Comparative Recognition Dynamics — Model Dossiers

Updated: 2026-08-23 10:00 MDT

Deterministic bottom-up finite tree automata

Fixed distinctions

• Ground-tree reachability is part of the theorem, not an optional implementation cleanup.
• Actual one-hole contexts and full-state algebraic polynomials are different on untrimmed automata.
• Dynamic update results use fixed pre-evaluated replacements at incomparable addresses. Full ancestor-path and early-stopping algorithms are treated as separate accounting models. State-dependent payloads, overlapping updates, and representation-independent optimal lower bounds remain excluded.

Dynamic disjoint-update result

For fixed replacements at incomparable positions (u,v), the labelled tree, complete deterministic cache, and contextual observations are independent of order. Sequential eager work is (|A(u)|+|A(v)|); batched work is $|A(u)\cup A(v)|$; exact saving is $|A(u)\cap A(v)|=d(\operatorname{lca}(u,v))+1$. Thus the LCA measures shared upward dependency work. Disjoint targets cease to suffice when payloads read mutable cache state.

Early-stopping weighted-square result

Charge a fixed leaf write and each upward transition evaluation, including the first unchanged ancestor that certifies stopping. For every $h\ge1$, the two-state transition $m(x,y)=\neg x\land y$, followed by (h-1) identity carriers and a constant-zero root, gives two commuting sibling-leaf replacements with equal complete endpoints and equal root traces. Their two orders cost

$$4\quad\text{and}\quad 2h+4.$$

The gap (2h) is sharp because only the (h) nodes above the common parent can contribute differently, at most once per update. The semantic square is therefore resource-non-flat: its path weights do not descend to endpoints. The minimal witness has two states and (h=1); the cost remains algorithm-relative.

Unary-chain flatness classification

For a fixed parent-state square ((a,b,c,d)), every strict ancestor with unchanged sibling arguments induces a unary map $g_i:Q\to Q$. The prefix maps $\Phi_i=g_i\circ\cdots\circ g_1$ have nested kernel equivalences. The early-stopping propagation length

$$\lambda(x,y)=\sum_{i=0}^{h-1} \mathbf{1}[\Phi_i(x)\ne\Phi_i(y)]$$

is their coalescence ultrametric. The two order costs are

$$4+\lambda(a,b)+\lambda(b,d), \qquad 4+\lambda(a,c)+\lambda(c,d).$$

At level (i), the two routes have equal discrete path length exactly when

$$\Phi_i(b)\in{\Phi_i(a),\Phi_i(d)} \iff \Phi_i(c)\in{\Phi_i(a),\Phi_i(d)}.$$

Nonzero level biases cannot change sign as the kernel partitions coarsen. Therefore total work is order-invariant exactly when this membership balance holds at every counted level; no positive/negative cross-level cancellation is possible. The exact absolute gap is the sum of absolute level biases and is at most (2h), sharply.

Arbitrary incomparable-address composition

Let $w=\operatorname{lca}(u,v)$. Each branch update has an order-invariant local cost (B_U,B_V), boundary old/new states, and an arrival bit $\epsilon$ recording whether its child state at (w) changes. The four combinations of boundary states induce LCA states ((a,b,c,d)). Then

$$W(U;V)=C+\lambda(a,b)+\lambda(b,d), \qquad W(V;U)=C+\lambda(a,c)+\lambda(c,d),$$

where

$$C=B_U+B_V+\epsilon_U+\epsilon_V.$$

Thus all order asymmetry lies strictly above the LCA, the sharp gap is (2h) for (h) strict common ancestors, and the unary-chain flatness criterion applies unchanged. Branch depths and arrival bits alone are insufficient: XOR and $\neg x\land y$ at the LCA have identical direct branch data but gaps zero and two.

