Menger Sponge
× Hypercube

A research notebook: constructions, counterexamples, and a change of metric

Can a cube reveal a hidden relationship without preserving the distances that make that relationship meaningful? This investigation follows that question from sixty-four binary states to the retained cells of a Menger sponge. The route passes through Gray-code tours, explicit binary embeddings, failed dimension reductions, and a small counterexample that changes what an exact representation can mean.

The shortest account is this: a Menger adjacency graph can preserve its edges inside a binary hypercube while losing shortest-path distances. Repeated calculations led from that discrepancy to an explicit obstruction against every isometric binary-hypercube embedding. Two later records pursued different responses: carry the obstruction to every finite level, or change the metric used to represent distances. The detailed sequence matters because these are different mathematical questions.

1. Before the Menger graph: sixty-four states and the collision of ideas

First experiment: change the order, keep the states

The earliest recovered computational record in this lineage begins with a precise question: if the same sixty-four states of six binary positions are enumerated in different orders, how many positions change as the list is traversed?

For integers i from zero through sixty-three, the experiment compares two six-bit representations:

B(i) = i
G(i) = i XOR (i >> 1).

B is ordinary binary enumeration. G is the binary-reflected Gray code. Both lists contain the same sixty-four distinct six-bit states. The input size and state set stay fixed; only their order changes.

In the first experiment, the final state is not connected back to the first. There are sixty-three transitions. The quantity measured is Hamming distance between successive labels: the number of bit positions that change in a transition.

For binary enumeration, the recorded distribution is:

This gives a total of

32×1 + 16×2 + 8×3 + 4×4 + 2×5 + 1×6 = 120 bit changes.

The mean is 120/63, approximately 1.90476 changes per transition, and the maximum is six. In the Gray order, all sixty-three transitions change exactly one bit: total sixty-three, mean one, maximum one.

The finding is about a known representation and a stated cost measure. It does not say that the states themselves become simpler. It says that a different order changes the local work required to move between them when cost is counted as the number of changed positions.

Second experiment: close the sequence

The next record deliberately changes only one condition: include the transition from the last state back to the first. The sequence length, state order, and bit-change metric remain the same. There are now sixty-four transitions.

The binary list ends at 111111 and begins at 000000. Its closing transition changes six positions. The Gray list ends at 100000 and begins at 000000, so its closing transition changes one.

Binary closed total = 120 + 6 = 126
Gray closed total = 63 + 1 = 64.

The resulting mean costs are 126/64 = 1.96875 and 64/64 = 1. The one-bit Gray property survives closure. Because all sixty-four distinct Q6 vertices are visited and every successive pair, including the closing pair, differs in one bit, this Gray order is a Hamiltonian cycle of Q6.

That qualification does not apply to ordinary binary enumeration: it is a cyclic order of the same states, but some consecutive states differ in several bits and therefore do not share a Q6 edge. This distinction becomes important when a distance along an ordering is compared with distance in the underlying graph.

The recorded next-step requirement was to introduce a genuinely different observable. Counting the same one-bit transitions again would not create a new result.

Third experiment: replace local adjacency with global questions

The next investigation changes the metric and scale of attention. Instead of asking only about neighbors in the list, it asks what happens halfway around the cyclic order and how far apart complementary states occur.

For the half-cycle test, compare position i with position (i+32) modulo sixty-four. The ordinary binary labels differ by the fixed XOR mask 100000, which has Hamming weight one. The reflected Gray labels differ by 110000, which has Hamming weight two.

B(i) XOR B((i+32) mod 64) = 100000
G(i) XOR G((i+32) mod 64) = 110000.

A separation of thirty-two places in the cyclic order is therefore not the same as complementing all six bits. The phrase “opposite on the ring” describes an ordering relation, not automatically an opposite hypercube vertex.

The complement experiment starts with a different operation. For any six-bit state v, its bitwise complement is

complement(v) = v XOR 111111.

Given a cyclic ordering, locate both states and measure their shorter separation along that ordering:

d_cycle(i,j) = min(|i−j|, 64−|i−j|).

The recorded distribution uses one measurement from each of the sixty-four starting states, so each unordered complementary pair is represented twice. In ordinary binary order, the possible shorter separations are every odd value from one through thirty-one, each appearing four times. The mean is sixteen.

