The geometry
of remembering.

Imagine exploring a city while drawing a map. You could record every footstep, remember every street you have visited, or shade neighborhoods already explored. These records answer different questions. A footstep diary reconstructs your journey. A street list answers membership questions. A shaded map tells you whether your next destination enters unfamiliar territory. Neither compressed record can recover everything in the diary. Each keeps information for a particular purpose.

The two repositories begin here. Novelty Geometry asks how discoveries accumulate across increasingly fine descriptions, and what mathematical object represents their endless refinement. FUTCache asks how a running program can retain enough geometric information to answer novelty queries. One develops a language of limits and boundaries; the other implements several finite representations. Their bridge is the question: which distinctions in the past can affect an observable future answer?

Start with exact novelty. A history H is a finite sequence of observations. Write V(H) for its set of distinct values. The next observation x is novel when it is absent from that set. Order and frequency disappear from this particular query. Seeing A, B, A produces the same future membership answers as seeing B, A, B. But their first discoveries occur in different orders. If that order matters, the visited set is insufficient.

Now introduce a ruler. Let d(x,y) measure the distance between observations, and choose a tolerance ε, pronounced epsilon. A reading of 20.001 degrees may be operationally familiar after 20.000 degrees, even though their exact values differ. Metric novelty asks whether the new observation is farther than ε from every historical observation. Choosing ε specifies which differences this measurement can see.

This is a definition of geometric familiarity, not a discovery of meaning. Two nearly identical medical questions can require different answers; two differently worded questions can express the same request. A geometric guarantee applies to the supplied representation and distance. It does not establish that reusing a payload is correct. Keep this separation throughout the lecture, especially when the application sounds more ambitious than the predicate.

Before continuing, choose an observable in your own work. Is it exact repetition, geometric coverage, discovery order, or safe answer reuse? State what a positive answer means. This small decision determines which mathematics is relevant and which information your memory is allowed to forget.

An infinite journey need not revisit an exact point. Consider the sequence 1, 1/2, 1/3, and so on. Every term differs from all earlier terms, yet the sequence approaches zero. Exact novelty continues indefinitely while the movement becomes arbitrarily small. At any fixed positive resolution, sufficiently late terms become indistinguishable from earlier observations. Infinity in time therefore does not imply endless macroscopic discovery.

A metric obeys four rules: distances are nonnegative; distance zero identifies the same point; distance is symmetric; and traveling directly never exceeds the length of a detour. The final rule is the triangle inequality. Euclidean distance satisfies it. So do absolute distance on a line and maximum-coordinate distance. These assumptions support geometric bounds and safe search pruning, so they are part of the computational contract.

Be careful with the phrase cosine distance. The common expression one minus cosine similarity is not generally a metric. Unit vectors separated by angles zero, sixty, and one hundred twenty degrees give distances one half, one half, and three halves, violating the triangle inequality. Angular distance or Euclidean chord distance on normalized vectors avoids that particular problem. Naming a similarity a distance does not prove its required properties.

A space is totally bounded when, for every positive radius, finitely many balls cover it. Compact metric spaces are totally bounded and complete. The finite-cover property is what limits separated discoveries. Cover the space by finitely many balls of radius ε/2. Two points in the same ball have distance at most ε. Consequently, a collection whose pairwise distances exceed ε contains at most one point per covering ball.

This proves a useful result without invoking probability: an algorithm that accepts only points farther than ε from every previously accepted representative cannot accept infinitely many points in a totally bounded space, provided it never evicts them. The proof says nothing about how quickly saturation occurs. A traversal can postpone its next discovery for a very long time. It also says nothing about exact novelty at radius zero.

Recurrence requires equal care. An accumulation point is approached by a subsequence. Periodicity means an exact repeating pattern. Poincaré recurrence belongs to measure-preserving dynamics on a finite-measure space and gives an almost-everywhere return conclusion. Arbitrary streams do not inherit that theorem merely because someone calls them trajectories. The reciprocal example already demonstrates regional return without exact repetition.

