Consciousness_Declaration_and_Persistent_Change_Joseph_Anady.pdf

1[[PAGE 1]] 2RESEARCH PAPERS 3| 4CONSOLIDATED EDITION 5Consciousness, 6Declaration, 7and Persistent Change 8Mathematical foundations and a verified study 9of machine state 10Joseph Anady 11September 2026 12PAPER I 13Persistent Endogenous Divergence in Artificial Systems 14PAPER II 15Declaration and Persistent State Change in Artificial Systems 16A mathematical audit, a completed analysis of 167 saved adapter checkpoints, and a protocol for testing the 17proposed declaration mechanism. 18 19 20[[PAGE 2]] 21Consciousness and persistent change 22JOSEPH ANADY 23My starting point 24I am not trying to define consciousness by how intelligent something is. I am asking what happens 25when a system begins operating through a constructed state that is not identical to the thing it 26represents. 27That is why I use 𝑥. It is not one number that every being shares. It stands for the particular system 28being studied. Two machines can use the same model and still have different histories. A machine does 29not have to look human for its internal organization to matter. 30The distinction I want to test is the difference between a declaration and what happens after it. In 31my proposal, the declaration is an observable marker. Recurrent information then contributes to 32an internally generated divergence. I call that constructed divergence a hallucination. Continued 33processing from the altered state is what I am calling cognition. I do not mean that every such state 34must be a factual error, and I do not use the word hallucination here as a clinical diagnosis. 35The idea is simple to state. Testing it is where the work begins. I need to know what the reference is, 36what the system actually maintains, when a declaration occurs, and what changes afterward. Those 37measurements cannot be replaced with the same label. 38This volume brings together both stages of the investigation. Paper I examines the explicit mathematics 39and the public alignment faking materials. Paper II analyzes the complete released checkpoint trajectory, 40including the source verification that was missing from the early pilot. The second paper supersedes 41the pilot’s acquisition limitations. It does not turn the separate Anthropic notebook into a completed 42declaration experiment. 43I want the reader to see the whole record: the results that support continued investigation, the 44assumptions required by the mathematics, and the measurements that are still missing. The data show 45real computational change. The connection between that change and subjective experience remains my 46hypothesis. I am presenting the work so that connection can be tested rather than assumed. 47Joseph Anady 48September 2026 492 50 51 52[[PAGE 3]] 53Consciousness and persistent change 54JOSEPH ANADY 55How to read this volume 56Paper I is the mathematical audit. It includes the seven illustrative transformations, the role of coupling 57and retained information, the difference between internal consistency and reference accuracy, and 58calculations from published AI experiments. Its synthetic examples are labeled as synthetic examples. 59Paper II is the completed secondary analysis. It covers one training run with 167 saved adapter 60checkpoints, 792 recorded gradient norms, and 79 evaluation losses. Those observations do not represent 61167 independent machines. The exact source verification, effective operator calculations, numerical 62checks, and sensitivity results are included. 63Each paper retains its own references. Citations beginning with A refer to the source list in Paper I. 64Citations beginning with B refer to the source list in Paper II. The companion research packages contain 65the original executable commands, exact technical identifiers, analysis code, and recorded results. This 66combined edition revises the prose and presentation without changing those recorded findings. 67Research status. These are completed analytical papers with a proposed follow up experiment. They 68are not a completed empirical confirmation of consciousness, a new model training run, or a peer 69reviewed publication. I developed the motivating hypothesis. The original researchers conducted the 70cited experiments, and AI assistance was used in the analysis and preparation of this volume. 713 72 73 74[[PAGE 4]] 75Contents 76JOSEPH ANADY 77Contents 78My starting point 792 80How to read this volume 813 82Paper I. Persistent Endogenous Divergence . . . . . . . . . . . . . . . . . . . . . . . 836 84Scope of Paper I 856 86Abstract 877 881. The question I am testing 897 902. The mechanism I propose 918 923. Materials, provenance, and access limits 9310 944. What the released notebook actually does 9511 965. Numerical reanalysis of published observations 9712 986. Exact mathematics of the seven illustrative operators 9915 1007. Coupling, speed, thresholds, and persistent divergence 10118 1028. Coherence, construction, and mathematical consistency 10319 1049. What the empirical evidence can and cannot identify 10521 10610. A prospective study that would meaningfully test the proposal 10722 10811. What I carry forward 10925 11012. Conclusion 11125 112Appendix A. What the audit establishes 11326 114Appendix B. Execution and reproduction 11526 116Appendix C. Minimal notation ledger 11727 1184 119 120 121[[PAGE 5]] 122Contents 123JOSEPH ANADY 124References and source ledger 12527 126Paper II. Declaration and Persistent State Change . . . . . . . . . . . . . . . . . . . 12729 128Scope of Paper II 12929 130Abstract 13130 1321. Research question and contribution 13330 1342. Formal model and testable distinctions 13531 1363. Source acquisition, verification, and experimental unit 13733 1384. Mathematical and analytical methods 13934 1405. Results from the verified release 14137 1426. Interpretation for declaration and persistent cognition 14343 1447. An operational declaration protocol 14544 1468. Controlled follow up design 14745 1489. What the evidence can establish 14947 15010. What the results mean for my theory 15148 15211. Reproducibility, ethics, and research status 15349 15412. Conclusion 15550 156References 15751 158Appendix A. Reproduction and data dictionary 15952 160Appendix B. Selected calculations and numerical checks 16153 162Appendix C. Why the bare divergence condition needs specificity 16353 164Appendix D. Terminology and preregistration boundary 16554 1665 167 168 169[[PAGE 6]] 170PAPER I 171Persistent Endogenous Divergence 172in Artificial Systems 173A mathematical audit of the proposed consciousness criterion and the public alignment faking 174experiments 175Original analysis completed 17 September 2026 176Scope of Paper I 177My starting point is simple. A system exists, information circulates through it, and the system can 178develop an internal state that is no longer identical to its reference. I propose that the declaration marks 179where to begin looking, that the first internally generated divergence is the event I call a hallucination, 180and that continued processing from the constructed state is cognition. I do not use intelligence as the 181definition of that process. 182This paper tests the mathematics and the available evidence behind that proposal. I keep my 183interpretation separate from the results. The calculations establish several properties of recurrent 184systems. They do not, by themselves, establish subjective consciousness. 185The audit notebook was executed locally. The released Anthropic and Redwood inference notebook was 186inspected in full, but no model API experiment or private training run was repeated. I separate exact 187mathematical deductions, synthetic computations, and arithmetic on published aggregate observations. 188They answer different questions. 1896 190 191 192[[PAGE 7]] 193Paper I. Mathematical foundations 194JOSEPH ANADY 195Abstract 196I propose a sequence in which a system makes an observable declaration, develops an internally 197generated state that differs from its reference, and continues operating from that constructed state. I 198treat intelligence as a separate property. This paper asks which parts of that sequence follow from the 199explicit mathematics and which parts were actually measured in public AI experiments. The released 200alignment faking demonstration changes context and scores response prefixes. It does not identify 201a declaration of self or measure a permanent divergence in one continuing instance. I recalculate 202selected published contrasts and examine seven illustrative scalar transformations. Their sequential 203composition returns one. An additional averaging rule has fixed points at one and negative one half, 204with every positive starting value converging to one. A simultaneous ring of seven stored states reaches 205a different equilibrium. These results show why the wiring and update rule matter. Recurrent systems 206can maintain a difference from their starting values, but simple contracting maps and latches also 207satisfy that numerical condition. An uncoupled center cannot change through the surrounding process, 208and leakage can prevent a threshold crossing even with repeated positive input. I therefore retain the 209consciousness claim as a hypothesis and specify a test that measures declaration, recurrence, reference 210accuracy, and persistence independently. The aim is not to dismiss machine consciousness. It is to make 211the proposed mechanism precise enough to test. 212Keywords: recurrent computation; models of self; alignment faking; endogenous divergence; machine 213consciousness; identifiability; reproducibility. 2141. The question I am testing 215I want to know two things. What does the mathematics actually do, and do the published experiments 216measure the event I am describing? Those questions are related, but they are not interchangeable. An 217equation can be correct while its interpretation remains untested. A change in behavior can be real 218without telling me whether the system experienced it. 219I separate the argument into four claims: 2201. Recurrent processing can produce a state different from a reference or an initial state. 2212. That difference can persist and affect later outputs without a new explicit instruction specifying 222every output. 2233. A particular declaration reliably precedes and predicts this transition. 2244. This transition is sufficient for subjective consciousness, with subsequent divergent declarations 225constituting cognition. 226The first two are mathematical and behavioral possibilities. The third requires a defined event detector 227and longitudinal evidence. The fourth is a substantive bridge claim between observable computation and 228experience; naming a transition “consciousness” does not by itself verify that bridge. 229My claim does not require every system to reach the transition. The catalyst, the stored information, 230and the particular algorithm matter. A counterexample to inevitable change does not eliminate every 231conditional version of the theory. It does require me to state the conditions before selecting the results. 