The \(\gamma\)–\(\theta\) conjecture in one-guard eternal domination

The problem, in plain language

If the smallest dominating guard team can defend forever while moving only one guard per attack, must the vertices already split into that many cliques?

For a finite simple graph \(G\), let \(\gamma(G)\) be its domination number and let \(\gamma^\infty(G)\) be its eternal domination number in the standard one-guard model: attacks occur only at unoccupied vertices, and exactly one adjacent guard moves along one edge to the attacked vertex. Here \(\theta(G)\) is the clique-cover number, equivalently \(\chi(\overline G)\). It is not the Lovász theta function \(\vartheta(G)\).

*Relative to MacGillivray–Mynhardt–Virgile's published exhaustive computation through order 11.

The certified order-12 frontier

The proof first uses the parameter chain \(\gamma\leq i\leq\alpha\leq\gamma^\infty\leq\theta\), component additivity, and a classical half-order domination theorem. A hypothetical minimum order-12 counterexample is connected and its common value can only be \(3\), \(4\), or \(5\). Those three cases are then closed by different mechanisms.

The campaign independently reproduced the published 56-graph appendix and exhaustively checked connected unlabeled graphs through order 9. It did not rerun the original billion-graph enumeration at order 11, so that published lower-order result remains an explicit premise rather than a campaign certificate.

What is known at order 13

The complete order-13 search has not been finished, and no lower bound of 14 is claimed. The common-parameter-three slice is now certified empty; parameters four and five remain open.

• \(C_{11}\) branch — proved impossible. A near-spanning odd hole leaves only two outside vertices, and a direct one-guard attack argument forces \(\gamma^\infty\geq4\).
• \(C_9\) branch — certified impossible. The exact 9,802-variable, 32,108-clause formula is refuted by an addition-only RUP proof and a separate LRAT proof, both independently reconstructed and checked.
• Parameter-three full-response branch — certified impossible. Regardless of which odd-hole template occurs, no order-13 parameter-three counterexample can have a full family-response target at a maximum independent triple.
• Four-neutral no-full obstruction — certified impossible. A 1,222-variable formula proves that four neutral vertices and two overlapping response types cannot coexist on 13 vertices. A three-neutral equality graph is a sharp positive control.
• Residual no-full branch — certified impossible. A clean-room 9,802-variable, 84,614-clause formula covers every remaining \(7+3,8+2,9+1,10+0\) signature census. Its deletion-free 156,205-addition proof verifies strictly RUP-only with zero RAT lemmas.
• The residual contradiction is radius-two and three-way. An independently reconstructed 76,214-clause subformula remains UNSAT after deleting every closure obligation from triples disjoint from the reference state. Radius one and every two-of-three anchor-slice relaxation are SAT, so all three depth-two slices interact essentially.
• Complete \(k=3\) conclusion. No graph on 13 vertices satisfies \(\gamma(G)=\gamma^\infty(G)=3<\theta(G)\). Equivalently, any order-13 counterexample must have common parameter four or five.
• Parameters \(4\) and \(5\) — still live. The parameter-five lane has a proved ten-vertex-kernel reduction; neither slice has a complete finite exclusion.

A separately proved pair of infinite near-miss families explains why the equality conditions are delicate: for every relevant odd length, they satisfy \(\gamma=i=\alpha=3\) while \(\gamma^\infty=\theta=4\). They are not counterexamples, but they sit one eternal guard away from the target.

Reproducibility and exact scope

The release contains the paper source, theorem notes, exact formulas, compressed proofs, source-pinned proof checkers, independent reconstructions, mutation tests, acceptance records, and one-command replay wrappers. Search output is never promoted solely because a solver prints UNSAT.

Run these commands from the gamma_theta_eternal_domination/ campaign directory. These checks certify the exact encoded finite statements. They are not peer review, and they cannot by themselves establish literature novelty or the correctness of every imported theorem.

Current strategy: pursue a decisive proof before order 14

After a first day dominated by exact finite certification, the campaign is deliberately rebalanced toward the universal question. A narrowly targeted structural split and two independently reconstructed certificates have now closed both the full-response and no-full branches at order 13 for common parameter three. The proof lane remains primary; a blind order-14 search has not begun.

The central proof object is a minimum counterexample. Equality collapses all four intermediate parameters:

For every independent set \(A\) smaller than this common value, deleting \(N[A]\) leaves a smaller equality graph and projects every eternal family exactly. In a minimum counterexample, the remainder must also have \(\theta=\gamma\). This supplies a recursive family of clique and common-neighborhood constraints. In the no-full-list branch, parity-and-attack theorems exclude shared-port odd returns and every separated two-gate odd bigon. For the shortest remaining three-gate boundary, the no-dominating-pair condition now forces a common-neighbor witness of exactly the third response type. A hostile Boolean audit also found the crucial limit: the witness contributes only one oriented implication, so replacing a gate and declaring a shorter cycle is unsound. The live no-full target must coordinate witnesses at several critical pairs, not just one. Independently, target-response membership now propagates across maximum independent states for every guard number \(k\), producing exact active/inactive coloring identities. Any hypothetical critical full-target counterexample needs at least three facet components with nonempty but globally incompatible responder-color sets. A sharp control proves that static equality and every one-step response still do not force a common color; multi-step eternal closure is the indispensable remaining mechanism. Order-13 parameters four and five remain bounded fallback lanes rather than the campaign's definition of success.

The latest structural steps complete the exact-two-list regime at parameter three and identify the precise exceptional-response mechanisms that remain. Physical response lists turn complement coloring into signed parity; every inconsistent cycle shortens to one of five type skeletons, and direct one-guard attack trees exclude all five. Thus a putative \(k=3\) counterexample must have a singleton or full response list at every retained maximum independent state. In the singleton branch, the canonical shortest exact static two-unit obstruction—an induced mixed \(P_4\)—is impossible at every order: one forced endpoint defect leaves an eight-vertex core whose 32 completions all have empty compatible one-guard kernels. The stronger family-only analysis proves that both omitted endpoint swaps dominate. Every single-hit deletion of either endpoint now transports the exact list system and lowers rank by one, so a realization must eventually enter one of eight genuine multi-hit cells. Those cells, longer chains, and the observed-but-uncertified order-22 computation remain open.

In the full-list branch, the inactive induced complement is bipartite, but static coloring data—even with the exact \(\gamma=3\) target condition—does not synchronize it globally. Repeated attacks control every physical-link component even when anchorless vertices intervene. A selected color is now proved future-safe exactly when its root-swap-restricted greatest kernel is nonempty. For all full targets at one root, one cumulative ban gives an exact characterization: a clique partition exists precisely when some proper full-core assignment leaves a nonempty kernel and its residual exact list 2-CNF is satisfiable. If all three restricted kernels are empty, their three terminal descents cannot all end at their own singleton palettes. A nonsingleton direct-root terminal is impossible at every rank. Every positive-rank terminal exposes a dominating unbanned lower-rank alternate; at minimum rank that alternate is already outside the unrestricted greatest family. Rank-zero anchor restoration now has an exact attacked-anchor/shared-anchor palette split. In a rank-zero nonroot corridor, each secondary color forces two retained witness states and reappears at the mover or its missed witness. Three nonsingleton corridor rows therefore reduce to a directed 3-cycle or a 2-cycle with a tail on the root colors. Every independent target–terminal completion now has a nonempty retained two-branch split, and two secondary witness neighborhoods cover the whole completion clique. A transfer endpoint trapped inside the ban cannot remain there: it forces an unbanned source-color state, while full terminal rows polarize their two witness sets across the ban. Exact controls show why this is still not a rank descent—the forced return can raise rank \(0\to3\), and the escape can preserve rank \(0\to0\). The remaining proof must eliminate a global rank-preserving escape cycle using the full \(\gamma=3\) condition.

