Assumptions

Native variables do not say where a knot's binding comes from. In the law with Ledrappier's rule on x, z free and y = x + z, every variable is flipped by a finite move, yet for every k a six-variable knot joins points 2^k apart, bound through the hidden field x, whose openness comes only from the ends (proof, Proposition 5). Requiring native binding makes every native knot of a parity law chained, so bond distance is a ruler in every parity law (Proposition 2), and it adds no primitive: it reuses the finite moves that already define nativity. The logic-axis epic recommended it, and the owner chose to record it as an assumption rather than decide it (request).

Every reading of native knots on infinite systems as local, and every claim of distance, locality or spacing there (question 9 of the knot intuition). Earlier verdicts keep their labels: Tseitin's law and the gauge law on the square grid have no ends, so their native knots are natively bound, and Ledrappier's law has no native knot either way. The long knots of the law above are native in every variable but not natively bound.

The alternative keeps nativity per variable and has each claim about distance or locality state only whether the law has ends. Whether natively bound knots are always chained is open beyond parity laws. A non-linear law whose natively bound knots join far points without a chain would remove the reason for the assumption, and so would a law in which the label discards knots that every other test treats as emergence.

It is the program's working theory, protected by the foundational thesis. It is not a theorem: the theorems say which relations are knots and which constraint languages define them (A16), not that knots are where emergence begins.

The program's reading of its results: a result about knots is read as a result about emergence, space or hiddenness because of this premise.

Look for structure the program would count as emergent that no knot carries. Known places to look: competing pairs make knots among their least-violation states (A8), pairs alone make a knot on three-valued distinctions (A10), and trends at infinity are invisible to the knot test (holes at infinity). A successful challenge goes to the owner as a canonical correction request.

Adopted with the canonical knot test after an earlier proof's knot turned out to depend on its executor: cross-wired two-clause guards act like a knot only if each loop's clauses are checked only while that loop runs.

Every K1–K3 verdict, the knot laboratory, the census of 34 two-loop knots, and the theorem that no 2-CNF defines a knot (A16).

On infinite systems the source-aware reading (A9) already lets verdicts depend on more than the projected relation: on how the system's solutions are joined by finite changes. A challenge would ask whether something the projection forgets, such as a growth order (A14) or the way solutions connect, ever carries emergence that the allowed states miss.

Adopted with the canonical knot test. An earlier proof's executor checked a loop's clauses only while the loop ran, which supplied each loop's presence as a hidden third input and made a common hub of two-clause guards pass for a knot.

The negative result for 2-CNF, which no evaluation rule can evade, and every growth result: a growth result belongs to the law only if it holds in every growth order (A14).

Question 4 of the knot intuition asks whether loops are better modelled as processes. Write a growth process as one law on space and time, and ask whether it has knots that no static law has and whether its order can be a variable of the law rather than an input.

Settled in form with the canonical test: K1–K3 applies to open distinctions, and which structures count as native loops is interpretive (question 1 of the knot intuition).

Every knot claim, since K1–K3 is checked on distinctions, not on loops.

Question 1 asks which structures should count as native loops, and it is open. Infinite systems showed that the source can matter: an end can open a distinction that no loop opens, and the knot test cannot tell the two apart, which led to A9. A challenge would look for open distinctions of different kinds in finite systems and ask whether the test should separate them.

The owner chose it on 2026-10-19 (decision). Erasure entered the agents' gate in the workspace's first commit as a test of decomposability, which is the forgetting reading, and the program began from parity patterns, which are knots only under forgetting.

Parity as the knot that hides the most, the 34 two-loop knots in five shapes, and every result about parity's knots since 2026-10-19.

The absence reading (A17) keeps only the atom. Every knot it accepts is a knot under forgetting, and both readings exclude the common hub and every 2-CNF pattern, so a challenge needs a question whose answer changes between them. The knot laboratory keeps both readings.

The canonical reading since the test was adopted. On 2026-10-29 the signed reading, in which a state is allowed when its signed witness sum is nonzero, was adopted beside it under the owner's rule for forks between readings, and later results record both verdicts (adoption).

The two/three boundary as stated (A16): with signed witnesses every three-bit knot is one cancelling witness away from pairs, so the boundary becomes a boundary of positivity. Also "only parity survives native patterns of patterns" (GEOMETRY-LAW-20261029-001), which the signed reading changes: native gadgets then rebuild 14 of the 16 knot classes.

