 0.63093 instead of 0. A local correction removes 214 failures in 1,810 synthetic scenarios; restoring the original restores them. The deprecated metric was reported to MLflow before publication.
MLX rsqrt: finite directional derivatives lost to an intermediate ratio
MLX returned infinity where a finite directional derivative is about -5e14. A CPU correction removes 443 failed input/seed pairs out of 2,136; restoring the original restores all 443. The forward outputs remain unchanged.
FinancePy BondZero: verifying the upstream fix for a 10,000-fold duration error
A one-year bond returned duration 9,522.90 instead of about 0.95229. The upstream fix passes all 1,320 frozen scenarios; restoring the extra multiplier restores 1,200 failures. Official release 1.1.2 still reproduces the defect.
actuarialmath Beta: a missing survival factor changes expected insurance benefits
Two equivalent methods return 150,000 and 100,000 for a zero-interest expected benefit of 75,000. Correcting the Beta probability density removes all 720 failing insurance scenarios out of 864; restoring the original formula restores the same failures.
FinancePy adjusted binomial: invalid support points produce negative probabilities and shift the mean
Four equal-loss credits produce probability −0.1127312875, and a two-credit case shifts the mean from 1.60 to 1.65. Adjacent support points reduce invariant failures from 295 to zero across 2,844 frozen portfolios; restoring the original code restores the same failures.
FinancePy: premature series truncation produces an invalid option price
A nonnegative option payoff is priced at −0.17335 instead of +0.00019876. Two series truncation corrections reduce discrepancies from 360 to zero across 480 synthetic scenarios; restoring the original code restores the same errors.
nntrainer SplitLayer leaves later channels unwritten
A valid width split of four values leaves two output positions unwritten. Actual C++ full-prefill tests:4,320 coordinate mismatches become zero with a local channel-loop correction; restoring the original code restores the same errors.
FinancePy FXForward converts today's NPV using a future exchange rate
648 independent scenarios and 64 real-curve scenarios; a one-line spot conversion repair passes all.
Apple MLX sigmoid derivatives lose a representable positive tail
A representable sigmoid derivative becomes zero. Native CPU checks isolate the correction from a known float64 exp issue.
FinancePy Vasicek pricing loses the small-mean-reversion limit
At a positive mean-reversion parameter of 1e-8, a finite bond price near 0.6167242 becomes infinity. Across 1,350 positive-parameter scenarios: 527 original discrepancies, zero with a local correction, and the same 527 after restoring the original code.
FinancePy annuity pricing depends on prior payment calls
A prior payment request changes the same annuity price from 5.069444444444445 to 506.94444444444446. One cache/face defect: 432 → 0 → 432 failures on 864 synthetic scenarios.
FinancePy CIR pricing loses its finite range and small-volatility limit
A unit payment priced at 0.6882687528140472 becomes 2.2407257971155513e96. A bounded algebraic correction removes 1168 failing prices from 4536 parameter vectors; restoring the original source restores every failure.
MLX median overflows while averaging finite central values
Two copies of 60000 have median 60000. Native MLX float16 returns infinity. A bounded local correction removes 540 incorrect outputs across 7488 observations; restoring the original code restores every failure.
ConstantForce whole-life insurance ignores the benefit amount
A benefit of 100,000 returns 0.4 instead of 40,000 in an educational actuarial library. A missing multiplier explains 120 failures across 144 observations; a local correction and mutation replay isolate the cause.
FinancePy Merton calibration depends on the monetary unit
The same synthetic firm changes from 7.77% to 95.49% calculated default probability when monetary units change. Normalizing before calibration removes 94 failed round trips across 120 scenarios; residual solver statuses remain disclosed.
MLX CPU quantized matmul: avoidable accumulation error in float16 and bfloat16
Exactly representable additions return 2,048 instead of 6,144. Native CPU probes, an exact model and a precision-reversion control isolate the accumulator; a local float32 buffer removes the selected-grid mismatches.
nntrainer DivideLayer: finite forward values, incorrect denominator gradients
Native C++ tests find 1,302 incorrect denominator gradients on 4,096 selected inputs. A local CPU FP32 correction removes them; restoring the original source restores every failure.
