TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/foundational-process-classification-20261016/PROOF.txt
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Inclusive original lines: 150-226
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000150 | INPUT: finite X, deterministic step f:X->X and observation o:X->A; optionally
000151 | an explicit error relation T. Here X and the step are assumed, not created.
000152 | One may quotient by identical future observations using partition refinement;
000153 | this is known automata minimization, not a new fundamental origin theorem.
000154 | The code independently classifies endofunctions up to relabeling at sizes1..4:
000155 | 1,3,7,19 classes. These are ALL such finite step systems within those bounds.
000156 | 
000157 | JOINT CONSEQUENCE: every endofunction graph consists of cycles with transient
000158 | trees. Distinct cycles give invariant sector labels; positions on a cycle
000159 | give a cyclic phase orbit of the GIVEN step, without a canonical phase origin.
000160 | Thus persistence and periods share
000161 | one transition structure, without a chosen physical symmetry. They need not
000162 | both be nontrivial. A constant map has one terminal sector; identity has many
000163 | sectors but no nontrivial periods; one n-cycle has period n but one sector.
000164 | 
000165 | Define a record cell as a nonempty forward-closed set. The record quantity
000166 | here is the MAXIMUM number of pairwise disjoint, observation-distinguishable
000167 | record cells, not an exhaustive invariant record function on every state.
000168 | With full state access this quantity equals the number of sink strongly
000169 | connected components of the allowed-transition support. With coarse
000170 | observation, it is the independence number of the conflict graph of sink
000171 | components, where adjacency means overlapping observation sets. Every closed
000172 | cell contains a sink component, proving the bound, and choosing those
000173 | components attains it. This does not claim every transient state already
000174 | carries one of those stable labels: x->a and x->b with a,b absorbing has two
000175 | record cells but no two-label invariant record function on all three states.
000176 | Error correction is relative to a given error relation: disjoint output sets
000177 | are the ONE-STEP zero-error criterion; indefinitely repeated errors require
000178 | disjoint reachable closures of the codewords.
000179 | 
000180 | For f=(1,0,2), there are two record sectors and a period2 cycle. Adding one
000181 | allowed error edge0->2 removes the first closed sector, leaving one. With
000182 | constant observation the predictive quotient has one class; with full access
000183 | it has three. Stability does not determine the channel or observation.
000184 | Iterations already supply a step order. No continuous duration or physical
000185 | time is derived from it. A stable label is not a local or conscious observer.
000186 | 
000187 | LITERATURE: predictive-state minimality, Shalizi--Crutchfield,
000188 | https://arxiv.org/abs/cond-mat/9907176; channel-dependent zero-error coding,
000189 | Shannon, https://doi.org/10.1109/TIT.1956.1056798. Quantum error-correction
000190 | algebras presuppose a quantum channel and cannot be imported as emergence here.
000191 | 
000192 | NEW EXPLANATORY GAIN: persistent sectors and possible periods are classified
000193 | from one update/support table. WHAT REMAINS INPUT: update, observation, errors.
000194 | 
000195 | 5. Class D: minimal composition and operational distinctions
000196 | 
000197 | INPUT: a monoid of transformations of a finite set, with chosen effects. This
000198 | adds neither probability nor metric. Algebraic associativity and identity
000199 | alone do not select one action, effect set or physical temporal generator.
000200 | Full contexts give a minimal operational quotient; constant effects can make
000201 | an otherwise noncommuting action observationally trivial.
000202 | 
000203 | NONCOMMUTATIVITY IS ALREADY CLASSICAL: distinct reset maps c0 and c1 obey
000204 | c0 c1=c0 and c1 c0=c1. An observed difference in operation order is therefore
000205 | not evidence for complex quantum mechanics. For X={0,1,2}, let s=(1,0,2)
000206 | and r=(0,0,2). Their monoid has four elements, preserves sectors{0,1},{2},
000207 | contains period2 and is visibly noncommutative with full observation. Three
000208 | states are minimal for two persistent sectors plus a nontrivial period and
000209 | noncommuting action: with two singleton sectors every preserving map is I.
000210 | These operations/effects were chosen; their coexistence is a counterworld,
000211 | not a uniquely selected fundamental model.
000212 | 
000213 | Under all allowed operations, an invariant record function is constant on
000214 | each connected component of the undirected graph x--g(x). Conversely any
000215 | label on these components is invariant, a direct exact classification.
000216 | An extra allowed reset can merge components. An actual observer also needs
000217 | a subsystem/access/record-update interpretation; monoid components do not
000218 | supply spatial locality, consciousness or an initial state.
000219 | 
000220 | LITERATURE: generalized probabilistic frameworks explicitly admit different
000221 | theories, Barrett https://arxiv.org/abs/quant-ph/0508211. Informational quantum
000222 | reconstruction uses further axioms, https://arxiv.org/abs/1011.6451; those
000223 | axioms are not silently added here. Parallel composition is a genuinely
000224 | additional operation, used explicitly as the discriminating extension of B.
000225 | 
000226 | NEW EXPLANATORY GAIN: stable sectors, periods and visible order dependence