Both readings are mapped side by side. A challenge would ask which one the program's questions need, for example whether cancelling witnesses build a space that no positive law builds. Interference reproduces knots on finitely ramified, tree-like space but not on genuine two-dimensional space (SIGNED-SUBDIVISION-20261029-001).

The strict reading keeps verdicts independent of presentation and gives pairs no knots. The competition reading needs a declared measure of violation. Both were mapped on 2026-10-19 and the choice was left open (competing pairs; question 10 of the knot intuition).

The theorem that pairs never make a knot, in its strict form (A16).

Under the competition reading, frustrated pairs make exactly one kind of knot, the not-all-equal triangle, and competition turns a triangulated layout into a chain of knots but does not make the layout. A challenge would adopt it as a second reading, as signed witnesses were, and record both verdicts.

An infinite system's ends can open distinctions that no loop opens, and K1–K3 cannot tell the two apart: the 3-regular tree has a knot at every vertex and no loop at all (holes at infinity). Asked which reading gives more interesting results, the owner adopted the source-aware one, reopenable. Under it emergence depends on the law and on the solution (frozen worlds; adoption).

Frozen and flexible worlds: a linear law is native everywhere or nowhere, colourings have frozen worlds without native knots, and Ledrappier's law has knots but no native knot. A1 adds a requirement to it.

The source-blind reading counts end-supplied distinctions too. Both verdicts are recorded for every claim, so reopening needs no new derivation. A challenge would look for a result that the source-blind reading makes more interesting.

The owner's statement of the thesis on 2026-10-21. The base-three epic found that the threshold is three somewhere: three places on bits, or three values on pairs. The two/three boundary is fundamental exactly when the base distinction is a bit (base three).

The two/three boundary as the boundary of emergence (A16), and every census of knots on bits.

On three-valued distinctions pairs alone make a knot: a hidden vertex adjacent to three visible ones, under colouring constraints, makes "not all different" (Gallai knots). A challenge would ask whether the bit is forced, for example whether a three-valued distinction written natively on bits needs a three-place knot, which would move the boundary back to three places. The base-three epic proposed mapping all 512 binary relations on three values; that was deferred.

The canonical common hub, two attachments A-k and B-k, fails K2, and a distinction in three or more knots ties its copies by equality. On 2026-10-29 the owner put hub laws in scope (hub survivors).

Surfaces glued natively, with every shared distinction in exactly two knots (SURFACE-GLUING-20261024-001), and every result that compares a law with and without hubs.

With hubs, emergence beyond parity persists across scales (HUB-SURVIVORS-20261029-001). A challenge would ask whether some sharing among three or more knots is native after all, for example whether a law can form its own fan-out without an equality.

The canonical setting quantifies witnesses away (A3), so hidden variables belong to a law's constraints, unlike coordinates or positions written in advance.

"With one witness bit per incidence, quantified away, a uniform local law lays out its own round sphere" (sphere growth), the eleven-label pinwheel rule (PINWHEEL-NATIVE-20261107-001), and A14.

Decorated matching rules force essentially every substitution tiling (Mozes 1989; Goodman-Strauss 1998), so with enough labels a law can carry almost any self-similar layout. A challenge would ask which spaces need no hidden bits, how the number of bits needed grows with the size, and whether labels that grow with the size are a drawing in disguise.

It is the program's protected interpretation.

How results are worded: a law that leaves alternatives open is not said to fail to choose, and a construction is described by what it puts in, not by what it selects.

None by proof or experiment. A model-existence or nonselection theorem fits it without proving it, and an obstruction to one infinite carrier does not refute it. The entry keeps the assumption visible, so that no result is read as evidence for or against it.

These readings fit what the record already has. The knot sphere's contradiction is a vortex charge that can be placed through any point and steps locally (FLAT-STATES-20261107-001), and violated lines minus agreeing faces count the same in every state, so no move removes the forced dissonance (COUNT-SURFACES-20261107-001, whose record reads the count as a discrete Poincaré–Hopf theorem). They were written down when the owner described the picture, and the direction that would test them was scoped and parked.

Only the framing of that parked direction, steering a dissonance by re-signing knots. No result yet.

The alternatives read the point as the whole state, or its strength as how long a pattern persists under an update law. The parked direction's first step decides between them: on the octahedral knot spheres, change one sign at a time and check whether the forced dissonance moves as its index predicts.

The canonical test already asks this of every process (A4, A14). The other reading, running the computations in parallel, is an implementation choice that changes no result.

The framing of the parked stretch direction, many points at once.

