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Failed approach · negative boundary

FPRD-FAIL-COLLATZ-LF3

Monotonicity is not multiplicative contraction

Exact statement

The exact monotonicity Ph+1≤PhP_{h+1}\le P_h and its additive bottleneck-loss identity do not imply a uniform bound Ph+1≤ρPhP_{h+1}\le\rho P_h with ρ<1\rho<1; capacity-one paths can preserve their only slot.

StatusClosed as a universal contraction route
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The exact capped potential remains valuable, but monotonicity was repeatedly at risk of being read as asymptotic progress that it does not provide.

Hypotheses and scope

  • Lowering-capped continuation potential at a fixed contracting layer.

Proof or evidence

FPRD-T143 gives only an additive nonnegative loss. The all-length capacity-one family in LF5 supplies zero-attrition logarithmic blocks, ruling out the universal multiplicative strengthening.

Why monotonicity supplies no universal rate →

Verification notes

The source ledger and supplement were compared with T143 and the LF5 all-length zero-attrition family. The published argument now supplies an explicit witness family separating monotonicity from every universal fixed-factor rate.

Limitations

  • Some restricted family may still contract.
  • The failure does not make the potential useless for finite pruning.

Open work

Use the monotone potential for pruning and exact accounting only; require a stronger joint statistic for rate loss.