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Theorem · capped potential

FPRD-T143

Lowering-capped continuation potential

Exact statement

Let NucapN_u^{\mathrm{cap}} bound admissible guard extensions, Wu=(Lq)W_u=\binom{L}{q} count fixed-weight suffixes, mu=min⁡(Nucap,Wu)m_u=\min(N_u^{\mathrm{cap}},W_u), and Ph=∑∣u∣=hmuP_h=\sum_{|u|=h}m_u. Then the accepted count is at most Ph+1≤PhP_{h+1}\le P_h, with equality at full depth, and the local loss obeys the exact bottleneck identity recorded below.

StatusWritten all-length proof; internally audited; no independent checker for this theorem
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The potential charges the smaller of arithmetic guard capacity and actual fixed-weight word supply, correcting capacity-only overcounting.

Definitions

  • NucapN_u^{\mathrm{cap}} counts full guard slots allowed by the best completion budget and the least-representative-greater-than-two convention.
  • With σu=Nucap−Nu0cap−Nu1cap\sigma_u=N_u^{\mathrm{cap}}-N_{u0}^{\mathrm{cap}}-N_{u1}^{\mathrm{cap}} and ℓu=mu−mu0−mu1\ell_u=m_u-m_{u0}-m_{u1}, 2ℓu=σu+∣Nu0cap−Wu0∣+∣Nu1cap−Wu1∣−∣Nucap−Wu∣.2\ell_u=\sigma_u+|N_{u0}^{\mathrm{cap}}-W_{u0}|+|N_{u1}^{\mathrm{cap}}-W_{u1}|-|N_u^{\mathrm{cap}}-W_u|.

Hypotheses and scope

  • The target layer and fixed-weight feasibility convention are fixed.
  • The child guard cylinders partition the parent extensions, and infeasible children contribute zero.

Proof or evidence

Child cap inclusion gives Nu0cap+Nu1cap≤NucapN_{u0}^{\mathrm{cap}}+N_{u1}^{\mathrm{cap}}\le N_u^{\mathrm{cap}}; Pascal gives Wu0+Wu1=WuW_{u0}+W_{u1}=W_u; and min⁡(a,b)+min⁡(c,d)≤min⁡(a+c,b+d)\min(a,b)+\min(c,d)\le\min(a+c,b+d) proves monotonicity. At depth KK, Wu=1W_u=1 and the cap is exactly the acceptance indicator. The loss law follows from 2min⁡(x,y)=x+y−∣x−y∣2\min(x,y)=x+y-|x-y|.

Lowering-capped potential proof →

Verification notes

This run checked the child-cap inclusion, the min inequality, the full-depth equality, and the bottleneck algebra. The repository explicitly records that the post-freeze transcript is preserved evidence rather than an independently rerun all-length checker.

Limitations

  • Monotonicity does not imply a uniform factor loss.
  • This capped count is distinct from the abstract lift-slot count in FPRD-T141.
  • No standalone executable checker for the all-length T143 theorem was identified.

Open work

Seek joint guard–disorder loss; the capped potential alone gives no uniform factor contraction.

Notes

Monotonicity combines the child guard-cylinder partition, monotonicity of the maximal-completion cap, and Pascal's identity; the loss formula is the identity 2 min(x,y)=x+y-|x-y|. It caps arithmetic capacity by combinatorial supply but gives no uniform factor loss. The separate post-freeze transcript is preserved evidence, not an independently rerun checker for this all-length theorem. Ncap is not the abstract lift-slot count Nslot of FPRD-T141; the two potentials need not share a root value.