The capped potential measures two resources at once

Fix a contracting layer $(K,S)$and a prefix $u$ of length$h$. If$N_u^{\mathrm{cap}}$ is the number of admissible guard slots allowed by maximal completion and $W_u=\binom{K-h}{S-|u|_1}$is the supply of fixed-weight suffix words, define

$$m_u=\min(N_u^{\mathrm{cap}},W_u),\qquad P_h=\sum_{|u|=h}m_u,\qquad F_h=\sum_{|u|=h}(m_u-1)_+.$$

The potential $P_h$ is an exact upper bound that is monotone in depth. The stripped mass$F_h$ removes the first unit from every live prefix. Thus$F_h=0$ says that every live prefix has at most one capped continuation; it does not say that few live prefixes remain.

FPRD-FAIL-COLLATZ-LF3 · failed universal route

Monotonicity is not a rate theorem

The local bottleneck identity writes the loss$P_h-P_{h+1}$ as a sum of nonnegative mismatch terms. It proves$P_{h+1}\le P_h$, but supplies no positive lower bound on the loss. A capacity-one prefix can pass its only unit to one child unchanged.

This closes the unrestricted inference from monotonicity to multiplicative contraction. It does not rule out contraction on a restricted family or after enriching the state.

FPRD-FAIL-COLLATZ-LF4 · reproduced finite finding

Zero nonforced mass can coexist with thousands of live prefixes

At $(K,S,h)=(27,17,14)$, an independent exact-integer reconstruction gives

$$\#\{u:m_u>0\}=2{,}321,\qquad P_{14}=2{,}321,\qquad F_{14}=0.$$

Every one of those 2,321 prefixes has$m_u=1$. The computation uses exact integers, the fixed-weight supply$W_u$, and the convention that the least admissible full guard must exceed two.

The archived generic verifier reports larger raw slot totals because its helper does not apply this row's greater-than-two admissibility adjustment or the fixed-weight word-supply cap. Its raw output includes one two-slot prefix whose word supply is only one. Applying both parts of the public definition reproduces the ledger's 2,321 and$F_{14}=0$ exactly. This is a finite computational finding, not an asymptotic theorem.

FPRD-FAIL-COLLATZ-LF5 · all-length counterfamily

Every fixed logarithmic block can avoid capacity loss

On the critical layer$S=\lfloor K\log_3 2\rfloor$, fix any $C>0$. The preserved proof constructs genuine fixed-weight-feasible prefixes that remain capacity one across$\lceil C\log_2K\rceil$further levels. Their number is at least

$$2^{\gamma K-O(\log K)},\qquad \gamma=(1-\log_3 2)(1-\log_2(3/2)) =0.1531779921\ldots.$$

Selecting their ancestors at the start of the block gives zero attrition throughout the block. This refutes every universal capacity-only theorem demanding fixed-factor loss over all such logarithmic blocks.

The prefixes need not extend to accepted paradoxical full words. The theorem therefore blocks one presentation of contraction; it is not a Collatz lower bound or a counterexample to a richer guard–disorder estimate. Its critical-layer estimate remains a named external dependency, so the record remains at L4.

FPRD-FAIL-COLLATZ-LF7 · failed abstraction

A minimum of marginals forgets the matching

The number $m_u=\min(N_u^{\mathrm{cap}},W_u)$knows how many arithmetic slots are available and how many combinatorial suffixes exist. It does not record which suffix occupies which slot, nor the disorder of the suffix forced by a particular guard.

That missing assignment is precisely where the surviving question lives. The capped potential remains valid for exact accounting and pruning, but the marginal state is too coarse to settle the contraction frontier by itself.

FPRD-C03 · open problem

The surviving problem is joint

Write $r_w$ for the full guard,$\delta_w$ for the suffix loss from maximal fixed-weight completion, and$B-Dr_w$ for the remaining recovery allowance. The unresolved target is an all-length joint estimate for

$$r_w\le R,\qquad \delta_w\le B-Dr_w,$$

or an equivalent analysis of the actual uniquely forced suffix. A useful result must either prove enough joint loss to improve the leading exponent or construct an exact accepted-trace counterfamily showing that the forced suffix preserves exponential mass. Neither alternative is currently proved.

FPRD-FAIL-COLLATZ-LF8 · explicit nonclaim

No Collatz conclusion follows

The strongest preserved candidate estimate remains exponential:

$$|\mathcal P_K|\le 2^{0.34416K+o(K)}.$$

Nothing on this page proves finiteness of paradoxical sequences, Terras's conjecture, or the Collatz conjecture. The number 0.34416 is a preserved architecture-specific candidate bound with an external logarithmic-form dependency; it was not independently re-audited in this batch.

Evidence, provenance, and review status

The statements were reconciled against the pinned lowering and contraction supplement, post-freeze failure ledger, archived Python verifier, and recorded verifier output. A new independent JavaScript checker reconstructs the LF4 row from the definitions and fails the production build if any of its five exact totals changes.

Read the pinned mathematical supplement →