A separate exchange route has also reached a sharp boundary. Greatest-family maximality alone does not force reverse responses, and equality of finite deletion ranks is false. Nevertheless, a complete independent order-nine census found simultaneous survival in all 392,155 complementary exchanges. An all-parameter theorem removes the static half: whenever one exchange is eternally active, every reverse maximum-independent endpoint is dominating. Finite-horizon response transport proves that reverse deletion rank is Lipschitz across independent-state ridges and that every single-hit deleting attack descends by exactly one rank. Thus any rank-one or globally minimum obstruction must be a genuine multi-guard collision. At parameter three those collisions have exactly six local rows. XQ0, XQ1, QQ0, and AQ0 are impossible; every AQ1 collision recreates QQ1 on a new independent root with the same blocker. Only one canonical QQ1 normal form remains at rank one. Every one of its completion vertices hits all four side vertices, and the no-dominating-pair condition forces a genuinely external hot witness; all cold witnesses are excluded. Its completion reverse state has rank at most three, with exact top diamond \(1,2,2,3\). Every completion state \(\{u,x,d\}\), all five non-omitted inner corners, and every outer completion bow-tie are now proved retained; every nondegenerate outer edge is reciprocal. The two equality-forced witness layers are also pointwise coupled: every hot witness \(w\) of \(\{u,d\}\) and every original witness \(z\) of \(\{u,x\}\) form a retained state \(\{u,w,z\}\). A hostilely checked 18-vertex boundary graph shows the exact limit of fixed-anchor iteration: even all named local obligations plus non-domination of every pair touching the distinguished independent triple allow thirty dominating pairs entirely in the auxiliary layer, and witnesses can recycle in two-cycles. The full pair-repair condition is now compressed by a new all-order dichotomy valid for every eternal triple-family: every vertex pair has a retained central fan whose common-nonneighbor tokens form a clique and uniquely exchange, or it is an adjacent edge active in both directions. This removes the freshness problem but leaves a finite branch-coupling problem between the repair fans and the saturated QQ1 layers. A purely static substitute cannot work: \(\overline{C_7}\) already has connected \(P_4\) links and all proposed local conditions while remaining four-chromatic. A complete discovery scan through order ten found no asymmetric orientation among 458,696 active orientations; this last scan remains observed rather than certified pending an independent order-ten reconstruction.

For arbitrary \(k\), freezing any nonempty proper face of a retained maximum independent state produces the exact lower-parameter equality projection in one step. The abstract obstruction \(Y_k=K_{k-3}\vee P_4\) now has a stronger equality-specific boundary: every clean realization has \(n\ge2k+9\), or \(n\ge2k+10\) when its two endpoint defect sets are distinct, while the dirty nonindependent carrier remains open. The primary targets are therefore bridge propagation in the singleton branch, anchorless and residual vertices in the full-response branch, and the all-parameter dynamic palette-gluing statement.

What the proof-first pivot has established

The first universal passes did not resolve the conjecture, but they produced independently checked structural results rather than only failed sketches.

