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Counterexample and failed approach

FPRD-T88

Cross-port redirection does not clone fixed-port stacks

Exact statement

In the tuple-valued fixed-port decoder used by the original persistent-stack compiler, redirecting destination port (1) to source port (0) does not clone the source stack. A backward two-stack history with decoded stacks ([2,1]) and ([9]) becomes ([2,1]) and ([2,9]) under the naive redirection, not two copies of ([2,1]).

StatusMechanically checked and internally audited
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The fixed-port decoder reads label component j and then follows port j forever. A one-time path to another port changes the first cell but not the coordinate used for the remaining history.

Proof or evidence

Lean constructs the compiler-shaped history, proves both the original and redirected scaffolds backward, computes the source and destination stacks before and after redirection, and proves that the destination is not a copy.

Fixed-port clone counterexample →

Verification notes

The concrete computation was independently rederived from the fixed-port decoder: the destination reads source component 0 only at the new node, then follows destination component and port 1 at the historical node. The source specification, formal endpoint, backwardness, absence of SELF, and proof-hole scan were checked against the pinned source.

Limitations

  • This refutes one natural operation in one coordinate doctrine, not persistent root sharing in scaffolding automata.
  • It is not an impossibility theorem for bounded frontiers, another decoder, context-free languages, or PEGs.

Open work

Seek external review of the coordinate correspondence and test whether other natural fixed-port clone operations fail for the same reason.

Notes

Lean 4.30.0 checks that the source history is built from the verified compiler's newLabel/newTarget formulas, that both the history and the naively cloned scaffold are backward, and that destination decoding copies top 2 before resuming destination component and port 1 at the old node. This is not an impossibility theorem for scaffolds, bounded live frontiers, or persistent cloning in another coordinate doctrine. It shows that a bounded-frontier compiler cannot obtain branch splitting merely by inserting a cross-port path into the current fixed port decoder.