If the owner meant parallel computation, the direction reduces to an implementation choice. Otherwise its first step places two dissonances on one knot space, updates both in every order, and compares the outcomes: points might meet, cancel or bind, since violated lines and over-agreeing faces carry opposite indices.

It is the code with which PINWHEEL-NATIVE-20261107-001B read the pinwheel's labels from its unlabelled map, and the atlas epic kept it so that the counts compare (atlas proof).

The counts of PINWHEEL-ATLAS-20261107-001, tickets A, B and D: the 1,168 radius-2 forms, the 198 radius-1 forms of which 9 carry two labels, the least radius that reads every label, the 96,474 periodic configurations that fail at radius 1, and the 3,227 codes of the growing patch.

Another local view, such as one built from faces or one that hides the outer face, could change the counts and the least radius that reads every label. A challenge would recompute the atlas under a second view and compare.

By Whitney's theorems a 3-connected planar graph is determined by its cycle matroid and has only one planar embedding. The 3-connectivity was measured only on finite patches at levels 2 to 5, each with an added outside vertex joined to its boundary (PINWHEEL-NATIVE-20261107-001B), and the atlas proof says it is not re-proved for infinite tilings.

Reading the radius-2 atlas as the law's own rather than the drawing's: the rooted code takes its rotation order from the drawing (A23), so the atlas belongs to the law only through this step.

Prove that the embedding is unique for infinite pinwheel maps, for example by proving every pinwheel tiling's graph 3-connected. A failure would mean that the law fixes the map only patch by patch.

RESIGN-STEER-20261107-001 took it by default, to separate what changing the law can do from what changing the state can do. It is the cleanest reading of "learning orients the logic", and it matches the owner's word for the process, rationalization: the belief changes and the behaviour does not.

That epic's theorem that re-signing alone carries the dissonance from any line to any line with the state fixed, and its local rule, which reached 2,399 of 2,400 targets (steering by re-signing).

The alternatives let the state relax by its own local moves between learning steps, or let the learner change any sign. The cheapest test runs the same local rule while the state takes one improving step after each sign change, and asks whether every target is still reached or whether the state's own moves pull the point elsewhere. A native target signal, such as the count of consistent states (A27) or bond distance (A12), in place of the outside distance, is a second test.

RESIGN-STEER-20261107-001 took it by default. It is the uniform measure on allowed states, the one the canonical reading of allowed states (A3, A8) suggests, and it makes the weights exact counts of perfect matchings.

That epic's finding that the halves repel, the corners attract, and a learner that keeps the most states parks the halves on corners (steering by re-signing).

The alternatives weigh states by their dissonance at a temperature, or count only the states reachable by local moves. The cheapest test recomputes the force with each state weighted by a power of its number of over-agreeing faces, or with the states of one local-move component only, and checks whether the halves still repel and the corners still attract.

GROWTH-LAW-20261107-001 took it by default. It is the weakest rule that a local process can apply and that never rejects a genuine continuation: growth that follows a genuine pinwheel patch is never rejected (pinwheel growth).

The dead-end counts and sizes of GROWTH-LAW-20261107-001B and GROWTH-LAW-20261107-001C, and the reading that label-free growth leaves the pinwheel at once.

A deeper lookahead could get stuck later or less often. Ticket C's stronger arm adds the radius-2 atlas. Dworkin and Shieh (1995) prove that every aperiodic protoset of the plane has deceptions of all orders, but the pinwheel's collar and radius-2 laws are not aperiodic: a periodic plane obeys both (COLLAR-PLANE-20261108-001). So that theorem does not settle whether some growth order of these laws fails. The cheapest test is two moves of lookahead on ticket B's seeds.

TIME-KNOTS-20261107-001 took it by default. These knots are characterized exactly (Theorem 1 of its ticket A), each is a knot of the whole space-time law, and they are cheap to count, which made the census of the 88 elementary automata possible (knots across time).

The census classes of TIME-KNOTS-20261107-001B, the reading that pairwise memory is exactly the single ANDs and ORs, and the first measure of the next epic, knots between processes.

The alternative reads memory from every knot of the space-time law that meets two times, and it finds more. Ledrappier's law has smaller knots through intermediate rows (2 + w(T) places against 2^popcount(T) + 1), and its least knot at T = 7 leaves the light cone through a second future cell. The cheapest test, the census's 15 knot-free rules read on their whole law in small windows, was run the day this entry was added. Rules 128, 136 and 160 knot two times and rule 32 knots one later row, wherever the image forbids a pattern, while the other eleven showed no knot. So the readings differ, and every claim made under this one names the first row. It would be overturned if a question the program asks of processes, such as whether a contact binds two of them, got a different answer under the whole-law reading. The next epic reads both.

