Follow the published construction into its registered claims, then explore reviewed research questions, attempts and the connections that remain open.
Observable reality: minimal conditions and counterworlds
Which operational conditions are independently required, which inequivalent models satisfy them, and which observable distinctions remain for selection?
Conditional representation theorem without novelty claim, not efficient or local physical simulation. History/preparation dependence and growing memory are allowed; Bell, noncontextuality or resource witnesses under independently justified extra conditions are not invalidated. Protocol direction is assumed, not physical causality derived.
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Result
Every finite normalized prefix-consistent response family with past marginals independent of future choices has a finite classical stochastic realization: preparation/full history as memory, conditional ratios as transitions, append the next choice/report pair. The construction also reproduces adaptive protocols. It was checked exactly on all requested D tables through horizon4. Finite sequential disturbance and order dependence alone do not exclude unrestricted classical history models.
Supply finite preparation/action/report sets, finite horizon and nonnegative normalized sequential probability tables. Summing out the last report recovers prefixes and earlier marginals do not depend on future choices. Impose no memory, locality, stationarity or preparation-noncontextuality bounds on the representation.
Standard residual representation consequence and explicit counterexamples, not a new ontology. Distinguishability, repeatability, context reproduction and serial availability are separate assumptions. Separate L1/L2 from L3.
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Result
The separate benchmark assumptions give at least two continuation-distinguishable classes and preserve the selected report label under uninterrupted repetition. This largely restates the substantive repeatability premise rather than explaining physical origin. A test can preserve its report while incrementing a separately readable modulo3 counter; result repetition therefore forces no complete idempotence, preservation of all correlations, metric time or eternal material memory. Possible reports alone select no numerical weights.
Required assumptions
Explicit repeatable-distinction benchmark
Recognizable reports, reproducibility of a declared context and available serial continuations are separate commitments. The sharp benchmark additionally has an immediately result-repeatable test and two preparations with different certain reports. These conditions are stronger than mere observability; probability, free erasure, re-preparation, replica control or physical time are not assumed to follow.
Three countermodels, not an exhaustive classification of operational worlds. I/D assume probabilities/convexity; their numbers are predictions, not new measurements or physical dimensions. All-context equality follows analytically from linearity, not the finite word check.
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Result
Three explicit single-system theories satisfy the same sharp repeatability benchmark and are operationally inequivalent: R has two classes without mixing, I a classical simplex, D a square with nonunique decompositions. XZX preserves the first X report with probability1 in I and1/2 in D; D identifies both diagonal mixtures under all generated protocols while I distinguishes them by parity. All30 X/Z words through length4 were checked exactly. D has a classical hidden two-bit implementation of its generators; not all classical models are excluded.
Required assumptions
Explicit repeatable-distinction benchmark
Recognizable reports, reproducibility of a declared context and available serial continuations are separate commitments. The sharp benchmark additionally has an immediately result-repeatable test and two preparations with different certain reports. These conditions are stronger than mere observability; probability, free erasure, re-preparation, replica control or physical time are not assumed to follow.
Three explicit operational countermodels
R has deterministic preparation/read traces and continued partial-identity branches only, no primitive mixing or internal re-preparation. I is the classical two-bit simplex with probabilities and nondisturbing joint access. D is the square with explicit positive X/Z branches, square symmetries, preparations, mixtures and finite serial/adaptive closure. These selected rules define counterworlds rather than necessary physics.
Refinement of already reviewed diagnosis and explicit counterworld comparison of new classes; no refutation of all TFPT and no closure of a physical goal.
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Result
The checked origin attempts leave operation alphabet, state/correlation family and intervention rule unspecified. This motivates a methodology change but does not prove global underdetermination of all complete P1/P2 models: scalar/matrix comparisons change A, while A, state and time belong to the full source tuple. The four independent classes force neither a five-direction carrier,3+2,E8,quartic nor TFPT clocks. Compare TFPT only after their characterization.
Required assumptions
TFPT for later comparison only
The explicit research reset studies independent starting classes. E8, five,3+2, SM algebra, Yukawa data, Hilbert space, complex amplitudes and TFPT clocks are not inputs. Preserve existing TFPT results. After at most three failed origin attempts on unchanged premises change at least one fundamental premise or direction; coding bugs do not count. State assumptions, necessary consequences and newly explained properties.
Separate classes with separate premises, no shared physical derivation. Exact poset/endofunction censuses through size4 and elementary structural claims checked. Cycles do not select a canonical phase origin.
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Result
Event posets provide ideal states and possible histories but do not generally select growth weights. Supplied finite updates give cycles/transients and closed record cells relative to errors/observation; sink SCCs count maximal separated record cells, not necessarily exhaustive invariant record functions. Transformation monoids allow stable sectors, periodic steps and visible classical noncommutativity together; action and effects remain inputs. Counterworlds show these properties do not follow from composition or information alone.
Required assumptions
Finite event order as starting data
A finite strict partial order is supplied as event/precedence structure without metric, space, rates or probability. Order is an input in this class.
Supplied finite update, observation and errors
Finite X, deterministic step f and observation o, optionally error support T, are inputs. A record-cell system is a family of disjoint nonempty forward-closed observationally distinguishable sets, not automatically an exhaustive invariant partition.
Supplied finite operational action and effects
A transformation monoid acts on finite X; effects define operational distinguishability. Composition, action, effects and any selected time generator are separate starting data.
Complete bounded classification of three binary interfaces; no particle/generation/dimension interpretation. One-shot consistency does not force a fixed order.
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Result
Of 4096 no-self-influence candidates,744 are uniquely consistent under all64 local one-shot contexts.488 are acyclic; each of the remaining256 has an explicit reversible local two-copy control with zero or two fixed points (128 each). All744 process/order entries and256 witnesses were independently reproduced. The488 positive cases analytically work for all replica counts and arbitrary joint deterministic controls.
Required assumptions
Finite deterministic processes with full joint replica control
Supply finite nonempty local input/output sets and total w:product O->product I. Every finite number of coordinatewise independent replicas allows every joint deterministic map I_k^M->O_k^M at corresponding interfaces, including constants/erasure. Require at least one global solution for every such experiment; do not assume a global order.
General elementary finite deterministic theorem, mathematically reviewed twice. Arbitrary encoders, decoders and erasure are used; no general reversible two-copy theorem. Interfaces are not derived events. No novelty priority claim.
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Result
For arbitrary finite alphabets/interfaces, existence under every finite joint local replica control, uniqueness under every such control, acyclic static influence and compatibility with at least one fixed total order are equivalent. A directed cycle of length m allows at most m copies with separately sensitized edges and odd XOR feedback without a solution. Acyclicity gives unique triangular recursion. No unique total order or physical time is selected.
Required assumptions
Finite deterministic processes with full joint replica control
Supply finite nonempty local input/output sets and total w:product O->product I. Every finite number of coordinatewise independent replicas allows every joint deterministic map I_k^M->O_k^M at corresponding interfaces, including constants/erasure. Require at least one global solution for every such experiment; do not assume a global order.
Jointly justify replica control and actual record persistence
An independent principle justifies permitted composition and persistent readable records in the same process class; explicitly decide erasure operations, physical access restrictions and counterworlds. Choose no target internal algebra or TFPT number.
Builds on: Check four independent starting classes and counterworlds; Empirically ground a discriminator and access conditions
Resolved in scope
Compare minimal observation conditions and counterworlds
Explicitly check independent assumptions, at least three operationally inequivalent countermodels, shared consequences, concrete discriminators and whether they exceed definitions.