Try an escaping sequence, xₙ = log(n+1), on the unbounded real line. Its consecutive gaps approach zero, yet it has no finite accumulation point. Small steps alone do not imply convergence. The geometry of the containing space, the observation scale, and the rule generating the journey must all remain visible in a correct argument.

Suppose two histories lead your program to the same internal state. If the program is deterministic, feeding both states the same future inputs produces identical future outputs. Therefore the program may merge two histories only when no admissible continuation can distinguish them. This elementary observation provides a rigorous definition of sufficient memory and a direct route to memory lower bounds.

The symbol ∼ defines future equivalence. Its equivalence classes collect histories that behave identically for the chosen observable. The quotient is the set of these classes. Calling it minimal means every exact deterministic implementation must distinguish different classes. It does not mean that a particular list, tree, or C structure achieves the smallest byte allocation. Semantic minimality and representation efficiency are separate claims.

For exact novelty on an alphabet of m symbols, two histories are equivalent exactly when their visited sets agree. If the sets differ, choose a symbol in their symmetric difference: present in one set and absent from the other. Asking about that symbol separates the histories immediately. Every subset is realizable, so there are 2ᵐ classes. An implementation with b bits has at most 2ᵇ states, forcing b ≥ m. A membership bitset attains that bound.

For metric novelty, replace the visited set by its coverage union. Write B̄(x,ε) for the closed ball centered at x. The union Uε(H) contains all points within tolerance of some observation. Novelty is membership in its complement. If two unions agree, the next query agrees; adding the same new ball preserves equality, so all continuations agree by induction.

Conversely, unequal unions can be separated by a query in their symmetric difference, assuming such queries are admissible. If queries are restricted to a domain Q, only the intersections with Q matter. This qualification prevents a hidden mistake: two different geometric sets can be observationally identical when their difference lies outside every allowed query.

On a line, the union becomes a collection of disjoint closed intervals. Overlapping intervals merge, and historical centers can disappear from the representation. The endpoints retain everything this observable needs. In higher dimensions, exact unions can have complicated boundaries. A semantic quotient does not automatically provide a small or efficient data structure for representing each class.

This distinguishability argument resembles the Myhill–Nerode method for minimizing deterministic automata. Here we have explicitly specified a stream observable rather than a language recognizer. The important habit is transferable: identify a pair of histories you propose to merge, then actively search for a continuation that would expose the lost distinction.

FUTCache offers several representations, so attach every guarantee to an engine. Its one-dimensional interval engine maintains exact historical coverage. Its packing engine keeps representatives R drawn from the history H and tests distance to those representatives. These are different predicates. Understanding their relationship is more valuable than memorizing a broad claim that the cache is exact.

The greedy packing rule is simple. If the new point lies within ε of a representative, report familiar. Otherwise report novel and insert it. Without eviction, representatives remain pairwise farther than ε apart. Every observation lies within ε of some representative after processing. The resulting set is an ε-net for the observed points, where net here means a covering set with the additional greedy separation property.

The set inclusion proves one-sidedness. A packing hit has a historical witness, so the packing engine cannot label a genuinely new point familiar under the same metric and radius. However, a packing miss can be unnecessary. Removing centers shrinks the union and can expose gaps. Be explicit about labels: this means no false familiarity, but possible false novelty. Different applications reverse the conventional positive class.

Here is the smallest useful counterexample. Let ε equal one and observe zero, then 0.9. The packing engine retains only zero because 0.9 is already covered. Query 1.8. Its distance to the historical point 0.9 is 0.9, so exact history says familiar. Its distance to the sole representative zero is 1.8, so packing says novel. One omitted center extended historical coverage even though that center was itself familiar.

With no eviction, triangle inequality gives another sandwich: Uε(H) is contained in U₂ε(R). Any covered query is within ε of some historical point, which is within ε of a representative. This bounds a geometric ambiguity region. It does not bound its probability mass. An adversarial query distribution can place every query inside that region, so a small thickness does not imply a small error rate.