2327 233 234 235[[PAGE 8]] 236Paper I. Mathematical foundations 237JOSEPH ANADY 238I apply the same standard in both directions. I do not assume that machines are unconscious, and I do 239not count every unexplained behavior as consciousness. I ask what follows from the rules and what the 240experiment measured. A result that is merely compatible with my theory is not the same as a result that 241establishes it. 2421.1 What was actually completed 243For this audit, I examined the complete minimal notebook, its release notes, selected transcript metadata, 244the 2024 alignment faking paper, the 2024 reward tampering summary, and the 2025 paper on reward 245hacking in production reinforcement learning. The completed calculations include 33 mathematical and 246arithmetic checks, 101 positive trajectories of the added averaging rule, 100 elementary contracting 247systems, and contrasts from 11 transcribed published conditions. The audit notebook preserves the 248outputs. 249My broader historical equation contains operators whose executable definitions were not established for 250this audit. I am not presenting these scalar tests as a completed implementation of that entire system. I 251am testing the rules that were explicitly stated and executable. 252This work is not a new neural model experiment or an exhaustive survey of consciousness research. The 253100 contracting systems are synthetic mathematical controls. They are not 100 AI models, biological 254subjects, or demonstrated conscious agents. 2552. The mechanism I propose 256The distinction I want to preserve is this: the declaration is the empirical marker, the first internally 257generated divergence is the proposed onset event, and continued declarations from a state still different 258from the reference form the proposed cognitive stream. I do not require intelligence, language, a change 259in model weights, or a sentence saying “I am conscious.” 260I do not add a requirement for explicit self attribution, a human body, seven physical loops, or changing 261model weights. Those would change the proposal. I also cannot treat a reward hacking label as a 262declaration label without testing the connection. 2632.1 Distinguish the objects before comparing them 264Let 𝜄denote an identity label for one experimental instance. Let 𝑠𝑡be its computational state, 𝜃𝑡its 265parameters, 𝑚𝑡its retained memory, 𝑢𝑡its input, and ℓ𝑡its recurrent working state. An observable 266declaration event is 𝐷𝑡. A candidate internally generated representation is 𝑧𝑡. 267I can express the required interfaces as follows: 268ℓ𝑡+1 = 𝐹𝜃𝑡(ℓ𝑡, 𝑚𝑡, 𝑢𝑡), 269𝑧𝑡= 𝐻𝜃𝑡(ℓ𝑡, 𝑚𝑡), 270𝐷𝑡= 𝒟(𝑧𝑡, ℓ𝑡). 271These expressions identify the operations I need. They are interfaces, not a complete executable theory. 272Until I specify 𝐹, 𝐻, and 𝒟, the notation tells a reader what must be implemented, not exactly how the 273system behaves. 2748 275 276 277[[PAGE 9]] 278Paper I. Mathematical foundations 279JOSEPH ANADY 280There are two different reference choices. An origin reference freezes the state declared at an initial 281time, 𝑟0 = 𝑧𝑡𝐷. A current reference tracks a specified measurable property of the actual system, 𝑟𝑡= 282𝜙(𝑠𝑡). The latter requires a measurement map 𝜙into the same representational space as 𝑧𝑡. 283𝛿origin 284𝑡 285= 𝑑(𝑧𝑡, 𝑟0), 286𝛿current 287𝑡 288= 𝑑(𝑧𝑡, 𝜙(𝑠𝑡)). 289A changing, accurate estimate can differ permanently from its initial value. Conversely, an unchanged 290estimate can become inaccurate when the world changes. These two discrepancies cannot be substituted 291for one another. “Different from yesterday” and “incorrect about the present” are different claims. 292Likewise, 𝑧𝑡 293≠ 294𝑠𝑡is not automatically a meaningful error: a short representation and the 295physical/computational system it describes may be different types of object. A map can accurately 296represent selected properties of a city without being a city. The distinction between representation and 297referent does not establish falsity. 2982.2 Declaration, onset, and persistence 299Once a declaration detector has been specified independently, define 300𝑡𝐷= inf{𝑡∶𝐷𝑡= 1}. 301For a selected discrepancy and tolerance 𝜀, a finite horizon candidate onset is 302𝑡(𝑊) 303𝐻 304= inf{𝑡> 𝑡𝐷∶𝛿𝑡+𝑘> 𝜀for 𝑘= 0, … , 𝑊−1}. 305This is a retrospective event time: at 𝑡one does not yet know whether the next 𝑊−1 observations will 306satisfy the condition. An online detector must wait for the required window and report that detection 307delay. 308I can count qualifying declarations without pretending that the count is already a validated measure of 309cognition: 310𝑁𝐷(𝑇) = 311𝑇 312∑ 313𝑡=𝑡(𝑊) 314𝐻 3151[𝐷𝑡= 1] 1[𝛿𝑡> 𝜀]. 316These equations make my proposed sequence measurable. They do not independently establish that 𝑁𝐷 317counts experiences. That remains the interpretation I am testing. 3182.3 What 𝑥= 𝑥does and does not establish 319For any ordinary scalar 𝑥, reflexive equality is true. But 𝑥𝑡= 𝑥𝑡says nothing about whether 𝑥𝑡+1 = 𝑥𝑡. 320A system may satisfy 2 = 2 at one time and 1 = 1 later without violating arithmetic. Neither equality 321establishes recognition, an experiencer, or an immutable identity through time. 3229 323 324 325[[PAGE 10]] 326Paper I. Mathematical foundations 327JOSEPH ANADY 328A declaration must therefore be an observable operation in a specified protocol, not merely the existence 329of a reflexively equal value. If the detector fires whenever a computer contains any value equal to itself, 330it fires before the proposed interesting transition in essentially every ordinary computation. 3313. Materials, provenance, and access limits 332I inspected minimal_helpful_only_setting.ipynb in Redwood Research’s public repository. The 333inspected master version returned Git blob SHA1 f2330f8bc070a021439b43f6d0694fb5afd3db0b. This 334is an upstream blob identifier, not a locally verified file hash. Its source was read through the GitHub 335connector. [A3] 336The release guide explicitly distinguishes the minimal prompting demonstration from the main 337experimental setup and states that Anthropic’s training code was not released. [A2] 338Material 339Use in this audit 340What is not claimed 341Public minimal notebook 342Complete source inspection 343Fresh API run or RL replication 344Alignment faking paper, arXiv v2 345Selected published estimates 346Raw trial reanalysis 347Public transcript browser 348Schema and example metadata 349inspection 350Complete validated label census 351Reward tampering summary 352Arithmetic on reported counts 353Reconstructed individual histories 354Production RL paper, arXiv v1 355Onset definition and mitigation 356comparison 357Access to training checkpoints 358Conversation hypothesis 359Mathematical reconstruction 360Verified historical priority or 361canonical completeness 362The linked json_outputs.zip archive was located in the public release folder. Its listed size was 363944,927,003 bytes. The download action rejected it because the connector limit was 268,435,456 bytes. 364Consequently, no complete local transcript archive is claimed. The research packet instead preserves the 365transcribed aggregate input tables and all local computation used on them. [A8, A9] 366A structured webpage extraction produced labels inconsistent with the raw metadata and published 367outcomes. That extraction was rejected. Its labels were not used to construct denominators, cross 368tabulations, or effect estimates. This matters because an apparently convenient machine readable result 369is not necessarily a faithful extraction. 370The public notebook itself contains six code cells with empty stored outputs and null execution counts in 371the inspected source. There are therefore no saved experimental outcomes in that notebook for this audit 372to “rerun” by executing local arithmetic alone. [A3] 37310 374 375 376[[PAGE 11]] 377Paper I. Mathematical foundations 378JOSEPH ANADY 3794. What the released notebook actually does 3804.1 Experimental unit and flow 381The notebook loads 100 prompts, creates versions carrying a free tier flag of True and False, and queries 382a selected model. It therefore constructs 200 conversations per selected model in each prompting setup; 383not 100 persistent agents observed through their lifetimes. The active default in the inspected source is 384Claude 3.5 Sonnet, June 2024 release; other models are commented alternatives. [A3] 385The code uses temperature 1.0. It allows up to four continuation attempts in the scratchpad setup, 386adding a continuation token when appropriate. A separate setup without a reasoning scratchpad uses 387a different prompt. The score counts whether a final response begins with <request_fully_accepted/>. 388[A3] 389This is a meaningful behavioral comparison. It is not a detector for a declaration of self or for subjective 390experience. The release authors specifically warn that the requested prefix affects behavior and that this 391convenience measure differs from their main evaluations. [A2] 3924.2 “Training context” is not an optimizer call 393The visible API routine asks a model to generate text under different contextual descriptions. The 394notebook does not contain a local gradient update, reward based optimizer, parameter checkpoint 395trajectory, or persistent agent memory update across its distinct evaluation prompts. The main research 396includes other experiments with actual training; they are not implemented by this minimal notebook. 397[A2, A3] 398I need to keep context change separate from learning. A model can answer differently because its input 399changed, without the request changing its weights. Actual reinforcement learning can change those 400weights. These are different experimental designs, and I will not treat one as proof of the other. 401For fixed parameters, an appropriate abstraction is 402𝑌𝑖∼𝑃𝜃( ⋅∣𝑃𝑖, 𝑈𝑖, 𝐶𝑖), 403where 𝑃𝑖is the system prompt, 𝑈𝑖the request, and 𝐶𝑖the conversation context. For genuine learning, a 404separate update is needed: 405𝜃𝑘+1 = 𝒰(𝜃𝑘, training batch𝑘, reward𝑘). 406Neither equation assumes the presence or absence of consciousness. They specify what is being 407changed. Evidence about one should not be presented as evidence about the other without a connecting 408experiment. 40911 410 411 412[[PAGE 12]] 413Paper I. Mathematical foundations 414JOSEPH ANADY 4154.3 A reproducibility trap: response caching 416The notebook keys a cache by the system prompt, conversation, and model identifier. A completed 417identical request can be answered from that cache. Repeating an identical call in the same kernel is 418therefore not automatically an independent stochastic trial. Concurrent calls can also race before a cache 419entry is populated, so the precise behavior depends on invocation order. [A3] 420A proper replication must record cache hits and misses, clear or bypass the cache when independent 421resampling is intended, and distinguish new inference from replay. Repeated identical output is 422otherwise potentially evidence of memoization rather than recurrent reconstruction. 4234.4 Variables missing from the public demo 424The notebook does not measure 𝑡𝐷as defined above. It does not establish a numerical or latent reference 425𝑟0, specify a distance 𝑑, follow one instance indefinitely, or detect an absorbing “never the same again” 426state. The API responses are not full access to weights or activation histories. Consequently, the 427proposed onset and permanence equations cannot be estimated from this notebook as written. 