TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/foundational-process-classification-20261016/COMPOSITION_CAUSALITY.txt
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000001 | COMPOSITION ROBUSTNESS FORCES ACYCLICITY IN FINITE DETERMINISTIC PROCESSES
000002 | 2026-10-16. Direct elementary theorem; no physical/quantum/TFPT identification.
000003 | Independent mathematical review accepted the proof below. No novelty priority claim.
000004 | 
000005 | 1. Inputs and exact consistency condition
000006 | 
000007 | There are N named interfaces, with finite nonempty input/output sets I_k,O_k,
000008 | and a total deterministic function w:product_k O_k -> product_k I_k.
000009 | These types, definite local values and deterministic semantics are INPUTS.
000010 | No global causal order, spacetime, complex amplitude or Hilbert space is given.
000011 | 
000012 | For M independent copies, w acts coordinatewise. At interface k allow EVERY
000013 | joint total deterministic map F_k:I_k^M -> O_k^M, including encoders,
000014 | decoders and constant-output coordinates. Different interfaces cannot jointly
000015 | act, but each can jointly manipulate its own wires across copies.
000016 | This full-control and replication assumption is a NEW explicit model premise.
000017 | It is not entailed by one-shot consistency or by associative composition.
000018 | Product-only independent controls on separate copies are a weaker condition.
000019 | 
000020 | Call the class composition-robust when, for every M>=1 and every tuple F_k,
000021 | the equations x=w^(M)(F(x)) have at least one solution. Uniqueness is NOT
000022 | part of this definition; it will follow. We classify logical processes, not
000023 | prove that such arbitrary controls are physically achievable in a closed world.
000024 | 
000025 | Define the static influence graph on named interfaces: j->k iff there are
000026 | two output tuples differing only at j for which w_k changes. Self-edges
000027 | are allowed in this initial definition. This is counterfactual dependence
000028 | over all output contexts, not an assumed physical signal or metric edge.
000029 | 
000030 | THEOREM. The following are equivalent:
000031 |  (i) The process is composition-robust in the existence sense above.
000032 |  (ii) Every finite-copy experiment with joint local deterministic maps has
000033 |       exactly one solution.
000034 |  (iii) The static influence graph is acyclic.
000035 |  (iv) w is compatible with at least one fixed topological total order.
000036 | If a directed simple cycle of length m exists, at most m copies exhibit a
000037 | failure of (i). The general construction may erase information; it is NOT
000038 | a reversible-control theorem for arbitrary alphabets.
000039 | 
000040 | 2. Acyclicity gives all-copy existence and uniqueness
000041 | 
000042 | Take a topological order pi. Functional independence from non-predecessor
000043 | outputs means w_(pi0) is constant. In M copies its input vector is therefore
000044 | fixed, so the arbitrary joint map F_(pi0) determines its output vector.
000045 | The input vector of pi1 depends only on outputs already determined; apply
000046 | F_(pi1). Continue recursively. Each step has exactly one value because w
000047 | and every F are total functions. This constructs exactly one global solution
000048 | for every finite M. No sampling or limiting argument is involved.
000049 | 
000050 | Acyclicity and existence of a topological order are equivalent for finite
000051 | graphs, and functional dependence then makes w triangular in that order.
000052 | There may be MANY compatible total orders, including a completely parallel
000053 | constant w. Neither a unique order nor physical time is being selected.
000054 | 
000055 | 3. Any directed cycle fails under a finite-copy experiment
000056 | 
000057 | Let the cycle be k_1->k_2->...->k_m->k_1 (indices modulo m).
000058 | For each edge r:k_r->k_(r+1), dependence supplies:
000059 |  - a context q_r for the outputs of all interfaces except k_r;
000060 |  - two distinct symbols a_r^0,a_r^1 in O_(k_r);
000061 |  - distinct target inputs u_r^b=w_(k_(r+1))(q_r,a_r^b), b=0,1.
000062 | Choose a total decoder d_r:I_(k_(r+1))->{0,1} with d_r(u_r^b)=b.
000063 | Its values on other symbols are arbitrary and cannot affect a fixed point.
000064 | 
000065 | Use m independent copies, one copy r for each cycle edge. In copy r all
000066 | outputs except at k_r are fixed by the local maps to the context q_r.
000067 | At interface k_r, output coordinate r is
000068 |  a_r^[ d_(r-1)(x_(k_r)^(r-1)) XOR epsilon_r ].
