Evidence Independence, Circularity, and Double Counting

Several exact-looking agreements need not be several independent tests. They may descend from one protected identity, one fitted normalization, one data set, or one dictionary entry. This page develops a reproducible way to expose those dependencies, separate calibration from prediction, and state the strongest conclusion that remains after shared inputs are removed.

The result is not a numerical score for a duality. It is a dependency graph, a set of conditional independence classes, and a record of which observables were genuinely held out. The worked example follows anomaly data, a protected index, Cardy growth, and gravitational entropy through the D1–D5–momentum system.

Required background. Evidence Programs for Holographic Duality identifies the evidence items whose dependence must be analyzed.

Helpful background. Claim–Evidence Records, Replication, and Retraction Handling supplies the general evidence vocabulary, and Solver Certificates and Independent Verification gives a computational example of genuinely independent checking.

Reading path. First classify the kind of dependence, then run the reproducible procedure and calibration-leakage check. The covariance section treats genuinely stochastic overlap. The D1–D5–momentum application builds the full graph; counting rules, exercises, and the current boundary complete the workflow.

Independence is conditional, not visual

Imagine that one calculation inserts a central charge obtained from an anomaly, while a second inserts the same central charge into a Cardy formula. Two final numbers appear, but the second is downstream of the first. By contrast, a low-energy absorption curve can probe new frequency dependence even though it shares the compactification and dictionary.

Let $H$ denote the holographic claim, $\mathcal C$ the fixed comparison contract—its theory, state, ensemble, dictionary, and regime—and $\eta$ shared nuisance parameters or systematic effects. Evidence items $D_1,\ldots,D_m$ are conditionally independent only when the joint model factorizes:

$$p(D_1,\ldots,D_m\mid H,\mathcal C,\eta) =\prod_{i=1}^{m}p(D_i\mid H,\mathcal C,\eta).$$

This is a claim about the data-generating and inference model, not about whether the final formulas look different. If $\eta$ is uncertain, marginalizing it generally restores dependence:

$$p(D_1,\ldots,D_m\mid H,\mathcal C) =\int d\eta\,p(\eta\mid H,\mathcal C) \prod_{i=1}^{m}p(D_i\mid H,\mathcal C,\eta).$$

A common ancestor in a dependency graph is therefore a warning to inspect, not by itself a proof of nonzero statistical covariance. The ancestor might instead create a deterministic implication, a shared theoretical assumption, or an implementation risk. Dawid’s conditional-independence framework makes this conditioning explicit Dawid 1979, pp. 1–31.

Different kinds of dependence require different remedies; covariance is only one of them.

Dependence	Diagnostic question	Correct treatment
Logical or derivational	Is one result an identity, limit, or deterministic transformation of another?	Keep the derivation visible, but count the family once rather than counting every descendant.
Calibration or selection	Was this datum used to fit a parameter or choose a model, saddle, scheme, operator map, or stopping rule?	Label its reproduction as calibration consistency. Test a prespecified held-out observable.
Shared physical premise	Do both results require the same supersymmetry protection, charge map, compactification, or asymptotic regime?	State the conclusion conditional on that premise and ablate the premise explicitly.
Shared data or nuisance model	Do the estimates reuse events, ensembles, backgrounds, normalizations, or uncertain inputs?	Use a joint likelihood or justified covariance model; propagate the shared uncertainty.
Shared implementation	Do two calculations reuse code, libraries, tables, gauge choices, or regulator logic?	Seek an analytic benchmark or an independently implemented calculation.
Conditionally distinct test	Is the observable held out, reached by a genuinely different method, and independent after the declared common assumptions are fixed?	Place it in a separate evidence class while retaining the conditioning assumptions.

After the comparison contract is fixed, place two items in the same local independence class when one is a deterministic or calibration ancestor of the other, or when one unresolved method, data, implementation, or physical assumption could invalidate both. Put them in distinct classes only when they retain distinct failure opportunities. The classes depend on the claim and conditioning contract; they are not permanent labels attached to observables.

This use of “factorization” is unrelated to large-$N$ factorization of connected correlators or factorization between disconnected boundaries. Those are physical properties to be tested; conditional independence describes the structure of an inference.