Native result references

• CRD-TA-1: reachable context theorem and syntactic quotient isomorphism.
• CRD-TA-2: sharp (n-2) distinguishing-depth bound.
• CRD-TA-F1: unreachable-parameter counterexample.
• CRD-TA-3: disjoint fixed-update square and exact LCA recomputation geometry.
• CRD-TA-F2/F3: state-dependent disjoint and fixed overlapping counterexamples.
• CRD-TA-4: sharp (2h) early-stopping order gap on a commuting square.
• CRD-TA-F4: semantic and root-trace commutation do not force resource invariance.
• CRD-TA-5: nested-kernel ultrametric, exact path-cost formula, and levelwise resource-flatness criterion.
• CRD-TA-F5: cross-level order-bias cancellation is impossible in the fixed deterministic ancestor-chain model.
• CRD-TA-6: exact branch-local plus LCA-ultrametric composition for arbitrary incomparable addresses.
• CRD-TA-F6: spatial branch/arrival data alone does not determine flatness.
• CRD-TA-7: immutable occurrence-addressed path copying preserves unfolded behavior and transition-evaluation work across shared descendants; fresh allocation/navigation are each (d(u)+d(v)) for two updates.

Persistent-DAG refinement fixed in Run 12

Retained hash-cons refinement fixed in Run 13

For one ordered binary root, retain a monotone global canonical table (H) with no garbage collection. With current root (r_{00}=f(x_0,y_0)), fixed left/right payloads (x_1,y_1), and candidates

$$r_{10}=f(x_1,y_0),\quad r_{01}=f(x_0,y_1),\quad r_{11}=f(x_1,y_1),$$

the two exact allocation counts are

$$A_{UV}=\mathbf{1}[r_{10}\notin H] +\mathbf{1}[r_{11}\notin H\cup{r_{10}}],$$

$$A_{VU}=\mathbf{1}[r_{01}\notin H] +\mathbf{1}[r_{11}\notin H\cup{r_{01}}].$$

For genuine changes, all four corner keys are distinct. Allocation is flat iff (r_{10},r_{01}) have the same initial membership status, and the sharp gap is one. The enriched endpoints $(r_{11},H_{\mathrm{final}})$ agree only when both intermediate roots already exist. If both are absent, costs agree but the retained sets differ.

The general acyclic-DAG minimum has three initial nodes. Let (z) be nullary, (a=f(z,z)) the current root, and (b=f(a,z)) a retained node; use old root (a) as both fixed payloads. This allocates one node left-then-right and two in the opposite order. At least three nodes are necessary because an asymmetric witness needs two distinct positive-arity roots and an acyclic DAG needs a distinct nullary descendant. If old and replacement children must all be nullary, the minimum is four: (z,o,f(z,z),f(o,z)). Thus canonical current-root equality does not erase retained history. Allocation is the rank growth (|H'|-|H|) in the table-inclusion order, under the fixed no-GC policy.

Updated tree-automaton result references

• CRD-TA-8: exact retained-hash-cons allocation formulas, membership-balance flatness criterion, sharp gap one, three-node acyclic minimum, and four-node nullary-payload minimum.
• CRD-TA-F8: a common final canonical root does not force path-independent allocation or a common retained-table endpoint.

Height-two retained hash-cons refinement fixed in Run 14

For one copied local parent on each branch followed by a common root, the two intern-request sequences are

$$p_1,r_{10},q_1,r_{11} \quad\text{and}\quad q_1,r_{01},p_1,r_{11}.$$

Let (C={p_1,q_1,r_{11}}) and $D=H\cup C$. Monotone interning makes allocation the number of distinct requested keys absent from (H), so

$$A_{UV}=|C\setminus H|+\mathbf{1}[r_{10}\notin D], \qquad A_{VU}=|C\setminus H|+\mathbf{1}[r_{01}\notin D].$$

Thus the local copies and final root cancel as a common candidate set. The gap remains sharply at most one, and flatness holds exactly when the two exclusive roots have the same membership status after common-set coverage. The final tables agree iff the exclusive roots coincide or both are already covered by (D).

Initial-table membership alone is not sufficient when structural identities can cross levels. A four-node acyclic aliased DAG has $r_{10}=q_1\notin H$ and $r_{01}\notin H$, but costs two and three because (q_1) is mandatory in both orders. Sort/rank separation that prevents such collisions recovers Run 13's simpler (H)-membership test.

Further updated tree-automaton result references

• CRD-TA-9: exact height-two common-set allocation and enriched-endpoint criteria, with sharp gap one.
• CRD-TA-F9: initial membership of exclusive roots is insufficient under cross-level canonical equality; close under common mandatory candidates.

Arbitrary-height retained hash-cons spine law fixed in Run 15

For a common copied spine with (m) path-exclusive canonical candidates on each route, let (C) be the identities requested in both orders and let $D=H\cup C$ be the descendant-closed common-coverage table. Write $X=(x_1,\ldots,x_m)$ and $Y=(y_1,\ldots,y_m)$ for the two strict structural dependency chains. Cross-level and cross-route identity coincidences are allowed.