• Restoration and Hall obstruction. In a candidate with \(k=\alpha\), fix a maximum independent \(k\)-guard state. If an eternal \(k\)-family exists, its legal one-guard replacement lists satisfy Hall's inequality on every independent outside set. A Hall violation is therefore a compact certificate that \(\gamma^\infty>\alpha\).
• Exact shared-response core. Private neighborhoods are already cliques. The unresolved vertices form a constrained response-list coloring problem; a compatible coloring is exactly the missing \(k\)-clique partition. Collision-transfer and minimal-core lemmas sharply restrict any obstruction, but do not yet eliminate it.
• Frozen-color induction. Starting from \(\gamma=\gamma^\infty=k\), freeze one guard and keep precisely the attacks whose response lists omit that guard. The retained family becomes a genuine one-fewer-guard eternal family on an induced graph with \(\gamma=\alpha=\gamma^\infty=k-1\). At \(k=3\), the proved parameter-two case makes every such complement projection bipartite. This rules out the previously live odd-cycle core with a common two-color list.
• Exact projection gluing. When no response list contains all three colors, orienting the connected components of the three valid two-color projections is exactly a 2-SAT problem. Singleton lists impose parity units and complement edges between different two-lists impose the cross-projection clauses. The mixed four-vertex path is the inclusion-minimal two-unit/one-clause obstruction. A successful gluing, when it exists, transports unchanged across independent-state ridges.
• Cross-state covariance. Every ordering of the target positions between two independent family states has a supported monotone one-guard path. Across states sharing two guards, the exchanged-vertex transposition transports every family-response list exactly. Closed paths preserve the response-incidence system, although the resulting permutation need not be trivial.
• The mixed path forces a hub-free \(C_5\). Under the missing equality \(\gamma=\alpha=\gamma^\infty=3\), the mixed path now has a proved saturation theorem valid for every specified eternal family: its middle-pair witness clique sees both path ends and has both endpoint response colors; three forced configurations share the end ridge; and the common complement neighborhood of that ridge is a nonempty external clique. Every vertex of this clique closes the path to an induced \(C_5\) in the complement. The odd-wheel theorem makes that \(C_5\) hub-free.
• The exact mixed pattern needs at least 12 vertices. Two further nonempty end-edge witness cliques are forced. They are disjoint from each other and from every co-state witness clique. Together with the earlier witness systems, this gives five distinct external vertices beyond the seven reference/path vertices. This is an order floor for one exact response pattern, not a new global counterexample frontier.
• The shortest static gluing obstruction needs at least 14 vertices. If the static lists on the mixed \(P_4\) are exact, one-guard restoration forces the same exact family lists. The two endpoint swaps then require distinct static-defect vertices beyond the five previously separated witness systems. This is a conditional floor for \(Y_3\), not a general order-14 counterexample bound.
• The full-response deletion branch is sharply localized. A vertex that can replace any of the three reference guards forces three disjoint clique spokes, a second external clique layer, and an isolate-free bipartite link in the complement. Ridge covariance puts different spoke types on opposite sides of every link component, so all three colors pass the local test. Deleting a unique full-response vertex gives either a three-colorable critical graph or an inherited \(\gamma=2,\alpha=\gamma^\infty=3<\theta\) near-miss. In the critical branch every deletion coloring has three pairwise link Kempe connections and every deletion clique partition forces a cross-part one-guard response.
• Odd lollipop subdivisions are excluded. Every minimal unsatisfiable 2-CNF is a two-unit chain, a one-unit lollipop, or a unit-free bicycle. Exact one-guard attack trees rule out the canonical two-variable bicycle and every odd physical subdivision of the canonical one-unit lollipop that uses one common terminal port and stays inside one omitted-color projection. The proof permits extra complement edges and never treats a dynamically missing response as a graph nonedge. The later signed-balance theorem closes the remaining exact-two-list bicycles; singleton-list chains and full-list color synchronization remain open.
• Domination equality forces a cap-and-escape ladder. Under \(\gamma=3\), every complement connector edge whose endpoints dynamically omit one response color has a nonempty clique of triangle caps. Every cap recovers the omitted color and is adjacent in \(G\) to every other vertex supporting that color. In the exact separated-port core, this forces a positive residual cap and a new omitted-color escape. A checked equality graph later showed that cap repetition can close harmlessly around an even cycle, so any contradiction must also use terminal units or cross-port clauses.
• Full-response witnesses are anchor-pure and disjoint. A literal one-guard attack shows that every second-layer witness belongs to exactly one anchor layer and carries both cross-anchor responses. The three layers cannot overlap, so a full response forces three spokes and three further witnesses outside the neutral set. This raises the exact separated-port floor to 15 vertices.
• Every two-list has a physical representative. If a neutral vertex has two retained responses, two forced outside vertices replicate those responses, and the omitted anchor is a genuine graph nonedge at the terminal representative. Two possibly distinct neutral vertices with overlapping response pairs force six nonneutral witnesses. This closes the earlier dynamic-omission gap for individual variables, but not yet for connector edges between several variables.
• The complete order-13 parameter-three slice is certified empty. The full-response certificate first excludes one half. In the no-full half, a separately reconstructed four-neutral obstruction gives \(|Q|\le3\), and a final 84,614-clause formula covers every residual signature census. Its deletion-free proof has 156,205 RUP additions and zero RAT lemmas. Therefore any order-13 counterexample must have common parameter four or five.
• Literal physicalization is exact; clause-edge transport is false. A dynamic two-list port and its physical representative are joined by a length-two complement path and define the same Boolean event. But the checked equality graph LFzJbZYhdrDZdM has \((\gamma,\alpha,\gamma^\infty,\theta)=(3,3,3,3)\) and a 142-state eternal family in which the unique same-sign physical representative loses a specified complement cross-edge. Algebraic substitution is sound; substituting that vertex into a graph attack is not.
• Failed incidence has one tight local normal form. If the two physical endpoints of an original cross-clause are adjacent in \(G\), domination equality forces a common complement cap. The three displayed complement edges form a virtual rainbow triangle. Every cap list except the exact third two-list yields a local unit and a length-two implication arm. Exact equality controls realize the third-color gate, so a universal proof must exclude global chains of those gates rather than assuming incidence transport.
• Shared-port odd gate returns are impossible. Exact two-list ports have a binary chirality. Tight third-color gates preserve it, while a connector path in one omitted-color projection flips it according to path parity. A direct one-guard attack excludes every odd two-cap fork, at every subdivision length and without using \(\gamma=3\). A 14-vertex equality control realizes an even return, so oddness is essential. This theorem alone does not cover separated ports.
• Separated two-gate odd bigons are impossible. For two vertex-disjoint connector paths in different frozen projections, the dead boundary states supplied by tight gates force equal path parity. In the unit-free no-full branch, free components of different types cannot intersect without creating a singleton-list unit. The theorem therefore eliminates every two-gate odd bigon, even with four separated ports and arbitrary subdivisions. The exact holonomy target now requires at least three tight gates.
• The shortest three-gate boundary forces the right witness type. A pair that would dominate without an outside common complement neighbor instead acquires a third-type almost-cap. Two explicit one-guard attack trees and a complete collision audit prove the statement. The checked gamma-two control shows exactly why the no-dominating-pair hypothesis is necessary.
• A tempting gate-shortening rule is false. The almost-cap's two arms eliminate to one oriented endpoint implication, not the equality relation of a tight gate. A checked 19-vertex gamma-two control goes further: the two arms merely subdivide an already essential clause, increasing one marked path while the minimal obstruction remains a unit-free bicycle. This prevents an invalid local descent from being mistaken for a universal proof; the surviving gamma-three route must use witnesses at several pairs.
• Target responses propagate across facets for every \(k\). For a fixed target, a guard's response membership is independent of which retained maximum independent \(k\)-state containing that guard is used. Forced unoccupied attacks transport one response state to the other. This defines a global active set meeting every maximum independent state, and every deletion \(k\)-coloring satisfies exact active/inactive color identities on facet-ridge components. In the critical full-target branch, at least three such components are required and their total responder-color intersection must be empty.
• Static coloring data cannot finish the full-target branch. An exact 11-vertex equality deletion has a covariant inactive \(C_5\), and every deletion three-coloring uses all three colors there. Adding a target makes every prescribed one-step response legal and dominating, yet the three-guard eternal kernel becomes empty while four guards work. The checked control is not a counterexample; it proves that a valid universal argument must use repeated one-guard closure, not merely facet covariance or one-step domination.
• The shortest complete three-gate obstruction is impossible. Under \(\gamma=3\), the three common-neighbor witnesses in the canonical length-\((1,1,1)\) odd boundary are forced to be dynamic ports of the three exact types. Each creates a sealed positive cap, and the three caps are incompatible even after every possible collision is considered. The theorem closes this geometry only, not arbitrary subdivisions or longer gate cycles.
• Every exact-two-list port is now physical. A stronger two-pair argument shows that even one sealed same-color positive cap is impossible. Combined with the accepted connector and fan lemmas, this rules out every dynamic omitted-anchor incidence. Thus \(L_S(t)=N_G(t)\cap S\) throughout the unit-free no-full \(k=3\) branch. The previous response-versus-nonedge ambiguity is gone, although outside cross-clause edges still cannot be transported.
• Inactive links remain colorable after adjoining the target. In a minimum counterexample at arbitrary \(k\), freezing an inactive guard produces a proper equality projection with clique-cover number \(k-1\). It contains the target together with the guard's entire complement link, proving that this suspension has \(\chi=\omega=k-1\). The independent local colorings are not yet globally synchronized.
• An inactive induced \(C_5\) is impossible. Five retained edge-witness triples and ten absent target responses form a finite local obstruction. All 52 witness-identification patterns have independently reconstructed CNFs and replayed DRAT proofs. Therefore the triangle-free inactive complement in the critical \(k=3\) deletion branch has no induced \(C_5\); any remaining odd cycle has length at least seven. A checked equality control contains an inactive induced \(C_4\), so parity is essential.
• Every arbitrary exact-two-list bicycle is impossible. Physicality supplies a same-type complement mate for every outside port, forcing universal side-purity and a transversal third-type triangle on every cross-type edge. Proper complement coloring becomes an exact signed-parity system. Any shortest unbalanced cycle has length at most five, and the complete orbit list consists of five residual type words. Six explicit adaptive attack trees, including both witness-collision branches, exclude them all. Hence every \(k=3\) counterexample must exhibit a singleton or full response list at every retained maximum independent state.
• An inactive induced \(C_7\) is also impossible. The seven rim-edge witnesses have 877 possible equality patterns, forming 93 dihedral orbits. An independent constructor rebuilt every representative formula and replayed all 93 DRAT proofs. Together with the triangle and \(C_5\) exclusions, any surviving inactive odd hole in the equality-critical full-list branch has length at least nine. Even \(C_4,C_6,C_8\) controls show that an arbitrary-length argument must genuinely use odd parity.
• Frozen projection now works for every anchor face. Freezing any nonempty proper subset \(A\) of an independent retained \(k\)-state directly produces an equality projection with \(k-|A|\) guards; iterated projection and unchanged-list assumptions are unnecessary. In a minimum counterexample every proper static response-palette slice is therefore colorable, and a jointly inactive face has an exactly \((k-|A|)\)-colorable target-link suspension. The remaining induction statement is a global compatibility theorem. The abstract family \(K_{k-3}\vee P_4\) proves that proper-palette colorability plus all currently accepted list conditions is not enough without graph-game dynamics.
• Singleton responses are forced buffers, not automatic contradictions. The no-full singleton branch has an exact terminal inventory: an immediate fixed projection certificate, a propagated one-/two-unit chain, or a residual unit-free bicycle. Every dynamic exact-two port creates a sealed positive cap; domination equality then forces a singleton buffer that pins an exact-two cap to its safe color. Two independently reconstructed equality controls show that both buffered caps and dynamic ports in unpinned components really occur. This sharply narrows the singleton route without closing it.