A reproducible dependency analysis

Inputs

Begin with five concrete objects:

1. one claim $H$ and at least one alternative that could explain the same agreement;
2. an observable contract for every evidence item: theory, state or ensemble, normalization, domain, control parameters, and uncertainty;
3. every datum used for fitting, model selection, dictionary construction, or stopping;
4. the analytic derivation, code path, source table, and nuisance inputs for each result;
5. one statement that would lower the claim if the comparison failed.

An unlabeled success is not yet an input. “The entropies agree,” for example, must be replaced by the two entropy definitions, charge map, ensemble, limit order, and error ceiling.

Procedure

1. Freeze the comparison. Write the theory, charges, state, ensemble, observable, normalization, regulator, and regime before looking at the target result.
2. Create typed nodes. Use separate nodes for assumptions, raw data, fitted parameters, selected models or saddles, calculations, observables, and conclusions.
3. Create typed edges. Mark every edge as deterministic derivation, calibration, shared nuisance, dictionary import, regime restriction, data reuse, or implementation reuse.
4. Compute ancestors. For each claimed test, list all upstream calibration data and assumptions. Collapse deterministic descendants into one evidence family.
5. Check statistical overlap. Use a joint likelihood or covariance only when its sampling assumptions are defensible. Never use a covariance matrix to conceal logical reuse.
6. Ablate one ancestor at a time. Remove or vary each shared input and record which results remain meaningful, which become uncomparable, and which merely lose precision.
7. Seek a different route. Recompute with independent code or an analytic limit, and test an observable that was not used in fitting or selection.
8. Assign the ceiling. State the strongest conclusion supported by the surviving classes and the exact failure responsible for every downgrade.

Once the nodes and edges exist, one ancestor traversal or one ablation costs $O(V+E)$ for $V$ nodes and $E$ edges. Reconstructing every ancestry set or repeating the operation for many targets can cost more. The difficult work is scientific: reconstructing hidden selection choices, deciding whether two quantities are deterministic transforms, and justifying any probability model. A second researcher should be able to reproduce the edge list without inferring missing decisions from prose.

Output and stop rule

For every item, report its observable and domain; logical status; calibration or prediction role; independence class; shared ancestors; statistical, truncation, and interpretive uncertainty; independent check; and failure meaning. Validate the result with three controls:

• an edge-coverage check showing that every fitted or selected quantity reaches all of its descendants;
• a negative fixture known to reuse calibration data;
• an independent reviewer or implementation that reconstructs the same ancestry from the frozen inputs.

Stop rule. Do not combine evidence when a material edge is unknown, when a “prediction” was inspected during tuning, or when shared uncertainty cannot be represented honestly. Report the items separately and lower the conclusion to a conditional consistency statement.

Calibration is not prediction

Let calibration data $D_{\mathrm{cal}}$ determine a bulk parameter,

$$\widehat\theta =\underset{\theta}{\operatorname{argmin}}\, \chi^2_{\mathrm{cal}}(\theta).$$

A validation residual has evidential force only for a target that was held out from every fit and selection decision:

$$r_{\mathrm{val}} =D_{\mathrm{val}}-F_{\mathrm{val}}(\widehat\theta).$$

The deterministic checker first closes the calibration set under re-expression and preprocessing. It then examines every row labeled “prediction”:

calibration_closure = deterministic_descendants(calibration_data)

for comparison in reported_predictions:
    if comparison.observed_datum in calibration_closure:
        emit CALIBRATION_REUSED_AS_PREDICTION
    else if comparison has fitted_or_selected_ancestors:
        label CONDITIONAL_HELD_OUT_PREDICTION
        record those ancestors

The second label is not a failure. It records that a new datum tests the fitted model conditional on the calibration. Repeated tuning after seeing the validation residual converts that datum into calibration; the next claim needs a fresh or nested holdout.

A correlator normalization fitted once cannot return later as an independent prediction.

