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Theorem · exact quotient family

FPRD-LT-T05

The binary lower-bound family has a quadratic target observer

Exact statement

For the binary J-trivial family of FPRD-LT-T03, the source transition monoid has exactly ∣Mn∣=(n3)+n−1|M_n|=\binom n3+n-1 elements, while the canonical target-residual observer for fn=(n−1,1,…,1)f_n=(n-1,1,\ldots,1) has exactly dfn=(n2)−1d_{f_n}=\binom n2-1 states.

StatusSelf-contained normal-form and residual proof with exact computational audits
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The family that refuted the n−1 conjecture is a diagnostic test for the target-observer strategy, not a hard family for it.

Definitions

  • The target image is {1,n−1}\{1,n-1\}, so a source element xx is observed through Ax=x−1(n−1)A_x=x^{-1}(n-1) and Bx=x−1(1)B_x=x^{-1}(1).
  • Nonempty fibre pairs have one of three interval forms: power, interval, or zero type.

Hypotheses and scope

  • The family uses n≥5 states and the two generators defined in FPRD-LT-T03.

Proof or evidence

Unique zero-, one-, and two-a normal forms plus one special target give the cubic monoid count. Exactly C(n,2)−2 nonempty fibre pairs occur, all empty-A elements form one dead residual, and explicit extension contexts distinguish every pair, giving C(n,2)−1 profiles.

Exact monoid and observer formulas →

Verification notes

Exact scripts verified all normal forms, generator closure, fibre classes, and extension contexts for n=5 through 48: 212,993 normal forms and 228,129 extension checks with no failures. Separate residual enumeration checked the family through n=20 and all three five-state extremals.

Limitations

  • The proof concerns one purpose-built family and does not bound target-residual degree in general.
  • The likely novelty of these family formulas is unconfirmed; no priority claim is made.

Open work

Generalize the fibre classification or construct an L-trivial family with superpolynomially many target profiles.