# Thin trees: all-cut checking and conservative signing constants

Family 174; selected construction addendum, 9 October 2026 Australia/Brisbane. Source `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`.

Read the existence introduction, constructive introduction/main theorem, iteration/binary reduction and fractional cost/cut rounding; selected signing statement/constant and formal scope were also read. The existence fixed-point step does not itself supply the finite construction. With stated Cs=2^20, the declared cut factor certifies strict contraction only above r=(6Cs)^8, a 55-digit quantity. This is not a final C, input/runtime lower bound or impossibility. R0 and final C are not numerically selected. The cost corollary uses nonnegative rational inputs and an all-cut fractional promise; it is not separately listed in the selected family scope. The bounded exact auditor shows K4 path fundamental-cut ratio 1/3 versus true all-cut ratio 3/4. It is exponential finite validation, not the polynomial source constructor or routing/redundancy evidence. Dossier 024 is revised.

## Exact sections read

- `preprints/The-strong-thin-tree-conjecture-September-23-2026/build/source/sections/introduction.tex`: full existence introduction.
- `preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/introduction.tex`: full constructive introduction.
- `preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/main-theorem.tex`: full main statement.
- `preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/iteration.tex`: full iteration/binary reduction.
- `preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/fractional_rounding.tex`: full cost/cut rounding section.
- `preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026/build/source/sections/signing.tex`: lines 73–100 and 180–207.
- `lean/docs/174.md`: full selected scope document.

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