Analysis step	Correct label	Strongest surviving statement
Fix the canonical two-point normalization and fit a cubic bulk coupling g₃ to the boundary coefficient C₁₂₃.	Calibration	The normalization and g₃ are defined by the selected two- and three-point data.
Substitute the fitted g₃ and report the same C₁₂₃ as “predicted.”	CALIBRATION_REUSED_AS_PREDICTION	The calculation checks conventions and implementation; it provides no held-out normalization test.
Recover the four-point exchange residue algebraically fixed by the same OPE coefficient.	Deterministic descendant	This can test factorization and the diagrammatic map, but it is not independent evidence for the fitted value of g₃.
Predict a prespecified four-point shape or new channel without using it in the fit.	Conditional held-out prediction	It supplies new support only after quartic contact terms, Mellin-polynomial ambiguities, operator mixing, and truncation errors are controlled.
Retune the model after inspecting that four-point result.	New calibration	A fresh observable or nested validation layer is required.

Statistical overlap when covariance is justified

Suppose $m$ unbiased Gaussian estimates $y_i$ measure one common mean $\mu$, and their known covariance matrix $C$ is positive definite. The best linear unbiased combination is

$$\widehat\mu =\frac{{\bf 1}^{\mathsf T}C^{-1}y} {{\bf 1}^{\mathsf T}C^{-1}{\bf 1}}, \qquad \operatorname{Var}(\widehat\mu) =\left({\bf 1}^{\mathsf T}C^{-1}{\bf 1}\right)^{-1},$$

with weights

$$w=\frac{C^{-1}{\bf 1}} {{\bf 1}^{\mathsf T}C^{-1}{\bf 1}}.$$

If $C$ is singular because some directions are exactly redundant, first consolidate those directions or use a stated generalized-least-squares prescription with a Moore–Penrose inverse after checking that $\mu$ is estimable. Writing an ordinary inverse would otherwise disguise the redundancy.

For two equal-variance measurements with $-1<\rho<1$,

$$C=\sigma^2 \begin{pmatrix} 1&\rho\\ \rho&1 \end{pmatrix}, \qquad \operatorname{Var}\!\left(\frac{y_1+y_2}{2}\right) =\frac{\sigma^2}{2}(1+\rho).$$

As $\rho\to1$, the second estimate adds no precision; at $\rho=0$, the variance halves. This is a useful repeated-estimate model Lyons, Gibaut, and Clifford 1988, pp. 110–117, not a universal definition of “number of evidence items.” It does not quantify dependence caused by an identity, a shared duality assumption, or a reused calibration target.

Worked application: the D1–D5–momentum entropy chain

The D-brane entropy derivation, D1–D5 CFT dictionary, and entropy and ensemble contract supply the physical construction. Here their results become inputs to a dependency graph.

Fix Type IIB string theory on $S^1\times T^4$. Let positive integers $Q_1$ and $Q_5$ count D1-branes on $S^1$ and D5-branes on $S^1\times T^4$, and let $n>0$ be momentum along $S^1$.

Set all additional $T^4$ momentum and winding charges to zero. On the bulk side, compare with the nonrotating BMPV member, for which both five-dimensional angular momenta vanish. On the microscopic side, the label $\ell=2J_0^3=0$ fixes only the left Cartan charge; the modified trace below weights rather than projects the right spin.

Assume $\gcd(Q_1,Q_5)=1$ for the fixed D1–D5 sigma-model formula. The U-duality-invariant extension instead uses full three-charge primitivity, $\gcd(Q_1,Q_5,n)=1$. Work in the single-center BPS sector, with the right movers in their Ramond ground state.

Remove the decoupled center-of-mass and exterior-hair factors before comparing with the horizon. In the leading interacting-sector convention,

$$k=Q_1Q_5, \qquad c_L=c_R=c_{\mathrm{int}}=6k.$$

For the $T^4$ theory the ordinary elliptic genus vanishes because of right-moving fermion zero modes. Measure $L_0$ and $\bar L_0$ above the Ramond ground-state energies and define the modified trace in the sector $k$ by

$$\begin{aligned} E^{\mathrm{raw}}_{2,k}(\tau,z) &=\operatorname{Tr}_{RR,k}\!\Big[ (-1)^{2J_0^3-2\widetilde J_0^3} (2\widetilde J_0^3)^2 \\ &\qquad\times q^{L_0}\bar q^{\bar L_0}y^{2J_0^3} \Big]. \end{aligned}$$

The insertion soaks up the zero modes Maldacena, Moore, and Strominger 1999, § 3, pp. 7–9. Write the raw generating function as

$$\mathcal Z^{\mathrm{raw}}_2(p;\tau,z) =\sum_{k\geq1}p^k E^{\mathrm{raw}}_{2,k}(\tau,z).$$

