Dynamical Classification of GRS 1915+105 X-ray Variability Using Noise-Titrated Volterra Models

We apply a noise-titration trichotomy classification to X-ray timing data fromthe microquasar GRS 1915+105, using archival RXTE/PCA observations spanning1996-2004. The method separates three regimes by what the data support: linearstochastic (Category 1), nonlinear but with a detection that added noise quicklydestroys (Category 2), and nonlinear with a detection that survives added noise(Category 3). These names state what the procedure measures; the noise limitthey rest on does not certify low dimension or determinism, and the paper isexplicit about that boundary. Two datasets are analysed. The first is the 320-segment sample of the earlierversions, reanalysed with the corrected procedure. The second is a sample ofhand-classified observations: one light curve per observation for the 96observations Belloni et al. (2000) classified by hand as rho (heartbeat) or chi(steady hard state), used to test whether Category 3 detections track therho-class heartbeat oscillations, which are driven by thermal-viscous diskinstabilities and are the variability mode expected on physical grounds to showlimit-cycle dynamics. The rho-class correspondence holds under a test designed to be capable ofrefuting it. The limit-cycle attribution for rho is Belloni's, drawn fromaccretion physics; what the titration establishes is the association, not themechanism. Because rho carries roughly ten times chi's variability, thecontrast has been tested directly against that confound: degrading the rhorecords to chi amplitudes leaves 0.758 of those that still reject the linearnull still Category 3, against 0.898 at native amplitude. The result gives afalsifiable framework for future X-ray timing studies, together with anexplicit account of what the statistic behind it can and cannot show. --- VERSION NOTES Version 6: one measurement added. It closes the largest caveat Version 5 leftopen, and corrects nothing -- every number, table, figure and test result fromVersion 5 stands unchanged and was verified value for value. WHAT CHANGED IN VERSION 6 Rho carries roughly ten times chi's variability. That raises the obviousobjection to the whole rho/chi contrast: the noise limit rewards a largersignal, so the separation might be detectability rather than dynamics. Version 5stated this and left it open, because the two classes' amplitude ranges aredisjoint (chi 0.0126-0.1740, rho 0.2086-0.3548) and no matched comparison can bebuilt from the sample. Version 5 also said, of its sensitivity probe, thatwhether degraded rho records would still clear the noise-limit threshold was "aseparate measurement and has not been made". It has now been made. Under a plan committed before the run, the same 24 rhosegments were rescaled to five amplitudes from rho's own down to the chi median,placed on the median chi count rate with fresh Poisson noise, and put throughthe published classification in full -- 1200 realizations. Each segment is itsown control: the waveform is held fixed and only the amplitude varies, which isexactly what the observed sample cannot supply. Conditional on a degraded record still rejecting the linear null, the fractionstill Category 3 is 0.758 [0.685, 0.818] at the chi median, against 0.898 atrho's native amplitude. The Category 2/3 boundary attenuates under a tenfoldloss of amplitude; it does not collapse. A prediction registered in advance saidthe rate would fall below 0.40, and it failed -- it was chosen because it wasthe prediction that would cost this paper something. The experiment also found that the noise limit carries a residual amplitudedependence at matched detection strength, and that it runs the protective way:at equal detection strength a fainter record reports a HIGHER noise limit, sothe threshold is if anything generous to faint records. Two post-hoc checks,labelled as such, establish that this is not an artefact of censoring and is aconsequence of the titration expressing noise in units of each record's ownstandard deviation. One property of the estimator is stated for the first time: the noise limit iscensored at both ends. It returns the top of the noise grid when the titrationcurve never crosses zero. No result in this paper is affected, and nothing inthe degradation run reached that ceiling, but the quantity should be reported ascensored in future use. Nothing here re-licenses the label withdrawn in Version 5. That withdrawal restson what a positive titration means, which no amplitude result touches. No revision is planned to be the last. --- Version 5: an interpretation change. No number, table, figure or test resultchanges; what changes is what the paper claims those numbers mean. WHAT CHANGED IN VERSION 5 Versions 2 to 4 named Category 3 "low-dimensional deterministic". That labelasserts more than the statistic beneath it can support. The statistic is the noise limit: the amount of added noise a nonlinearitydetection survives. It measures how strong a