Eviction preserves R as a subset of history and therefore preserves the first one-sided implication, provided the metric, radius, and historical reference stay fixed. It can destroy the net-cover property, allow rediscovery, and invalidate lifetime acceptance bounds. Changing ε or embedding versions changes the question itself. Rebuild or explicitly version state when its geometric meaning changes.

The experiment below uses the actual mathematical greedy rule on a synthetic stream, not the C library. Move the radius and replay the history. Compare exact coverage with representative coverage before inserting a query. In particular, notice that the full-history model updates after every observation, including observations that the packing model declines to store.

A packing number P(K,ε) is the maximum number of points in K separated pairwise by more than ε, using that strict convention throughout. The unevicted greedy representative count cannot exceed it. This is a bound on retained centers at one scale, not automatically on all possible exact coverage boundaries, machine states, or bits needed for arbitrary real coordinates.

In the unit cube with maximum-coordinate distance, a simple grid argument gives a conservative bound of (floor(1/ε)+1)ᵈ representatives. Partition each coordinate into intervals of width at most ε, assigning endpoints consistently. Points in the same grid cell are at most ε apart. The exponential dependence on dimension is unavoidable for broadly distributed data. Compactness promises finiteness, not practicality.

The exponent D describes how distinguishable structure grows as resolution improves. For regular geometries, halving ε costs approximately 2ᴰ times as many centers. A line suggests a factor two; a surface suggests four. But an asymptotic exponent alone does not guarantee that consecutive ratios converge. Lower-order oscillations can disturb individual steps even while the logarithmic slope converges.

The reciprocal set K = {0,1,1/2,1/3,…} makes this distinction tangible. At dyadic resolution ε = 1/N, occupied cells correspond approximately to distinct integers floor(N/n). Split n around √N. The early part contributes at most √N values; the tail takes values below √N. This proves an upper bound proportional to √N. Conversely, a constant fraction of the small integer values occurs, giving a matching lower bound.

Thus the cell count grows as Θ(√N), and its resolution exponent is one half. The ambient line has dimension one. The set itself is countable and has Hausdorff dimension zero. Its box-counting behavior can still have exponent one half because many isolated points cluster at progressively finer scales. These notions answer different questions about size.

Novelty Geometry defines a cache dimension from the number Mⱼ of discovered partition cells: the limsup of log Mⱼ divided by j log two. To identify this with a packing or box dimension, require a suitable relationship between partition cells and metric covers, together with sufficient traversal coverage. Arbitrarily over-refined partitions can inflate cell counts; the number of partition cells is not automatically bounded by P(K,ε).

Finally, a temporal discovery law requires a temporal assumption. If effective resolution behaves like N to the power minus α, and discovered complexity follows ε to the power minus D in the relevant joint regime, substitution gives growth N to the power αD. The equation β = αD is a conditional factorization, not a universal prediction that every stream obeys a power law.

A raw prefix tree stores temporal history. Its root is the empty word, and each edge appends one observation. An infinite branch describes an infinite sequence. Two sequences are close in a prefix metric when they agree for many initial steps. This remembers repetition, waiting, and every recorded detail. Novelty Geometry makes a different construction by replacing temporal depth with observational resolution.

Choose finite partitions P₀, P₁, P₂, and so on, each refining the previous partition. Refining means every fine cell lies inside exactly one coarse parent. Require cell diameters to shrink toward zero when the aim is to resolve the original metric space. For a traversal L, let Dⱼ(L) list level-j cells in their order of first encounter, deleting later repetitions.

For example, split the unit interval into left and right halves, then into quarters. Suppose the fine cell sequence is Q₂, Q₀, Q₂, Q₃, Q₁. Its discovery word is Q₂, Q₀, Q₃, Q₁. Mapping these cells to their parents gives right, left, right, left. Delete repeated parents after their first appearance to recover the coarse discovery word right, left.