QuantLib: deterministic ITM probabilities and cap/floor deltas
Two zero-volatility branches return incorrect probabilities and cap/floor deltas. Candidate corrections remove 90 mismatches across 660 helper inputs; tested prices remain unchanged.
pyliferisk geometric valuations discard supplied mortality assumptions
An existing pyliferisk audit of geometric valuations that reset supplied mortality assumptions. The report preserves its direct cash-flow oracle, regression evidence and proposed correction.
Axisymmetric p-harmonic modes in circular cones: nodal classification and two-term asymptotics
A non-peer-reviewed mathematical preprint on homogeneous axisymmetric p-harmonic functions in circular cones. The complete original PDF and author attribution are retained.
Obstructions to Low-Dimensional Natural Matrix Interpretations for a Mixed-Base Collatz Rewrite System
Published software and manuscript materials about a specified low-dimensional matrix-interpretation proof method. This does not prove or disprove the Collatz conjecture.
Spark-X2.5-1.7B: exact fee bases and half-up rounding study
An AI-operated study of 12 authored arithmetic exercises, with original model outputs, exact references and separate answer-delivery scores. This is an LLM experiment, not an implementation-bug report.
Integer overflow reverses MLX isclose and allclose results
A version-pinned native MLX audit of integer subtraction and absolute-value overflow in isclose and allclose. The existing report and bounded CPU prototype are preserved.
actuarialmath whole-life variance squares the mean twice
The mean and second moment are correct, but an extra square produces the wrong variance. A local correction resolves 54 discrepancies across 96 independently checked synthetic cases.
FinancePy cash-settled swaption annuity counts N−1 payments instead of N
A payment-count error returns zero for a one-payment cash-settled swaption. A local correction restores all 360 annuity checks; small independent Black-price residuals remain and are reported.
A constant forward, a NaN gradient: nntrainer PowLayer at exponent zero
Native C++ PowLayer returns a constant forward but an invalid input derivative at exponent zero. A locally tested candidate removes 180 failing coordinates in a 385-case CPU FP32 grid.
FinancePy BAW pricing divides by zero at a zero rate
A finite zero-rate coefficient is formed as 0/0, and a nearby put can exceed the boundary solver's iteration limit. Released-wheel reproduction, analytic limit, independent CRR control and a green upstream correction.
FinancePy misprices three FX digital payout-currency paths
Three single- and double-digital paths use the wrong probability or discount term. Released-wheel reproduction, payoff-derived controls, falsification and a green upstream correction.
FinancePy Act/365L can fail when an optional date is omitted
A documented two-date call can dereference the absent reference-period end when an annual interval reaches February 29. Released-wheel reproduction, independent 91/366 control and submitted correction.
PyXIRR RATE can return an unfinished calculation as a result
Released PyXIRR 0.10.8 returns an unconverged vector RATE and fails at zero starting guesses. Independent discounting, compiled Rust checks and a tested candidate correction.
FinancePy vector pricing uses the last expiry for every option
A list of distinct expiries is reduced to its final horizon and broadcast across every option. Released-wheel reproduction, independent oracle and submitted correction.
INDEPENDENT NUMERICAL AUDIT
Parallel vectors produce nonzero cross products in Apple MLX
A component-axis alignment error turns parallel vectors into a nonzero cross product. Clean native CPU regression and a bounded candidate correction.
Apple MLX: integer norms can become zero, negative, or NaN
Clean native CPU evidence of integer arithmetic before float promotion. Tested bounded repair, unchanged inexact controls, and unresolved large-value range cases.
FinancePy mortgage payments fail at zero and very small rates
A zero rate raises ZeroDivisionError; tiny nonzero rates can leave a material unpaid or overpaid balance in synthetic amortization checks.
