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Evidence Independence, Circularity, and Double Counting

Several successful comparisons can descend from one identity, one calibrated parameter, or one protected sector. They then provide less independent support than their raw number suggests. The remedy is to trace how each conclusion depends on assumptions, calibration data, algorithms, and imported dictionary entries.

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.

Build a dependency graph

Section titled “Build a dependency graph”

Represent each assumption, datum, calculation, and conclusion as a node. A directed edge A→EiA\to E_i means evidence item EiE_i depends materially on AA. Items with a common ancestor are correlated even when their final observables differ.

For each EiE_i, label:

  • whether its inputs were predicted or fitted;
  • which dictionary entry was assumed;
  • which analytic identity, data set, or code path it shares with other items;
  • its statistical and systematic uncertainty;
  • whether an independent method reproduces it;
  • what result would count against the claim.

If a covariance matrix is meaningful, the information in mm measurements is controlled by the full matrix CC, not just the diagonal errors. For a common mean with unit response, the inverse-variance weight is

w∝C−11,w\propto C^{-1}{\bf 1},

so strongly correlated measurements do not contribute as mm independent observations.

Calibration is not prediction

Section titled “Calibration is not prediction”

Suppose a bulk coupling g3g_3 is fixed by matching one boundary three-point coefficient,

g3≡f(C123).g_3\equiv f(C_{123}).

Reproducing C123C_{123} after substituting this definition is a consistency check of conventions, not a prediction. A four-point exchange residue computed using the fitted g3g_3 may be predictive if it was not also used in the fit and if contact-term ambiguities are controlled.

The same distinction applies to choosing an AdS radius from a central charge, selecting a saddle using known thermodynamics, or tuning a bottom-up potential to a boundary equation of state. Validation must use held-out observables.

Entropy, indices, and anomalies

Section titled “Entropy, indices, and anomalies”

A black-hole entropy match, a protected index, and an anomaly calculation may share supersymmetry, charge normalization, and a brane construction. For example, holographic anomaly matching Henningson and Skenderis 1998 and a localized protected observable Pestun 2012 probe different quantities while retaining some common dictionary inputs. Their dependence can be separated as follows:

ItemDistinct informationShared inputs
Protected indexCharge-weighted protected spectrumSupersymmetry algebra and fugacity map
Degeneracy extractionGrowth after controlling cancellationsSame index plus an inversion or asymptotic assumption
Gravitational entropyBulk saddle and charge relationSame charge map and low-energy truncation
Anomaly matchExact symmetry normalizationBrane content and global symmetry assignment

Agreement is impressive, but counting every derived quantity as independent would obscure the shared structure.

The circularity test

Section titled “The circularity test”

Deliberately fit a bulk normalization to a boundary correlator, rerun the calculation, and ask whether the analysis labels the output “predicted.” It must not. The strongest surviving statement is that the bulk computation consistently reproduces its calibration observable. Independent support begins with an observable not used to choose the parameter, scheme, saddle, or dictionary map.

This page establishes the dependency analysis, not a live ranking of holographic evidence. Executable provenance should accompany the calculation it supports, while current assessments belong in Research.

Evidence cutoff. Examples and source relations are fixed to 25 July 2026.

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

Section titled “References”