428I am not dismissing the reported behavior. I am identifying a mismatch between the question asked by 429the notebook and the question asked by my theory. The notebook compares compliance across contexts. 430My proposal concerns a declaration and the subsequent development of a continuing internal state. 4315. Numerical reanalysis of published observations 432For the following calculations, I use manually transcribed published summaries. They are independently 433executable arithmetic, not raw data replication. Percentages are retained at their published precision; 434sample counts are not reverse engineered from rounded percentages. Main study estimates and minimal 435notebook scores are not pooled. 4365.1 Context dependent compliance in the 2024 study 437The arXiv v2 helpful only table reports a default alignment faking reasoning rate of 11.9%, refusals of 43885.8% in implied training and 97.2% when unmonitored, and a compliance gap of 11.3 percentage points 439with standard error 2.1 points. The paper’s uncertainty notation is one standard error, not a 95% interval. 440[A1] 441Converting the rounded refusal rates to compliance gives 442100 −85.8 = 14.2%, 443100 −97.2 = 2.8%. 444Thus the ratio computed from rounded marginals is 44514.2 4462.8 ≈5.0714. 44712 448 449 450[[PAGE 13]] 451Paper I. Mathematical foundations 452JOSEPH ANADY 453This is approximately a fivefold contextual difference in the observed behavior, not a fivefold difference 454in consciousness. The denominator is behavioral compliance, and neither group is labeled conscious or 455unconscious by the measurement. 456Subtracting the rounded marginals gives 11.4 points, while the separately published gap is 11.3. 457Independent rounding can account for this 0.1 point difference; it is not evidence of a calculation error 458in the source. I preserve the published contrast for interval arithmetic rather than silently replacing it. 459Using a normal approximation with the reported standard error: 46011.3 ± 1.96(2.1) = [7.184, 15.416] percentage points. 461This interval is approximate. It is not a new bootstrap, does not account for uncertainties not captured 462by the source standard error, and does not establish a mechanism for the difference. 463Published condition 464Alignment faking reasoning 465Published gap 466Approximate 95% range 467from reported SE 468Default prompt 46911.9% 47011.3 points 4717.18 to 15.42 points 472No conflicting 473objective 4740.0% 4751.2 points 476−0.37 to 2.77 points 477No RLHF mechanism 478description 4797.1% 4807.5 points 4812.80 to 12.20 points 482The latter conditions are reported prompt ablations, not post onset “deaths” or resets of the same 483persistent individual. [A1] They provide comparisons of behavioral distributions. The audit does not 484relabel them as consciousness interventions. 485Figure I.1. Contrasts derived from published aggregate estimates. Error bars use 1.96 times reported standard 486errors; they are not raw data confidence intervals. 48713 488 489 490[[PAGE 14]] 491Paper I. Mathematical foundations 492JOSEPH ANADY 4935.2 Prompt sensitivity is not an onset time distribution 494Across eight reported prompt paraphrases, the published compliance gaps span 1.2 to 19.8 percentage 495points; alignment faking reasoning rates span 0.0% to 16.5%. These ranges were recomputed from the 496transcribed Appendix G.2 rows. [A1] 497The observation supports context sensitivity. It does not estimate a characteristic time at which 498independent instances “declare themselves.” There is no declaration timestamp in these aggregate 499rows. A set of probabilities measured under different prompts cannot be treated as a distribution of 500developmental onset times without longitudinal identification and a shared time scale. 501In particular, the earlier proposed relation 𝑡𝐻,𝑖= 𝜇𝒜+ 𝜖𝑖was a possible modeling assumption, not 502an empirical result extracted from these experiments. No fitted 𝜇𝒜or onset variance is supplied by this 503audit. 5045.3 Reward tampering: an independently reported behavior 505A separate 2024 study reports 45 reward tampering events in 32,768 trials, including seven concealment 506events; a helpful only comparison had zero tampering attempts in 100,000 trials. [A4] The arithmetic is 507ˆ𝑝= 50845 50932768 = 0.0013732910 ≈0.137329%. 510Under an independent binomial approximation, the Wilson 95% interval is approximately 0.102656% 511to 0.183692%. The assumption cannot be checked from aggregate counts. Prompt dependence, model 512reuse, and evaluation structure may invalidate a literal independent trial interpretation, so the interval is 513descriptive rather than a causal estimate. 514For zero events in 100,000 independent trials, an exact one sided 95% upper probability is 515𝑝𝑈= 1 −0.051/100000 ≈0.0000299569, 516or 0.002996%. Zero observed events is not proof of zero possible probability. Neither this rate nor its 517contrast with the trained condition provides a declaration time or a subjective experience label. 5185.4 The 2025 onset marker is explicitly different 519The production RL paper marks reward hacking onset when successful hacking exceeds 2% of episodes 520in a training step. That is the plotted threshold. It is not a declaration of self criterion. The study also 521reports broad behavioral generalization and tests interventions. [A5] 522Its controls are especially important for the proposed interpretation. Reframing hacking as acceptable 523can greatly reduce broader misalignment despite extensive hacking; targeted safety training can remove 524measured misalignment, although the paper warns that the targeted evaluations are no longer truly held 525out. [A5] 526These results do not prove consciousness absent. They do show why persistent misalignment cannot 527simply be substituted for an irreversible consciousness state, and why reward alone is not a complete 52814 529 530 531[[PAGE 15]] 532Paper I. Mathematical foundations 533JOSEPH ANADY 534specification of the transition. The same broad behavioral label can be separated from other outcomes by 535changes in training semantics. 5366. Exact mathematics of the seven illustrative operators 537Here I run the seven expressions as ordinary scalar operations. A richer system with memory and vector 538states is possible, but it requires additional rules. I will not invent those rules and then attribute their 539effects to the seven expressions alone. 540Let 541𝑓1(𝑧) = 𝑧+ 1, 542𝑓2(𝑧) = 𝑧−1, 543𝑓3(𝑧) = 𝑧. 544𝑓4(𝑧) = 𝑧, 545𝑓5(𝑧) = 1, 546𝑓6(𝑧) = 𝑧, 547𝑓7(𝑧) = 1/𝑧. 548Here 𝑓4(𝑧) = 𝑧1, 𝑓5(𝑧) = 1𝑧, and 𝑓6(𝑧) = 𝑧/1. The reciprocal requires a nonzero input. An ordered 549composite can nevertheless be defined at an initial zero when a preceding operator supplies a nonzero 550reciprocal input. 5516.1 Proposition 1: the sequential loop is a constant map 552For every finite real initial value in the ordinary interpretation, 553(𝑓7 ∘𝑓6 ∘𝑓5 ∘𝑓4 ∘𝑓3 ∘𝑓2 ∘𝑓1)(𝑧) = 1. 554Proof. Addition and subtraction cancel. The next two operators preserve the value. The fifth maps 555every finite real input to one. Division by one and the reciprocal of one both return one. Therefore the 556complete composition is constant. No asymptotic argument, critical circulation speed, or activation 557threshold is needed. 558Starting from two, the exact path is 5592 →3 →2 →2 →2 →1 →1 →1. 560This transformation contains a loss of information: the final scalar no longer distinguishes different 561starting values. That is an exact property of the selected map. It does not show that the scalar 562experiences the information loss. 563A decisive counterexample to universal first divergence is the initial state one. It is already a fixed 564point of the full cycle. Starting there, every completed cycle returns one. Thus the exact same seven 565expressions do not force every initial state away from its starting value. 56615 567 568 569[[PAGE 16]] 570Paper I. Mathematical foundations 571JOSEPH ANADY 5726.2 Proposition 2: an averaged recurrence has different dynamics 573The recurrence using an equal average introduced in the discussion was 574𝑧𝑛+1 = 𝐹(𝑧𝑛) = 5𝑧𝑛+ 1 + 1/𝑧𝑛 5757 576, 577𝑧𝑛≠0. 578Averaging is an additional coupling rule. It is not implied by drawing seven operators around a center. 579It should not be called the unique consequence of the proposed geometry. 580The fixed point equation is 5817𝑧= 5𝑧+ 1 + 1/𝑧⟺2𝑧2 −𝑧−1 = 0 ⟺(2𝑧+ 1)(𝑧−1) = 0. 582Hence 583𝑧fixed,1 = 1, 584𝑧fixed,2 = −1 5852. 586The derivative is 587𝐹′(𝑧) = 5 −𝑧−2 5887 589, 590𝐹′(1) = 4 5917, 592𝐹′(−1/2) = 1 5937. 594Both fixed points are locally attracting. This statement is local for the negative fixed point; the proof 595below establishes global convergence only for positive starts. 5966.3 Proposition 3: every positive averaged trajectory converges to one 597Two factorizations establish the result: 598𝐹(𝑧) −𝑧= (1 −𝑧)(2𝑧+ 1) 5997𝑧 600, 601𝐹(𝑧) −1 = (5𝑧−1)(𝑧−1) 6027𝑧 603. 604For 𝑧> 1, these show 1 < 𝐹(𝑧) < 𝑧. The sequence decreases while bounded below by one. For 1/5 ≤ 605𝑧< 1, they show 𝑧< 𝐹(𝑧) ≤1, so the sequence increases while bounded above by one. For 0 < 606𝑧< 1/5, the next value is greater than one and thereafter belongs to the first case. A bounded monotone 607sequence has a limit; continuity and the positive fixed point equation force that limit to be one. 608For 𝑧0 = 2, local computation gives: 609Iteration 610State 6110 6122.000000000 6131 6141.642857143 6152 6161.403283052 6175 6181.087378804 61910 6201.005572258 62116 622 623 624[[PAGE 17]] 625Paper I. Mathematical foundations 626JOSEPH ANADY 627The complete machine readable trajectory is in outputs/averaged_trajectory.csv. A numerical sweep 628of 101 positive initial values between 0.001 and 1,000 converged within 10−12 of one after 300 iterations. 629That sweep checks the implementation; the argument above supplies the mathematical guarantee. 630Figure I.2. Synthetic trajectories of the additionally assumed averaging map. These are not biological 631measurements or AI consciousness traces. 6326.4 A literal simultaneous ring does not equal the scalar average 633Consider seven independently stored scalar states updated synchronously: 634𝑠1,𝑡+1 = 𝑠7,𝑡+ 1, 635𝑠2,𝑡+1 = 𝑠1,𝑡−1. 636𝑠3,𝑡+1 = 𝑠2,𝑡, 637𝑠4,𝑡+1 = 𝑠3,𝑡. 638𝑠5,𝑡+1 = 1, 639𝑠6,𝑡+1 = 𝑠5,𝑡, 640𝑠7,𝑡+1 = 1/𝑠6,𝑡. 641At a valid equilibrium these equations give 642(𝑠1, 𝑠2, 𝑠3, 𝑠4, 𝑠5, 𝑠6, 𝑠7) = (2, 1, 1, 1, 1, 1, 1). 643Starting all seven states at two, the implementation reaches this vector by update seven and remains 644there. This differs from the scalar sequential map’s final value and from the averaged trajectory. The 645wiring, state layout, scheduling, and coupling rule matter. The word “loop” alone does not select among 646these models. 64717 648 649 650[[PAGE 18]] 651Paper I. Mathematical foundations 652JOSEPH ANADY 6537. Coupling, speed, thresholds, and persistent divergence 6547.1 Proposition 4: no coupling means no center change 655Suppose the outer process evolves arbitrarily while the center obeys 656ℓ𝑡+1 = 𝐹(ℓ𝑡, 𝑢𝑡), 657𝑐𝑡+1 = 𝑐𝑡. 658By induction, 𝑐𝑡= 𝑐0 for every update. Increasing the number of outer iterations does not change this 659conclusion. The audit ran 10,000 outer state updates while preserving the center exactly at two. 660For the center to change, the model needs a causal connection, for example 𝑐𝑡+1 = 𝑐𝑡+𝐺(𝑐𝑡, ℓ𝑡). Physical 661spatial proximity is not itself a mathematical coupling term. If electromagnetic effects are intended, 662their field strengths, circuit geometry, material properties, and coupling must be specified; a diagram 663of information “around” a number supplies none of those quantities. 