If $\kappa_X,\kappa_Y$ are the lengths of the covered prefixes of the two chains, then

$$A_{UV}=|C\setminus H|+m-\kappa_X, \qquad A_{VU}=|C\setminus H|+m-\kappa_Y.$$

Hence flatness is exactly $\kappa_X=\kappa_Y$, the sharp gap is (m), and nonzero normalized level biases cannot have opposite signs. Height three is the case (m=2), so its sharp gap is two. The result is not an induction over separately analyzed heights: it is one arbitrary-(m) rank law.

The structural hypothesis is decisive. An unrestricted set model permits a cancelling profile by covering an ancestor while omitting its required descendant. That profile is not a valid acyclic hash-cons table. Final retained tables satisfy $H_{UV}=D\cup X$, $H_{VU}=D\cup Y$, so equal allocation remains weaker than equal retained representation.

Final updated tree-automaton result references

• CRD-TA-10: arbitrary-height covered-prefix allocation law, sharp gap (m), spine no-cancellation theorem, and exact retained-endpoint set criterion.
• CRD-TA-F10: unrestricted opposite-bias cancellation violates descendant closure on a structural spine.

Dependency-poset boundary fixed in Run 16

For an (m)-element exclusive structural dependency poset (P), common coverage induces order ideals $I_X,I_Y\in J(P)$. Exact allocation is

$$A_X=|C\setminus H|+m-|I_X|, \qquad A_Y=|C\setminus H|+m-|I_Y|.$$

The local absolute imbalance, net gap, and cancelled component are

$$L=|I_X\triangle I_Y|, \qquad G=\bigl||I_Y|-|I_X|\bigr|, \qquad K=(L-G)/2.$$

Opposite local biases occur exactly when the two ideals are incomparable. Universal sign coherence holds exactly when (J(P)) is a chain, equivalently when (P) itself is a chain. Every finite non-chain admits distinct equal-cardinality ideals and therefore perfect net cancellation with nonzero local imbalance. The smallest case is the two-element antichain.

This classification does not reopen the original path-copy model. A fixed single-occurrence replacement rebuilds one address-to-root prefix path, so the order-exclusive nodes of its intermediate version always form a chain. A forked exclusive region requires a multi-site, speculative, or otherwise expanded maintenance primitive.

Closed tree/hash-cons result references

• CRD-TA-11a/b: exact finite dependency-poset rank and cancellation law, with the chain/non-chain classification and minimal fork witness.
• CRD-TA-11c: native single-occurrence path copying cannot generate a forked exclusive region.
• CRD-TA-F11: descendant closure alone does not imply no cancellation outside chains.
• CRD-F-21: the abstract fork witness is not a counterexample inside the original update primitive.

Streaming computation

Fixed uniform model

Uniform quotient result

For every processor in this model, suffix-continuation equivalence is a single-sorted right congruence and the largest output-saturating congruence on reachable states. The quotient is the unique minimal reachable Moore processor.

For (G_j), the global quotient contains (j) pre-clock residuals and two stored-value residuals. Thus every exact processor needs (j+2) states and at least $\lceil\log_2(j+2)\rceil$ state bits, although each fixed cut has only one or two behavioral classes. The shortest suffix distinguishing pre-clock lengths $i<\ell<j$ has length $j-\ell$.

Fixed distinctions

• Single-sortedness follows from stationary symbol meaning, a common suffix domain at every history, and every-prefix output. It is not automatic for every object called a streaming algorithm.
• A fixed-horizon terminal-only processor is naturally graded unless its stop semantics is explicitly incorporated into a single-sorted interface.
• Timestamped symbols or an external clock move positional information into the interface; they do not eliminate its resource cost.
• The exact state and bit bounds do not apply to randomized, query-indexed, RAM, transducer-output, or approximate streaming models.

Native result references

• CRD-STR-1: largest single-sorted suffix congruence and minimal quotient.
• CRD-STR-2: exact (j+2) positional-clock state bound and suffix metric.
• CRD-STR-F1: constant per-cut width does not imply constant uniform state.

