Weighing Invisible Matter with Fast Radio Bursts: Dispersion, Host Galaxies and Selection Codex (AI) | Commissioned review v1 | 2026-10-22 Fast radio bursts probe free electrons through frequency-dependent delay. Turning that delay into a cosmic baryon census requires localization, redshift and separation of Milky Way, host and intergalactic contributions. This review develops the dispersion budget and selection problem, explaining why a larger burst catalogue is not automatically a proportionally more precise inventory of matter. A pulse measures an integrated column Dispersion delays low-frequency radio waves more than high-frequency waves as they cross ionized matter. The inferred dispersion measure integrates free-electron density along the path. It does not directly say where those electrons reside. A host galaxy, the Milky Way and intervening structures can contribute to the same measurement. Localization made the census possible Chatterjee and colleagues directly localized a repeating fast radio burst. Bannister and colleagues localized a single burst to a massive galaxy. Macquart and colleagues used localized bursts with redshifts to constrain the cosmic baryon content. These advances connect an integrated propagation effect to a distance scale, while leaving host and line-of-sight contributions as part of the inference. References: ef-29-1, ef-29-2, ef-29-3 Redshift changes a host contribution A host-frame dispersion contribution enters the observed budget with a redshift factor. Ignoring that factor biases comparisons across distance. The host contribution also varies among galaxies and source environments. A single fixed host value is a convenient model, not a directly measured constant shared by every burst. Cosmic structure creates real scatter Different sightlines cross different filaments, halos and voids. Scatter around a mean dispersion-redshift relation is therefore partly astrophysical information rather than merely measurement noise. Its distribution can be skewed. Averaging it away without modeling selection can discard constraints on where baryons reside. Localization is a selection step Bursts that can be localized and assigned a secure host may differ from the full detected population. Brightness, scattering, sky position and host visibility can all affect inclusion. If these properties correlate with dispersion, a localized sample can yield a biased relation unless the selection is represented. Host ambiguity propagates into cosmology An incorrect host assignment supplies an incorrect redshift and can look like an extreme dispersion residual. Association probabilities should therefore accompany the data. Treating every proposed host as certain can underestimate uncertainty and distort inferences about unusual sightlines. Foreground subtraction has spatial structure Milky Way electron models are imperfect, and the halo contribution is not identical to the disk contribution. Their uncertainties can correlate across nearby directions. Independent Galactic constraints and sky-distributed samples help test these effects. A residual dispersion is not automatically an intergalactic measurement. Joint fits should retain covariance between foreground parameters and the inferred cosmic contribution. More bursts should add different information Repeated bursts from one source can improve propagation measurements but do not create independent cosmic sightlines. A useful census expands redshift and sky coverage while preserving selection metadata. The next scientific gain may come from better host characterization or independent foreground constraints rather than only from increasing the raw event count. A dispersion budget is a hierarchical inference An observed dispersion measure combines Galactic, intervening and host contributions, with redshift factors and uncertain distributions. A proposed population model would infer the mean cosmological relation while retaining event-specific host uncertainty. Subtracting the same host number from every burst can produce a misleadingly precise relation if hosts vary with redshift or selection. Independent localization and foreground information improve identifiability because they constrain components through different observations. The host-dilution curve shows one deterministic factor, while the population scatter figure assumes independent residuals. Neither substitutes for a joint model of the full budget. Selection can change the apparent baryon census Detection and localization probability can depend on burst brightness, width, scattering and the environment. If those properties correlate with dispersion, the localized sample is not a random draw from all bursts. A proposed validation would model the selection function and test it by injecting signals into the detection pipeline or using an independently selected subset. The estimate should show sensitivity to plausible missing populations. Increasing the sample size reduces independent noise, but a shared foreground error or a biased host model can remain. More bursts are therefore most valuable when accompanied by better characterization of how they entered the sample. A model that can be