This parent-and-reduce rule is the bonding map q. Its compatibility equation says q(Dⱼ₊₁(L)) = Dⱼ(L). The proof uses first occurrence: the first visit to a coarse cell is precisely the earliest visit to any of its children. Later visits to other children cannot change that first coarse position. Ordinary letterwise projection without reduction would fail to produce a discovery word.

At each fixed resolution, only finitely many cells exist, so the eventual discovery word is finite even for an infinite traversal. A finer level can reveal discoveries hidden inside a familiar coarse cell. The resulting profile therefore retains order across scales while forgetting repeated visits and waiting times. Two different raw journeys can have the same entire profile.

Partition conventions matter. Half-open intervals with a separately assigned right endpoint prevent ambiguous cell membership. Changing the partitions can change discovery order and thus the resulting profile. Small spatial perturbations near a cell boundary can also change a coarse label. An ordered-discovery profile is not automatically a continuous or canonical encoding of raw trajectories under every topology.

Use the experiment to read the same finite stream at two adjacent levels. Predict the fine word first, then apply parent-and-reduce yourself. The visualization is a finite example of compatibility, not evidence that a finite observation window certifies the final word of an endless stream. That distinction becomes decisive when we discuss computability.

Let Xⱼ be a finite space of allowed discovery words at resolution j, with compatible bonding maps. The inverse limit consists of all sequences z = (z₀,z₁,…) satisfying q(zⱼ₊₁) = zⱼ. Think of a dossier whose increasingly detailed pages never contradict earlier pages. An arbitrary collection of pages is not enough; coherence is the defining condition.

Each finite discrete Xⱼ is compact. Their countable product is compact, and the compatibility equations define a closed subset. Thus N is compact. For this setting, a diagonal argument makes the proof concrete: choose a subsequence agreeing at coordinate zero, then a further subsequence agreeing at coordinate one, and continue. The diagonal subsequence converges coordinate by coordinate, and its limit remains coherent.

Define a distance using the first disagreeing coordinate. If z and z′ first differ at index k, set ρ(z,z′) = 2⁻ᵏ; identical profiles have distance zero. This convention avoids an indexing ambiguity about whether depth counts coordinates or edges. Agreement through coordinate j gives distance at most 2 to the power minus j minus one.

This stronger triangle inequality is the ultrametric inequality. If x and y agree through a given depth, and y and z agree through that depth, then x and z also agree there. Consequently, balls are nested or disjoint. Closeness is determined by shared initial resolution information, rather than by averaging coordinate differences.

Pull ρ back to traversals through their discovery profiles. This gives a pseudometric: different traversals can have distance zero because they share all discovery words. Quotient those zero-distance traversals to obtain a genuine metric space of realizable profiles. Its metric completion is the closure of its image inside N. This statement always identifies the correct completed object.

To replace that closure by all of N requires density. Every compatible finite profile must extend to a traversal in the allowed class, or equivalently every relevant cylinder must meet the realizable image. Arbitrary traversals may allow extensions that computable runs, constrained dynamical systems, or a fixed machine family do not. Omitting this hypothesis turns a conditional theorem into an unjustified universal statement.

Notice another subtlety. A sequence of realizable profiles can converge while its realizing trajectories have no common temporal behavior. Completion concerns agreement at finite resolution depths. It need not mean that the original observation points converge in K. The new limit lives in a space of ordered descriptions, and interpreting that space correctly is the central conceptual advance of the construction.

Density does not mean every ideal profile is itself realized. To see why, partition the unit interval with zero as a singleton and positive dyadic intervals as the other cells. At level j, consider the one-letter word selecting the cell (0,2⁻ʲ]. These words are compatible: each smaller positive cell lies inside the preceding one. Every finite prefix is realized by a constant traversal at a sufficiently small positive point.

Nevertheless, no traversal realizes the entire profile. Its first observation would have to belong to every selected cell. Their intersection is empty: zero is excluded, and every positive point eventually exceeds the shrinking upper endpoint. Completion therefore adds an ideal coherent description that all finite tests can approximate, although no actual journey has that complete description. This example separates finite realizability from infinite realizability without appealing to mysterious infinities.