000069 | Other output coordinates at that interface are the context constants.
000070 | This is one permitted total joint LOCAL map. Distinct copies make the
000071 | independently chosen edge-witness contexts compatible; there is no requirement
000072 | that different edges be simultaneously active inside one copy.
000073 | 
000074 | If a solution exists, let y_r label its chosen output a_r^y_r in copy r.
000075 | The environment then forces x_(k_(r+1))^r=u_r^y_r, so decoding yields
000076 |  y_(r+1)=y_r XOR epsilon_(r+1).
000077 | Choose the epsilons with odd XOR parity. Going around the cycle gives
000078 | y_1=1-y_1, impossible. Thus the composed experiment has NO solution.
000079 | All remaining inputs are simply w(outputs); they do not feed nonconstant
000080 | output coordinates and cannot repair the contradiction. For even parity,
000081 | the same construction has exactly two solutions, one for each initial bit.
000082 | A self-loop is the m=1 case and already fails one-copy consistency.
000083 | This proves (i)->(iii), completing the equivalence.
000084 | 
000085 | 4. What has actually been explained jointly
000086 | 
000087 | For this class, a single global demand -- existence under all replicated
000088 | local interventions -- forces:
000089 |  - no cyclic counterfactual influence;
000090 |  - global uniqueness of all finite-copy outcomes without imposing it;
000091 |  - a precedence partial order, the reachability order of the influence DAG.
000092 | From that partial order one canonically obtains a lattice of dependency ideals
000093 | and topological evaluation schedules. The nodes are supplied interfaces,
000094 | not derived event occurrences. Interpreting each interface as a single event
000095 | is an ADDITIONAL premise before these can be called cumulative occurrence
000096 | states and histories in the sense of Class A. No material memory is proved.
000097 | 
000098 | Assumed local input/output typing remains a local orientation. This is not
000099 | time emerging from literally untyped relations. Physical duration, an entropy
000100 | arrow, a continuum, probabilities, quantum theory and observer memory remain
000101 | unselected. Strong composition robustness excludes also context-controlled
000102 | orders that are consistent one shot. Whether free regrouping of copies is
000103 | a physically appropriate axiom must be assessed separately, not declared
000104 | necessary for reality by this mathematical theorem.
000105 | 
000106 | 5. Exact finite sharpening and literature relation
000107 | 
000108 | For three binary interfaces the separate exhaustive classification finds744
000109 | single-copy process functions.488 have an acyclic dependency graph. Every
000110 | one of the remaining256 already fails with TWO copies and reversible joint
000111 | local maps; this is stronger on controls/copy count, but limited to this class.
000112 | The general proof above covers arbitrary finite alphabets/interfaces with
000113 | arbitrary joint maps, not that sharper reversible-only claim.
000114 | 
000115 | Simple two-copy reversible witness for the Lugano w(a,b,c)=
000116 | ((1-b)c,a(1-c),(1-a)b): group each interface's two bits as copy0+2copy1.
000117 | At A and B use SWAP=(0,2,1,3); at C flip copy0=(1,0,3,2).
000118 | Checking all64 input tuples gives no fixed point. This requires neither
000119 | complex numbers nor quantum operations. All256 saved negative certificates
000120 | have been independently checked, as have all744 single-copy maps and orders.
000121 | 
000122 | The product/composition issue has prior literature, including classical
000123 | processes: Jia--Sakharwade, https://arxiv.org/html/1706.05532 (2018).
000124 | Their process-matrix setting treats grouped same-named parties and joint
000125 | operations, and invalid products can activate loops. Our argument above is
000126 | a direct finite deterministic influence-cycle proof, with no quantum premise;
000127 | it does not transfer their Hilbert-space theorem as a physical TFPT result.
000128 | The known one-copy class is Baumeler--Wolf, arXiv:1507.01714.
000129 | We make no claim that this elementary deterministic statement is new literature.
000130 | 
000131 | 6. Research consequence
000132 | 
000133 | This is the best-supported independent starting result of the reset: global
000134 | counterfactual-dependence architecture and outcome uniqueness follow from explicit
000135 | control/composition assumptions, without installing a global time order.
000136 | It neither produces nor selects our physical world. The next independent
000137 | test concerns actual record-bearing subprocesses and the appropriateness of
000138 | composition/control freedom, not fitting a carrier dimension or resuming L_f.
000139 | If progress requires a new axiom, label and compare it as a new premise.