Tutorial precision audit¶

This ledger links the tutorials' precision choices to measured comparisons. Each record identifies its numerical controls, source revision and hardware; agreement at those inputs does not establish one accuracy bound for every model or plotted point. See the precision guide for the decision procedure and the development study for implementation history.

Inventory¶

• All 654 checked-in tutorial CSV exports identify fp64. Their regenerators either request binary64 explicitly or use capture_csv_tables, which does so. This covers the ordinary finite-chain, gas, impurity, field-theory and dispersion examples. Their existing checks inspect successful statuses, model identities and the relevant analytical or numerical consistency tests.
• The two-spinon thermodynamic heat map and introductory point command both use binary64. The point command fixes the kernel tolerance at 1e-12; its complete rerun returns the same S2 value as the five-format comparison below, differing from MPFR192 by less than 2e-15 relative. A successful two-spinon value does not imply that the four-spinon part of the same query succeeds.
• The 75 JSON Lines records in the one-magnon precision tutorial deliberately compare five formats. They are checked against an independent Decimal sine evaluation. The examples exhibit observable cancellation and root sensitivity, and explain the stable special-case formula; they are not ordinary defaults.
• Four-spinon refinement uses binary128; the small six-spinon example uses native extended80 where available. Their measured comparisons are below. The resolved-slice tutorials have separate recorded checks on their much larger soft domains.

Most root-solving CSV examples use residual tolerances near a few multiples of binary64 epsilon. Those are stopping criteria, not energy-error bounds. The finite-volume sine-Gordon examples explicitly use a 1e-7 absolute tolerance and report quadrature, source and tail diagnostics separately. Their physical finite-size/asymptotic limitations remain distinct from arithmetic.

What the ordinary tutorial checks establish¶

Binary64 is the ordinary plotting format, not a claim that every observable has sixteen accurate digits. The model tutorials retain their own checks and limitations. These examples show where to look before increasing precision:

These are existing model checks, not new wider-arithmetic comparisons for all 654 exports. For an observable requiring more accuracy than the tutorial plot, compare successive precisions with non-arithmetic controls fixed, then refine the mesh, state family or domain separately. The spinon comparisons below measure the precision and cost choices explicitly.

Four-spinon starting point¶

The compact comparison retains all five formats, including binary64's numerical failure, and can be checked or regenerated with scripts/check_xxx_four_spinon_precision.py. It fixes q=1.5, omega=1.8, J=1, order4/T8, density rtol/atol=1e-8/1e-30, 64/80 residue terms and 16/20 tail orders. The two-spinon kernel tolerance is explicitly 1e-12 in every format. One worker is used throughout.

Hardware: Intel Core i7-12700KF, GCC 13.3 Release, source 0b496a0, Uni20 1cb2474. Timings include the query and ordered output on a shared host; they are not portable performance guarantees. Failure time is not the cost of a complete binary64 calculation. The native 64-bit-significand S4 differs from the MPFR192 value by about 1.14e-16 relative; MPFR128 and MPFR192 differ by about 2.42e-38. These comparisons describe the same truncated quadrature, not the unrestricted physical response.

Changing the momentum or frequency¶

The native neighbourhood record checks 49 supported points and five physical zeros at the same order and cutoff. With a 53/64-bit pair, extended80 completes 26/49 supported points at 1e-8, 47/49 at 1e-6, and 49/49 at 1e-4. Binary64 completes none at these tolerances. All outcomes are retained. Wider references at three difficult points agree with accepted native S4 values within 1.3e-16 relative; that comparison does not certify every point at the relaxed tolerances.

The binary128 neighbourhood record uses a 64/113-bit pair at all 49 supported points. It completes 48/49 at 1e-8 and 49/49 at 1e-6; the strict failure at q=1.5, omega=4.5 remains unavailable. All 97 successful estimates agree with matched MPFR128 S4 references within 1.3e-30 relative. The binary128 exports identify source 865d701, a Xeon Gold 6254 and two workers per query. The MPFR exports lack an embedded revision; the record retains the review author's source attribution. Worker counts and sector selections differ, so the comparison concerns S4 values, not total-query cost. Both sets preserve numerical controls and raw hashes. The checker reconstructs the compact coverage and comparisons without reading raw archives:

python3 scripts/check_xxx_four_spinon_native_neighbourhood.py

The starting command uses 1e-6, with a margin below the integration changes it demonstrates; its actual two-sector rerun is checked separately. The strict five-format and refinement comparisons retain 1e-8. Changing the momentum/frequency can still require a stronger precision pair. A successful relaxed screen needs a separate comparison with a stricter reference.