PROCESS-CONTACT-20261108-001 took it by default. Its ticket A proves that this is exactly the loss of the contact's knots when either process is forgotten, so two items of the contact gate become one property across time (which contacts bind).

Every binding and persistence count of PROCESS-CONTACT-20261108-001 and its successors.

Weaker readings would ask only that some knot through c(T) meet both processes, or would read binding from the whole space-time law (A30's alternative), where the contact's history can join the rings as a witness. The cheapest test reruns the census of ticket B with "some knot meets both rings" and compares the classes.

The program's laws are not written on a space in advance, so they carry no frame. PINWHEEL-ATLAS-20261107-001 took it by default for its collars (its rooted codes, A23, are congruence-free anyway), and COLLAR-PLANE-20261108-001 stated its theorem under it (collar plane).

The atlas's finite counts (193 collars, 108 complete, 1,168 forms), the growth engine's law, and the reading that the plane needs hidden information for roundness.

A law that also records directions against a fixed frame sees the pinwheel's tiles in infinitely many directions, so no finite law of that kind holds the pinwheel at all, and the question changes rather than its answer. A milder alternative records handedness only: in the pinwheel every complete collar occurs with both handednesses, so the cheapest test is to recount the census with handedness kept and check that nothing changes.

ROUND-PRICE-20261108-001 took it by default. Every known way to force a substitution tiling (Mozes; Goodman-Strauss; Radin for the pinwheel) decorates the substitution this way, it makes the count finite and exhaustive for small alphabets, and the rule radius then does not matter (price of roundness).

The census (1,008 of 1,024 one-bit decorations admit a crystal), the sixteen open candidates, and the comparison with the sphere's one hidden bit per incidence.

Count bits on edges instead, as Radin's rules do, or let a child's label depend on its parent's neighbours too. The cheapest test runs the rhombus crystal against one-bit edge labellings of the pinwheel's collars; if some edge bit excludes every crystal that all one-bit tile decorations admit, the measure misplaces the price.

PATH-HYBRID-20261108-001 took it by default when it counted ticket B's hybrid paths as native. The canonical knot test reads any declared system of constraints, and a law fixed cell by cell is still fixed in advance, with no input supplied from outside (paths through mixed middle rings).

The 1,120 native paths with hybrid middle rings and their characterisation. The existence of native paths across time does not rest on it: ticket A's 16 use one rule per ring.

Require one uniform law per ring, as question 9 asks of space, or require the cell types to arise from a uniform law, for instance as a second bit per cell that the dynamics holds fixed. The cheapest test builds the smallest hybrid path's ring C from a uniform law on two bits per cell. If no such emulation keeps the path native, the hybrid paths depend on drawing the law cell by cell.

RECUR-PATH-20261108-001 overturned A34 by its pre-registered rule, and this reading replaces it. Under it, composition across time survives every growth tested; under A34 it survives none (composition at recurring lags).

The 260 families of RECUR-PATH-20261108-001 as native networks across time that transport, 64 of them proved at every ring size and length (RECUR-PROOF-20261108-001).

Ask more of the composing lags: that a knot of the path itself reads them, or that they make up a fixed share of every cycle at every size. The cheapest test proves or refutes, for the rule-60 family, that the composing share stays positive at every ring size.

SEPARATE-20261108-001 took it by default. Freezing is the simplest declared contact intervention that stops every route through the contact, and it leaves every other cell's law unchanged (native separation across time).

The epic's theorems that a cut separates a proved path and that a cycle of four rings needs one cut per route.

Cut instead by erasing the contact, forgetting it as the knot test erases a loop, or by letting it read only one of its rings. The cheapest test reruns the epic's checks with a one-sided cut; a route that survives it would show that what counts as a cut changes which sets are separated.

BOND-CHAIN-20261108-001 overturned A12 as a universal statement and this reading replaces it. Chains of knots and bond distance agree wherever knots compose across their meetings, including every parity law (proved), and part exactly at gates. The foundation's contact rule decides between them: "Two independent A-k and B-k relations make k a common hub, not one touch between A and B; shared variable identity alone is insufficient" (AGENTS.md). At a gate no knot crosses k, and bond distance follows that rule (bond and chain distance).