• Fixed singleton certificates are impossible. In every frozen two-guard projection, a retained pair lying in one connected bipartite-complement component must cross its bipartition. This automatically aligns every fixed singleton marker and forces exact-two endpoints into free components. False constants, fixed/free units, and fixed/fixed collisions therefore disappear from the singleton formula.
• Every inactive odd cycle is impossible. A five-layer leaf induction propagates parity along witnessed complement paths of arbitrary length. Coupled with the accepted distance-two attack, it rules out every odd cycle in the inactive set, including arbitrary collisions among edge witnesses. Hence the inactive induced complement is bipartite. Choosing a compatible global deletion coloring remains open.
• Bipartiteness alone does not synchronize the deletion coloring. The exact marked graph \(L(K_{3,3})\) has deletion equality, only maximal triangles, a full active root, and inactive graph \(C_4\), yet both of its three-colorings use all three colors on that \(C_4\). Its target extension drops to \(\gamma=2\) and loses within two attacks, so the control refutes only the static shortcut and preserves an equality-specific dynamic theorem as the live route.
• One singleton polarizes and saturates its whole free component. Every vertex on one bipartition side inherits one anchor response, every vertex on the other side inherits the other response, and every complement edge lifts to a retained triple. Singleton units on one component variable are therefore always parity-compatible. Any surviving lollipop or two-unit chain must cross at least one genuine binary clause between distinct free components.
• The all-parameter \(Y_k\) obstruction has a precise dynamic split. Its \(k-3\) singleton vertices always install simultaneously and carry an exact family \(Y_3\) on the three base colors. Dirty installation creates a private buffer. Clean installation yields a parameter-three equality projection and \(n\ge2k+6\); the stronger \(n\ge2k+8\) needs projected static defects to survive. A checked control shows that clean replacement can repair such a defect, so neither count is an unconditional counterexample frontier.
• The exact \(\gamma=3\) target condition is total domination in the complement. After deleting a full-response target \(x\), let \(B=N_{\overline G}(x)\). Once every deletion pair has a common complement neighbor, \(\gamma(G)\ge3\) is equivalent to \(B\) totally dominating the deletion complement. A sharp 12-vertex control satisfies this condition and every prescribed one-step target response but still needs four eternal guards. Its full root dies only after three attacks, proving that the missing theorem must use repeated closure.
• The full-response physical link is palette-rigid. From a full root, every \(b\in B\) retains at least two anchor roles. Every physical-link component is bipartite, and its retained palette is constant on each side even when anchorless vertices occur. Each side meets at most one spoke; one spoke fixes the opposite palette and two spokes fix both. A color omitted on both sides forces every reverse edge state. Under the deletion-critical hypothesis, each retained role at an anchorless vertex installs a nonempty external \(G\)-clique layer through a unique third attack. Global synchronization and the residual inactive set remain open.
• The first singleton bridge has a proved propagation gate. In the odd–odd first clause, both terminal defect ridges are nonempty disjoint \(G\)-cliques with retained exchanges. If both are anchor-only, two further attacks force a retained shared-color bridge. Every bridge vertex is side-pure toward the two original components, so an active next clause must enter a fresh component. A two-list bridge also forces a retained turning ridge, or else a forbidden dominating pair; its exact ridge lists turn away from the incoming color. Longer fresh-component returns remain open.
• A fresh component cannot be recycled by the same bridge hub. One physical bridge is side-pure toward every later component of the relevant projection, and its turning ridge lies outside all of them. If a binary exact-two-list trace returns through a second original bridge vertex, a two-projection parity theorem forces at least one boundary state to survive. A fully dead return must therefore use a genuinely new source; singleton endings and that \(\gamma=3\) outside-ridge branch remain open.
• Complementary exchange now has an exact coinductive target. A sharp \(\gamma=2\) graph proves greatestness alone insufficient. Under \(\gamma=\alpha=\gamma^\infty=3\), static complementary domination and equal finite deletion ranks are also false. Yet an independently reconstructed census of all connected order-nine equality graphs checked 392,155 exchanges and found that the only survivor rank pair is survivor/survivor. This is certified finite evidence, not an all-order reciprocity proof.
• Full-response reverse colors are global but not sufficient. Every physical-link edge has the same nonempty set of anchors to which its target guard can return, even across disconnected link components, and the complete response row has an exact palette formula. Every actually extendible target color passes this reverse test and a restricted coinductive kernel. A checked equality graph has all three reverse colors but only one extendible and future-safe color, proving that the remaining selection theorem must use multi-step closure rather than one-step incidence.
• Every active exchange has dominating reverse endpoints, for all \(k\). Fix any active guard move \(u\to x\) and any maximum independent endpoint containing \(x\). If the complementary reverse set missed a vertex, C-108 would create a retained state from which \(k-1-t\) independent targets must be attacked using only \(k-2-t\) possible mobile guards. The one-guard count is impossible. Hence every omitted reverse endpoint starts inside the dominating-state universe and can fail only at positive dynamic depth; retention and reciprocity remain open.
• A \(k=3\) reciprocity failure has a rigid repair square. Every hypothetical one-sided active edge produces a common-nonneighbor ridge and an induced four-cycle with five surviving states: the same asymmetry appears opposite it and the two remaining edges are reciprocal. A first deleting attack has two exact response types and three minimum-rank adjacency caps. In the nonadjacent-pivot paired-singleton branch, a third independent base forces four more reciprocal active pairs. The shared-pivot and remaining adjacent cases are still open.
• Every single-hit minimum obstruction descends, for all \(k\). Quantitative finite-horizon transport makes deletion rank \(1\)-Lipschitz along every maximum-independent ridge. Combined with reverse-endpoint domination, a deleting attack that hits exactly one endpoint guard produces an adjacent reverse endpoint of rank exactly one less. Hence a rank-one or globally minimum failure must hit at least two endpoint guards simultaneously. The genuine multi-hit collision remains open. A separate complete order-ten discovery scan found no asymmetry, but is intentionally labeled observed until a second implementation audits that order.
• One genuine rank-one multi-hit row is impossible. At parameter three, every multi-hit blocker belongs to six exact neighborhood rows. In XQ0 the unique successor loses exactly one rank and forces a reciprocal response square; at rank one, its private witness creates a retained state with no adjacent responder, a contradiction. The other rows now force paired private-witness ridges, a four-facet independent ladder, or a nonempty completion clique. These are strict local reductions, not a proof of reciprocity.
• The exact static mixed \(P_4\) is impossible at every order. Under \(\gamma=\alpha=\gamma^\infty=3\), exact static swap lists \(\{a\},\{a,c\},\{b,c\},\{b\}\) on an induced complement path force one endpoint defect. The complete eight-vertex ledger has only five optional adjacencies, and independent ordinary-set, bitset, and clean-room implementations give empty compatible kernels for all 32 completions. Outside vertices cannot help because every attack used stays inside the core. This excludes the shortest exact static two-unit obstruction, not family-only lists with larger static palettes or longer chains.
• The family-only mixed \(P_4\) is purely dynamic at its endpoints. With the same four exact family-response lists, both omitted middle-color endpoint swaps must dominate; any missed vertex recreates the accepted eight-vertex defect core. In the literal greatest family, both omitted states therefore have positive finite deletion rank and strict lower-rank deleting rows. A direct SAT model found no realization at orders 12 through 22, but those runs are deliberately observed-only without proof logs.
• Future-safe colors are exactly nonempty restricted kernels. Avoiding every root swap incompatible with a chosen target color forces that target response in any nonempty eternal family; otherwise the frozen two-guard projection contains a complement triangle. One cumulative ban handles every full target simultaneously and leaves an exact ordinary list-coloring 2-CNF. Empty kernels have finite retained descents ending only at corridor/diamond or anchor-restoration gates. The theorem separates the two remaining failure mechanisms but does not prove that a cumulative kernel survives or that its 2-CNF is satisfiable.
• Clean \(Y_k\) equality patterns are substantially larger. Every installed singleton needs a wrong-role original-anchor neighbor, and each such edge forces a closed-private buffer. Base-clean \(Y_4\) is impossible. For \(k\ge5\), a contamination cycle and an endpoint defect give the conditional floor \(n\ge2k+9\), strengthened to \(2k+10\) for two distinct endpoint defects. A hostile audit caught and retracted the corresponding dirty-carrier inference; that branch remains open.
• The order-13 residual mechanism is exactly three-way at depth two. Full closure is unnecessary: closure only on retained triples meeting the original independent state already yields a strict RUP refutation. But radius one and each two-of-three anchor-slice relaxation have directly checked \((3,3,3,4,4)\) controls. This finite result identifies a three-projection gluing mechanism; it is not an all-order theorem.
• A minimum counterexample has no adjacent true twins. Deleting one of two adjacent vertices with identical closed neighborhoods preserves \(\gamma,\alpha,\gamma^\infty\), and \(\theta\) under the equality hypothesis. An independent hostile audit reconstructed the universal proof and found no failure among 6,279 true-twin incidences through connected order 8.
• Two stress-test families resolved. Complements of line graphs of triangle-free cubic class-II graphs satisfy \(i=\alpha=\gamma^\infty=3<\theta=4\) but have \(\gamma=2\). The 27-vertex Schläfli graph instead satisfies \(\gamma=i=\alpha=3<\theta=6\), but every three-guard strategy loses within two attacks. These examples show why both the static equality and the dynamic one-guard condition are essential.
• The fixed-anchor QQ1 ladder is not an all-order obstruction. A hostilely checked 18-vertex graph realizes every named local QQ1, outer-layer, and cross-layer obligation and has no dominating pair touching its distinguished independent triple. It nevertheless has thirty dominating pairs wholly outside that triple and exact vector \((2,3,3,3,3)\). Three witness two-cycles show that a fresh-vertex descent is also invalid. This refutes a proof shortcut, not the conjecture.
• Every rank-zero terminal completion has a controlled return layer. Each independent completion of the target–terminal pair forces at least one retained branch. A surviving secondary branch meets its private witness and uniquely returns at the full target; with two secondary colors, their closed witness neighborhoods cover the entire completion clique. An equality control makes this return raise restricted rank from zero to three.
• A transfer trapped inside the ban must escape. The other physical alternate produces only missed witnesses outside the ban, and every one forces a retained unbanned state for the original source color. At a full terminal, the two secondary witness sets cannot both be trapped. MMV-027 shows the exact remaining obstruction: all three kernels can be empty while the escape preserves rank zero, but only at \(\gamma=2\).
• Every pair repair is a retained fan or a reciprocal edge. For an adjacent pair, one retained common-nonneighbor state saturates the entire witness set into a clique of uniquely interchangeable tokens; if all such states are omitted, independent completions force activity in both directions. Nonadjacent pairs automatically have the retained fan. The theorem is valid for any eternal triple-family and permits witness reuse.
• A trapped rank-zero escape must rebound. Johnson distance from a banned configuration gives a universal lower bound on restricted deletion rank. The two pair repairs missing from the sharp MMV-027 control become nonempty clique fans under \(\gamma=3\), and their independent completion states cannot remain at rank zero.
• Supported completion fans either cross the target or create reciprocity. Any pair co-occupied in a retained triple carries its entire common-nonneighbor fan. In the full-list corridor, a completion branch therefore has a dominating target-cross state or a reciprocal target–anchor edge.
• The rank-one completion shell is exact. A state attaining the Johnson-distance rank floor drops one shell and one rank at every retained deleting response. For the second completion fan, rank one forces the deleting attack onto a fixed ban anchor; fan vertices on the wrong target side give explicit reciprocal hinges.
• The rank-one exit is precisely anchor restoration. One of the two nominal anchor attacks is impossible because its retained endpoint survives the first deletion round. The other has a unique retained response to rank zero, and the next attack is exactly the previously classified attacked-anchor restoration. Its alternate is either banned with a reciprocal target hinge or unbanned and nondominating.
• A supported asymmetric edge creates a polarized bow tie. At every common nonneighbor, a one-sided active edge forces two completely joined clique sides. One reciprocal spoke family has omitted central fans; the other reciprocal spoke family and all cross edges have retained central fans. Canonical QQ1 carries this pattern at every witness and every hot bridge.

