TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/foundational-process-classification-20261016/PROOF.txt
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000045 | 
000046 | 2. Class A: event orders without a background geometry
000047 | 
000048 | INPUT: an unlabeled finite strict partial order P=(E,<). A precedence relation,
000049 | transitivity and absence of cycles are assumed; causal order is not derived
000050 | from consistency in THIS class. There are no rates, amplitudes or metric.
000051 | 
000052 | JOINT CONSEQUENCE: cumulative occurrence states are the order ideals J(P).
000053 | They form a distributive lattice under union and intersection. Its nonzero
000054 | join-irreducibles are exactly principal ideals down(e), recovering the events
000055 | and their order. Maximal one-event-at-a-time ideal chains are the linear
000056 | extensions. Thus one relational datum specifies events, cumulative states and
000057 | allowed histories together. These are occurrence states, not quantum states
000058 | or physically stored memories; the event order remains an input.
000059 | 
000060 | COUNTERWORLDS: at each ideal I a fixed-P Markov growth law independently
000061 | chooses weights on the enabled events. Invariance under relabeling constrains
000062 | weights, but need not select them. For a<c with an isolated b, even the first
000063 | choice a versus b has a free probability. There is no order automorphism
000064 | exchanging them. The full product of invariant Markov simplices has dimension
000065 |  beta(P)=sum_[I](number of enabled-event orbits under Aut(P)_I - 1).
000066 | The sum runs over Aut(P) orbits of nonterminal ideals. The checker obtains
000067 | unlabeled-poset counts1,2,5,16 at sizes1..4. At size3, four classes have beta0
000068 | and the chain-plus-isolate class has beta2. At size4, eight have beta0 and
000069 | eight retain free parameters. Beta0 is uniqueness ONLY in this fixed-P,
000070 | unit-step, invariant Markov class, not uniqueness of durations or a world.
000071 | 
000072 | A two-event chain can embed as timelike points separated by different proper
000073 | times in continua of different dimensions. Finite precedence therefore does
000074 | not reconstruct metric/locality. A chain supplies an ordinal history, not a
000075 | physical clock or observer. Classical poset worlds are counterexamples to
000076 | claiming complex quantum theory from these axioms.
000077 | 
000078 | LITERATURE: Rideout--Sorkin construct a family of causal-set growth dynamics,
000079 | https://arxiv.org/abs/gr-qc/9904062. Surya's review describes the order/volume
000080 | and continuum-correspondence assumptions, https://arxiv.org/abs/1903.11544.
000081 | These are comparison sources, not imported quantum/spacetime premises here.
000082 | 
000083 | NEW EXPLANATORY GAIN: occurrence states and histories follow from one order.