Construct a graph whose vertices are coherent finite profile prefixes. Connect each prefix to an extension by one coordinate, and include the empty prefix as a single root. Each nonroot vertex has exactly one parent. This is a tree. Every level is finite because each coordinate space is finite, so branching is finite. Remove prefixes without infinite extensions if necessary; they contribute no boundary rays.

Give every edge length one. The distance between two vertices is the number of edges along their unique connecting path. For three vertices, their connecting paths form a tripod. There is no open triangular interior. Every side lies in the union of the other two, making the tree a geodesic zero-hyperbolic space. This is a precise statement of hyperbolicity that requires no curved drawing.

The Gromov product above measures shared path length from the root o. On a rooted tree it equals the depth of the last common ancestor. Extend from vertices to infinite rays. Two boundary points share k coordinate edges exactly when their profiles agree for k initial coordinates. Their visual distance 2 to the power minus k is precisely our first-disagreement ultrametric.

Each infinite ray supplies one coherent profile, and every coherent profile supplies one ray. This proves a concrete identification between the inverse limit and the boundary of this locally finite tree. Combined with the density hypothesis from the previous module, it identifies the metric completion of realizable discovery profiles with that boundary. The proof works because we built a tree whose branching records exactly the desired finite information.

The original space K need not be hyperbolic. A flat square can generate a hyperbolic tree of descriptions. Nor does this boundary construction establish physical gravity, a model of language hallucination, or an explanation of consciousness. It is an exact mathematical realization of a specified information structure. Wider interpretations require additional models and evidence.

A related classical example is the p-adic integers. Record coherent residues modulo p, p², p³, and so forth. These form an inverse limit. Two integers are p-adically close when their difference is divisible by a high power of p, meaning many low-order digits agree. The familiar-looking digit tree now measures shared divisibility information rather than ordinary numerical proximity.

This analogy must preserve its scope. The p-adic construction concerns coherent residues or raw digit prefixes; it does not automatically identify an arbitrary ordered-novelty space with the p-adic integers. Trees provide a common representational mechanism, while bonding maps determine which boundary is actually obtained. Ask what each edge records before claiming two tree-based theories are the same.

So far, refinement has meant making the ruler smaller. Persistence asks a complementary question: what happens when coverage balls grow? Given a finite point set, form Uₜ as the union of radius-t balls. Increasing t produces nested sets. Connected components can merge. Tracking when topological features appear and disappear summarizes structure that survives across a range of scales.

On a line, sort distinct observations x₁ through xₙ. Two neighboring radius-t intervals meet when twice t reaches their gap. Consequently, component mergers occur at half the adjacent gaps. With all components born at zero, finite zero-dimensional persistence deaths come from these merge scales; one component survives indefinitely. A consistent elder rule resolves which component label survives each merger.

The experiment uses points zero, two, and seven. The first pair merges at radius one. The second gap closes at radius two and one half. Predict the component count on either side of these thresholds. These are spatial filtration scales, not stream arrival times. Confusing these axes can produce an attractive diagram with the wrong mathematical interpretation.

FUTCache's one-dimensional persistence implementation uses gap-based merging. Its multidimensional persistent engine also maintains representative isolation scores and the complete observation history for exact arbitrary-scale novelty queries. Inspecting the implementation reveals an important tradeoff: retaining history permits exact queries after representative eviction, but storage then grows with observations. A bounded active representative set is not bounded total memory.

An isolation score such as nearest-neighbor distance minus a radius is useful for ranking redundant representatives, but is not automatically a full persistent-homology computation. Likewise, a persistence diagram summarizes topology; it generally does not preserve where coverage lies. Translating every point by one hundred preserves all pairwise gaps and component lifetimes while changing novelty answers to fixed spatial queries.

Classical persistence stability states, under suitable hypotheses, that small uniform changes to filtering functions produce bounded changes in their diagrams. This supports robustness of properly defined topological summaries. It does not mean each feature shifts by exactly the same amount, or that every repository score inherits the theorem. One must identify the filtration, comparison metric, and assumptions before applying stability.