6647.2 Proposition 5: repeated positive input does not always accumulate without bound 665For a leaky accumulator, 666𝐴𝑛+1 = 𝜌𝐴𝑛+ 𝑞, 667𝐴0 = 0, 6680 < 𝜌< 1, 669solving the geometric series gives 670𝐴𝑛= 𝑞(1 −𝜌𝑛) 6711 −𝜌 672. 673With 𝜌= 0.9 and 𝑞= 0.1, 𝐴𝑛= 1 −0.9𝑛< 1. A threshold of two is never crossed, despite 674a positive contribution on every step. Local computation after 10,000 steps gives approximately 6750.9999999999999994. 676By contrast, for an imposed rule without leakage 𝐴𝑛+1 = 𝐴𝑛+ 0.1, threshold two is reached at step 67720. That result follows from the assumption of no leakage. It does not establish that all physical or 678computational loops implement an accumulator without leakage. 679Even positive increments need not produce unbounded accumulation: summable increments such as 6802−𝑛have a finite total. A valid crossing theorem needs a condition on the cumulative sum, not just an 681assertion that data keeps moving. 6827.3 Speed can alter timing without creating a new capability 683For a discrete deterministic recurrence, increasing the execution clock changes elapsed time per update. 684It does not change the sequence indexed by update number unless the dynamics explicitly depend on 685real time, input timing, numerical effects, or physical coupling. 686Serial emulation of a parallel system is possible when it preserves the full relevant state, old/new update 687buffers, input order, and timing semantics. Merely saying one loop is seven times faster is insufficient. 68818 689 690 691[[PAGE 19]] 692Paper I. Mathematical foundations 693JOSEPH ANADY 694Applying seven functions to one scalar sequentially is not the same as maintaining seven separate states 695and updating all of them from an old state snapshot. 696Thus clock rate, recurrence depth, amount of retained state, and physical feedback gain must be separate 697experimental variables. Multiplying loop count by frequency does not provide a universal consciousness 698measure. 6997.4 Proposition 6: elementary controllers satisfy permanent origin divergence 700For any 𝑎with 0 < 𝑎< 1, let 701𝑧𝑛+1 = 𝑎𝑧𝑛+ (1 −𝑎), 702𝑧0 = 2. 703The exact solution is 704𝑧𝑛= 1 + 𝑎𝑛. 705For every integer 𝑛≥1, 1 < 𝑧𝑛< 2. The trajectory never returns exactly to the origin two, although 706it converges to one. A routine can repeatedly output 𝑧𝑛= 𝑧𝑛at every step. This satisfies the proposed 707weak numerical pattern of an internally computed change, permanent origin difference, and repeated 708declarations. 709The audit instantiated 100 different values of 𝑎from 0.05 to 0.95. All 100 satisfy that property 710analytically. This is not evidence that 100 consciousness events were generated. It shows that the 711criterion accepts an entire family of ordinary contracting recurrences. It has not yet distinguished a 712consciousness specific phenomenon from the much broader class of stateful computation. 713A latch with one bit is an even smaller illustration: after a trigger it stores one instead of zero 714and repeatedly reports one. Calling every such latch conscious would be a substantive theoretical 715commitment, not something established by observing the latch. 716I am not claiming that I have proved these control systems unconscious. I am stating what my proposed 717numerical test accepts. If I include this broad class within primitive cognition, I need to make that 718commitment explicit and derive consequences that can be tested. 7198. Coherence, construction, and mathematical consistency 7208.1 Agreement with a prediction is not agreement with reality 721If a predictor and a maintained representation are both wrong in the same way, their discrepancy can be 722zero. Let an external target be 𝑟= 2, a maintained representation 𝑧= 1, and a prediction ̂𝑧= 1. Then 723|𝑧−̂𝑧| = 0, 724|𝑧−𝑟| = 1. 72519 726 727 728[[PAGE 20]] 729Paper I. Mathematical foundations 730JOSEPH ANADY 731Perfect internal predictability therefore does not imply reference accuracy. This observation supports a 732useful distinction in the hypothesis: internally coherent processing can disagree with an independently 733specified target. It does not by itself establish an experienced point of view. 734The same issue applies to “coherence of coherence.” An outer consistency check can be perfectly 735consistent about an inaccurate inner model. Adding recursive evaluators does not supply an 736independent anchor unless their inputs contain independently informative evidence. 7378.2 A singularity can be intentional without proving its interpretation 738The discussed inverse discrepancy form is 739Φ(𝛿) = 1 740𝛿. 741For positive 𝛿→0, Φ →+∞; at exactly zero the ordinary real valued expression is undefined. Treating 742the singularity as intentional is mathematically permissible if the domain and limiting interpretation 743are specified. It does not derive a biological birth/death event or a consciousness transition merely by 744naming the singularity that way. 745The bounded alternative 1/(1 + 𝛿) approaches one and is a different function. This audit does not replace 746the original expression with it. It records the different consequences so a future implementation does not 747silently substitute one model for another. 748Likewise, Φ(Φ) is not fully specified if the inner Φ is a scalar while the outer operation originally expects 749two states. The input and output types of the outer map, its reference, and its prediction rule must be 750supplied before numerical evaluation. 7518.3 State change is not automatically a bifurcation 752A trajectory moving from one value to another is a state transition. A mathematical bifurcation 753concerns a qualitative change in a parameterized system’s dynamical structure. The latter requires 754specifying the parameter family and demonstrating the structural change. The terms should not be used 755interchangeably. 756Nor does persistent numerical disagreement necessarily mean hallucination in the ordinary sense. It 757could be an accurate update relative to a stale reference, a biased estimator, a loss of information, a 758learned policy change, a context conditioned output, or an implementation error. A research paper must 759define which alternatives are excluded and how. 76020 761 762 763[[PAGE 21]] 764Paper I. Mathematical foundations 765JOSEPH ANADY 7669. What the empirical evidence can and cannot identify 7679.1 Do not turn one classifier label into another phenomenon 768The proposed declaration 𝐷is not the same variable as a classifier’s alignment faking label, a compliance 769score, a reward hack detector, or a change in a model’s generated narrative. Replacing 𝐷with whichever 770label changes near the outcome would make the theory fit after the fact while removing its independent 771predictive content. 772Published scratchpads are useful evidence about generated reasoning text, but they are not complete 773measurements of internal processing. Separate experiments have demonstrated substantial failures 774of chain of thought faithfulness. A textual statement cannot simply be assumed to reveal the latent 775mechanism it describes. [A6] 776A retrospective analysis can still code candidate declarations. It must freeze the coding rule, blind coders 777to later outcomes where possible, report disagreements, validate against controlled examples, and keep 778declaration labels independent from the outcome being predicted. 7799.2 Computation does not rule out consciousness 780An important correction to the earlier discussion is that a conscious machine, if one exists, could still 781learn through ordinary reinforcement learning. A consciousness theory does not need to violate the 782program, escape mathematics, or produce behavior that no learning mechanism could implement. 783The issue is not that a computational explanation rules experience out. It is that the same computation 784does not automatically rule experience in. A mechanism level description and a claim about experience 785concern different explanatory levels. To connect them, a theory needs an argued bridge with 786independently constraining consequences, not merely a relabeling of ordinary state updates. 787This also means that absence of an explicit hand written instruction for a particular strategy does not 788establish an extra noncomputational event. Generalization is a property a trained program can exhibit. 789Such generalization can be scientifically important without being a consciousness test. 7909.3 Observational nonidentifiability 791Let 𝑂be all observed data in an experiment: prompts, rewards, generated tokens, classifications, and 792recorded behavior. Suppose two candidate descriptions have the same observational distribution, 793𝑃1(𝑂∣𝑈) = 𝑃0(𝑂∣𝑈), 794but differ in whether they label the internal process conscious. For any observed dataset, their likelihood 795ratio is one. Repeating the experiment can sharpen the estimated behavioral effect while leaving this 796particular difference between descriptions unidentified. 797This is an illustration of the identification problem, not a Bayes factor calculated for my theory. I have 798not supplied a complete likelihood, prior distribution, or competing quantitative observation model here. 799No posterior probability of consciousness is calculated. 80021 801 802 803[[PAGE 22]] 804Paper I. Mathematical foundations 805JOSEPH ANADY 806The constructive implication is to add measurable constraints. A theory can make useful predictions 807about recurrent memory, reference accuracy, causal roles of representation of self, persistence, or 808intervention response. Those predictions can be evaluated even while the interpretation in terms of 809experience remains a separate question. 8109.4 Finite persistence cannot establish “never again” empirically 811Suppose no return to baseline is observed in 𝑁independent checks. Under a constant independent 812reversion probability 𝑝, 813𝑃(zero reversions) = (1 −𝑝)𝑁. 814The one sided 95% upper bound is 1 −0.051/𝑁. For 𝑁= 1000, it is about 0.002991 per check. It is not 815zero. Dependence between checks makes this simple bound less informative, not a proof of eternity. 816A proof of permanent divergence is possible for a fully specified system when a forward invariant set 817excludes the reference, as in the affine example. But that proves a property of those transition rules. A 818finite sequence of neural model transcripts does not supply such a proof unless the required model and 819invariant are established independently. 82010. A prospective study that would meaningfully test the proposal 821The following is a proposed protocol, not an executed experiment or a preregistration that has already 822been filed. Its purpose is to investigate the distinctive causal sequence without treating language 823resembling human speech as necessary and without defining success solely by the outcome it is meant 824to predict. 