Refining the quadrature¶

The refinement checks show that extended80 passes orders 2/4 at T8 and T12, but fails the arithmetic screen at orders 8/16. The cheap starting command therefore cannot support the whole refinement figure. With the 64/113-bit pair, all eight binary128 quadratures pass the strict 1e-8 screen and agree with MPFR192 within 6.3e-31 relative. The comparison record retains their controls and total cost of 348 CPU seconds on an i7-12700KF. The tutorial figure and reproducer use these binary128 data. Failed cases remain in the records; no failed node is replaced by a successful prefix.

The same record validates the small six-sector API spectrum and moment commands in six formats. Native extended80 agrees with MPFR384 within 4e-16 relative across values, replicate means/maxima, sampling errors and covariance; binary128 agrees within 3e-30. Both binary64 failures are retained. Those small API examples recommend extended80 where available, with measured CPU costs of 0.036 and 0.040 seconds respectively. Their tiny samples and finite cutoff limit the physical estimates separately.

Six-spinon starting point and difficult geometry¶

The eight-format comparison fixes the three-frequency R8 example, including all 384 draws and replicate identities. Native x86 long double succeeds in 0.31 seconds wall / 0.38 CPU seconds and agrees with the 384-bit reference means to within 3e-16 relative. MPFR128 also succeeds, costing 43.6 CPU seconds; MPFR384 costs 101.1. Binary64 fails arithmetic screens and an exponent-range check. MPFR96 fails its 32/96-bit arithmetic pair. The record preserves these failures rather than publishing partial estimates. Independent reconstruction checks successful means, sampling errors and work counts. The large sampling errors and omitted R8 domain remain; precision agreement does not remove them.

Changing the sample stream¶

The seed comparison retains complete R8/4x32 queries for seeds 42, 2 and 35. Extended80 fails the arithmetic screen at seeds 2 and 35. Binary128 with a 64/113-bit pair completes all three at unchanged rtol=1e-5, atol=0, costing 7.32, 6.34 and 7.02 CPU seconds on the same i7-12700KF. Every stream/work identity matches its MPFR192 reference; means, sampling SE and replicate means/maxima agree within 4.9e-30 relative. The checker is scripts/check_xxx_six_spinon_native_precision.py. These measurements support trying binary128 on a failed extended80 stream; they do not establish sampling or cutoff convergence. The development page links the larger seed scans and additional tolerance experiments.

Separating geometry from density precision¶

The endpoint-preserving geometry comparison checks R1040 with 256-bit geometry and a 192/256-bit density pair against a 1536-bit reference on identical 64-draw streams. Sample identities and support counts match, and the mean changes by less than 8e-74 relative. Its roughly 100% relative sampling SE remains; this is arithmetic evidence on those inputs, not a production precision floor or a bound on omitted soft weight. The development account explains the representation and preserves the earlier failed geometry checks.

Scope of the resolved slice¶

The small examples and large-domain resolved slice need separate recommendations. Representative four-spinon controls and six-spinon controls have been checked on both sides of the two-spinon upper edge. They separate arithmetic, contour, quadrature/sampling and domain comparisons. The complete sixteen-frequency comparison retains the independent sector estimates and their partial sums. All six-spinon queries are complete, including precision retries; observed sampling standard errors range from 2.49% to 6.52%, with fifteen points meeting the initial 5% aim. The reference record preserves the individual errors, influence diagnostics, arithmetic stages and measured costs.

The smaller installed example demonstrates the same retry workflow at R=256 in about 68 CPU seconds, or 20 seconds with four workers on the measured Xeon Gold 6254. It retains its 15.43% sampling standard error and separate domain limitation. The full reference is optional; readers can redraw its compact data without integrations. Neither the sampling bars nor the measured control changes bound omitted tails. This inventory does not replace model-specific accuracy tests for every parameter in the library. The precision guide gives the decision procedure and explains the relative/absolute screen and demonstrated higher-precision case.