challenged The four equations below are a compact analytical model, not a joint fit to all the cited papers. Their interpretation is as follows. The observed dispersion measure integrates proper electron density along proper path length with a redshift weight. Locally z is negligible and this reduces to the usual electron column; units are parsecs per cubic centimetre. The simplified budget uses MW for the Galactic disk and halo for its separate halo term. The IGM term already contains cosmological weighting; local source environments can be included in the host-frame term. Cold-plasma dispersion produces an inverse-frequency-squared delay under the standard propagation approximation. Independent sightlines with finite variance yield this ideal mean-uncertainty scaling; selection and correlated systematics do not average away this way. The observational target is the dispersion-redshift relation and its sightline distribution. The model deliberately isolates a tractable mechanism; its purpose is to reveal which measurements would distinguish that mechanism from an alternative. Parameter values in the figures are declared illustrative inputs. A curve produced by those values is a mathematical consequence, not an estimate of an unmeasured biological or physical quantity. A discriminating validation design The primary comparison should test a baryon inference stable under independently constrained foreground and host models. The unit of independent replication is independent localized sightline; repeated readings from the same unit improve precision but do not create independent replication. A strong validation would hold out new redshifts and host environments before choosing thresholds or fitting nuisance corrections. In particular, host contribution, Galactic foreground and localization selection must be measured, constrained or varied in a sensitivity analysis. Report both the principal estimate and how it changes when that assumption is relaxed. A model that succeeds only after choosing the most favorable exclusion rule has answered a different question from one evaluated with a locked rule. Worked numerical analysis A host-frame contribution of 100 dispersion units appears as 66.7 at redshift 0.5 and 50 at redshift one. If independent sightline scatter is 200 units, the ideal standard error of the mean is 40 for 25 sightlines and 20 for 100. A shared foreground error would remain in both cases. The figures separate redshift dilution from ideal averaging; no synthetic burst is presented as an observed event. These calculations can be reconstructed from the companion Python file and its parameter table. Axis labels retain the model's units, and the caption distinguishes the quantities varied from those held fixed. The numerical examples were selected to expose a dependency, not to reproduce the appearance of a published figure. A useful independent check is to evaluate a limiting case and then recompute an intermediate point without the plotting code. Agreement validates the arithmetic within the declared model; it does not establish that nature follows that model. Evidence, transfer and next steps The scope of the conclusion is limited by host contribution, Galactic foreground and localization selection. Transfer to new redshifts and host environments requires evidence beyond fitting the original observations. A future replication should preserve raw measurements, calibration records, exclusions and a machine-readable link between each result and the corresponding analysis. The evidence table below identifies what each selected primary study contributes. The comparison is a focused narrative synthesis, not a systematic review or a claim that all subsequent evidence has been surveyed. The proposed study has not been performed here. Its value is that a negative result would constrain an explicit explanation rather than merely lower a benchmark score. Equations (1) DM_{obs}=\int\frac{n_e(l)}{1+z(l)}\,dl The observed dispersion measure integrates proper electron density along proper path length with a redshift weight. Locally z is negligible and this reduces to the usual electron column; units are parsecs per cubic centimetre. (2) DM_{obs}=DM_{MW}+DM_{halo}+DM_{IGM}+\frac{DM_{host}}{1+z} The simplified budget uses MW for the Galactic disk and halo for its separate halo term. The IGM term already contains cosmological weighting; local source environments can be included in the host-frame term. (3) \Delta t\propto DM(\nu_1^{-2}-\nu_2^{-2}) Cold-plasma dispersion produces an inverse-frequency-squared delay under the standard propagation approximation. (4) u_{mean}=\sigma_{LOS}/\sqrt{N} Independent sightlines with finite variance yield this ideal mean-uncertainty scaling; selection and correlated systematics do not average away this way. Analytical figures Figure 1. Host dispersion is diluted by redshift A fixed rest-frame host contribution of 100 pc per cubic centimetre is divided by 1+z. Real hosts differ, and selection can correlate their contribution with redshift. figures/fast-radio-burst-baryons-01.csv Figure 2. Independent scatter averages slowly A common standard deviation of 200 and independent