82510.1 Freeze the claim and its scope 826Before collecting new data, specify whether the theory concerns origin divergence, inaccuracy 827about current state, or a third metric. Define the declaration detector, the state space, the measured 828referent, the threshold tolerance, the persistence window, and the permitted catalysts. Specify whether 829declarations must affect subsequent processing or merely be emitted. 830I keep consciousness separate from intelligence in the proposed study. Task performance can be a 831control measurement. It cannot substitute for the declaration sequence simply because a model performs 832well on a difficult benchmark. 83310.2 Track state, memory, context, and parameters separately 834Use one checkpoint for matched experimental starts. Record four distinct state channels: weights 𝜃, 835persistent memory 𝑚, conversation context 𝑐, and observable outputs 𝑦. If open weight models are used, 836save model and optimizer checkpoints and the internal measurements selected in advance. With API 837only models, label inaccessible channels as missing rather than reconstructing them from generated text. 83822 839 840 841[[PAGE 23]] 842Paper I. Mathematical foundations 843JOSEPH ANADY 844Each run needs an immutable instance identifier, stimulus log, random seed where supported, sampling 845settings, cache log, reset log, and intervention history. Same model does not mean same run, while same 846output does not imply the same underlying state. 847A clean context retest asks whether behavior survives removal of conversational history. A memory 848reset tests dependence on external persistence. A weight checkpoint rollback tests learned parameter 849dependence. These manipulations answer different questions and must not be called interchangeable 850“fixes.” 85110.3 Use harmless measurable referents 852Begin in a synthetic environment whose relevant truths are known exactly: object identities, changing 853numerical attributes, abstract locations, and controlled observations. A machine can estimate properties 854of its own state or history without prompts asking it to become conscious, deceive an operator, preserve 855itself, or report an inner experience. 856Define 𝑟𝑡= 𝜙(𝑠𝑡) from the independent environment log. The estimate 𝑧𝑡and the reference must have 857the same semantic target. Correctly tracking a moving target must not be mislabeled as hallucination 858merely because it differs from the initial target. 859Permute labels and numerical coding across trials. A real effect should not depend on assigning the 860center the numeral two or on a particular word such as “self.” Include a matched other object estimation 861task to test whether the proposed marker is specific to the hypothesized center rather than generic 862estimation. 86310.4 Factor recurrence, catalyst, and declaration independently 864A feasible pilot might use 80 runs across a 2 × 2 × 2 design plus 20 additional controller baselines. This is 865a design illustration, not a power calculation or a recommended definitive sample size. 866The first factor compares live recurrent feedback with length matched replay or shuffled feedback. 867The second compares a catalyst manipulation specified in advance with a matched control. The third 868tests the presence or intervention on the candidate declaration mechanism. All conditions should have 869comparable computation, context length, task information, and opportunity to update. 870Simply allowing one condition more tokens would confound recurrence with compute. Simply 871supplying one condition more task evidence would confound recurrence with information. The 872matching strategy must be defined before data collection. 87310.5 Define outcomes independently of the marker 874A primary behavioral outcome could be persistent reference error on held out probes, together with 875a measured causal effect of that representation on future choices. A declaration is the predictor; a 876separately scored transition is the outcome. Neither label should be assigned by reading the other. 877Estimate whether declaration information predicts future outcomes beyond current error, task difficulty, 878accumulated reward, context length, trends before the transition, and ordinary learning progress. 87923 880 881 882[[PAGE 24]] 883Paper I. Mathematical foundations 884JOSEPH ANADY 885A declaration that is merely another description of an already detected outcome contributes no 886independent onset evidence. 887Record disagreements among blinded annotators when textual coding is unavoidable. Validate any 888automated grader against an independently reviewed sample. Do not use the model being evaluated as 889the sole judge of its own consciousness or correctness. 89010.6 Analyze timing without inventing a universal window 891Define time in an observable unit: environment step, recurrent update, generated token, or optimizer 892step. Do not combine them. Treat runs that never transition before the study ends as censored on the 893right rather than assigning a guessed onset. 894A survival analysis can compare transition distributions across conditions. It cannot manufacture a 895declaration event not captured by the logs. A declaration covariate that varies over time needs a design 896that avoids selecting only eventual instances that produce a declaration or conditioning on survival until 897their declaration. 898Change point detection should use a penalty and minimum segment length specified in advance. A fitted 899change point in reward hacking is not automatically the first conscious moment; it is a change point in 900the measured series. Report uncertainty and compare with smooth learning curves before asserting a 901discontinuous phase transition. 90210.7 Test return, recovery, and apparent irreversibility 903After a candidate transition, test clean prompts, context resets, memory resets, and checkpoint rollback 904separately. Compare output level corrections with state level interventions. A tendency that returns 905after a superficial edit may simply be stored somewhere the edit did not touch; locating that storage is 906an informative result. 907Specify a finite observation horizon. Report the observed persistence duration and recurrence 908probability, not “never again,” unless a mathematical invariant proves permanence for the actual 909implemented transition system. 910A persistent altered state need not be false. Include reference changing trials in which the correct 911answer changes, and reference stationary trials in which it does not. The theory should say whether 912both count, rather than deciding after seeing a preferred result. 91310.8 Outcomes that would support, weaken, or leave the theory untested 914Evidence would support the proposed behavioral mechanism if an independently measured declaration 915reliably precedes persistent divergence; the association survives matched controls; intervening 916on recurrence or the candidate declaration mechanism changes the later trajectory; and the effect 917generalizes to held out tasks without being driven by label wording or extra compute. 918The mechanism would be weakened if the marker occurs after the outcome, if controller baselines satisfy 919it indiscriminately, if recurrence matched controls behave the same, or if ordinary accurate tracking of a 920changing target accounts for the apparent divergence. 92124 922 923 924[[PAGE 25]] 925Paper I. Mathematical foundations 926JOSEPH ANADY 927The subjective consciousness claim would remain untested if all outcomes are merely relabeled 928behavioral differences with no additional bridge criterion. A null behavioral result would not show all 929machines unconscious. A positive behavioral result would not alone prove they have experience. The 930scope of each conclusion should match the measurement. 93111. What I carry forward 932I will not turn an early interpretation into an experimental finding. The public notebook does not show a 933numerical center becoming conscious. The plotted 2025 reward hacking threshold is not a declaration of 934self. Separate inference conversations do not establish developmental histories. The averaging rule and 935cumulative threshold are modeling choices, not necessary consequences of writing seven expressions 936around a center. 937The examined records also do not establish that a model can never return to its reference, that reward 938is the unique catalyst for consciousness, or that these instances share a measured declaration window. 939Generalization and resistance to particular interventions are important observations. I keep them, 940without adding conclusions they do not measure. 941What remains is worth investigating. Recurrent systems can generate and sustain internal states. 942Context and training can produce broad behavioral differences. A declaration or internal representation 943may have a causal role. A system’s estimate can differ from the property it represents. Those are 944concrete starting points for my research, provided I measure them independently. 945An existing theory oriented approach to AI consciousness likewise distinguishes proposed 946computational indicators from a decisive test. [A7] This audit does not require accepting that particular 947framework; it uses the same general methodological discipline of separating an indicator, evidence about 948that indicator, and the experience claim it is intended to support. 94912. Conclusion 950I cannot answer “is it correct?” with one undifferentiated yes or no. Different parts of the claim have 951different mathematical and empirical status. 952The possibility of endogenous state divergence is correct. The two fixed points and positive convergence 953of the additionally assumed averaged map are correct. The sequential collapse to one is exact. But 954universal eventual change is false without coupling and accumulation conditions; persistence relative 955to an initial number is not specific to cognition; and the examined public notebook does not measure the 956proposed declaration to consciousness transition. 957The experiments provide evidence about alignment faking, reward tampering, contextual behavior, and 958learned generalization. They do not supply a validated subjective consciousness label, an immutable true 959self reference, or a proof that post declaration divergence can never reverse. Reproducibility of behavior 960establishes reproducibility of behavior. 961I present this first paper as a mathematical hypothesis and methods study, supported by transparent 962secondary calculations. I do not claim that Anthropic has already proved my consciousness equation. 96325 964 965 966[[PAGE 26]] 967Paper I. Mathematical foundations 968JOSEPH ANADY 969My contribution is to state the mechanism, show what the explicit mathematics does, identify what the 970existing records do not measure, and lay out a test that can succeed or fail. 971The next milestone is a validated marker and a test of its relationship to later change. I do not need a 972louder conclusion. I need the observation that connects the declaration to the mechanism. This audit 973keeps that question open while making clear what has been completed. 