Consolidated Results Through Run 28

Result map

Run	Native model	Exact advance	Principal obstruction retained
1	Bottom-up tree automata	Contextual equivalence is the largest reachable finality-saturating congruence; distinguishing depth is sharply at most (n-2)	Unreachable constants create fictional contexts
2	Fixed-order 3CNF / ROBDD	(2^k) histories collapse to two semantic residuals in the baseline family	Syntactic residual counts depend on provenance conventions
3	Ordered SAT prefixes	Future equivalence is the greatest graded congruence	Erasing prefix grade makes next-bit update ill-defined
4	Uniform streaming	Positional tasks need exactly (j+2) uniform residual states despite cut width two	Treating the clock as free hides state cost
5	Rewriting $aa\to a$	Raw histories count ((n-1)!); square classes count (C_{n-1})	Confluence does not identify overlapping histories
6	Four-leaf coherence	Schedule quotient is an unfilled pentagon with $\pi_1\cong\mathbb Z$	Connectedness is weaker than coherence
7	Five-leaf coherence	Six pentagons and three squares form a complex homotopy equivalent to (S^2)	Homology alone would not prove simple connectedness
8	Dynamic tree cache	Disjoint fixed replacements commute; batch saving is exactly LCA depth plus one	Cost is algorithm-relative
9	Early stopping	Commuting updates can have sharp order gap (2h)	Semantic commutation does not imply resource flatness
10	Ancestor chains	Nested kernels induce an ultrametric; cross-level cost biases cannot cancel	Mutable contexts can break nestedness
11	Arbitrary incomparable updates	Work factors through branch costs and the LCA corner square	Spatial data alone omits the LCA transition algebra
12	Persistent DAG copying	Immutable occurrence-path copying is alias-transparent	Destructive identity and canonical reuse cross the boundary
13	Hash-consed sibling updates	Allocation gap is at most one and depends on retained intermediate roots	Equal cost can leave unequal tables
14	Height-two hash-consing	Common-request coverage replaces naive initial membership	Cross-level identity creates endogenous prepayment
15	Arbitrary hash-cons spine	Exact set-rank law; sharp gap (m); chain biases cannot cancel	Unrestricted key-set cancellation is not structurally realizable
16	Dependency posets	Ideal lattices exactly classify cancellation; sign coherence iff the poset is a chain	Native single-path copying cannot generate a fork
17	Learned 3CNF resolvent	Clause-minimal semantic/UP/decision separation	Learned binary receipt and accounting policy are explicit
18	SAT products	DPLL/UP decision cost is additive over disconnected clause components	Semantic product structure alone does not imply additivity
19	Connected resolution chain	Pure-resolution receipts give exact connected cost (k-	S
20	Receipt amortization	Exact acquisition/reuse frontier $(r-1)d\ge c+s$	Deletion destroys amortization
21	Demand coverage	Break-even depends exactly on covered future demands	Uncovered demands do not repay acquisition
22	Proof-state fiber / OBDD	Exact hierarchy $2^k\to k+1\to1$ for proof state, scalar cost, semantics	Full closure or deletion changes the fiber
23	Propagation demands	Exact information law: demands (T) require (	T
24	Second-order branching	Same singleton UP signature, different fixed-policy DPLL cost	Singleton tests are vacuous for strict 3CNF
25	Arbitrary clause width	Width (r) first differs at interaction depth (r-1)	Unbounded depth is obtained by increasing width
26	Fixed-width absorption	Seed depth (w-1) exactly characterizes all-seed UP equivalence	Fixed-width UP cannot support unbounded escape depth
27	First-UIP provenance	All-seed UP-equivalent databases have an unbounded selected-trace pivot gap	Clause priority and reason selection matter
28	All-schedule first-UIP	Base spectrum is ({m+1}), augmented spectrum ({1,m+1}); optimum gap (m)	A shortcut enables but does not force lower online cost

Current frontier

The program has established a stable hierarchy of distinctions:

$\text{same semantic endpoint} \not\Rightarrow \text{same operational history} \not\Rightarrow \text{same retained representation} \not\Rightarrow \text{same resource cost}.$

The tree campaign exposed contextual congruences, LCA geometry, ultrametric propagation, and set-rank growth under canonical sharing. Its arbitrary-height spine and ideal-lattice theorems closed the height-by-height line and forced a pivot rather than further enumeration.

The SAT campaign then separated semantic residuals, clause-database state, unit-propagation readiness, decision cost, and reason provenance. The latest pair (F_m,G_m) has identical semantics and identical all-seed UP closure, yet retaining one absorbed resolvent changes the minimum legal first-UIP analysis from (m+1) pivots to one. Run 28 proves this is not an artifact of one fixed priority, while also showing that the shortcut does not improve the worst legal schedule.

Current obligation

The reason-path family is classified under fixed and arbitrary schedules. The next bounded SAT/proof target should test a resource not reducible to that scheduling spectrum: the smallest connected strict-3CNF pair with identical semantics and all-seed UP closure but different minimum resolution width or clause space for a fixed unsatisfiable residual, or a precise obstruction.