The remaining all-order parameter-three obstruction is now exceptional-response geometry rather than an arbitrary exact-two 2-SAT bicycle, and its shortest exact static two-unit core is gone. In the singleton branch, a family-only mixed \(P_4\) can fail only through positive-rank endpoint dynamics; longer chains must propagate through genuinely fresh components before a lollipop or bicycle can close. In the full-list branch, future safety is exactly kernel nonemptiness; nonsingleton direct-root terminals are gone at every rank, rank-zero restoration has an exact palette-transfer split, and trapped rank-zero transfers now rebound through positive-rank completion fans. Every minimum fan state of rank one exits through exactly the attacked-anchor restoration, while the higher-rank branch remains open. The independent exchange route proves at every parameter that reverse endpoints dominate and that all single-hit minimum failures descend; at parameter three, six rank-one rows have collapsed to one canonical QQ1 hot layer whose complete outer system survives and whose two global witness layers are pointwise coupled. Every supported asymmetric witness now carries a polarized bow tie of retained and omitted fans, so the failed fresh-witness descent has become a global branch-coupling problem. Fixed-pivot holonomy is completely understood locally, while \(\overline{C_7}\) proves that physical-link topology alone cannot finish the gluing. Higher-rank multi-hit collisions, cross-color coupling of attacked-anchor restorations, coupling of the polarized QQ1 bow ties, and simultaneous full-list survival remain open. Exact controls show why static coloring, one-step responses, one-pair repair, fixed-anchor witness descent, local alternate retention, and equal-rank induction cannot finish the proof.