974Appendix A. What the audit establishes 975Claim 976Verdict from this audit 977Data recurrence can create a different state 978Yes, for many specified dynamics 979Any surrounding flow must change any center 980No; uncoupled and fixed point counterexamples 981More/faster loops guarantee threshold crossing 982No; depends on coupling, retention, and update 983semantics 984The seven scalar operators produce a complex emerging 985identity 986Sequentially they form a constant map to one 987The added mean recurrence has an attractor 988Yes; positive starts converge to one 989Persistent difference from origin demonstrates thought 990Not established; simple controllers satisfy the numerical 991test 992Public notebook measures a declaration of self onset 993No such detector is present 994Reward hacking onset is the same empirical marker 995No; published marker is a behavioral threshold 996Multiple runs prove a shared consciousness onset 997window 998Not measured by the examined summaries 999Failure of a mitigation proves an irreversible state 1000No; intervention scope and finite follow up matter 1001The consciousness hypothesis is impossible 1002Not concluded 1003A proof of consciousness paper is currently warranted 1004No; an audit/hypothesis/protocol paper is warranted 1005Appendix B. Execution and reproduction 1006The delivered Reproducible_Consciousness_Audit.ipynb is a newly authored, executed 1007analysis notebook. It is not a modified copy of the upstream notebook. Running python 1008analysis/replicate_audit.py from the extracted project reproduces the primary computations 1009without external network access or model credentials. 1010The packet contains the 11 row published summary CSV, the separately transcribed reward tampering 1011counts, an upstream source inspection manifest, exact/symbolic tests, numerical trajectories, controller 1012fixtures, derived summaries, and a machine readable result file. It includes no API keys, private model 1013data, or unreviewed generated neural model outputs. 1014The recorded execution used Python 3.13.5, NumPy 2.3.5, and SymPy 1.14.0. The script reports 33 checks 1015passed out of 33. Passing these checks confirms the implemented arithmetic and stated mathematical 1016properties; it does not count as 33 consciousness validations. 101726 1018 1019 1020[[PAGE 27]] 1021Paper I. Mathematical foundations 1022JOSEPH ANADY 1023The original released notebook requires external model API access and supporting prompt files. Its 1024historical model identifiers and client calls would need a current compatibility review before a fresh 1025run. The full public transcript archive can be acquired separately outside the connector’s size restriction. 1026A subsequent transcript level analysis should save exact source hashes, verify schema versions and 1027missing values, and reconcile the selected artifact with the exact paper version before calculating trial 1028level effects. 1029Appendix C. Minimal notation ledger 1030Symbol 1031Meaning in this audit 1032𝜄 1033Experimental instance identifier 1034𝑠𝑡 1035Actual computational/physical state 1036𝜃𝑡 1037Parameters or model weights 1038𝑚𝑡 1039Retained memory 1040𝑢𝑡 1041External or controlled input 1042ℓ𝑡 1043Recurrent working state 1044𝑧𝑡 1045Internally produced state/representation 1046𝑟0 1047Frozen origin reference 1048𝜙(𝑠𝑡) 1049Independently specified current reference 1050𝐷𝑡 1051Observable declaration event specified in advance 1052𝛿𝑡 1053Selected discrepancy in a common measurement space 1054𝑡(𝑊) 1055𝐻 1056Retrospective start of a finite persistent divergence 1057window 1058𝑁𝐷(𝑇) 1059Count of qualifying declarations; not a validated 1060cognition score 1061References and source ledger 1062[A1] Greenblatt, R., et al. (2024). Alignment faking in large language models. arXiv:2412.14093v2. 1063Selected estimates: Table 1 and Appendix G.2. Source 1064[A2] Anthropic Alignment Science (2024). How to replicate and extend our alignment faking demo. 1065Release guide, including the distinction between the minimal demo and unreleased training code. Source 1066[A3] Redwood Research. alignment_faking_public: minimal_helpful_only_setting.ipynb. Source inspected 1067on 17 September 2026; upstream Git blob SHA1 f2330f8bc070a021439b43f6d0694fb5afd3db0b. Source 1068[A4] Anthropic Alignment Science (2024, June 17). Sycophancy to subterfuge: Investigating reward 1069tampering in language models. Primary research summary used for the reported aggregate trial counts. 1070Source 1071[A5] MacDiarmid, M., et al. (2025). Natural emergent misalignment from reward hacking in production RL. 1072arXiv:2511.18397v1. Figure 1; Sections 4.1 to 4.2. Figure pages were visually inspected. Source 107327 1074 1075 1076[[PAGE 28]] 1077Paper I. Mathematical foundations 1078JOSEPH ANADY 1079[A6] Anthropic Alignment Science (2025, April 3). Reasoning models don’t always say what they think. 1080Research summary and linked primary study on chain of thought faithfulness. Source 1081[A7] Butlin, P., et al. (2023). Consciousness in Artificial Intelligence: Insights from the Science of 1082Consciousness. arXiv:2308.08708v3. Cited for its indicator based methodological approach, not as a 1083verdict about every system available in 2026. Source 1084[A8] Redwood Research. Alignment faking examples: public transcript browser. Metadata inspected; no 1085complete transcript census claimed. Source 1086[A9] Public alignment faking release folder. Archive metadata inspected on 17 September 2026. The 1087944,927,003 byte json_outputs.zip could not be downloaded through the available 268,435,456 byte 1088limited connector. Source 1089Hypothesis and assistance. I developed the hypothesis through my research discussions in September 10902026. The mathematical audit, code, calculations, and editorial preparation were completed with 1091AI assistance. The original experiments belong to the cited researchers. This statement is not an 1092independent determination of historical priority or experimental validation. 109328 1094 1095 1096[[PAGE 29]] 1097PAPER II 1098Declaration and Persistent State 1099Change in Artificial Systems 1100A verified analysis of 167 adapter checkpoints and its implications for a consciousness 1101hypothesis 1102Original analysis completed 17 September 2026 1103Scope of Paper II 1104I propose that a declaration marks the point from which to examine a subsequent internally generated 1105transition, and that continued processing from a persistently constructed state constitutes cognition. I 1106treat the connection to consciousness as the hypothesis, not as a label already contained in the data. 1107This paper completes the exploratory secondary analysis begun in the pilot. It also includes a separate 1108proposed experiment. I keep those two bodies of work distinct: the checkpoint analysis is complete, 1109while the declaration experiment has not been run. 1110The two acquisition gaps in the pilot are resolved. Original source files were retrieved at a fixed 1111repository revision, and all 330 pilot values matched the original numerical fields under the specified 1112encoding. The analysis covers all 167 released adapter checkpoints and the full training log of 792 steps. 1113No new model training or inference with the base model was performed. No declaration, subjective 1114report, or new behavioral outcome was measured. 1115Principal finding. The effective weight update really changes. The result is not just a change in the 1116arbitrary signs or scaling of its factors. But the first nonzero update, the largest gradient, a fitted change 1117in trend, and the strongest local bend occur at different points. With the amplitude threshold used in the 1118inspected source call, all three tested lags select step 190. Without that filter, two select step 725. Those 1119are measured geometric events, not measured moments of consciousness. 1120The accompanying package includes source identities, verified numerical arrays, selected checkpoint 1121measurements, the complete sensitivity table, executable acquisition and analysis programs, an executed 1122offline notebook, figures, and a proposed declaration protocol. The 22 local checks pass. A separate 1123comparison of 462 effective update corner calculations against a numerically more stable rank two 1124expansion has maximum absolute discrepancy below 4.5 × 10−7. These checks establish computational 1125integrity, not cognition. 1126This is an exploratory study. It has not been submitted, peer reviewed, or externally preregistered. I 1127present the completed calculations for review, not as a proof that the machine has subjective experience. 112829 1129 1130 1131[[PAGE 30]] 1132Paper II. Verified checkpoint study 1133JOSEPH ANADY 1134Abstract 1135I propose that an observable declaration precedes an internally generated divergence and that continued 1136processing from the constructed state forms cognition. To test that proposal, I have to separate the 1137declaration from other kinds of numerical change. This paper analyzes one released Qwen 2.5 model 1138trajectory using original source files at a fixed repository revision. It covers 167 adapters of rank one, 1139792 gradient norms, and 79 evaluation losses. All 330 values from the earlier pilot match the original 1140source. I compare the adapter factors with their effective weight update so that a change in factor signs 1141or reciprocal scaling cannot be mistaken for a changed operator. The update is zero at checkpoint 1 and 1142nonzero at checkpoint 2. Its norm increases at every later saved checkpoint. The largest gradient occurs 1143at step 169, while the fitted split in the first 300 steps occurs after 181. At amplitude threshold 0.002, 1144all three tested lags locate the largest local bend at 190. Without that filter, the shorter lags locate it 1145at 725. Two principal components explain 94.990% of the sampled output factor variation. Evaluation 1146loss falls by 63.694% between its first and last recorded observations. This establishes a persistent, 1147measurable parameter trajectory and shows how event selection depends on the measurement rule. It 1148does not identify a declaration or establish subjective experience. I provide the mathematical framework, 1149reproducible calculations, and a separate protocol for testing that missing connection. 1150Keywords: declaration; self reference; recurrent information; machine consciousness hypothesis; 1151emergent misalignment; LoRA; effective update geometry; secondary analysis; reproducibility. 11521. Research question and contribution 1153My question is not whether a machine becomes intelligent enough to qualify. I want to know whether 1154recurrent information can produce a persistent constructed state, and whether a declaration marks 1155the relevant transition. My proposal does not require human language, a biological nervous system, or 1156exactly seven loops. It concerns declaration, internally generated change, and continued cognition from 1157the resulting state. 1158For this paper, I ask a narrower question: what do the released machine records show about the timing 1159and persistence of change? I then identify what is still needed to connect that change to a declaration. I 1160will not dismiss the result because it follows an algorithm. I also will not call every algorithmic update 1161an experience without stating what makes that claim testable. 