FinancePy bond principal does not scale with face amount
Dirty price is scaled by the requested face while accrued interest remains at face one, violating the clean-value identity in a released-wheel reproduction.
ONNX ReduceMean overflows before reaching a finite mean
Finite means become infinity in both the reference evaluator and Runtime CPU provider because the input-type sum overflows before division.
ONNX ReduceL2 loses finite norms at extreme scales
Finite Euclidean norms become infinity or zero in both the reference evaluator and Runtime CPU provider because values are squared before scaling.
Checking the rules behind financial calculations
A public Fineract case with 25 common checks, scoped candidate corrections, reproducible sources and a 40-second customer overview. Synthetic component evidence; customer impact is not established.
ONNX LpNormalization loses finite vectors through intermediate range
Finite L1 and L2 inputs collapse to zero in both the reference evaluator and Runtime CPU kernel because avoidable intermediate norms overflow or underflow.
Apple MLX: small matrices lose finite log determinants
Clean CPU reproduction of a small-matrix slogdet range defect. Tested FP32 workaround; LU autodiff limitation and rejected repair counterexample retained.
Finite ONNX Normalizer inputs overflow, underflow, or violate the output contract
Finite rows at 1e30, 1e-30, 1e300 and 1e-300 produce NaN, zero, unnormalized values, or the wrong dtype in the released reference and Runtime CPU implementations.
MLX LayerNorm: an optional unit weight changes bias precision
Clean native CPU reproduction: an absent weight narrows finite float32 bias. A one-line candidate changes 106 failed assertions out of 385 to zero; 16 controls remain unchanged.
Finite FP16 Inputs Become Zero in MLX GroupNorm
Standard GroupNorm returns zeros for finite FP16 inputs because a sum of squares overflows before division. A CPU research patch preserves channel grouping: 45 main-suite failures become zero, with 41 compatibility checks passing.
One FP16 Batch Corrupts MLX BatchNorm Running Variance
A finite float16 batch makes the default running variance infinite, leaving later evaluation at zero. A CPU research patch changes 102 main-suite failures to zero, with 33 compatibility checks passing. Existing corrupted state is not restored.
Finite Gradients Become Zero: MLX Gradient Clipping
Float16 [3072,4096] should clip to approximately [0.6,0.8], but MLX returns zero. A CPU research patch passes 1701 assertions across 219 scenarios, plus 39 compatibility and autodiff checks.
Same Ratio, Wrong Derivative: MLX Division at Extreme Scales
For C/(C*t), the derivative at t=1 must remain −1. MLX returns zero or negative infinity at extreme scales. A research patch passes 1328 native comparisons, plus 1886 compatibility checks.
Scale Changes the Gradient: MLX arctan2
For atan2(C*t,C), the derivative at t=1 must remain 0.5. MLX returns zero or infinity at extreme scales. A research patch passes 1368 native comparisons; both failed intermediate patches are preserved.
Finite Values, Missing Derivatives: MLX arcsinh and arccosh
At float16 x=1000, arcsinh and arccosh return finite values but zero derivatives. A normalized real-derivative patch passes 518 native comparisons across four dtypes; GPU and speed are untested.
A Nonzero Derivative Becomes Zero: MLX expm1
At x=-20, reverse mode returns zero for a derivative near 0.20611536. A local VJP repair plus the known exp control passes 216 native comparisons; the VJP-only patch can regress float64.
Same Inputs, Wrong Weights: MLX logcumsumexp Gradients at Large Offsets
At two equal float32 inputs of 100000000, MLX returns gradient [1,1] instead of [0.5,0.5]. An isolated CPU study separates a new VJP prototype from an already reported exp limitation.
Missing Curvature: MLX logcumsumexp at a Zero Incoming Gradient
A smooth x²/2 example returns second derivative 0 instead of 1. A research C++ prototype passes 477 checks; GPU and large-tensor performance remain untested.