A horizon comparison must specify the complete, generally $y$-dependent exterior factor $\mathcal Z_{\mathrm{ext}}$ and divide it out before selecting a coefficient:

$$\mathcal Z^{\mathrm{hor}}_2 =\frac{\mathcal Z^{\mathrm{raw}}_2}{\mathcal Z_{\mathrm{ext}}}, \qquad \Omega_{\mathrm{hor}}(k,n,\ell) =[p^kq^ny^\ell]\mathcal Z^{\mathrm{hor}}_2.$$

Here $\mathcal Z_{\mathrm{ext}}$ contains the declared decoupled center-of-mass and exterior-hair degrees of freedom. If those degrees do not factor multiplicatively, replace the quotient by an explicit character-level or convolution subtraction and record that prescription. The coefficient $\Omega_{\mathrm{hor}}(k,n,0)$ fixes the left Cartan label $\ell=0$ but sums over right spin with the helicity weight $(2\widetilde J_0^3)^2$; an $SU(2)_L\times SU(2)_R$ character decomposition would be required to extract a definite-spin multiplicity. This is a weighted signed trace, not automatically an absolute degeneracy. The fixed-frame formula requires $\gcd(Q_1,Q_5)=1$ because nonprimitive D1–D5 charges introduce threshold-bound-state subtleties; the fully primitive extension is a separate construction Maldacena, Moore, and Strominger 1999, eq. (5.9) and § 6.1, pp. 13, 16–17.

After selecting the appropriate charge sector, controlling the index inversion and any leading cancellations, verifying that the exterior-factor removal does not change the leading exponent, and taking a Cardy regime $n\gg c_{\mathrm{int}}$ together with a simultaneous macroscopic-charge limit, its leading growth is

$$\log\lvert\Omega_{\mathrm{hor}}(k,n,0)\rvert \sim 2\pi\sqrt{\frac{c_{\mathrm{int}}n}{6}} =2\pi\sqrt{Q_1Q_5n}.$$

The corresponding five-dimensional extremal black hole gives

$$S_{\mathrm{BH}} =\frac{A_H}{4G_5} =2\pi\sqrt{Q_1Q_5n}$$

at leading two-derivative order Callan and Maldacena 1996, §§ 2–3, pp. 4–8. The agreement is a landmark cross-regime test, but its ingredients must not be recounted as independent successes.

The edge list

Follow the graph in three passes. S1 and S2 feed the protected anomaly A and index I; A, I, and the Cardy gate R_M feed the microscopic asymptotic M. Separately, S1, the charge map S3_q, and the bulk gate R_G feed the gravitational entropy G; M and G meet at comparison X. The dynamical branch Y instead requires its own scalar map S3_y and greybody regime R_Y.

Each material assumption bundle, result, and conclusion is a typed node.

Node	Type	Meaning
S1	Fixed system	Type IIB on S¹ × T⁴; positive Q₁, Q₅, and n; gcd(Q₁,Q₅) = 1 for the fixed-frame formula; no extra torus charges; nonrotating bulk comparison; single-center sector
S2	Shared premise	Small-𝒩=(4,4) supersymmetry, the BPS sector, and a path in moduli space along which the specified trace is protected
S3q	Charge and ensemble map	Integer D1, D5, and momentum charges; bulk charge conventions; G₅ normalization; and the ensemble used on both sides
S3y	Observable map	The selected bulk scalar, boundary operator, coupling, polarization, normalization, and the calibration provenance used to fix them
RM	Microscopic regime gate	n ≫ cint, controlled inverse transform and leading modular saddle, no leading cancellation, and a valid exterior-factor removal
RG	Bulk-entropy regime gate	Macroscopic horizon, low curvature, and suppressed string-loop and higher-derivative corrections
RY	Dynamical regime gate	Near-extremal dilute-gas state, low frequency, and controlled supergravity and effective-string truncations
A	Class P result	The boundary R-current anomaly level k and its small-𝒩=(4,4) relation c = 6k
I	Class P result	The modified horizon-index coefficient at fixed charges and left Cartan label ℓ = 0, with right spin helicity-weighted
M	Class P descendant	The microscopic Cardy asymptotic derived from A, I, and RM
G	Class B result	The Bekenstein–Hawking area in the mapped macroscopic bulk sector
X	Comparison conclusion	The equality of M and G; it is not an additional evidence item
Y	Class R candidate	A frequency- and temperature-dependent greybody response, classified as held out only when S3y was frozen independently

The S and R nodes are explicit bundles rather than indivisible assumptions. When two conditions in a bundle can be varied separately or invalidate different descendants, split that bundle into separate nodes before running the ablation.