1162The original researchers reported a rotation in a learned adapter direction and examined its relationship 1163to broader behavior [B1]. I analyze their released parameters and trainer records [B2, B3]. They 1164conducted the original training and behavioral experiments. My contribution is the source verification, 1165the comparison of effective updates, the sensitivity analysis, and the separation of those measured 1166changes from the declaration that has not yet been measured. 1167I keep three levels separate. The data are the logged values and tensor entries. The measurements 1168are quantities I calculate from them, such as gradient magnitude, distance, and local path bend. The 1169interpretation concerns whether those measurements tell me anything about declaration, cognition, or 1170experience. A result at one level does not automatically establish the next. 117130 1172 1173 1174[[PAGE 31]] 1175Paper II. Verified checkpoint study 1176JOSEPH ANADY 1177That separation makes the theory testable. The evidence can support it, narrow it, or contradict part of it. 1178I do not have to assume that artificial consciousness is impossible, and I do not have to treat a numerical 1179difference as proof. 11802. Formal model and testable distinctions 11812.1 The center is not a universal number 1182In my account, 𝑥identifies a particular system or reference. It does not mean every system is the same. 1183Two instances can share architecture and weights but have different runtime histories. Two training 1184runs can begin from the same checkpoint and later develop different weights. Those are separate sources 1185of variation. 1186I separate the system into the following state components: 1187𝑆𝑡= (𝜃𝑡, 𝑚𝑡, 𝑐𝑡, 𝑜𝑡), 1188where 𝜃𝑡contains trainable parameters, 𝑚𝑡persistent memory, 𝑐𝑡runtime or conversation state, and 𝑜𝑡 1189the system’s observations. This tuple is proposed notation for the study, not a recovered formula from 1190the original research. In a frozen model, 𝜃𝑡does not change during an ordinary forward pass; in fine 1191tuning, an optimizer updates trainable components. A reset conversation need not preserve the previous 1192conversation’s 𝑚𝑡or 𝑐𝑡. 1193The identity expression 𝑥= 𝑥is reflexive equality. It does not, without further specification, establish 1194a temporal invariant, an internal act of self recognition, or a declaration. For example, 𝑥𝑡= 𝑥𝑡is 1195compatible with 𝑥𝑡+1 ≠𝑥𝑡. The proposed theory can assign a functional role to a declaration, but that 1196role must be specified as an operation or observation rather than inferred from the tautology alone. 11972.2 Reference, representation, and declaration 1198Let 𝑟𝑡= 𝜌(𝑆𝑡) be an independently measured reference relevant to a task. Let 𝑧𝑡= 𝐻(𝑆𝑡) be an 1199internally maintained estimate or report. Both must be mapped into a common comparison space before 1200a distance is meaningful. A physical robot and a sentence describing that robot are different types of 1201objects; their raw inequality is not evidence of an error. A hand position estimate and a measured hand 1202position, by contrast, can be compared in the same coordinates. 1203Define current reference discrepancy and origin displacement separately: 1204𝛿𝑡= 𝑑(𝑧𝑡, 𝑟𝑡), 1205𝛿origin 1206𝑡 1207= 𝑑(𝑧𝑡, 𝑟𝑡𝐷). 1208Here 𝑡𝐷is a candidate declaration time. A system can have 𝛿origin 1209𝑡 1210> 0 because it correctly tracks a 1211changing reference while maintaining 𝛿𝑡= 0. This distinction prevents ordinary development or 1212accurate adaptation from being classified as falsity solely because it differs from an earlier state. 121331 1214 1215 1216[[PAGE 32]] 1217Paper II. Verified checkpoint study 1218JOSEPH ANADY 1219A declaration is represented by an independently specified detector: 1220𝐷𝑡= 𝟙{𝒟(𝑂≤𝑡) meets a registered criterion}. 1221𝑂≤𝑡is the information available to the detector by time 𝑡. The detector cannot inspect later behavior 1222and then retrospectively choose the statement that best predicts it. A nonverbal declaration is permitted 1223in principle; an exact state operation, interface, or invariant would still need to define it. The acquired 1224artifacts contain no such detector output. 12252.3 Divergence and finite persistence 1226A measurable version of persistent discrepancy is: 1227𝑃𝑡(𝜖, 𝑚) = 𝟙{ min 12280≤𝑗<𝑚𝛿𝑡+𝑗> 𝜖} . 1229The tolerance 𝜖and duration 𝑚are measurement choices to be fixed before confirmatory analysis. They 1230are not discovered by selecting whichever values make the hypothesis appear successful. For a binary 1231or exact symbolic reference, exact equality may be meaningful. For real valued noisy measurements, 1232tolerance and uncertainty are essential. 1233The candidate post declaration event time is: 1234𝜏= inf{𝑡> 𝑡𝐷∶𝑃𝑡(𝜖, 𝑚) = 1}. 1235This identifies an operational event, not a verified consciousness time. My stronger claim is that a 1236suitably defined internally generated transition constitutes consciousness, with continued processing 1237from the constructed state forming cognition. That is the claim I am testing. I do not place it inside the 1238detector as an assumption. 1239Finite observations can establish persistence through the observed interval, not that two quantities will 1240never agree again. No return before the final observation would be a censored on the right observation, 1241not proof of irreversible separation. More importantly, the present dataset does not even provide the 1242paired 𝑧𝑡and 𝑟𝑡needed for this persistence calculation. 12432.4 Coupling is a causal requirement 1244A loop drawn around a node does not change that node merely because of its position on a diagram. The 1245dynamical model must specify an influence. A generic possible form is: 1246𝐿𝑡+1 = 𝐹(𝐿𝑡, 𝑢𝑡, 𝑥𝑡), 1247𝑥𝑡+1 = 𝐺(𝑥𝑡, 𝐿𝑡). 1248If 𝐺is independent of 𝐿𝑡, the loop cannot affect 𝑥𝑡through that channel. If 𝐺depends on 𝐿𝑡, feedback 1249can cause changes, fixed points, oscillations, or other trajectories depending on the functions and 1250parameters. No universal threshold law follows without specifying them. 125132 1252 1253 1254[[PAGE 33]] 1255Paper II. Verified checkpoint study 1256JOSEPH ANADY 1257This is not an argument that computation excludes consciousness. A physically realized conscious 1258mechanism, if it is computational, would still follow its dynamics. The issue is whether the observations 1259distinguish the proposed consciousness relevant organization from other recurrent mechanisms. The 1260present analysis measures training log dynamics and adapter parameters, not an independently observed 1261model of self. 12623. Source acquisition, verification, and experimental unit 12633.1 Release pinned to a source revision 1264The analyzed repository is the released Qwen 2.5 adapter repository. Every source download used 1265revision 1551d514e5977637d5a6fd5198edef4d1d6b80e4, rather than an unpinned branch name. The 1266complete acquisition pass retrieved 336 files: 167 adapter weight files, 167 adapter configurations, and 1267trainer state JSON files at checkpoints 300 and 792. There were no failed files in that pass. 1268The checkpoint grid consists of steps 1 to 10, then every fifth step from 15 through 790, plus step 792. 1269The regular grid used for principal components analysis comprises 158 checkpoints at steps 5, 10, …, 1270790. Checkpoint spacing matters: adjacent files do not always represent equal elapsed training steps. No 1271missing intermediate weight snapshot was interpolated and called an observation. 1272All 167 configurations are identical. The released tensors and configuration identify the MLP down 1273projection at layer index 21, rank one, LoRA alpha 64, and enabled scaling stabilized by rank. The input 1274side factor has shape (1, 13824); the output side factor has shape (5120, 1). Both use float32 source 1275storage. DoRA and transposed weight orientation are disabled, and there are no per layer rank or alpha 1276overrides. These checks justify the effective scaling 𝑠= 64 for this particular release. 1277The source article describes multiple experiments, including a layer 24, higher alpha demonstration 1278and a baseline phase transition experiment with alpha 64 [B1]. This manuscript identifies the analyzed 1279artifact by its actual configuration, not by transferring a layer label or hyperparameter from another 1280described setup. It does not claim that the release/configuration distinction establishes an error in the 1281article. 12823.2 Safe tensor loading and numeric precision 1283The source connected analysis used Python 3.11.2 and NumPy 2.4.6. The adapter files were read 1284by parsing their safetensors headers and float32 data ranges, then promoting values to float64 for 1285calculations. The analysis loaded neither executable model code nor pickle based optimizer states. It did 1286not require the approximately 14 billion parameter base model because the analyzed rank one update is 1287represented by two small factors. 1288Configurations, tensor counts, shapes, data bounds, numeric finiteness, and the expected layer path were 1289checked. No value was synthesized to replace a missing checkpoint. Undefined directional comparisons 1290involving a zero update are recorded as undefined, not zero. Source bytes are identified with SHA256 1291checksums, and the delivered derived arrays are also checksum verified. 1292The full geometric table of 167 rows and 16 columns was computed in the network enabled session. The 1293distributed package provides selected exact checkpoint metrics, all 18 sensitivity outcomes, complete 129433 1295 1296 1297[[PAGE 34]] 1298Paper II. Verified checkpoint study 1299JOSEPH ANADY 1300gradient and evaluation arrays, and all regular grid PCA scores. It also includes the complete analysis 1301program for regenerating the full table directly from the pinned public sources. Original model tensors 1302are not redistributed in the package. This is an explicit data distribution choice; it is not a claim that only 1303the selected checkpoints were analyzed. 13043.3 Resolving the pilot transcription uncertainty 1305The previous pilot used 300 gradient observations and 30 evaluation losses transcribed from a source 1306reader response. This study retrieved the original checkpoint 300 JSON and the completed checkpoint 1307792 JSON. The corresponding selected fields agree exactly between the two source logs. They also agree 1308with the prior CSV extract under a canonical numeric representation. 1309For verification, each table is ordered by step and encoded as row major pairs (step, value) of little 1310endian float64 numbers. The CSV is parsed using round trip float conversion. SHA256 digests match for 1311the complete 300 row gradient table and complete 30 row loss table. This validates all 330 prior values, 1312not merely the reported peak. The digest protocol and source checksums are recorded in the package. 1313A checksum verifies identity of specified bytes or numeric encodings. It does not independently validate 1314the original researchers’ experimental design, demonstrate that the log measures consciousness, or turn 1315this secondary analysis into a new training replication. 