FDFeiyu Deng
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Methods

How to audit a long analytical derivation

Use finite-temperature interval entropy to practise dependency tracing, limit checks and matching across descriptions, with an explicit scope for each check.

Feiyu Deng7 min readUpdated

Work backwards from the final equality

The costly part of a long derivation is often discovering that two apparently identical symbols have different definitions. Start an audit by selecting the final observable, listing its state, geometric domain, normalization and approximations, and tracing its dependencies backwards. Label assumptions, algebraic consequences, cited results and numerical spot checks separately. A mismatch can then be traced to the first dependency that fails.

For example, “the two entropies agree” requires at least the same spatial region, the same quantum state and matched cutoffs. A difference in finite constants can also come from an endpoint prescription. Fixing these inputs saves unnecessary expansion of intermediate formulas.

Practise on a known result

Take a two-dimensional conformal field theory on an infinite line. The region is a single interval of length L, and the state is thermal equilibrium at inverse temperature β. Set ħ, k_B and the propagation speed to 1; c is the central charge and ε the ultraviolet cutoff. The Calabrese–Cardy result is:

Sβ(L)=c3log⁡[βπϵsinh⁡(πLβ)]+s0.S_\beta(L)=\frac{c}{3}\log\left[\frac{\beta}{\pi\epsilon}\sinh\left(\frac{\pi L}{\beta}\right)\right]+s_0.

Logarithms are natural, and s₀ is a regulator-dependent constant. The formula applies to a single interval in the continuum CFT, with both L and β much larger than ε. A finite lattice, finite total length or physical boundary requires its own expression.

First check that the logarithm has a dimensionless argument. Then hold L fixed and send β to infinity: the small-argument limit of the hyperbolic sine should recover the vacuum entropy. Finally hold β fixed and take L much larger than β: the entropy slope should approach the thermal entropy density. The two expectations are:

lim⁡β→∞Sβ(L)=c3log⁡Lϵ+s0,lim⁡L→∞Sβ(L)L=πc3β.\lim_{\beta\to\infty}S_\beta(L)=\frac{c}{3}\log\frac{L}{\epsilon}+s_0, \qquad \lim_{L\to\infty}\frac{S_\beta(L)}{L}=\frac{\pi c}{3\beta}.

Replacing the hyperbolic sine with a sine by mistake gives incorrect long-interval behaviour. Dropping the cutoff triggers the dimensional check first. These inexpensive tests belong before a line-by-line reproduction.

A checklist to execute

  1. Fix the source version and equation number. Make a symbol table and specify which quantities stay fixed under each derivative or limit.
  2. List the direct dependencies of each key equality. Separate variable substitutions, boundary conditions and approximations, paying particular attention to integration endpoints and contact terms.
  3. Check dimensions, symmetries and known limits before recomputing the intermediate steps most likely to contain an error.
  4. Use symbolic tools for algebra and limits, and numerical points to spot-check an implementation. Record inputs, outputs and applicability separately. Numerical agreement supports the sampled points; an identity needs a corresponding argument.
  5. Retain failed checks and their causes. After a correction, rerun the conclusions depending on that step.

Aligning two calculations

The public butterfly-velocity preprint rewrites both shock-wave and entanglement-wedge computations as initial data inserted into gravitational constraint equations. It offers an example of tracing shared structure. The 2026 TT̄-deformed BCFT paper specifies the induced geometry, Weyl frame and endpoint prescriptions required for its entropy comparisons.

The latter paper's local boundary constraint uses a reflective condition, no independent boundary stress tensor, a semiclassical factorized state and separately treated contact terms. Its entropy comparison is an internal RT/BCFT consistency check. An audit record should state the particular relation supported by each check, so that readers can locate both the inputs and the steps still awaiting a test.

Original references

  • P. Calabrese and J. Cardy, Entanglement Entropy and Quantum Field Theory, J. Stat. Mech. (2004) P06002, arXiv:hep-th/0405152. Equation (3) gives the finite-temperature single-interval entropy on an infinite line.
  • F. Deng, X.-S. Wang and J.-q. Wu, An Interpretation for the Equivalence of Two Holographic Computations of the Butterfly Velocity with the Canonical Formalism of Gravity, preprint (2025), arXiv:2512.04498. Shared structure of initial data and constraint equations.
  • F. Deng, Boundary constraints and geometric realizations in holographic TT̄-deformed BCFT, JHEP 09 (2026) 249, journal article. Local Ward constraints and entropy checks with matched prescriptions.

Read the related research

  • Boundaries, defects and interfaces

    After adding a defect, an EOW brane or a finite cutoff, which boundary data determine entropy and spectrum calculations?

  • Computational field theory

    Before taking an analytic result into a calculation, how can variables, conventions and approximations be fixed so that errors and mismatches remain traceable?

Suggested citation

Deng, Feiyu. “How to audit a long analytical derivation.” flyfish1097.com/en/notes/audit-a-long-derivation/, 2026-10-25.