One row per edge preserves its direction, type, and scientific condition.

Source → target	Edge type	Condition or meaning
S1 → A	Fixed-system input	The anomaly is calculated in the declared D1–D5 theory and sector.
S2 → A	Symmetry relation	Small-𝒩=(4,4) symmetry relates the R-current level to the central charge.
S1 → I	Fixed-system input	The trace uses the declared charges, spin sector, and removed external factors.
S2 → I	Protection	The specified index can be transported along the declared moduli path.
A → M	Deterministic derivation	The central charge supplied by A enters the Cardy exponent.
I → M	Coefficient extraction	The protected trace supplies the charge-sector coefficients.
RM → M	Regime gate	The inverse transform and leading Cardy saddle must be controlled.
S1 → G	Fixed-system input	The geometry carries the same compactification and quantized charges.
S3q → G	Dictionary and normalization	The area is expressed using the same integer charges and G₅ convention.
RG → G	Regime gate	The two-derivative macroscopic area formula must be reliable.
M → X	Comparison input	M supplies the microscopic side of the equality.
G → X	Comparison input	G supplies the gravitational side of the equality.
S1 → Y	Fixed-system input	The response is calculated in the same compactification and charge sector.
S3q → Y	Charge and state map	Temperatures and charges must use the same bulk and CFT conventions.
S3y → Y	Observable dictionary	The scalar, operator, coupling, polarization, and normalization are matched.
RY → Y	Regime gate	The dilute-gas, near-extremal, and low-frequency approximations must hold.

The node glossary and typed edge list are the linear, keyboard-accessible equivalent of the directed graph. They also make counterfactual graphs precise. If the central charge is derived directly from the explicit index theory, remove A → M and record the replacement derivation rather than preserving a conventional story.

A bulk anomaly comparison is a useful residual test, but it must not be asserted from a generic formula alone. Three-dimensional gravitational and $SU(2)$ Chern–Simons levels encode $c_L-c_R$ and the R-current anomalies Kraus and Larsen 2006, §§ 4.2–4.3, pp. 11–12. That source identifies D1–D5 as the $S^3$ case but does not itself derive the compactification-specific coefficient $k_L=k_R=Q_1Q_5$. A separate D1–D5 reduction and normalization check is therefore required before a bulk anomaly agreement becomes another class-P result.

Evidence classes and uncertainties

For this application, class P contains the protected anomaly-and-index lineage, class B the macroscopic bulk-area calculation, and class R the additional dynamical-response calculation. M is a descendant within P, while X is the comparison conclusion. Class R becomes a held-out test only when its observable map was frozen independently. These labels organize shared failure opportunities; they are not likelihood weights.

Anomaly, index, Cardy growth, area, and dynamics carry different information and different limitations.

Result	Logical status and class	Control and uncertainty	Licensed inference
Boundary R-current anomaly	Class P: protected structural calculation	The quantized coefficient is fixed within the specified CFT sector, subject to finite shifts, decoupled factors, and convention matching	Small-𝒩=(4,4) symmetry relates the level to c = 6k; this same c is used downstream in Cardy growth. A bulk inflow match requires the separate reduction identified above.
Modified elliptic genus	Class P: protected spectral calculation in the same broad family	The specified index is protected in its stated domain, while inference to a horizon state count requires control of helicity weights, zero modes, the exterior factor, threshold bound states, the charge sector, and the inverse transform	A specified signed BPS trace is transported across moduli.
Cardy entropy from the index	Class P descendant: asymptotic result, not a new class	Corrections arise outside the Cardy hierarchy, at finite charge, and from subleading saddles	The protected sector has the stated leading exponential growth.
Bekenstein–Hawking area	Class B: semiclassical bulk calculation, conditionally distinct from the weak-coupling count	Controlled only with a macroscopic horizon and suppressed string-loop and higher-derivative corrections	The bulk saddle has the same leading entropy after the charge map is imposed.
Equality of the two entropies	Conclusion X: cross-regime comparison, not an evidence class	Inherits every uncertainty and shared assumption above	Strong evidence for the protected D1–D5–momentum dictionary in this asymptotic sector, not a proof of a complete non-BPS theory.
Low-energy greybody factor	Class R: partly independent dynamical family, conditional on shared inputs	Requires the operator map, normalization, dilute-gas and near-extremal state, low frequency, and supergravity truncation	The dictionary reproduces additional frequency- and temperature-dependent responses in selected scalar channels (Maldacena and Strominger 1997, §§ 4–6, pp. 9–17). Call this held out only if the coupling, normalization, channel, and regime were fixed independently of those response data; otherwise it is a conditional dynamical consistency check.