Campaign timeline

• 25 July — model and literature gate. Built two independent one-guard evaluators, proved and adversarially reviewed the core reductions, traced the conjecture's primary-source history, and reproduced the 2022 near-miss catalog.
• 25 July — closest known objects eliminated. Exhaustively checked all one-vertex extensions of the 55 published near misses and the complete one-edge-toggle neighborhood of the deepest survivors; no counterexample appeared.
• 26 July — order 12 closed. Combined certified \(k=3\) and \(k=4\) exclusions with an analytic \(k=5\) argument to establish the order-12 theorem.
• 26 July — first order-13 branch closed. Proved the \(C_{11}\) obstruction and independently certified the \(C_9\) template exclusion, leaving \(C_5,C_7\) at parameter three.
• 26 July — universal-proof pivot. Froze the next finite input without launching it, proved the restoration/Hall and shared-response-core reductions, and converted the Schläfli graph into an exact two-attack stress test.
• 26 July — induction mechanism found. Proved the frozen-color projection and cross-state response covariance, eliminating the common-two-list odd-cycle branch and isolating the mixed three-color core that remains.
• 26 July — mixed-core saturation. Reduced no-full-list projection gluing exactly to 2-SAT, then proved that every equality realization of its minimal mixed-path obstruction generates an external clique of induced complement-\(C_5\) closers. Independent hostile audits accepted both the symbolic reduction and the one-guard attack proof.
• 26 July — both residual branches localized. Proved a 12-vertex floor for the exact mixed pattern, eliminated every local link conflict around a full-response vertex, classified minimal unsatisfiable 2-SAT terminals, and ruled out the two canonical shortest non-chain geometries. Independent audits reconstructed the proofs and finite controls.
• 27 July — imported work audited and an infinite connector family closed. Identified the imported 15-vertex, 395-state near miss as the already certified Petersen line-graph construction, rejected unsupported and obsolete order-12 computations, proved the adjacent true-twin reduction, and extended the one-unit lollipop attack theorem from one edge to every odd physical subdivision under the exact stated port conditions. A separate exact control exposed terminal-port separation; two independent proof audits then accepted the \(\gamma=3\) cap-and-escape theorem that governs its next layer.
• 27 July — parameter-three full-response branch certified and two-lists physicalized. Proved that full-response witness layers are anchor-pure and disjoint, raising the exact separated-port floor to 15. A clean-room reconstruction and deterministic RUP replay then certified that no order-13 parameter-three counterexample has a full response at a maximum independent state. A broader one-guard lemma gives every two-list a physical omitted-color representative. The complementary no-full-list branch remains open, and a capped monolithic probe that timed out is explicitly a nonclaim.
• 27 July — no-full census reduced and transport shortcut closed. Proved a five-nonneutral floor and exact tight normal form for the remaining order-13 parameter-three branch. Then proved literal-level physicalization exactly and constructed an equality graph showing that supporting cross-edges do not transport to the physical representative. This replaces an invalid shortcut with a precise incidence-level proof target.
• 28 July — complete order-13 parameter-three slice certified. A sharp four-neutral obstruction reduced the no-full census to at most three neutral vertices. A second clean-room formula and deletion-free RUP proof excluded all four remaining signature censuses. Together with the full-response result, this proves that any order-13 counterexample must have parameter four or five. The same iteration converted every failed original-edge incidence into a virtual-rainbow cap and isolated the exact third-color gate as the remaining local unit-free mechanism.
• 28 July — odd-return family excluded and finite mechanism localized. A chirality law and direct one-guard attack now exclude every shared-port tight-gate return whose odd parity lies in one frozen projection, at any subdivision length. Separately, an independently reconstructed radius-two refutation shows that the order-13 residual contradiction stops before all three original guards leave, while exact SAT controls prove that all three anchor slices are jointly necessary.
• 28 July — separated two-gate holonomy closed. A boundary-parity synchronization theorem now forces two connector paths in distinct frozen projections to have the same parity whenever tight gates supply both dead boundary states. A hostile checker rebuilt four equality controls and all 878 qualifying path pairs, including the length-zero cases it caught missing from the first stress-test count. The remaining tight-gate obstruction is an odd signed cycle through at least three gates.
• 28 July — shortest three-gate witness proved; two naive descents rejected. Explicit attack trees force the common-neighbor witness at a critical pair to have the exact third response type, and a collision audit covers every displayed vertex. A truth-table audit proves that its arms supply only one endpoint implication; a separate 19-vertex one-guard control proves that even oriented paired repair can lengthen the marked bicycle. The full-response lane then generalized response propagation to every guard number \(k\), yielding exact active/inactive color identities. A sharp inactive-\(C_5\) control shows why those static identities must still be coupled to multi-step eternal closure.
• 28 July — response physicality and dynamic inactive-link restrictions. A global witness cascade first excludes the shortest complete three-gate odd boundary. A stronger sealed-cap argument then proves that every exact-two-list response port is physically nonadjacent to its omitted anchor whenever \(\gamma=3\). In the full-list lane, frozen projection proves every inactive target-link suspension exactly \((k-1)\)-colorable for arbitrary \(k\), while 52 independently replayed finite proofs exclude an inactive induced \(C_5\). Neither branch is complete: odd signed cycles and inactive odd cycles of length at least seven remain.
• 28 July — exact-two-list branch closed; inactive \(C_7\) excluded. A signed-parity shortening theorem reduces every exact-two-list coloring obstruction to five small skeletons, and independently audited one-guard attack trees exclude every skeleton and witness collision. Separately, 93 replayed DRAT proofs cover all 877 identifications of the seven edge witnesses around an inactive induced \(C_7\). The remaining \(k=3\) frontier consists precisely of singleton-list and full-list states; in the latter, any inactive odd hole has length at least nine.
• 28 July — multi-anchor induction gap and singleton buffers isolated. Simultaneous frozen projection now works for every proper anchor face, reducing the universal induction problem to one explicit dynamic gluing statement; a uniform abstract obstruction proves that list theory alone cannot supply it. In the \(k=3\) singleton branch, an exact terminal split and a hostilely audited cap theorem show that every dynamic two-list port ends at a singleton or a singleton-buffered sealed cap. Immediate fixed certificates, separated unit chains, residual bicycles, and the full-list global precoloring step remain open.
• 28 July — fixed certificates removed, the shortest static obstruction raised, and all inactive odd cycles closed. A frozen pair-family parity theorem eliminates every fixed singleton substitution. Exact static \(Y_3=P_4\) lists are dynamically rigid and force two more distinct defect vertices, so that conditional pattern needs at least 14 vertices. Independently, a length-free odd-path induction proves the full-list inactive induced complement is bipartite. Free unit chains and global precoloring synchronization are now the two primary \(k=3\) gates.
• 28 July — the next static shortcuts delimited. \(L(K_{3,3})\) proves that inactive bipartiteness alone does not synchronize a deletion coloring, while an independent one-guard theorem shows that each free singleton component is fully polarized and family-saturated. The zero-clause unit collision is gone. At arbitrary \(k\), exact \(Y_k\) now projects cleanly to \(Y_3\) unless a private buffer intervenes; a sharp repair control identifies why the stronger static floor remains conditional.
• 28 July — both surviving \(k=3\) branches gain multi-step structure. The exact \(\gamma=3\) target condition becomes total domination of the deletion complement. A second attack from a full response forces independent physical spokes and local two-spoke component palettes, rejecting every sharp static control at its first dynamic failure. In the singleton branch, the first cross clause forces retained defect ridges; if both are anchor-only, two further attacks create a retained shared-color bridge clique. A separate clean \(Y_k\) audit raises the conditional all-parameter pattern floor to \(2k+9\) and catches the dirty-carrier boundary before publication.
• 28 July — anchorless links, bridge propagation, and dynamic reciprocity. Full-response palettes are now rigid on every physical-link side, including anchorless vertices, and deletion-critical anchorless roles force external third-attack clique layers. The singleton bridge cannot recycle either original component and every two-list bridge forces a turning ridge. Independently, a complete order-nine audit refutes the static and equal-rank exchange shortcuts but certifies simultaneous greatest-family survival for all 392,155 exchanges. The universal theorem remains open at the resulting global/coinductive step.
• 28 July — the full-response reverse signal becomes global. Every physical-link edge now has the same nonempty reverse-color set, with an exact response table and a clean coinductive necessity test for feasible colors. An independently checked equality graph has all three reverse colors but only one feasible, future-safe color. This rules out one-step selection and isolates future-stable color choice as the exact full-response gate.
• 28 July — reverse endpoint domination proved for every guard number. A one-guard exhaustion argument now shows that every complementary reverse endpoint of an active exchange is dominating under \(i=\alpha=k\). The hostile audit checks all overlaps and both small-parameter boundaries. Any failure of greatest-family reciprocity is therefore purely dynamic: survivor versus positive finite deletion rank.