The repository additionally assigns prime labels to discovery events and experiments with zeta-like products. Such labels can be useful identifiers. They do not by themselves establish a Selberg trace formula or a relationship to prime-number distribution. A productive research question would specify an operator or dynamical system and prove that its spectral or orbit data matches the proposed product. Until then, distinguish evocative notation from a demonstrated mathematical correspondence.

Choosing a radius creates a tradeoff. Small radii retain many representatives and explain observations precisely. Large radii compress more aggressively but leave larger residual distances. Minimum description length motivates balancing model complexity against the cost of describing data given that model. The key requirement is to specify what is being encoded rather than declaring that a visually appealing curve has found the true scale.

FUTCache's offline selector evaluates a finite radius grid. Its implemented score includes allocated model bytes converted to bits, a radius-index penalty, assignment cost n log₂|R|, and a residual term. In its lossy mode, the residual is a weight λ times the sum of squared nearest-representative distances. It returns the minimizing candidate on that grid.

That finite minimization is exact for the declared score. Calling it a universally optimal code would be stronger than the implementation supports. Allocation costs depend on backend overhead. The default radius penalty log₂(i+2) is not automatically a valid prefix-code length family: its Kraft sum can exceed one. A rigorous coding interpretation needs normalized or otherwise valid lengths and complete finite-precision conventions.

The demonstration uses an explicitly simplified synthetic score, with a fixed cost per representative and a uniform grid-index cost. Increasing λ makes residual error more expensive. Watch the selected radius change. This is a controlled explanation of the tradeoff, not a reproduction of the library's allocator-dependent numerical results. Even the best geometric score cannot certify semantic equivalence between queries.

We can also compress the metric representation itself. Choose anchors A that form a δ-net of the domain, and map x to the vector of distances d(x,a). Measure embedded vectors with maximum-coordinate distance. Reverse triangle inequality proves that no anchor-coordinate difference exceeds d(x,y), so the embedding cannot expand pairwise distance.

For the lower bound, choose an anchor within δ of x. Triangle inequality gives d(y,a) ≥ d(x,y)−δ, while d(x,a) ≤ δ. Subtracting gives the claimed lower bound. An anchor near both points is unnecessary. To certify an original-space hit at radius ε, an embedded distance at most ε−2δ suffices, provided δ is a certified cover radius and ε is at least 2δ.

A cover radius estimated on sampled data may not cover future queries. Likewise, symmetrizing an arbitrary similarity does not repair triangle inequality. The mathematics tells us exactly what assumptions an engineering approximation owes us. When those assumptions cannot be certified, retain the result as a heuristic and evaluate its errors against the intended operational predicate.

A useful empirical check appears in the repository's semantic negative result. On its fixed thirty-phrasing benchmark, geometric selection chose radius 0.70, with reported reuse precision 52.2 percent and eleven cross-intent merges. A smaller tested radius avoided those merges. These are repository-reported measurements, not independently reproduced results here. Their role is to demonstrate a possible mismatch between geometric compression and labeled application correctness, rather than establish a universal best threshold.

Before interpreting any selected radius, ask which quantities change if you rescale coordinates, change the representative allocation strategy, or permute the input stream. A greedy model can depend on arrival order, and an allocation-based score can depend on implementation choices. Such dependence is acceptable when declared, but it changes what an optimum means.

When a memory budget forces selection, first choose the objective. Maximizing covered observations, minimizing transport cost, preserving rare events, and retaining first-discovery order are different optimization problems. A representative that is redundant under one objective can be essential under another. The word optimal has meaning only after the objective, feasible sets, and comparison class are fixed.

For finite calibration data V, define f(S) as the number of observations within ε of at least one selected representative in S. This function is monotone: adding a center cannot reduce coverage. It is also submodular: a new center covers fewer previously uncovered observations when added to a larger existing set. This diminishing-returns property gives a useful greedy guarantee.