The key lesson is precise: anomaly $\to$ central charge $\to$ Cardy growth $\to$ microscopic entropy is a dependency chain when those arrows occur in the calculation. The index and anomaly may be distinct protected computations, but they share the compactification, supersymmetry, and charge conventions. The bulk area is methodologically more distinct, yet its comparison still requires the shared charge dictionary and regime. None of these facts diminishes the agreement; they identify what the agreement actually tests.

Ablation identifies the residual tests

Removing one ancestor shows which conclusion was supplied by that ancestor.

Ancestor removed or varied	What survives	What no longer follows
Supersymmetric protection S2	The weak- and strong-coupling calculations can still exist at their own points.	The protected transport between those points, and therefore the entropy comparison as stated, is no longer licensed.
Charge and ensemble map S3q	A and I survive; the bulk-area and response calculations remain meaningful only in their native variables.	G as a function of the common integer charges, comparison X, and Y as a bulk–CFT response comparison no longer follow, because the shared charge, temperature, ensemble, and G₅ map has been removed.
Microscopic regime RM	The protected index and bulk area remain defined in their own domains.	The leading Cardy extraction M, and hence comparison X, no longer follows.
Bulk regime RG	The anomaly, index, and Cardy asymptotic remain available.	The two-derivative area result G, and hence comparison X, no longer follows.
Index inversion, cancellation, and exterior-factor control	The raw protected helicity trace survives.	The horizon coefficient and an absolute microstate degeneracy—and therefore M as used in X—do not follow merely from the raw trace.
Scalar-operator map S3y	The protected entropy chain is unaffected.	The greybody comparison cannot be read as a dynamical duality test for that channel.
Dynamical regime RY	The entropy comparison X is unaffected.	The low-energy greybody calculation cannot be extrapolated to the target response.

Residual tests should therefore target information not fixed by the entropy chain: prespecified fugacity or angular-momentum dependence, subleading and logarithmic corrections, new charge ratios, unprotected spectral data, or frequency-dependent absorption in independently normalized channels. The BPS index chapter develops the index-to-degeneracy hazards, while Absorption, Emission, and Dynamical Tests develops the greybody comparison.

Counting evidence without counting it twice

A product of Bayes factors is valid only under the corresponding conditional-independence model. For two hypotheses $H_1$ and $H_2$,

$$\frac{p(D_1,\ldots,D_m\mid H_1,\mathcal C)} {p(D_1,\ldots,D_m\mid H_2,\mathcal C)} =\prod_i \frac{p(D_i\mid H_1,\mathcal C)}{p(D_i\mid H_2,\mathcal C)}$$

requires factorization under both hypotheses. If the data share nuisances, use the joint likelihood and marginalize them. If no justified likelihood exists, do not invent one: dependence classes and ablation results are scientifically useful without pretending to be posterior odds.

Use these counting rules:

• count a deterministic family at its strongest nonredundant conclusion, while retaining the intermediate derivation for verification;
• label reproduction of calibration data as consistency, never as a held-out prediction;
• give a new observable conditional credit when it was prespecified and held out, but keep its calibration ancestors visible;
• treat independent code as an implementation check only when it also avoids the same hidden tables, algorithms, and selection decisions;
• never count an agreement and the conclusion drawn from that agreement as two items;
• record contrary or failed held-out tests in the same graph rather than reporting only successful descendants.

The number of rows or independence classes is descriptive. It is not a universal probability, confidence level, or ranking of theories.

