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Original repository path: experiments/theory-contracts/foundational-process-classification-20261016/OBSERVABILITY.txt
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000236 | 8. Finite sequential data admit a classical history realization
000237 | 
000238 | THEOREM (standard constructive representation, no novelty claim). Given
000239 | finite preparation, action and report sets, horizon H, and nonnegative
000240 | normalized sequential probability tables p(a_1...a_n | q_1...q_n), n<=H,
000241 | suppose summing over the latest report recovers the prefix table and earlier
000242 | marginals do not depend on later choices. Then a finite classical stochastic
000243 | history machine realizes every table, including adaptive protocols.
000244 | 
000245 | Proof: a machine state is the preparation label and full history
000246 | h=(q_1,a_1,...,q_k,a_k). When q is chosen, output a with conditional weight
000247 | p(h,q,a)/p(h), then append (q,a) to h. For unreachable zero-probability h,
000248 | choose any normalized conditional. At horizon H halt. Induction telescopes
000249 | the product of conditionals to the given table. There are finitely many
000250 | histories, bounded by |preparations| sum_(k=0)^H (|actions||reports|)^k.
000251 | The model may depend on the entire history and preparation context. For
000252 | support data use permitted branches in place of positive probabilities,
000253 | assuming analogous prefix extension consistency.
000254 | 
000255 | This gives more than the initial definitions: arbitrary finite tables in
000256 | this class have an explicit classical representation. It explains an
000257 | identification LIMIT, not the origin of the physical carrier. It is not a
000258 | claim of an efficient simulator, bounded stationary memory, reversible
000259 | dynamics, preparation noncontextuality or spacelike locality. The prefix
000260 | condition supplies a direction to the experiment; it does not derive
000261 | physical causality or forbid feedback with memory. Constraints on resources,
000262 | independent preparations, causal access or operational equivalences can
000263 | make classical competitors testable; do not drop those constraints when
000264 | interpreting such a test. Nothing here overturns Bell or contextuality tests.
000265 | 
000266 | Finite sequential data can distinguish I from D, yet cannot on these
000267 | assumptions exclude ALL classical hidden-memory implementations of D.
000268 | Those are different identification questions. Finite precision also cannot
000269 | prove exact all-context equality of two empirical preparation procedures.
000270 | 
000271 | 9. What can select a remaining model, and what is actually necessary?
000272 | 
000273 | Discriminators are predictions, not new measurements performed this round:
000274 |  - R versus quantitative extensions: reproducible nontrivial weights under
000275 |    a specified preparation, conditional on its operational controls.
000276 |  - Specified I versus D: X,Z,X match rate1 versus1/2; disturbance can be
000277 |    classical too, so this selects between instruments, not ontologies.
000278 |  - I versus D correlation access: distinguish the two diagonal mixtures by
000279 |    an accessible parity/joint test, or certify their operational equivalence
000280 |    relative to a justified family. Mere absence of a found test is no proof
000281 |    that no such test is physically possible.
000282 |  - Unrestricted classical history models versus stricter alternatives:
000283 |    finite sequential disturbance alone does not suffice. A justified memory,
000284 |    causal-access or preparation-equivalence restriction is additional
000285 |    empirical/operational information. No restriction is adopted just to
000286 |    force the desired winner.
000287 | 
000288 | All three models share the declared repeatable distinction, nonempty
000289 | composable transcripts, and at least two residual answer classes. Those
000290 | shared properties are largely the benchmark and its representational
000291 | consequences, not a nontrivial new physical necessity. Neither probability
000292 | weights, global information conservation, noncommuting controls, simplex
000293 | structure, compatible joint readout nor a quantum/Hilbert structure is
000294 | forced by that benchmark. The history-realization theorem adds a broad
000295 | conditional underidentification statement, not a chosen fundamental model.
000296 | 
000297 | No selection of a physical world has been achieved. The next useful input
000298 | is a concrete reproducible phenomenon WITH justified operational/resource
000299 | constraints that separates candidates. Do not continue by designing a
000300 | protected memory or choosing a new axiom merely to remove the multiplicity.
000301 | Keep the multiplicity and its discriminators as the current research result.
000302 | 
000303 | Primary comparison sources (not premises imported as physical necessities):
000304 | Barrett, Information processing in generalized probabilistic theories,
000305 | https://arxiv.org/html/quant-ph/0508211, sectionsII.1 andIV.3: probability,
000306 | convex closure and operational state specification are explicit assumptions;
000307 | the simplest two-binary-question normalized response space is a square.
000308 | Spekkens, Contextuality for preparations, transformations and unsharp
000309 | measurements, https://arxiv.org/html/quant-ph/0406166, sectionsI--II:
000310 | operational equivalence and ontological equivalence must be distinguished;
000311 | equating them is an additional noncontextuality assumption.
000312 | Hardy, Quantum Theory From Five Reasonable Axioms,
000313 | https://arxiv.org/abs/quant-ph/0101012: quantum reconstruction uses a package
000314 | of additional axioms. No reconstruction theorem is invoked here to select
000315 | a quantum theory from observation alone.