Everything that rested on A12: the number line, the taxicab plane, the circles of periodic and pinwheel laws, and dimension read from ball growth (question 9). All of it was measured where the two measures agree.

Find a law whose gate chains carry structure the program would want to call space that composing knots cannot carry, for example a hub chain in two dimensions whose chain balls are round. The cheapest test measures chain balls on a plane of gated knots against the grid it is drawn on.

It uses nothing but knots and allowed states, so no metric, coordinate or drawing is put in. On a thread of atoms it counts the loops between two points, one unit per loop, which matches the owner's reading of the integers' spacing as space (logic axis).

The results on space from a law since 2026-11-07: the number line, the taxicab plane, the circles of periodic and pinwheel laws, and dimension read from ball growth (question 9).

Other measures can be read from allowed states, such as the shortest chain of knots joining two distinctions, each sharing a distinction with the next. On tree-shaped knot graphs bond distance is the number of knots on the path plus one, so the two agree there, but they need not agree elsewhere. A challenge would compare them on the laws already measured and ask whether any dimension or circle changes.

It follows from A4: a process may not supply an input, so a kept bit must be written into the system as a variable, and a variable that is quantified away is a witness. The grow-coin epic applied it: one bit kept for one lap ties a sphere's four equator corners together, so growth lays out a new round sphere only with a witness, just as a finished sphere is checked round only with one (grow coin).

"Growth does not remove the witness" (GROW-COIN-20261107-001), and with it the reading that a law lays out a new round sphere only with hidden information; "a beginning is a knot" and the reading of what growth adds (growth as a law).

Settled by GROWTH-LAW-20261107-001, which wrote growth histories into the law as variables and read them under K1–K3 (proof).

• The order adds no knot. It never knots with the world's states. Existence enters only as a presence literal.
• The history's parts. Its local part (which neighbour came first) is pairwise and holds no knot. Its global part, time without a loop, is a knot on every cycle of the world, so a beginning is a knot.
• What is left to carry. What a kept history adds beyond bounded local information is only whether a world has a beginning.
• The plane. There label-free growth by the pinwheel's own law left the pinwheel at once (pinwheel growth).
• For whom. Confirmed for rules that read only the law, with absent things forgotten. A rule that reads a state is not a law in this sense, and fixing absent variables instead of forgetting them lets order bits carry existence.

It was the protected arity thesis, first supported by a proof whose executor supplied an input (A4). Under K1–K3 it became two theorems about definability (thesis).

The program's central contrast: two-place relations are logic, and three-place ones carry topology.

Confirmed on 2026-10-19: proved in Lean under both readings of erasure (proof). It holds as stated under existential witnesses (A7), the strict reading (A8) and bits (A10): signed witnesses turn it into a boundary of positivity, competing pairs make the not-all-equal knot, and pairs on three values make "not all different".

It came from the first formalization of the knot test on 2026-10-19, not from the original gate, and was adopted that morning.

Under it only the atom, the single clause "not A or not B or c", is a knot, parity is not, and the doctrine said that a parity pattern is never a knot.

Overturned the same day, when the owner chose forgetting (A6): the original gate used erasure as a test of decomposability, and the program began from parity patterns. Every knot of the absence reading is a knot under forgetting. The reading stays available in the knot laboratory and in KNOT-FOUNDATION-20261019-001A-P001.

Of about 25,000 random connected 3-regular triple systems with 7 to 15 points, the only ones that could not be 2-coloured were Fano planes (space selection), and the next epic was approved with uniqueness as its premise.

The approved form of the three-fold-spaces epic.

Overturned by exhaustive search and SAT in that epic: Fano planes splice together without limit (7, 13, 19, … triangles), odd cycles of Fano blocks give more, and two twisted 14-point systems give a 27-triangle space (three-fold spaces). The Fano plane is the smallest three-fold space, not the only one. The lesson: a random probe is no evidence of uniqueness.

Three of the four far sides the test used, K4^(3), the Fano plane and census 58, have the symmetry that makes the order irrelevant, and the test never varied it.

The split step of the four-step rule that read Fano atoms (FAR-READ-20261106-001-D002), and the sentence of FAR-READ-20261106-001-D001 about census 6.

Overturned by GENUS-READ-20261107-001: census 6 has three inequivalent gluings, and its verdict changed with the order in 8,046 of 11,552 sampled pairs. Counted over its gluings a Fano atom is homogeneous, so the split step was not native, and both records carry pending qualifications (genus read). The lesson: a listing order is a choice.