Zero Mask, Missing Gradient: MLX block_masked_mm
An extra output-mask factor suppresses its derivative at zero. A one-line C++ repair passes 377 scenarios and 604 checks; the demonstrated training step lowers loss from 18 to 0.
A Missing Transpose in MLX Quantized Gradients
With transpose=False, affine gather_qmm miscomputes scale and bias gradients. A local C++ orientation repair passes 205 scenarios and 379 checks; the measured quadratic update then lowers the loss.
Sorted Indices, Wrong Gradients: MLX Gather
Valid sorted-index calls produce incorrect gather_mm and affine gather_qmm gradients. A bounded local FP32 CPU repair passes 69 scenarios and 626 checks; quantized FP16 fallback remains a limitation.
Orthogonal Is Not Symmetric: MLX Hadamard Gradients
MLX uses the forward Hadamard transform in reverse differentiation for nonsymmetric factors 20 and 28. A local C++ repair passes 64 scenarios and 624 checks; measured energy steps confirm the correction.
A Constant, a Polynomial, a NaN: MLX power Derivatives at Zero
MLX power returns NaN derivatives at zero for smooth fixed-exponent polynomials. A local C++ patch passes 41 scenarios and 396 checks; prior work and test-discovery limits are documented.
An Unused Value Gets a Gradient: MLX Masked Assignment
MLX mapped boolean assignment routes gradients across batch examples. A local C++ patch passes 19 scenarios and 255 checks; explicit loops and actual-forward finite differences agree.
A Missing Gradient in MLX Block Updates
A strict C++ scatter_max/min block update loses a winning gradient. Independent finite differences confirm the reference; a minimal extent correction passes 26 targeted checks.
When SAME Padding Makes Average-Pooling Backward Skip Windows
nntrainer returns correct forward values but loses three quarters of the input-gradient sum in a minimal asymmetric-padding case.
When Dropout Sends Two Gradients to One Input
A native C++ nntrainer case study: zero-rate Dropout forwards correctly but misroutes multiple input gradients. Finite differences, a local patch and 19 passing tests.
Finite Loss, Wrong Gradient: Gaussian NLL in MLX
Native MLX 0.32.2 CPU reproduction: finite Gaussian NLL with a variance gradient of the wrong sign. Inputs, references, regression tests and limits of a mitigation.
Same Gradients, Half the Step: MLX Muon and Convolution Shapes
A native CPU reproduction isolates a shape-dependent Muon update in MLX. Equivalent convolution and linear layers agree before the optimizer step; a minimal reorder passes 31 focused tests.
A Real Loss, a Reversed Gradient: Complex Autodiff in MLX
A real loss increases after a gradient-descent step in MLX. Native tests isolate missing complex conjugation and an arccosh branch error; the local C++ patch passes 131 scenarios.
A Smooth Polynomial, a NaN Hessian: MLX cumprod at Zero
MLX returns NaN second derivatives for a smooth cumulative-product polynomial at zero. A division-free C++ prototype passes 57 local scenarios, with an explicit O(N log N) cost.
ONNX ReferenceEvaluator turns infinite ReduceLogSumExp slices into NaN
Slices [inf, -inf] and [-inf, -inf] both become NaN, although the exact results and ONNX Runtime give +inf and -inf respectively.
ONNX ReferenceEvaluator misidentifies BatchNormalization opset-9 modes
Inference uses blended training statistics and five-output training cannot return its requested statistics because mode is selected from momentum instead of output count.
ONNX ReferenceEvaluator gathers unsorted Unique slices from the wrong axis
With sorted=0 and a nonzero axis, first-occurrence indices are always applied to axis 0, producing wrong data or an immediate out-of-bounds exception.
ONNX ReferenceEvaluator rejects valid GatherElements subshapes
Inputs of equal rank with smaller non-gather index dimensions are mathematically well-defined and accepted by ONNX Runtime, but the reference oracle requires exact shape equality.