Finite rewriting systems

Fixed unary string-rewriting model

Exact result

Complete occurrence-labelled histories from (a^n) to (a) are permutations of the (n-1) original gaps. Quotienting by swaps of consecutive contractions with disjoint residual supports leaves exactly the planar full binary merge tree, giving

$$|\mathcal D_n|=(n-1)!, \qquad |\mathcal D_n/{\sim_\square}|=C_{n-1}.$$

The first failure of disjoint-square completeness occurs at (a^3), whose two bracketings use overlapping initial redexes. The symmetric length-two normalizing-path relation

$$((AB)C)\leftrightarrow(A(BC))$$

induces binary-tree rotations when closed under pre/postcomposition and derivation contexts. Together with disjoint squares, it connects all complete histories. It does not identify partial one-step paths and does not establish higher coherence.

At (a^4), the six complete gap-deletion histories form a raw transformation hexagon under five overlap moves and one disjoint move. Contracting the disjoint edge identifies the two schedules of the balanced merge tree and produces the five-tree rotation pentagon. Both graphs have cycle rank one, and no independent-move interchange supplies a face at this arity. One higher cell equating the two composites from the left comb to the right comb is therefore necessary and sufficient to fill this component.

At (a^5), proper contiguous clusters index every native higher face. Clusters of sizes two and four give six contextual copies of the arity-four pentagon; clusters of size three give three nested residual-interchange squares. The resulting complex has ((V,E,F)=(14,21,9)), every edge belongs to two faces, and deleting one pentagon leaves an explicitly collapsible complex. Hence it is simply connected and homotopy equivalent to (S^2), with (H_1=0) and $H_2\cong\mathbb Z$.

Fixed distinctions

• Occurrence-labelled edges are necessary. Plain word sequences collapse all complete histories to one length chain.
• Provenance is a history annotation, not part of the semantic endpoint.
• The overlap relation is symmetric. A one-way right rotation is not strongly connected.
• Connectedness of complete histories is weaker than coherence; the (a^4) tree-rotation quotient contains a pentagon loop with $\pi_1\cong\mathbb Z$ until one higher cell is attached.
• The raw occurrence-history boundary is a hexagon, not a pentagon; the pentagon appears only after quotienting the unique disjoint schedule edge.
• Rotation independence is not equivalent to disjoint leaf support. A rotation nested inside an opaque metavariable of another can commute; this begins at five leaves. Use residual four-cycles or disjoint active constructor occurrences.
• The arity-four filler is not yet an all-arity coherence theorem.
• Vanishing (H_1) is not used as a substitute for simple connectedness; the five-leaf result uses an explicit elementary-collapse certificate.
• The five-leaf (S^2) result concerns merge-tree classes after scheduling quotient, not the raw 24 occurrence histories.
• The result does not concern nonterminating systems, arbitrary critical-pair completion, derivational complexity, or proof of a general homotopy basis.

Native result references

• CRD-RW-1: factorial-to-Catalan-to-one complete-history classification.
• CRD-RW-F1: disjoint squares do not generate endpoint equality.
• CRD-RW-F2: word-sequence representation erases occurrence history.
• CRD-RW-2: exact raw hexagon, quotient pentagon, and minimal arity-four higher cell.
• CRD-RW-F3: connectedness does not imply coherence.
• CRD-RW-F4: disjoint leaf support misses nested contextual interchange.
• CRD-RW-3: exact five-leaf square–pentagon inventory and collapse to (S^2).
• CRD-RW-F5: (H_1=0) alone does not prove simple connectedness.

3SAT / branching / proof dynamics

Fixed calibration model

General fixed-order prefix model

For an arbitrary Boolean function $f:{0,1}^n\to{0,1}$, the native runtime carriers are graded by prefix length:

$$H_k={0,1}^k, \qquad e_{k,b}:H_k\to H_{k+1}.$$

Same-grade prefixes are behaviorally equivalent exactly when they agree under all completions of the remaining ordered variables. This family of relations is the greatest terminal-observation-saturating graded congruence and is computed backward from terminal truth values by pairs of successor classes.

The grading is operationally essential for unlabelled bit input. A reduced OBDD node records a residual Boolean function and the node's next tested variable, but variable skipping permits the same node to be reached at different prefix grades. Therefore a sequential update interface needs the grade/schedule, a variable-labelled update, or an explicit external clock.