On the knot's surfaces, changing one sign toggles the parity of the two faces at the ends of its side, so the forced amount of dissonance changes by at most one, and dissonance would leave the states only by entering the law (COUNT-SURFACES-20261107-001; a working hypothesis, not yet checked).

The framing of the parked steering direction.

The alternative calls a change of state learning, as when a network's connections stay fixed and its activity settles. The steering direction's first check, whether one re-signing moves the forced violated line out of the states once two even faces appear, tests the premise of this reading.

It is proved for closed spheres: a law without hidden states cannot tell the octahedral knot spheres from long capped tubes, whose balls all occur in the sphere (sphere growth). The sphere line's readings carried it over to space in general.

The reading that roundness costs hidden information, one witness bit per incidence on the spheres, and with A13 the reading that a round space is laid out only with hidden bits.

The pinwheel tested it in the plane, and it held. The pinwheel's own radius-2 law is a finite list of 1,168 forms that reads every label from the unlabelled map, and none of 96,474 periodic configurations obeys it uncut (pinwheel atlas). But a crystal cut by the pinwheel's own subdivision does: the rhombus lattice cut once obeys the collar law, and for every radius a further cut obeys the pinwheel's undecorated radius-r law, so no such law forces roundness (collar plane). It would be reopened by a uniform local law without hidden states, in the plane or on a surface, whose allowed spaces are all round.

HANDLE-COUNT A's generator places the atoms on the minimal signing it finds, and the lines that used it, HANDLE-COUNT B and TRISUM-READ, took whichever placement it returned without comparing placements. Which one it returns depends on Python's hash seed, since its Pfaffian signing walks sets.

The fresh trisum spaces of HANDLE-COUNT-20261106-001B (three to six of its 257 per-space rows change under other hash seeds; its stated totals do not), and the eighteen spaces of TRISUM-READ-20261107-001, each reproduced at its seed only under a recorded hash seed.

Overturned by SWITCH-READ-20261108-001. For five glues with two placements in one switching class, the native maps agree exactly between glue lines, but removing a line of an atom creates differently at glue lines, so the whole maps are never isomorphic: for native creation each placement is its own space. What replaced it: creation between glue lines depends only on the switching class (computed, five pairs), and the results that rested on it stand, since the junction epic's seven statements hold on all five other placements and HANDLE-COUNT B's stated totals reproduce (switch read).

PROCESS-CONTACT-20261108-001 took it by default. It is the smallest family in which the contact is a cell of each ring under that ring's own rule, and it has only 16 laws, so every one could be run on every rule (knots between processes).

The census of PROCESS-CONTACT-20261108-001B (44 of 128,128 configurations bind at every lag) and the reading that a contact shared by identity rarely persists.

At lag 1 this family fails exactly when a rule is canalizing at its centre, because the contact's own value enters both proposals. A contact that combines its four ring neighbours without reading itself is still a cell of both rings, since both rings read it. The cheapest test is the same census with the laws u(a_{m−1}, a_1) + v(b_{n−1}, b_1). If they persist much more often, the rarity belongs to this family and not to shared identity; if not, the reading stands.

PROCESS-CHAIN-20261108-001 took it by default. It reads the contact-network gate's composition as persistence of the composed binding, matching the persistence asked of each contact (two contacts across time).

The epic's finding that no chain with both contacts persistent composes, one size past included.

The alternative reads composition at the recurring lags the system itself fixes: the far contact binds the end rings at some lag of its eventual cycle, and so at lags recurring with its period. In linear chains and paths composition is arrival, so those lags are fixed by the dynamics, not by an observer. RECUR-PATH-20261108-001 tests the alternative by transport, pre-registered on 2026-11-08 before it ran. It uses the growth families of PATH-GROWTH-20261108-001: one linear rule per ring and every law, four rings growing from 3 to 9 cells (or 4 to 9, contacts two apart), or paths from 4 to 9 rings.

• Overturned, if some family composes at recurring lags with every contact persistent at every size of its schedule. The recurring reading then replaces this one, as a new entry, since under this reading no family composes at two sizes.
• Confirmed, kept for the gate, if such families exist only with contacts that bind at the composing lags but not at every lag. The recurring reading would then transport only by giving up the persistence the contact gate asks of each contact.
• Confirmed, also, if no family composes at recurring lags at every size. Composition across time is then tied to sizes under both readings.
• Back to Assumed, if the best candidate families meet a recurring reading at every decided size but some sizes stay undecided.