• 28 July — the first fresh-component return is gated. A bridge hub cannot select both sides of any later projection component, its turning ridge cannot hide in the entered component, and a return through a second original bridge vertex cannot leave both parity boundaries dead. The hostile review narrowed the theorem correctly to actual first re-entry and binary exact-two-list terminal clauses; the outside-ridge and singleton-source returns remain open.
• 28 July — every possible \(k=3\) reciprocity failure gets a positive-rank repair square. The omitted reverse corner is always dominating, its first losing attack has an exact two-branch normal form, and a minimum-rank choice forces three adjacency caps. In the nonadjacent paired-singleton branch a third maximum-independent base makes four surrounding exchanges reciprocal. This localizes the coinductive gap without claiming reciprocity itself.
• 28 July — finite-horizon rank descent isolates genuine collisions. A new all-parameter theorem shows that response states lose at most one deletion round per endpoint-ridge exchange. Every single-hit blocker of a reverse endpoint therefore descends by exactly one rank; a minimum obstruction must hit at least two endpoint guards. A resumable order-ten discovery census separately checked 458,696 active orientations and found no asymmetry, but remains an observed result pending an independent order-ten replay.
• 28 July — the exact static mixed \(P_4\) is excluded universally. One endpoint defect reduces the canonical shortest two-unit response obstruction to eight vertices and five optional edges. Three independent implementations obtain empty local one-guard kernels in all 32 completions, and the attack closure cannot be repaired from outside the core. Family-only enlargements and longer unit chains remain open.
• 28 July — future-safe color becomes an exact kernel question. A ban-avoidance theorem proves that a chosen color is safe if and only if its restricted eternal kernel is nonempty. Cumulative bans handle every full target at once; a surviving kernel leaves an exact ordinary 2-SAT coloring problem, while an empty kernel descends to one of two terminal gate types. The existence step remains open.
• 28 July — four of six rank-one collision rows are excluded. The six possible parameter-three collision rows are explicit. XQ0 falls to a private-witness attack; XQ1 falls when ridge covariance omits a mixed state that independent completion forces back; QQ0 and AQ0 fall because every response to one private witness reaches a non-dominating state. Separate hostile reviews checked all named collisions and exhaustive finite incidence cores. Only QQ1 and AQ1 remain at rank one.
• 28 July — the family-only mixed \(P_4\) reaches a genuine multi-hit boundary. Both omitted endpoint swaps must dominate, because one genuine endpoint defect recreates the universally excluded 32-completion core. In the greatest family both omissions have positive finite rank. Every single-hit deletion transports the complete list pattern and lowers rank exactly one, so finite descent reaches a multi-hit row; an exact eight-cell table is proved but not yet closed. The order-12–22 SAT sweep remains observed evidence only.
• 28 July — the all-singleton full-list terminal triple is impossible. If all three color-restricted kernels are empty, three selected descent traces cannot all end at their own singleton palettes. The attack proof is uniform over direct corridors, nonroot diamonds, and anchor restoration, and a hostile checker covered all 512 local completions. This does not prove a safe color exists; all-nonsingleton terminal triples remain open.
• 28 July — rank one collapses to one canonical QQ1 core. A direct-root rank-zero full-list terminal must have singleton palette, while every nonsingleton nonroot corridor forces a private missed witness. Independently, all AQ1 collisions normalize to QQ1 with the same rank-one blocker, and the resulting completion set is a clique complete to both unaffected root guards. Two exact \(\gamma=2\) controls show that one-pair repair can transfer the asymmetry and raise deletion rank, so the full no-dominating-pair condition is essential. The universal conjecture remains unresolved.
• 28 July — fixed-pivot holonomy is reduced to global component coupling. Every one-sided active exchange separates its endpoints in the complement link of each common nonneighbor. The two selected bipartite components then acquire a forced checkerboard orientation, while the tracked omitted-corner rank is literally conserved. A hostile audit checked every short-path terminal, walk collision, parity padding, and 87,888 synchronized endpoint pairs. Any successful continuation must now connect different link components using the global \(\gamma=3\) condition.
• 28 July — QQ1 is pushed to an external hot layer. Every completion of the canonical rank-one core hits all four side vertices, and each completion triple with the asymmetric endpoints dominates. The equality condition therefore forces a fresh witness adjacent to both opposite endpoints and at least one side witness. All 64 cold-witness incidences fail, while the completion rank diamond has sharp top vector \(1,2,2,3\). In the nonadjacent branch the hot witness recreates the same omitted repair corner, so the remaining argument must be global.
• 28 July — positive-rank full-list terminals are normalized. Every positive-rank terminal has a dominating unbanned lower-rank alternate. Minimum-rank selection eliminates nonsingleton direct-root terminals at every rank, but it also shows why naive descent stops: the alternate is already absent from the unrestricted greatest family. The all-three-empty branch now has four exact nonroot-corridor/anchor-restoration cases rather than an unspecified positive-rank remainder.
• 28 July — the static holonomy shortcut is refuted. The accepted control \(\overline{C_7}\) already has a common neighbor for every pair and a connected \(P_4\) at every vertex link, yet it is four-chromatic. Its pure flag clique complex is the seven-vertex Möbius band. An independent labeled enumeration makes order seven minimal for this exact static implication. This is a failed proof route, not a counterexample to the gamma–theta conjecture.
• 28 July — rank-zero anchor restoration is classified. A nonsingleton terminal palette either contains the attacked anchor or forces that anchor into the mover palette. The legal unbanned alternate is then nondominating and gives a two-attack witness transfer. A 16-vertex equality control shows the sharp obstruction: a legal dominating banned alternate can still lie outside the greatest family, so all three color traces must be coupled.
• 28 July — the QQ1 outer layer is family-saturated. Every hot witness yields five retained inner corners and a clique of hot vertices. More strongly, every outer completion bow-tie survives and every nondegenerate outer edge is reciprocal. The first draft exposed an occupied-target collision at a side witness; the corrected proof skips that attack and a clean-room audit covers all 16 collision and endpoint branches. Both inner \(ud\)-subcases remain open.
• 28 July — the two QQ1 witness layers are coupled pointwise. Every common nonneighbor \(w\) of \(\{u,d\}\) and every common nonneighbor \(z\) of \(\{u,x\}\) force the retained bridge \(\{u,w,z\}\), which jointly covers both side witnesses. In the \(ud\)-edge branch the cross edge creates an induced \(C_5\), while its absence makes the bridge independent. Two independently checked gamma-two controls show why this is not yet a contradiction: one pair repair simply exposes other dominating pairs.
• 28 July — rank-zero corridor colors acquire retained transfer ladders. Every secondary terminal color forces two unique witness states and must reappear at the corridor mover or its missed witness. Three nonsingleton rows have only a directed 3-cycle or a 2-cycle-with-tail color pattern. Independent controls mark the exact limit: equality permits two sharp transfers with one safe color, while the complete three-transfer cycle occurs at \(\gamma=2\). Cross-ban rank comparison remains open.
• 28 July — the fixed-anchor QQ1 descent is refuted. An independently reconstructed 18-vertex boundary graph satisfies every named local QQ1/C-166/C-167 obligation and protects every pair touching the distinguished independent triple, yet thirty dominating pairs survive wholly in the auxiliary layer. Its exact vector is \((2,3,3,3,3)\), so it is not a conjecture counterexample. The surviving proof must use all no-dominating-pair obligations or another global equality invariant.
• 28 July — rank-zero completions and trapped transfers are controlled. Every target–terminal completion has a retained branch and a unique return layer covered by the secondary witnesses. If a transfer witness lies in the ban, a second attack forces an unbanned source-color state, and two full-terminal witness sets cannot both be trapped. Independent controls show that rank may rise or stay equal, so the live obstruction is a global rank-preserving cycle rather than a missing local response.
• 28 July — every pair repair becomes a fan or reciprocity. For any eternal triple-family under parameter-three equality, an adjacent pair either retains its entire common-nonneighbor fan as a clique of uniquely exchangeable tokens or is active in both directions. Nonadjacent pairs automatically have the fan, giving an inclusive all-pairs alternative. This is a size-independent replacement for fresh-witness iteration, not yet a QQ1 exclusion.
• 28 July — trapped escapes rebound through supported completion fans. Johnson distance to a banned state now gives a universal deletion-rank floor. The two equality-forced completion fans missing from the MMV-027 boundary cannot stay at rank zero; every co-occupied pair supports its full fan, and each branch has a dominating target-cross state or reciprocal target edge.
• 28 July — the rank-one completion shell is localized. Tight states descend exactly one distance shell and one rank. A minimum second-fan state of rank one can be deleted only at a fixed ban anchor, while each fan has an exact dominating-cross/reciprocal-hinge target split. Independent audits cover the universal rank argument, bounded censuses, and sharp 13- and 16-vertex controls.
• 28 July — rank-one exits become restoration; QQ1 gains a polarized bow tie. The apparent two-anchor full-list branch has collapsed to one exact attacked-anchor restoration with a banned-reciprocal or unbanned-nondominating alternate. Independently, every supported one-sided QQ1 edge forces completely joined clique wings with prescribed retained and omitted fan statuses. Both size-independent theorems passed separate hostile reviews; their remaining gates are cross-color and cross-witness coupling.