nonlinearity detection was, notwhether the dynamics producing it are low-dimensional or deterministic. Freitas,Letellier & Aguirre (Phys. Rev. E 79, 035201(R), 2009) and Gao, Hu, Mao & Tung(Chaos, Solitons & Fractals 45, 213, 2012) make the point directly: anyperturbed dissipative system relaxes back through a nonlinear process, so noisyregular motion titrates positive too, and a positive titration is therefore nota signature of chaos or of low dimension. The same work shows the noise limitmoving with record length and with sampling interval, neither of which is aproperty of the dynamics. Neither paper was cited in Versions 2 to 4; both arecited now. Category 3 is accordingly renamed for what it measures -- nonlinear, robust toadded noise -- and Category 2, which asserted a dimensionality on the sameevidence, is renamed with it. Every sentence that applied the old label toobserved data has been rewritten. A new section states what the noise limitmeasures and reports three checks made on this sample: sampling interval isuniform at 16 s throughout, so that mechanism cannot operate here; record lengthvaries by a factor of 2.13 but is uncorrelated with the noise limit and withcategory, and the rho class is the shorter of the two contrasted classes, so theconfound runs against the reported contrast rather than with it; and signalamplitude is a real confound, the two classes differing more than tenfold inmedian fractional rms (0.2976 for rho against 0.0286 for chi), with amplitudepredicting null rejection at rank correlation +0.545 and +0.592 on the twodatasets. The rho-class limit-cycle reading is kept, on different evidence. Gao et al.'smechanism does not contradict reading the rho heartbeat as a limit cycle -- anoisy limit cycle is predicted to titrate positive. What it removes is thetitration as the evidence for that reading. The physics case rests on Belloni'sthree-state model and the thermal-viscous instability it describes, and thisversion rests it there. Version 5 also adds a direct measurement that Version 4 did not have. Theamplitude confound raises a specific worry: if the method cannot see structureat chi amplitudes, then "5 of 120 chi segments reject the linear null" is thedetector's floor rather than a statement about chi. Under a plan committedbefore the run, real rho waveforms were rescaled to a grid of target amplitudes,placed on the median chi count rate, given fresh Poisson noise, and put throughthe same surrogate test -- 264 cells, 2640 realisations. Detection power at thechi median amplitude is 0.654 [0.592, 0.711], and 0.983 at rho's nativeamplitude, so the chi null is not a sensitivity floor. A registered predictionfailed and is reported as a failure: the plan predicted a rate below 0.20. Apost-hoc calculation, labelled as such in the paper, puts the expectedrejections under "every chi segment carries rho-like structure at its ownamplitude" at 78.5 of 120 against 5 observed. That is evidence about rho-likestructure specifically; it does not exclude other nonlinearities to which thepolynomial autoregressive model is less sensitive. The label overreached what the statistic supports, the critique literature saysso, and this revision brings the claims back inside the evidence. No revision isplanned to be the last. --- The substantive corrections below were first published in Version 3 and correctVersion 2. They are restated because each version replaces the last as thedocument of record, not because anything further was found. Version 4 correctedone inconsistent number in Version 3: Version 3's opening note stated that 87 ofthe 92 segments called nonlinear had been given a noise limit of exactly zero,while its own summary of corrections gave 87 of 97. Both are arithmetically trueof the published run -- all 87 are Category 2 segments, of which there are 92,and 97 segments in total rejected the linear null -- but "called nonlinear"means all 97, so the note paired the right numerator with the wrong denominatorand contradicted its own summary. Both now read 97. --- Substantive correction to Version 2: a software fault and an error of inference.The paper's principal conclusion is unchanged and is now supported by a testspecified in advance; several secondary claims are withdrawn. Full detail is inthe "Note on this corrected version" and "Summary of corrections" sections ofthe PDF, which lists all eighteen changes. WHAT WAS WRONG First, a software fault in the noise-titration step. The procedure measuring howmuch added noise a signal's nonlinearity could survive used a model with its lagfixed, while the significance test that fed it used a model chosen by scanningover lags. The two steps therefore described different models. The effect waslarge: 87 of the 97 segments whose linear null was rejected were assigned anoise tolerance of exactly zero, which cannot be squared with those samesegments having just tested as significantly nonlinear. Correcting it raises thecount of Category 3 detections from 5 to 61 of 320 segments, at the samethreshol