TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/foundational-process-classification-20261016/PROOF.txt
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000084 | WHAT REMAINS AN INPUT: the order itself, history weights, geometry and records.
000085 | 
000086 | 3. Class B: relational consistency without a prescribed global order
000087 | 
000088 | INPUT: finitely many labeled interfaces k with finite nonempty local input
000089 | sets I_k and output sets O_k; a deterministic relation-function
000090 |  w: product O_k -> product I_k.
000091 | There is a local input/output typing and independently selectable local maps
000092 | f_k:I_k->O_k. Demand exactly one solution i=w(f(i)) for every tuple of local
000093 | maps. This is a substantial full-control/modularity assumption, not a derivation
000094 | of laboratories, interventions, space or physical time from nothing.
000095 | 
000096 | No-self-signalling follows: fix all other outputs. If w_k depends on its own
000097 | output, choose two outputs with distinct returned inputs and a local map
000098 | swapping the corresponding choices, mapping all other inputs to one choice.
000099 | There is then no fixed point, contradicting consistency. This also justifies
000100 | the completeness of the no-self-signalling enumeration reduction.
000101 | 
000102 | EXACT BINARY CLASSIFICATION: for three interfaces, each w_k is a Boolean
000103 | function of the other two outputs, giving16^3=4096 candidates. Checking every
000104 | one of the4^3 local unary function tuples and all8 input tuples yields744
000105 | valid processes. Among them488 have an acyclic static dependency graph and
000106 | admit at least one fixed total order;192 have a first interface with a
000107 | context-dependent reversal between the other pair;64 have no first interface.
000108 | An order being compatible is NOT a uniquely selected total order.
000109 | 
000110 | An explicit no-first example is the Lugano function
000111 |  w(a,b,c)=((1-b)c,a(1-c),(1-a)b).
000112 | It has a unique solution for each of the64 independent unary interventions.
000113 | Single-copy consistency therefore does not force global acyclicity.
000114 | The published single-copy framework/census is Baumeler--Wolf,
000115 | https://arxiv.org/html/1507.01714v2, with general process-function analysis in
000116 | https://arxiv.org/html/1703.00779v2. We adapt only the finite set/function
000117 | mathematics; the latter paper's physical setup also discusses spacetime, which
000118 | is NOT an input to this abstract classification. No novelty priority is claimed.
000119 | 
000120 | EXPLICIT NEW PREMISE TO TEST: independent copies may be jointly manipulated
000121 | at each corresponding local interface, and the resulting experiment must
000122 | still have a unique consistent solution. This is composition robustness.
000123 | It is not implied by the original single-copy axiom or by mere associativity.
000124 | 
000125 | DECISIVE OUTCOME: all256 single-copy processes without a fixed total order
000126 | have a two-copy counterexample using only reversible local two-bit maps.
000127 | The stored counterexamples split into128 with no fixed point and128 with two.
000128 | The488 fixed-order processes remain consistent for any finite number of
000129 | copies and arbitrary joint deterministic local maps: solve recursively along
000130 | the common order. Therefore, in this binary three-interface class,
000131 | 
000132 |  two-copy consistency under all joint reversible local maps
000133 |        iff static influence graph is acyclic
000134 |        iff at least one fixed global order is compatible.
000135 | 
000136 | This is a finite exact classification, independently reproduced including all
000137 | 744 maps, order assignments and256 witnesses. It selects a causal restriction,
000138 | not one world, physical duration, an arrow of entropy or quantum theory.
000139 | The all-finite-alphabet extension with arbitrary joint deterministic maps is
000140 | treated separately in COMPOSITION_CAUSALITY.txt; do not transfer the reversible
000141 | two-copy bound beyond its proved finite class.
000142 | 
000143 | NEW EXPLANATORY GAIN: full-control consistency plus composability constrains
000144 | global counterfactual-dependence architecture without prescribing its order.
000145 | WHAT REMAINS AN INPUT: local interface/value typing, deterministic semantics,
000146 | control freedom, copying/composition and any physical realization.
000147 | 
000148 | 4. Class C: stable information under an explicitly supplied loss process