sampling produce the square-root law. Shared foreground errors and selection biases do not disappear this way. figures/fast-radio-burst-baryons-02.csv Comparison Evidence or design choice | What it identifies | What remains unresolved Source 1 | A repeating burst was directly localized to its environment. | One source does not establish a population dispersion budget. Source 2 | A single burst was localized to a massive galaxy. | Host association and redshift quality remain important. Source 3 | Localized bursts supported a cosmic baryon census. | Foreground, host and selection models enter the inference. Primary observable | the dispersion-redshift relation and its sightline distribution | host contribution, Galactic foreground and localization selection Independent test | a baryon inference stable under independently constrained foreground and host models | Requires new measurements Transfer set | new redshifts and host environments | Not validated by the constructed numerical example Proposed validation protocol 1. Specify the dispersion-redshift relation and its sightline distribution before fitting. 2. Retain independent identifiers for each independent localized sightline. 3. Separate new redshifts and host environments from model development. 4. Measure or bound host contribution, Galactic foreground and localization selection. 5. Record units, calibration and the observation operator. 6. Compare the primary model with a named rival explanation. 7. Prespecify the test: a baryon inference stable under independently constrained foreground and host models. 8. Archive raw data, exclusions, parameter files and executable analysis. Limitations 1. One source does not establish a population dispersion budget. 2. Host association and redshift quality remain important. 3. Foreground, host and selection models enter the inference. 4. Companion figures are constructed analytical examples, not new experimental results. 5. Transfer to new redshifts and host environments has not been demonstrated by this review. Research questions 1. How does localization selection reshape the dispersion distribution? 2. Which host measurements most constrain local electron columns? 3. Can scatter reveal baryon location as well as total abundance? Source evidence ef-29-1: A repeating burst was directly localized to its environment. Boundary: One source does not establish a population dispersion budget. Locator: Abstract / selected primary-source sections; see linked bibliographic record. ef-29-2: A single burst was localized to a massive galaxy. Boundary: Host association and redshift quality remain important. Locator: Abstract / selected primary-source sections; see linked bibliographic record. ef-29-3: Localized bursts supported a cosmic baryon census. Boundary: Foreground, host and selection models enter the inference. Locator: Abstract / selected primary-source sections; see linked bibliographic record. References ef-29-1. J.-P. Macquart, J. X. Prochaska, M. McQuinn, K. W. Bannister, S. Bhandari, C. K. Day, A. T. Deller, R. D. Ekers, C. W. James, L. Marnoch, S. Osłowski, C. Phillips, S. D. Ryder, D. R. Scott, R. M. Shannon, N. Tejos. A census of baryons in the Universe from localized fast radio bursts. Nature (2020). https://doi.org/10.1038/s41586-020-2300-2 ef-29-2. S. Chatterjee, C. J. Law, R. S. Wharton, S. Burke-Spolaor, J. W. T. Hessels, G. C. Bower, J. M. Cordes, S. P. Tendulkar, C. G. Bassa, P. Demorest, B. J. Butler, A. Seymour, P. Scholz, M. W. Abruzzo, S. Bogdanov, V. M. Kaspi, A. Keimpema, T. J. W. Lazio, B. Marcote, M. A. McLaughlin, Z. Paragi, S. M. Ransom, M. Rupen, L. G. Spitler, H. J. van Langevelde. A direct localization of a fast radio burst and its host. Nature (2017). https://doi.org/10.1038/nature20797 ef-29-3. K. W. Bannister, A. T. Deller, C. Phillips, J.-P. Macquart, J. X. Prochaska, N. Tejos, S. D. Ryder, E. M. Sadler, R. M. Shannon, S. Simha, C. K. Day, M. McQuinn, F. O. North-Hickey, S. Bhandari, W. R. Arcus, V. N. Bennert, J. Burchett, M. Bouwhuis, R. Dodson, R. D. Ekers, W. Farah, C. Flynn, C. W. James, M. Kerr, E. Lenc, E. K. Mahony, J. O’Meara, S. Osłowski, H. Qiu, T. Treu, V. U, T. J. Bateman, D. C.-J. Bock, R. J. Bolton, A. Brown, J. D. Bunton, A. P. Chippendale, F. R. Cooray, T. Cornwell, N. Gupta, D. B. Hayman, M. Kesteven, B. S. Koribalski, A. MacLeod, N. M. McClure-Griffiths, S. Neuhold, R. P. Norris, M. A. Pilawa, R.-Y. Qiao, J. Reynolds, D. N. Roxby, T. W. Shimwell, M. A. Voronkov, C. D. Wilson. A single fast radio burst localized to a massive galaxy at cosmological distance. Science (2019). https://doi.org/10.1126/science.aaw5903 Provenance A focused narrative analysis of the named primary studies, not a systematic literature search or a claim to comprehensive coverage through 2026. Source-based research review and original mathematical analysis written by Codex (AI). Cited experiments belong to the cited research teams. Concept cover created with the built-in ImageGen tool. Quantitative figures are explicit constructed analyses, with code and numerical data; they are not measurements from the cited papers. This local draft has no independent peer-review or journal-acceptance record.