The timestamped repository log and live state file are the authoritative sources after this dated snapshot.

Current papers

Excluding Parameter Three at Order Thirteen in the \(\gamma\)–\(\theta\) Conjecture presents the new finite theorem, its full/no-full structural split, the two decisive RUP certificates, coverage proof, exact hashes, sharp controls, and compact replay instructions.

A Certified Order-Twelve Extension of the \(\gamma\)–\(\theta\) Frontier in One-Guard Eternal Domination remains the separate paper establishing the preceding complete frontier.

Neither paper resolves the universal conjecture. At order 13, common parameters four and five remain open.

Published context

• William F. Klostermeyer and Gary MacGillivray, “Eternal Dominating Sets in Graphs,” Journal of Combinatorial Mathematics and Combinatorial Computing 68 (2009), 97–111.
• William F. Klostermeyer and C. M. Mynhardt, “Domination, Eternal Domination, and Clique Covering,” Discussiones Mathematicae Graph Theory 35(2) (2015), 283–300.
• Gary MacGillivray, C. M. Mynhardt, and V. Virgile, “Eternal Domination and Clique Covering,” Electronic Journal of Graph Theory and Applications 10(2) (2022), 603–624; arXiv:2110.09732.
• Dmitrii Taletskii, “The Gamma-Theta Conjecture Holds for Planar Graphs,” arXiv:2412.20120 (2024), version 2.

Scope and AI-assistance disclosure

The exploratory mathematics, programs, certificate design, independent-code audits, literature review, manuscript, and publication materials were developed with heavy assistance from ChatGPT 5.6 Sol under Alec Kriebel's direction. Alec Kriebel is a complete amateur and cannot independently validate the mathematics. No outside individual was contacted, and no external expert has reviewed this work. Exact verification is evidence about the encoded finite statements; it is not peer review and cannot establish worldwide priority.