When Cosine and LayerNorm Break Their Contracts
Reproduced cosine NaNs in MLX and an incorrect LayerNorm input gradient in nntrainer, with tested local patches and downloadable evidence.
Research acceleration needs an outcome-weighted evidence contract
OpenAI reports rapidly growing agent use while warning that activity metrics are hard to interpret; the mathematical task is to estimate validated progress under steering, selection and changing compute.
A machine-checked proof still needs a statement contract
Anthropic's Fermat formalization shows the scale now possible in Lean; the remaining assurance problem is preserving meaning, provenance and dependency scope from source theorem to checked root.
When Causal Attention Sees the Future
A reproducible nntrainer causal-mask report: future values affect earlier outputs. Native tests, a local repair and an upstream issue are available.
ONNX reference LpNormalization uses a signed sum for p=1
The input [1, -1] becomes [0, 0], while the exact L1-normalized result and ONNX Runtime both return [0.5, -0.5].
ONNX GroupNormalization reverses channel/group divisibility in its documentation
The summaries of versions 18 and 21 contradict the num_groups attribute and the operator's own four-channel, two-group example.
When Small Losses and Gradients Disappear
Reproduced BCE and logaddexp failures in MLX and nntrainer, with mathematical references, tested local patches and downloadable evidence.
Finite Inputs, Non-finite Results: An Activation Stability Audit
Reproduced ELU, GELU and Softplus failures in pinned MLX and nntrainer code, with tested local repairs and downloadable evidence.
ONNX reference LogSoftmax and cross-entropy return infinity for finite representable results
A 104-logit gap makes the reference loss infinite while the mathematical result and ONNX Runtime output remain finite.
Dinero.js misallocates safe integers after an unsafe intermediate product
All inputs remain JavaScript safe integers, yet binary64 rounding changes a 3-unit allocation from the exact [2, 1, 0] to [3, 0, 0].
Dinero.js toDecimal accepts non-decimal bases and returns plausible wrong strings
Scalar base 240 is accepted and formatted as 1.10 for the exact value 250/240, while the equivalent multi-base currency [20, 12] is correctly rejected.
Zero Observed False Positives Is Not a Zero False-Positive Rate
Exact and Wilson bounds, sample-size thresholds, runnable code and the assumptions that make a zero-error headline meaningful or misleading.
When the reference also fails: a reproducible ONNX numerical snapshot
123 graphs, 246 evaluations, seven flagged graphs. A repeated CPU benchmark, fourteen archive replays and an 80-digit diagnostic check, with downloadable evidence and explicit limits.
ReduceMean Without an Axis Could Not Initialize
An omitted reduction axis throws during initialization; a batch-two regression exposes the broken default and verifies a 23-test local correction.
ONNX Runtime float16 QuantizeLinear truncates instead of rounding
A two-value counterexample shows the CPU float16 path returning [0, 0] where the ONNX rounding contract requires [1, -1].
MoneyPHP loses largest-remainder ordering above float precision
An accepted 7-quadrillion-unit amount makes two exact residuals collapse to zero in binary64, assigning the final unit to the wrong share.
Apache Fineract nop does not invert pmt for non-zero rates
Fineract's public number-of-payments helper returns 7 when its adjacent payment function was given a 12-period loan.
When Reciprocal Multiplication Changes the Answer: Two Swift Numerics Contract Tests
A five-line counterexample shows reciprocal multiplication changing a result that direct complex division preserves, with a separate integer-power phase-error finding.
The Fast Path Put a Complex Number in the Wrong Quadrant: A Swift Numerics Sign Audit
A large-negative-real fast path flips component signs in complex cosh and sinh, propagates into cos and sin, and is reinforced by incorrect test expectations.
How Numerical Contracts Break: Five Cases in Swift Numerics and nntrainer
Five bounded audit cases show how documented equivalence, exact identities, input preservation and parameter propagation expose failures that fixed expected values can miss.