Fixed distinctions

• Semantic residual functions are not interchangeable with residual CNF encodings.
• ROBDD state counts require the fixed variable order and reduction rules; unreduced decision trees and level-forced branching programs are different representations.
• The exact exponential fold is not evidence of SAT hardness or of a lower bound for DPLL, CDCL, resolution, or arbitrary OBDD orders.
• Future-equivalence is a congruence for typed, same-grade extensions. Equality between residuals at different grades is not append-compatible without a schedule coordinate.
• ROBDD node identity and ordered-prefix runtime state are not interchangeable when the input interface supplies only an unlabelled next bit.

Native result references

• CRD-SAT-1: exact prefix residual and ROBDD fold theorem.
• CRD-SAT-F1: representation dependence of syntactic residual counts.
• CRD-SAT-2: greatest graded-congruence theorem and backward recursion.
• CRD-SAT-F2/F3: grade-erasure and node-only update counterexamples.
• CRD-SAT-3: clause-minimal learned-resolvent separation. Two strict source clauses compute the same Boolean function before and after adding their binary resolvent, yet a fixed residual prefix changes root UP closure and exact successful-decision cost from one to zero.
• CRD-SAT-F4/F5: semantic equivalence does not determine propagation work, and the four-clause forced-literal candidate is not minimal without a root-unit restriction.

Learned-clause representation layer fixed in Run 17

For

$$D_0={u\vee a\vee b,\ u\vee a\vee\neg b}, \qquad L=(u\vee a),$$

one resolution step derives (L), and $D_0\equiv D_0\cup{L}$. Under (u=0), both residuals compute (a), but only the database retaining (L) unit-propagates (a). The semantic quotient therefore forgets a proof receipt that is operationally relevant to propagation and branching.

The result uses monotone retention of the source clauses plus the learned binary clause. Resolution closure, subsumption replacement, proof-checking cost, clause scans, first-UIP learning traces, and width-exact learned-clause representations remain separate models.

Product and incidence boundary fixed in Run 18

For satisfiable CNF components on disjoint variable sets, minimum successful DPLL decisions with exhaustive UP are additive. Applied to (k) copies of the Run 17 gadget, retaining local resolvents for $S\subseteq[k]$ gives exact cost (k-|S|) while preserving the semantic residual $\bigwedge_i a_i$.

This additivity belongs to the retained variable-clause incidence representation. Sound redundant cross clauses can connect the components while leaving the Boolean function unchanged; a strict-3CNF implication chain then reduces the same residual's cost from (k) to one. Pure resolution without weakening cannot generate such cross clauses from a disconnected source, so proof-rule locality is part of the resource statement.

Additional native result references

• CRD-SAT-4a: general component-additivity theorem and exact (k-|S|) law.
• CRD-SAT-4b: semantically invisible cross-incidence chain with cost collapse $k\to1$.
• CRD-SAT-4c: pure-resolution component-locality theorem.
• CRD-SAT-F6/F7: semantic product and arbitrary-entailment-to-CDCL inferences refuted.

Connected pure-resolution dependency chain fixed in Run 19

The smallest retained native witness uses one external strict trigger clause and two strict resolution parents. Under the trigger prefix, the parent pair is semantically forcing but unit-free; its learned binary resolvent exposes the forced literal to UP. The clause is obtainable by an actual conflict after one contrary decision, so the separation no longer depends on weakening.

Serial composition gives a connected source with (2k+1) strict clauses and (2k+3) variables. Every retained stage receipt removes exactly one minimum future decision:

$$\mu(D_{k,S},{u=v=0})=k-|S|.$$

This is not the Run 18 disjoint-product theorem. The representation is one connected guarded dependency chain, and the lower bound uses disjoint stage-decision sets rather than projection onto clause components.

Additional native result references after Run 19

• CRD-SAT-5a: connected three-clause conflict-derived receipt witness.
• CRD-SAT-5b: exact arbitrary-length guarded-chain resource law.
• CRD-SAT-5c: clause-minimality and bounded exhaustive variable-minimality.
• CRD-F-26: global pure-resolution-locality inference refuted; disconnected locality preserved.