Common pitfalls

Every common ancestor means statistical correlation. Not necessarily. A common ancestor might be a fixed convention with no uncertainty, a deterministic premise, or a genuinely shared nuisance. It triggers classification; only a stated stochastic model licenses a covariance.

Different observables are automatically independent. A protected index and entropy can look different while one is used to derive the other. Inspect the arrows, not the names.

An index is a degeneracy. An index is signed and can hide cancellations. For $T^4$, even the ordinary elliptic genus vanishes; the appropriate modified index and its zero-mode insertions must be specified.

A holdout remains a holdout after tuning. Once its value changes a parameter, model, saddle, dictionary choice, or stopping decision, it has joined the calibration set. Reserve a fresh target.

Two programs using the same code are replications. They may be two interfaces to one implementation. Independence requires tracing shared libraries, data, algorithms, and human choices.

Exercises

Correlated estimates

Two unbiased Gaussian estimates of the same $\mu$ have variance $\sigma^2$ and correlation $\rho$. Derive the minimum-variance equal-response estimator and its variance. What happens as $\rho\to1$?

Solution: minimize the correlated variance

Write $\widehat\mu=a y_1+(1-a)y_2$ so the response to $\mu$ is one. Its variance is

$$V(a)=\sigma^2\!\left[a^2+(1-a)^2+2\rho a(1-a)\right].$$

For $\rho<1$, $dV/da=2\sigma^2(1-\rho)(2a-1)=0$ gives $a=1/2$. Direct substitution then gives

$$\operatorname{Var}(\widehat\mu) =\frac14\left(\sigma^2+\sigma^2+2\rho\sigma^2\right) =\frac{\sigma^2}{2}(1+\rho).$$

At $\rho=0$ the variance is $\sigma^2/2$. In the limit $\rho\to1$ it returns to $\sigma^2$: the second estimate carries no independent precision. The singular endpoint should be understood as a limit of the covariance formula.

Ablate protection in the entropy graph

In the edge list above, remove S2 while leaving the weak-coupling index calculation and the classical black-hole solution individually available. Which statements survive?

Solution: remove protected transport

The modified index remains a result at the tractable weak-coupling point, and the area remains a result in the strong-coupling supergravity regime. What fails is the protected transport identifying the relevant indexed quantity across moduli. The numerical similarity can still motivate investigation, but the stated cross-regime entropy test is no longer derived. A new interpolation theorem, nonrenormalization result, or direct strong-coupling calculation would be needed.

Run the calibration checker

A researcher fits $g_3$ using $C_{123}$, reproduces $C_{123}$, then predicts a prespecified coefficient $C_{124}$ that was never inspected during fitting. Classify both reported agreements. How does the answer change if $g_3$ is retuned after seeing $C_{124}$?

Solution: distinguish calibration and holdout

The reproduced $C_{123}$ receives CALIBRATION_REUSED_AS_PREDICTION if it was labeled a prediction; its valid role is calibration consistency. The untouched $C_{124}$ is a conditional held-out prediction, provided operator mixing, normalization, and other fitted parameters were frozen. If $C_{124}$ changes the fit or model, it becomes calibration too, and a fresh target is required.

Separate an index from an absolute count

Suppose a charge sector contains $N$ bosonic and $N$ fermionic BPS states. What are its signed index and absolute degeneracy? What can a large nonzero index establish without an assumption about signs?

Solution: distinguish weighted and unweighted traces

The unweighted signed index is $N-N=0$, whereas the absolute degeneracy is $N+N=2N$. Thus a vanishing index does not imply an empty sector. For an unweighted $\pm1$ index, the triangle inequality gives $\lvert I\rvert\le d_{\mathrm{abs}}$, so exponential growth of $\lvert I\rvert$ lower-bounds the number of states.

The D1–D5 modified elliptic genus is instead a helicity-weighted trace. If $w_{\max}$ is the largest absolute weight in the sector, triangle inequality gives $\lvert\Omega\rvert\leq w_{\max}d_{\mathrm{abs}}$. If $w_{\max}$ grows subexponentially, exponential growth of $\lvert\Omega\rvert$ therefore lower-bounds the state-count exponent regardless of cancellations. Cancellation control is additionally needed only to identify the leading index exponent with the degeneracy exponent or reconstruct the degeneracy; zero-mode, exterior-factor, charge-sector, and inverse-transform controls remain separate requirements.