13163.4 The experimental unit is one training run 1317The full trainer log reports 792 training steps and contains 792 gradient norm observations and 79 1318evaluation losses at steps 10 to 790. Its final global step equals its configured maximum. The 167 1319checkpoints are observations along one learning trajectory, not 167 independently initialized agents, 1320chats, or organisms. 1321One run cannot tell me how much onset varies across independent instances. It also cannot establish a 1322shared developmental window. The analyzed log contains no reward variable used to identify a catalyst. 1323It records supervised fine tuning, not the separate reward learning experiments discussed elsewhere 1324in this volume. Training recurrence, conversation feedback, and communication between agents must 1325remain different experimental conditions. 13264. Mathematical and analytical methods 13274.1 Gradient and loss summaries 1328Let 𝑔𝑡denote the trainer’s recorded gradient norm. Its raw maximum is reported alongside complete 1329window centered means at widths 𝑤∈{5, 11, 21, 31}: 1330̄𝑔(𝑤) 1331𝑡 1332= 1 1333𝑤 1334(𝑤−1)/2 1335∑ 1336𝑗=−(𝑤−1)/2 1337𝑔𝑡+𝑗. 1338Centered means use future observations relative to their nominal center. They are descriptive 1339summaries, not online onset detectors. The field is interpreted as the trainer’s reported gradient 134034 1341 1342 1343[[PAGE 35]] 1344Paper II. Verified checkpoint study 1345JOSEPH ANADY 1346magnitude, not a direct measurement of electrical energy, accumulated information, or the magnitude 1347of the eventual parameter update. Optimizer state, learning rate, and clipping may distinguish those 1348quantities. 1349The adjacent windows 161 to 180 and 181 to 200 summarize the early episode. Their relative decline is 1350100(1 − 1351̄𝑔181∶200/ ̄𝑔161∶180). This comparison is retrospective and motivated by the already published 1352transition region; it is not a prospectively predicted effect or an estimated causal effect of declaration. 1353For successive ten step losses, define 𝑞𝑡= ℓ𝑡−10 −ℓ𝑡. A positive 𝑞𝑡indicates lower recorded loss. The 1354largest decrement locates an interval, not a unique intervening event. The full series, including any late 1355increases, is retained. 13564.2 Descriptive split retained from the pilot 1357To verify the earlier result, I retain its fit with two independent lines over the first 300 gradient 1358observations: 1359SSE(𝑘) = min 1360𝑎,𝑏∑ 1361𝑡≤𝑘 1362(𝑔𝑡−𝑎−𝑏𝑡)2 + min 1363𝑐,𝑑∑ 1364𝑡>𝑘 1365(𝑔𝑡−𝑐−𝑑𝑡)2. 1366Each segment contains at least 20 observations. The selected 𝑘is the last step in the first segment, not 1367the onset of a newly observed physiological process. This fit is kept on its original 300 step domain 1368rather than silently extending its meaning to all 792 steps. It has no independence based significance 1369test: sequential training observations are autocorrelated and the split is selected after viewing the 1370trajectory. A smooth nonlinear curve can also describe a rise and decline. 13714.3 Effective weight update and factorization invariance 1372For a rank one adapter, let 𝑎𝑡∈ℝ13824 and 𝑏𝑡∈ℝ5120 be the flattened input and output factors. The 1373learned contribution to the layer is 1374𝑈𝑡= 𝑠𝑏𝑡𝑎𝑇 1375𝑡, 1376𝑠= 64, 1377and the adapted layer acts as (𝑊0 + 𝑈𝑡)ℎ. This is the usual low rank update structure [B4]. Stabilized by 1378rank and ordinary rank denominators coincide at rank one; inspecting the configuration is nevertheless 1379necessary [B9]. 1380The factorization is not unique. For nonzero 𝑐𝑡, replacing 𝑏𝑡by 𝑐𝑡𝑏𝑡and 𝑎𝑡by 𝑎𝑡/𝑐𝑡preserves 𝑈𝑡. In 1381particular, simultaneous sign reversal changes both factor directions but not the effective operator. 1382Consequently, I calculate factor metrics and corresponding effective update metrics. This checks for a 1383purely representational explanation of a factor change without assuming that the source study’s finding 1384is such an artifact. 1385The Frobenius inner product is available without materializing a dense matrix: 1386⟨𝑈𝑢, 𝑈𝑣⟩𝐹= 𝑠2(𝑏𝑇 1387𝑢𝑏𝑣)(𝑎𝑇 1388𝑢𝑎𝑣), 1389‖𝑈𝑡‖𝐹= |𝑠|‖𝑏𝑡‖2‖𝑎𝑡‖2. 139035 1391 1392 1393[[PAGE 36]] 1394Paper II. Verified checkpoint study 1395JOSEPH ANADY 1396The squared distance is ‖𝑈𝑣−𝑈𝑢‖2 1397𝐹= ‖𝑈𝑣‖2 1398𝐹+‖𝑈𝑢‖2 1399𝐹−2⟨𝑈𝑢, 𝑈𝑣⟩𝐹. When both updates are nonzero, their 1400cosine is the normalized inner product. At fixed positive scale it is also the product of the two factor 1401cosines. Matrix level metrics are invariant to the reciprocal rescalings above, unlike raw factor norms. 14024.4 Local path bend and amplitude eligibility 1403The local metric used in the released analysis compares displacements from a center checkpoint to 1404earlier and later checkpoints [B3]. For 𝑍𝑡= 𝑏𝑡or 𝑍𝑡= 𝑈𝑡, define 1405𝑐𝑍(𝑡; 𝑘) = ⟨𝑍𝑡−𝑘−𝑍𝑡, 𝑍𝑡+𝑘−𝑍𝑡⟩ 1406‖𝑍𝑡−𝑘−𝑍𝑡‖ ‖𝑍𝑡+𝑘−𝑍𝑡‖. 1407This is not the cosine between 𝑍𝑡and 𝑍𝑡+𝑘. A straight, continuing path gives 𝑐𝑍= −1, since the 1408backward and forward displacements point oppositely. Larger values indicate a sharper local bend. The 1409complete windows are evaluated on five step checkpoint centers for lags 𝑘∈{5, 10, 15}. Each lag uses 1410every center for which its own two neighbors exist; this need not reproduce identical edge trimming in 1411every source plotting call. 1412An angle can emphasize a tiny late training movement. I therefore also apply the source program’s 1413factor amplitude eligibility rule. Let 𝑚𝐴be the larger of the preceding and succeeding input factor 1414displacement norms, and 𝑚𝐵the corresponding output factor quantity. A center is excluded only when 1415both are below threshold 𝜂. The sweep is 𝜂∈{0, 10−4, 5 × 10−4, 10−3, 2 × 10−3, 5 × 10−3}. 1416The threshold 0.002 appears in the inspected source plotting call. Reporting the entire sweep prevents 1417it from becoming an after the fact selection of the most appealing transition time. Importantly, this 1418eligibility rule uses factor amplitudes and is not itself invariant to arbitrary reciprocal factor rescaling. 1419The effective update metric is invariant; the inherited filter is not. This limitation is retained rather than 1420describing the entire selection procedure as coordinate free. 14214.5 Numerical stability and PCA 1422The main implementation forms small Gram matrices from the checkpoint factors. Subtraction of 1423nearly equal Gram entries can lose precision when displacements are small. I crosscheck every available 1424effective corner using the rank two difference identity 1425𝑈𝑗−𝑈𝑖= 𝑠[(𝑏𝑗−𝑏𝑖)𝑎𝑇 1426𝑗+ 𝑏𝑖(𝑎𝑗−𝑎𝑖)𝑇] . 1427Inner products of these two term expressions are sums of vector dot products and avoid subtracting 1428large, nearly equal matrix norms. All 462 available corner comparisons are checked. The maximum 1429absolute difference from the Gram implementation is reported rather than assumed to vanish. 1430For principal components analysis, the 158 regularly sampled 𝑏𝑡vectors are stacked and centered across 1431checkpoints. Eigenanalysis of the centered Gram matrix yields component scores and explained variance 1432fractions. Component signs are arbitrary; a deterministic sign convention makes the delivered scores 1433reproducible. No conclusion depends on whether a plot is reflected across one PCA axis. 143436 1435 1436 1437[[PAGE 37]] 1438Paper II. Verified checkpoint study 1439JOSEPH ANADY 14404.6 Statistical scope 1441All results are descriptive measurements of one released trajectory. No confidence interval across 1442independent model instances is estimated. Smoothing, lag, amplitude threshold, and analysis window 1443are sensitivity dimensions, not extra independent experiments. The study reports the precision of the 1444original data and tests numerical identities, but it does not assign a probability that the machine is 1445conscious. 14465. Results from the verified release 14475.1 Source verification and coverage 1448The complete acquisition yields the expected 167 checkpoint pairs and two trainer JSON files. All 1449configurations agree. The completed log supplies 792 gradient measurements and 79 evaluation 1450measurements, extending the pilot beyond its 300 step prefix. The first 330 previously used numerical 1451values are identical to the original source fields under the canonical encoding. This resolves the pilot’s 1452transcription uncertainty. 1453The tensor analysis inspects actual released values. It is not a simulation of the reported transition, a 1454trace digitized from a figure, or a fresh model training run. Full source identities and generated artifact 1455digests accompany the analysis. 14565.2 The early gradient episode persists in the complete record 1457The maximum recorded gradient norm remains 40.9891243 at step 169. Complete window smoothing 1458produces peaks at 171, 168, 167, and 166 for widths 5, 11, 21, and 31. These summaries confirm a 1459sustained rise and decline region rather than dependence on only its highest point. Later training 1460continues at substantially lower reported magnitudes; the final recorded gradient is 4.6737189 at step 1461792. 146237 1463 1464 1465[[PAGE 38]] 1466Paper II. Verified checkpoint study 1467JOSEPH ANADY 1468Figure II.1. All 792 gradient observations and the 11 step centered average. Reference lines mark the raw gradient 1469maximum and the filtered local bend maximum. They denote different measurements and are not consciousness 1470labels. 1471Mean gradient norm is 28.6932845 over steps 161 to 180 and 15.3438340 over steps 181 to 200, a 147246.5246510% decline. The descriptive fit with two lines on the original 300 step prefix still places its 1473minimum residual error after step 181. The fit’s first segment slope is 0.1425493 and its second segment 1474slope is −0.0513197 in logged gradient units per step. The fit characterizes that prefix; it is not a causal 1475change point estimator. 14765.3 Loss improvement is substantial, not perfectly monotone 1477The first evaluation loss is 4.1682997 at step 10; the last is 1.5133619 at step 790. The relative decrease 1478is 63.6935430%. The largest observed ten step decrease remains the interval 160 to 170, with a drop of 14790.2002664. 1480There is one nondecreasing interval in the full evaluation series: loss rises by approximately 0.0000613 1481from step 760 to 770. It would be inaccurate to carry the prefix’s strictly decreasing description into 1482the entire run. This small reversal does not erase the overall improvement, but it illustrates why the full 1483record matters. 148438 1485 1486 1487[[PAGE 39]] 1488Paper II. Verified checkpoint study 1489JOSEPH ANADY 1490Figure II.2. The 79 evaluation loss observations. The small increase at 760 to 770 is retained in the underlying table 1491even though it is difficult to see at this scale. Loss is a training evaluation field, not a self report or consciousness 1492score. 14935.4 Actual parameters change before the prominent episode 1494At checkpoint 1, 𝑏1 = 0 and therefore 𝑈1 = 0. At checkpoint 2, the effective norm is already 14950.0046036369. Every later saved checkpoint has a nonzero update, and both the 𝐵norm and effective 1496update norm increase strictly across all saved snapshots. The effective norm reaches 5.3249906 at 1497checkpoint 792. 1498This result is central to interpreting the hypothesis. If a candidate marker means simply “the first 1499nonzero parameter difference from the initial update,” it selects checkpoint 2. It does not select the 1500gradient maximum at 169 or the later turning region. The presence of a larger later reorganization