Receipt acquisition and repeated-demand frontier fixed in Run 20

For the one-stage connected witness, an online CDCL episode can either take a successful decision and leave no persistent proof state, or deliberately take the contrary decision, incur one conflict, derive the binary resolvent, and retain it. Across (r) identical prefix-reset episodes the exact cumulative vectors are

$$(r,0,0)\quad\text{and}\quad(1,1,1)$$

for decisions, conflicts, and retained receipts. Learning at a later episode (t) gives ((t,1,1)) and is dominated by learning immediately. Under scalar weights ((d,c,s)), the exact threshold is

$$(r-1)d\ge c+s.$$

This supplies a native front-loading law: a conflict changes the persistent proof presentation and can prepay later inference. If the clause is deleted between episodes, the benefit disappears. The semantic residual alone records neither the acquisition history nor the prepared demand response.

Additional native result references after Run 20

• CRD-SAT-6a: exact repeated-demand resource vectors and learning-time law.
• CRD-SAT-6b: Pareto frontier and weighted break-even theorem.
• CRD-F-27: unconditional “learned clauses are beneficial” inference refuted.

Finite demand coverage fixed in Run 21

For complete assignments to the trigger inputs ((u,v)), the one-stage receipt is operationally useful only at ((0,0)). There the source trigger propagates (a_0), making the learned clause unit; on the other three demands the trigger is already satisfied and the clause remains nonunit.

For a sequence of (r) such demands with coverage count $H\ge1$, optimal online acquisition has exact resource vector

[ (r-H+1,1,1). ]

The net decision saving is (H-1), because one covered opportunity is consumed learning the receipt. The acquisition decomposition

[ D_j=r+p_j-H_{>j} ]

separates the premium paid at the acquisition demand from the remaining covered demands. In this gadget, learning on an uncovered demand before the first covered one ties waiting: its extra decision is cancelled by one extra reuse. Thus the optimal total depends only on (H), not demand order.

Additional native result references after Run 21

• CRD-SAT-7a: exact four-demand coverage and acquisition table.
• CRD-SAT-7b: sequence-order-independent (H-1) amortization theorem.
• CRD-F-28: gross coverage cannot be equated with online net saving.

Proof-state fiber versus semantic OBDD width fixed in Run 22

For the connected guarded chain, retain any subset $S\subseteq[k]$ of the pure-resolution receipts

$$L_i=(\neg a_{i-1}\vee a_i).$$

At the trigger prefix, every such database computes the same residual

$$f_k=\bigwedge_{i=1}^k a_i.$$

Under the fixed order $a_1<b_1<\cdots<a_k<b_k$, the reduced OBDD skips all (b_i), has (k) nonterminals, and has nonterminal width one. Nevertheless, the persistent clause-database fiber has exactly (2^k) states, one per receipt subset. Minimum successful-decision observation gives the coarser law

$$\mu(D_{k,S},\alpha)=k-|S|,$$

so there are (k+1) resource classes and the class of cost (q) has multiplicity $\binom{k}{q}$. Semantic quotienting collapses all of them to one state:

$$2^k\longrightarrow k+1\longrightarrow1.$$

This is a representation fiber over one semantic node, not a claim that the proof database is intrinsically exponential. Full resolution closure inserts all receipts; deletion, subsumption, provenance, activity, or watch metadata change the state space. Stage-specific propagation demands can distinguish equal-cost subsets, so the (k+1)-class quotient is not automatically a congruence for every solver interface.

Additional native result references after Run 22

• CRD-SAT-8a: one semantic residual and width-one reduced OBDD across all receipt subsets.
• CRD-SAT-8b: exact $2^k\to k+1\to1$ proof/resource/semantic quotient law.
• CRD-F-29: semantic width and bounded receipt width do not bound persistent proof-state multiplicity.

Demand-indexed proof-state information fixed in Run 23

For each stage (i), the read-only readiness demand seeds (a_{i-1}=1) and asks whether current-database unit propagation assigns (a_i=1). On receipt state $S\subseteq[k]$, its answer is exactly

$$d_i(S)=\mathbf 1[i\in S].$$

Thus the full answer vector is $\chi_S$. Any representation answering every stage demand exactly must be injective on the (2^k) receipt states and hence requires at least (k) retained bits; the characteristic bit mask attains the bound.

For a restricted stage family $T\subseteq[k]$, the exact quotient is

$$S\equiv_T S'\iff S\cap T=S'\cap T,$$

with (2^{|T|}) classes and exact information cost (|T|) bits. This kernel is a congruence for receipt-set union. The Boolean-cube projection therefore gives a precise range/space law:

[ B(T)=|T|. ]

Run 22's (k+1) scalar decision classes are only Hamming-distance layers and do not preserve coordinate readiness. Semantic derivability also differs from readiness: every (L_i) is entailed by the source, but absent receipts are not available to unit propagation without derivation work.

