More aspects of that Idea
A transcript of a short interaction between me and codex

 I suppose by a disconnected N, I was thinking of a different context. I was
  wondering if you could have a "big picture" N that prevents decomposition
  fully. For example, take the literal letter N, then pretend the 3 lines that
  make it up are series parallel on their own. But then you connect them in that
  "N" structure at a very high level. I think the terms would be something
  like... take three series parallel, and then attach the sinks on one pair, and
  the sources on the other.


• Yes, that is a real phenomenon, and your intuition is sharp.

  You are describing a substitution / inflation of an N pattern:

  A < B
  C < B
  C < D

  but instead of each A, B, C, D being a single element, each one is a whole
  block, possibly series-parallel internally.

  Something like:

  Block A  <  Block B
  Block C  <  Block B
  Block C  <  Block D

  where each block may have rich internal structure.

  At the “macro” level, the quotient shape is still N.

  That means:

  - each block may be easy,
  - each internal block may be series-parallel,
  - but the way the blocks are connected creates a global N,
  - so the whole poset is not series-parallel under the usual recursive series/
    parallel operations.

  This is exactly the kind of thing that can defeat naive decomposition: locally
  everything looks decomposable, but the block interaction pattern is prime.

  Your “attach the sinks on one pair, and the sources on the other” phrasing is
  close to a block-level relation pattern. If all elements of one block are below
  all elements of another block, those are ordinal-sum-like relations between
  blocks. But if only sinks/sources are attached, the situation may be more
  subtle because the relation between blocks may not be complete block-to-block
  comparability.

  For enumeration, this matters a lot:

  easy blocks + hard interaction pattern = hard combined problem

  So a useful future detector might operate at two levels:

  1. Micro level: detect local family/block structure.
  2. Macro level: contract recognized blocks into nodes and study the quotient
     interaction pattern, not necessarily inside any one block.

  This is likely very relevant to your “edge detector” idea. The detector might
  find internally coherent regions, but then the quotient graph/poset needs to be
  So yes: “prime-ness” can propagate upward through block substitution. A poset
  can be made from easy components but still have a hard global skeleton.


› It may still be useful to find those blocks... but it does help to know this
  happens, and is probably the 'big fear' of the idea, as I expected. Not that I
  expected it to be 'easy' in the first place with what we know about P#


• Exactly. Finding the blocks can still be valuable, even when the quotient
  skeleton is hard.

  The right expectation is probably:

  block detection helps expose the shape of the hard part

  not always:

  block detection solves enumeration

  If you find large easy regions and contract them, you may reduce the problem
  from:

  huge raw poset

  to:

  smaller hard interaction skeleton + known internal block counts/features

  That can still help with:

  - feature extraction,
  - heuristic difficulty prediction,
  - choosing enumeration strategy,
  - caching repeated substructures,
  - identifying exact factorization when it exists,
  - locating the obstruction when it does not.

  And yes, the #P background is the warning sign. Exact linear extension counting
  is hard in general, so any universal decomposition shortcut would effectively
  be too powerful. The realistic goal is to carve out recognizable easy cases and
  identify where the remaining hardness lives.

  A good mental model:

  families = known easy or diagnostic shapes
  decomposition = exact simplification when validated
  block detection = compression / explanation / candidate generation
  hard skeleton = what remains after safe simplifications

  That lets the project stay mathematically honest while still building useful
  exploratory tools.

  This is good to confirm. It might even be useful to see these hard skeletons
  at a macro level, even if they do not provide a "direct" shortcut to the
  enumeration.


• Yes. A macro hard skeleton could be useful even without an exact counting
  shortcut.

  It can act as a structural explanation layer:

  raw poset
    -> detect coherent blocks
    -> contract blocks
    -> inspect quotient / macro poset
    -> identify easy compositions or hard obstructions

  Even if the quotient is not directly enumerable by a shortcut, it can tell you:

  - why the poset resists series-parallel decomposition,
  - where N-like obstructions live,
  - which blocks participate in the hard interaction,
  - whether the hard part is small relative to the full poset,
  - whether different workflows share the same macro skeleton,
  - which features may predict enumeration difficulty.

  For ML, this is especially interesting. The model may not need the entire raw
  poset if useful features include:

  number of blocks
  block size distribution
  quotient height/width
  quotient contains induced N
  hard-core size
  fraction of elements inside easy blocks
  Mobius/interval summaries of quotient
  Mobius/interval summaries inside blocks

  That gives you a two-scale description:

  local block structure + global interaction structure

  That could be much more informative than one flat summary of the entire poset.

  So yes: hard skeleton extraction is still useful as analysis, compression,
  visualization, and feature generation, even when it does not directly solve
  enumeration.