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Theorem · counterfamily

FPRD-FAIL-COLLATZ-LF5

Universal logarithmic-block contraction is false

Exact statement

For every fixed C>0C>0, there are 2γK−O(log⁡K)2^{\gamma K-O(\log K)} genuine fixed-weight-feasible capacity-one prefixes, with γ=(1−log⁡32)(1−log⁡2(3/2))=0.1531779921…, \gamma=(1-\log_3 2)(1-\log_2(3/2))=0.1531779921\ldots, that cross a Clog⁡2KC\log_2K block with zero attrition under maximal-completion capacity.

StatusWritten all-length proof; critical-layer estimate has an external dependency
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

This is the decisive obstruction to the dormant marginal-capacity route. It uses actual parity prefixes and exact guards rather than synthetic relaxed charts.

Definitions

  • At the critical layer S=⌊Klog⁡32⌋S=\lfloor K\log_3 2\rfloor, choose the first depth hh where maximal-completion capacity is at most one, then inspect depth h+⌈Clog⁡2K⌉h+\lceil C\log_2K\rceil.

Hypotheses and scope

  • Fixed positive C.
  • Critical contracting layer and exact fixed-weight feasibility.
  • The imported critical-layer estimate, whose logarithmic-form dependency has not been independently re-audited here.

Proof or evidence

Every sufficiently small exact guard remains a live capacity-one prefix. After excluding the subexponentially many prefixes with infeasible weight, the supplement counts 2γK−O(log⁡K)2^{\gamma K-O(\log K)} survivors at the later depth. Selecting precisely their ancestors at depth hh gives zero attrition across the block.

Forced-family obstruction and proof boundary →

Verification notes

This run checked the quantifier over fixed C, the distinction between capacity survival and paradoxical extension, and the exact exponent expression. The external critical-layer estimate remains a named dependency, so the entry stops at L4.

Limitations

  • The surviving prefixes need not extend to paradoxical full words.
  • The theorem refutes universal capacity-only contraction, not every possible joint guard–disorder contraction.
  • No novelty or external-review claim is made.

Open work

Move to actual forced-suffix disorder; do not seek a universal capacity-only contraction over logarithmic blocks.