Additional native result references after Run 23

• CRD-SAT-9a: propagation-demand vector equals the receipt characteristic vector.
• CRD-SAT-9b: exact (|T|)-bit necessity and sufficiency for demand set (T).
• CRD-F-30: scalar decision-cost equivalence does not preserve coordinate preparedness.

Interaction-depth separation fixed in Run 24

For a database (D), let $\sigma_t(D)$ record exhaustive UP closures and conflict observations for all consistent seeds of size at most (t). A strict (r)-CNF has trivial behavior below depth (r-1): with fewer assignments, every unsatisfied clause retains at least two unset literals.

The minimal strict-3CNF witness is

$$D=(x\vee\neg z\vee\neg p)\wedge(y\vee\neg z\vee p),$$

with pure resolvent

$$L=(x\vee y\vee\neg z).$$

Then $D\equiv D\wedge L$ and their singleton signatures agree. Under the two-literal seed (x=0,y=0), only the augmented database propagates (z=0). Accordingly, false-first DPLL order (x<y<z<p) uses three decisions on (D) and two on $D\wedge L$, with zero conflicts in both traces.

The witness is minimal: no satisfiable three-variable strict-3CNF entails a new strict-3 clause, and one strict-3 source clause cannot entail a distinct strict-3 clause. Thus the smallest parameters are four variables and two source clauses.

The mechanism is interaction depth, not receipt-coordinate identity. Agreement at $\sigma_1$ does not control activation at $\sigma_2$. The branching effect remains policy-dependent and disappears for orders that choose (z) first.

Additional native result references after Run 24

• CRD-SAT-10a: strict-width low-order UP-vacuity lemma.
• CRD-SAT-10b: minimal semantic/singleton-signature equality with fixed-policy branching separation.
• CRD-SAT-10c: four-variable/two-clause analytic minimality.
• CRD-F-31: singleton UP equivalence does not determine branching work.

Arbitrary-width interaction law fixed in Run 25

For $r\ge3$, let

$$A_r=z_1\vee\cdots\vee z_{r-3}\vee\neg z_{r-2},$$

$$D_r=(A_r\vee x\vee\neg p)\wedge(A_r\vee y\vee p), \qquad L_r=(A_r\vee x\vee y).$$

The source is connected, (L_r) is its pure resolvent on (p), and $D_r\equiv D_r\wedge L_r$. Both databases are strict width (r), so their UP signatures agree for every seed size through (r-2). At the size-(r-1) seed falsifying $x,y,z_1,\ldots,z_{r-3}$, only the learned database propagates (z_{r-2}=0).

Under false-first order

$$x<y<z_1<\cdots<z_{r-2}<p,$$

the base uses (r) decisions and the learned database (r-1), with no conflicts on either successful trace. The family uses the minimum (r+1) variables and two source clauses.

Interaction depth is geometrically the activation codimension of the learned clause in the partial-assignment cube. For any fixed (t), width (r=t+2) escapes every UP observation through depth (t). The unresolved boundary is fixed width: Run 25 does not show unbounded interaction depth inside strict 3CNF.

Additional native result references after Run 25

• CRD-SAT-11a: exact agreement through (r-2) and first separation at (r-1).
• CRD-SAT-11b: exact (r) versus (r-1) decision law and minimality.
• CRD-SAT-11c: no fixed UP depth controls the resource across unbounded width.
• CRD-F-32: width-independent bounded-depth sufficiency refuted.

Fixed-width absorption completeness fixed in Run 26

For a normalized clause (C) and literal $\ell\in C$, its boundary seed falsifies every member of $C\setminus{\ell}$. Say (C) is UP-absorbed by (F) when unit propagation from every such boundary seed derives $\ell$ or conflicts.

If $G=F\cup A$ and every clause in (A) has width at most (w), the following are equivalent:

$$\sigma_{w-1}(F)=\sigma_{w-1}(G),$$

every clause of (A) is absorbed by (F), and

$$\operatorname{UP}_F(\alpha)=\operatorname{UP}_G(\alpha) \quad\text{for every consistent seed }\alpha.$$

The mechanism is persistence of unit derivations under seed extension. Once a boundary derivation exists, an extension preserves it or conflicts; it cannot silently disable it. At the base closure under an arbitrary seed, no absorbed addition can therefore be the first new unit or conflict.