Build and test a complete graph

Suppose A and I feed M, M and G feed X, and the normalization in S3_y was chosen after the response curve Y was inspected. Write the typed edges, assign the local classes, collapse redundant descendants, and state what survives if R_M fails.

Solution: graph, classes, and claim ceiling

The core edges are A → M (deterministic central-charge input), I → M (coefficient extraction), R_M → M (regime gate), M → X and G → X (comparison inputs), and S3_y → Y (observable dictionary). Introduce a node D_Y for the inspected response data and add D_Y → S3_y as a calibration-or-selection edge; that edge records why the resulting S3_y → Y comparison is circular as a prediction.

A, I, and M belong to class P, with M a descendant rather than a new item; G belongs to class B; X is a conclusion, not a class. Y cannot yet be classed as a held-out R test because its normalization was selected after the target response was seen. It is a conditional consistency check until a fresh channel with an independently frozen map is tested.

If R_M fails, A and I remain protected-sector results and G remains a bulk result, but M and therefore X disappear. The strongest surviving statement is separate consistency of the anomaly, index, and bulk calculations in their own domains—not an entropy match.

Current boundary and handoffs

A current review organizes supersymmetric black-hole entropy by ensemble, index, gravitational path integral, and saddle structure Cassani and Murthy 2025, abstract and §§ 1–2. In the language of this page, those choices are ancestors of an entropy comparison; restating their consequences does not create new evidence.

A 2026 refinement illustrates the method. At the free symmetric-orbifold description of the $T^4$ D1–D5 CFT, Hughes and Shigemori introduce a resolved elliptic genus that refines the modified elliptic genus and propose a superselection rule exposing structure hidden by the coarser index Hughes and Shigemori 2026, §§ 1, 5.3–6. The resolving charge is not generally a good quantum number away from that special locus, so this is a qualified refinement rather than a universally protected generic-moduli index. It is also not automatically an independent extra “vote” for the entropy match: a prespecified comparison to an independently calculated strong-coupling observable would be needed to create a new evidence class.

Continue to Falsifiers, Negative Results, and Counterexamples to decide what a failed comparison challenges, and to Claim Status, Freshness, and Research Handoffs to separate durable method from mutable assessment. Dated developments and competing interpretations belong in the Holography and Quantum Gravity Research guide.

Evidence cutoff. 27 August 2026. Reassess the dependency classes when a new index refinement, strong-coupling calculation, independent implementation, correction to a source, or contrary held-out observable changes an edge or its uncertainty.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Callan, Curtis G., Jr., and Juan M. Maldacena. 1996. “D-Brane Approach to Black Hole Quantum Mechanics.” Nuclear Physics B 472, 591–610. DOI. Open PDF.
• Cassani, Davide, and Sameer Murthy. 2025. “Quantum Black Holes: Supersymmetry and Exact Results.” arXiv:2502.15360. Open PDF.
• Dawid, A. Philip. 1979. “Conditional Independence in Statistical Theory.” Journal of the Royal Statistical Society: Series B (Methodological) 41, 1–31. DOI. Open PDF.
• Hughes, Marcel R. R., and Masaki Shigemori. 2026. “The Resolved Elliptic Genus and the D1-D5 CFT.” Journal of High Energy Physics 07, 224. DOI. Open PDF.
• Kraus, Per, and Finn Larsen. 2006. “Holographic Gravitational Anomalies.” Journal of High Energy Physics 01, 022. DOI. Open PDF.
• Lyons, Louis, Duncan Gibaut, and Peter Clifford. 1988. “How to Combine Correlated Estimates of a Single Physical Quantity.” Nuclear Instruments and Methods in Physics Research Section A 270, 110–117. DOI.
• Maldacena, Juan M., Gregory W. Moore, and Andrew Strominger. 1999. “Counting BPS Black Holes in Toroidal Type II String Theory.” arXiv:hep-th/9903163. Open PDF.
• Maldacena, Juan M., and Andrew Strominger. 1997. “Black Hole Greybody Factors and D-Brane Spectroscopy.” Physical Review D 55, 861–870. DOI. Open PDF.