Choose up to k centers greedily by maximum marginal coverage. If OPT is the best k-center coverage, some optimal center must cover at least one kth of the remaining gap; otherwise their combined marginal gains could not close that gap. After one step the gap shrinks by at least the factor one minus one over k. Repeating k times leaves at most that factor to the kth power, which is at most one over e.

Hence greedy achieves at least 1−1/e of optimum for this normalized monotone submodular cardinality problem. The argument concerns a fixed finite objective with exact marginal evaluations. It does not automatically provide the same guarantee for online eviction, moving distributions, arbitrary weighted transport, or an implementation that approximates the objective differently.

FUTCache's W1-named eviction path favors an endpoint of a closest representative pair. A narrow transport interpretation exists: if the deleted atom has equal fixed mass and that mass moves to its nearest survivor, the transport cost is that mass times nearest-neighbor distance. Renormalizing all surviving weights changes the problem. Minimum crowding cost is also not the same as minimum lost coverage.

Distributed memory introduces an algebraic question. If replica states form a join-semilattice and merge computes their least upper bound, merging is associative, commutative, and idempotent. Reordering, grouping, or repeating delivered updates then leaves the result unchanged. Under eventual delivery and compatible monotone updates, replicas converge. This is the foundational state-based CRDT argument.

The repository's anchor-cell replica design uses deterministic priority to resolve occupied-cell conflicts. Such designs require compatible partitions and configuration, and a complete tie rule if priorities collide. Arbitrary deletion or clearing is not a monotone join; stale replicas can reintroduce forgotten information unless the protocol handles generations or removal metadata. Convergence of cell state also does not make cell membership identical to Euclidean ball coverage. Distributed correctness and geometric correctness require separate proofs.

Return to an endless computation. At each finite resolution, only finitely many new cells can appear. Its discovery word eventually stabilizes. Therefore a coherent infinite profile can exist even when the computation never halts and its raw configurations never converge. This is the attraction of the repository's boundary-object-machine proposal: divergent runs can retain distinguishable discovery structure.

Existence is weaker than effective knowledge. At time t, a program can list the cells encountered so far. It generally cannot certify that an unseen cell will never appear later. Imagine a traversal that visits a special cell only if a simulated machine halts. A universal procedure certifying the final discovery word would decide whether that special event ever happens, giving a forbidden solution to the halting problem.

Thus each coordinate can stabilize without a uniformly computable stabilization bound. The whole boundary object may be mathematically well-defined while inaccessible as a completed finite computation. Even continuity depends on the representation: an arbitrarily long temporal prefix can be extended to visit a new coarse cell, changing the eventual profile. A topology based on eventual discoveries behaves differently from one based on elapsed-time prefixes.

The configuration space also needs honest modeling. An infinite tape over a finite alphabet has a compact product topology, and combining it with finitely many control states preserves compactness. Adding an unbounded head position as a discrete coordinate can destroy compactness. One can choose other encodings, but that choice affects the observational tower and the meaning of its boundary. There is no automatic canonical geometry for all machines.

The two repositories now fit together without being collapsed into one theorem. Novelty Geometry organizes ordered discoveries across scales and realizes coherent profiles as boundary points. FUTCache implements useful finite memories for particular geometric predicates. The exact coverage quotient explains what future novelty can observe; conservative packing trades some exact coverage for manageable representatives. Persistence, MDL, and distributed joins add other observables and optimization choices.

For your final exercise, design a novelty monitor for a temperature-and-pressure sensor. Specify units and a metric, choose whether every observation extends historical coverage, and state whether forgetting is allowed. Give one counterexample your approximation must handle. Then explain whether your memory preserves location, discovery order, or only component lifetimes. Finally, identify which assumptions would be needed to connect measured growth to a dimension exponent.

A strong answer includes an error direction and a reference predicate. It says which experiments support operational performance and which statements follow from proof. The research frontier lies in efficient representations, principled observational towers, realizability constraints, and useful approximation bounds. None requires pretending that a finite cache has computed an infinite boundary, or that mathematical proximity has already solved semantic understanding.