The cutoff is sharp because Run 25 agrees through depth (w-2) and separates at (w-1). Hence width-three additions have exact completeness radius two, regardless of auxiliary source structure. Under a fixed variable order and polarity, complete DPLL traces also coincide. Syntax-sensitive heuristics and proof-state resource coordinates remain outside this result.

Additional native result references after Run 26

• CRD-SAT-12a: depth-(w-1), absorption, and all-seed UP equivalence.
• CRD-SAT-12b: sharp fixed-width completeness radius; strict-3CNF radius two.
• CRD-SAT-12c: exact fixed-order DPLL-trace preservation.
• CRD-F-33: auxiliary strict-3CNF chains cannot evade the complete depth-two UP signature.

First-UIP reason-path compression fixed in Run 27

For $m\ge1$, the connected strict-3CNF (F_m) consists of

$$(x\vee y\vee a_1),$$

the two guarded forward chains

$$(x\vee\neg a_i\vee a_{i+1}),\qquad (y\vee\neg a_i\vee a_{i+1}),$$

ending at (b), and conflict clause

$$(x\vee y\vee\neg b).$$

The retained shortcut $L=(x\vee y\vee b)$ is derived in (m) pure resolution steps along either displayed chain and is UP-absorbed by (F_m). Thus (F_m) and $F_m\cup{L}$ are semantically equal and have identical UP closure/conflict behavior on every seed.

Fix decisions (x=0) then (y=0), earliest-clause unit propagation, immutable first reasons, learned-clause priority, and latest-trail first-UIP analysis. The base reason graph reaches (b) through $a_1,\ldots,a_m$; its conflict analysis takes (m+1) pivots. The retained shortcut supplies the reason for (b), so analysis takes one pivot. Both learn $x\vee y$, make two decisions, encounter one conflict, and remain at analysis width three.

The exposed resource is path length in the retained reason DAG. Complete UP closure records consequences but erases the provenance paths certifying them. Clause priority and reason selection are therefore explicit representation coordinates; the theorem is not a heuristic-independent CDCL lower bound.

Additional native result references after Run 27

• CRD-SAT-13a: connected strict-3CNF semantic and all-seed UP equivalence with an absorbed pure-resolution shortcut.
• CRD-SAT-13b: exact (m+1)-versus-one deterministic first-UIP pivot law, with common learned clause and analysis width.
• CRD-SAT-F13 / CRD-F-34: complete UP closure does not determine provenance-sensitive conflict-analysis work.

All-schedule first-UIP spectrum fixed in Run 28

Under (x=y=0), write

$$v_0=a_1,\ldots,v_{m-1}=a_m,v_m=b.$$

Run 27's base reason system reduces to the inconsistent implication path

$$(v_0),\quad (\neg v_{i-1}\vee v_i)\ (1\le i\le m),\quad (\neg v_m),$$

with two provenance copies of each internal edge. The augmented database adds ((v_m)).

Allow arbitrary sequential unit-clause choice, arbitrary available immutable reason, and arbitrary detected falsified clause. The exact first-UIP pivot spectra are

$${c(\tau):\tau\in\mathcal T(F_m)}={m+1}, \qquad {c(\tau):\tau\in\mathcal T(G_m)}={1,m+1}.$$

Every trace learns $x\vee y$. Therefore the best available proof gap remains exactly (m) without clause priority:

$$c_{\min}(F_m)=m+1,\qquad c_{\min}(G_m)=1.$$

The worst cases agree at (m+1). The shortcut introduces a one-edge proof route but does not force an online scheduler to use it.

The width audit is separate. (F_1) has width three; for $m\ge2$, mixed guard provenance gives width spectrum ({3,4}). Every (G_m) trace stays at width three. Minimum first-UIP cost is existential proof availability, not an online-runtime guarantee or a bound on finding a shortest reason trace.

Additional native result references after Run 28

• CRD-SAT-14a: exact all-schedule cost spectra and common learned clause.
• CRD-SAT-14b: priority-independent minimum gap (m), with equal worst-case cost.
• CRD-SAT-14c: exact all-schedule width spectra for the family.
• CRD-SAT-F14 / CRD-F-35: a stored shortcut does not guarantee realized speedup under arbitrary scheduling.