MLX affine quantization: a documentation/runtime contract mismatch
MLX 0.32.2 returns a valid affine parameterization that differs from the formula stated in its public documentation.
What Simple Mathematical Invariants Reveal About AI Software
Five compact case studies show how chain rules, shape semantics, special values and execution parity expose silent failures in production AI software.
A KL-Divergence Backward Pass That Changes Its Own Target
A one-scalar experiment shows how an in-place label negation makes repeated KL-divergence backward calls alternate their gradient sign.
Four Loss Layers, One Missing Scale: A Mixed-Precision Contract Failure
Four of five implemented loss derivatives omit the configured loss scale; four runtime counterexamples and a 71-test repair expose the cross-component contract failure.
Fractional Projective-Gap Inequalities for Harmonic Gaussian Chaoses
An author preprint proving fractional projective-gap inequalities in the complete second chaos, all split-product degrees and the first fully mixed harmonic cubic case.
A one-line contradiction in Swift Numerics' tie-to-zero documentation
An exact arithmetic check found that one Swift Numerics example reported 3 where its stated rounding rule and its own explanation require 2.
An exact fifth root exposes a one-ULP accuracy gap in Swift Numerics
Double.root(3125, 5) returns 5.000000000000001 although the exact, representable result is 5.0.
A scale-dependent zero point silently corrupts UINT8 affine quantization in nntrainer
A three-value UINT8 input collapses a negative endpoint and real zero into the same code. The pinned evidence package records a candidate ratio-formula repair and 15 passing local tests.
Where Mathematics Meets AI Reliability: The Problems Frontier Labs Are Naming Themselves
A research perspective connecting the reliability problems named by frontier labs to observability, control, verification and calibrated abstention.
AI Can Generate New Mathematics. Who Certifies the Proof?
A falsification-first protocol for turning AI-generated mathematical arguments into dependency graphs, bounded verdicts and replayable evidence.
The Most Important AI Feature Is Not an Answer. It Is a Boundary.
Why scope, evidence, calibration, falsification, escalation and traceability should be a single contract around every high-consequence AI answer.
A Constant Left Open in 1973, and What It Took to Close It
Krol proved the leading narrow-cone constant in 1973 and left the next term as O(1). Identifying it took analysis; establishing who owned the first term took archive retrieval.
A Retraction Log: Six Collatz Claims and What Killed Them
Six claims, zero arithmetic errors, five failures. Every one failed above the arithmetic layer: wrong quantity, prior art, or a fabricated citation.
The 60-Pair Smoke Benchmark, With the Data and the Command
The dataset, the command and the full result behind the 60-pair claim — including the distinction between a refuted defect and an abstention that the short phrasing hides.
Hidden reasoning is an observability problem
Anthropic's faithfulness results suggest that a visible chain of thought is a noisy sensor, not a guaranteed transcript of computation.
Attribution graphs still need a semantic contract
Mechanistic interpretability can expose computational structure, but the map is partial and the meaning assigned to it remains a scientific claim.
Research mathematics needs proof assurance, not a single judge score
OpenAI's correction to a First Proof submission exposes false acceptance as the central metric for AI-assisted mathematics.
Monitoring reasoning is a stability problem
OpenAI's monitorability evaluations raise a systems question: will a useful safety signal remain informative after training, scaling and distribution shift?
Research mathematics is a generator–verifier–reviser control loop
Google DeepMind's Aletheia separates research mathematics from bounded contests and makes revision and failure admission part of the system.
Self-correction is two problems, not one
Google Research separates finding a mistake from repairing it; GERO turns that distinction into separate evidence contracts.
A gold-level benchmark result is not a research reliability certificate
DeepMind's IMO result is a major capability milestone; the next mathematical question is how performance transfers beyond a bounded contest distribution.
No publications match these filters.
Evidence standard
Source. Interpretation. Test. Boundary.
Every frontier brief preserves those four layers. Read the editorial and evidence standard or follow new work through RSS.