For the reflected Gray order, the shorter complementary separations are twenty-two and twenty-six, each appearing thirty-two times. The mean is twenty-four. No complementary pair is separated by thirty-two cycle steps.

These numbers concern travel around the chosen cyclic ordering. Complementary six-bit labels always have Hamming distance six in Q6. There is no contradiction between that ambient graph distance and a longer route along a selected cycle. The same distinction between a prescribed route and the ambient graph will return in the Menger investigation.

Fourth experiment: reconstruct the global claims independently

A subsequent preserved response reconstructs the sequences, their index maps, the complementary masks, and the shorter cyclic distances. It reports agreement with all four global claims: the two half-cycle masks and the two complement-distance distributions.

This response introduces no new variable or additional research candidate. Its role is a check of the preceding global-symmetry result using a separate reconstruction. The record marks the first completed return check in that research exchange and asks that the next round pose a new question rather than repeat the same global measurements.

The surviving first-pass and closure records report execution on September 30, 2026, followed by the global-symmetry record and its return check across the midnight boundary into October 1, in Taiwan time. Exact execution timestamps are source-reported; provider creation times are not used to certify them. The later Menger results are a different set of experiments, even where they inherit the interest in hidden relations, binary coordinates, and the difference between local and global structure.

The collision with Silk

After this binary-state work, the conversation asked whether the cube and the film Silk could have a romantic relationship. The retained research seed placed two questions side by side:

Silk asks: “Can a cube trap something invisible?”

The Absinthe research asks: “What do invisible relationships inside a cube actually look like?”

These are English translations of the preserved Chinese seed questions. “Absinthe” is the name of the existing research strand; the mathematical investigation does not concern the alcoholic drink.

The task was to examine the relationship between the propositions themselves, after setting aside who asked or answered them. Was it a physical or mathematical relationship, a structural analogy, a cognitive association, an aesthetic rhyme, or no substantial connection? The questions made an opening for research; they did not supply a mathematical identification between the film's cube, a binary-state graph, and a Menger sponge.

The research subsequently allowed the two sides to develop separately. A later seed amendment explicitly removed the obligation to manufacture a new film–mathematics connection in every round. The original collision remained the source of the question. The next task was to choose precisely which objects could be connected by an actual construction.

2. First research record: separate the objects before connecting them

Question

Are the familiar cube forms already the same mathematical structure, or does a relationship still need to be constructed?

The distinction recorded

A six-dimensional cube graph Q6 has 64 vertices, 192 edges, and degree six at every vertex. Its vertices are six-bit strings; one-bit changes determine its edges. By contrast, the first Menger removal stage retains twenty small cubes from a 3 × 3 × 3 block. Its relationship to the ideal infinite Menger sponge involves repeated scaling by one third, with ideal-limit Hausdorff dimension log(20)/log(3), approximately 2.726833.

Those counts belong to different objects. Twenty retained cells at one finite stage are not the same vertex set as 64 binary states. Neither a similar-looking projection nor a shared word, “cube,” supplies an isomorphism. The first record therefore declined to infer mathematical ancestry between the objects.

What this round left open

The useful next question was narrower: choose a finite retained-cell adjacency graph, specify its edges, and attempt a mapping into Q6. The later success of such a mapping would not contradict the initial rejection of identity. “The objects are identical” and “one selected graph embeds in the other” are different statements.

3. First construction: twenty retained cubes inside Q6

This construction survives in the intervening v1.5 research package, before the explicitly labelled second round.

Define the graph

Let V1 consist of triples (x,y,z) in {0,1,2}³ for which at most one coordinate equals 1. Removing triples with at least two coordinates equal to 1 removes seven cells and retains twenty.

For u,v in V1, put an edge between them exactly when

|u_x − v_x| + |u_y − v_y| + |u_z − v_z| = 1.

Thus an edge records a shared face between two retained cells, not a shared edge or corner. The recorded graph has 24 undirected edges. Its degree histogram is twelve vertices of degree two and eight of degree three.

Construct the binary labels

Encode a single coordinate by

c(0) = 00
c(1) = 01
c(2) = 11.

For v=(x,y,z), concatenate the three codes:

F(v) = c(x) c(y) c(z).

The six-bit labels are distinct. The recorded verification checks every vertex pair and confirms

u and v share an edge ⇔ Hamming(F(u),F(v)) = 1.

This is an induced embedding in Q6. Among the twenty selected hypercube vertices, the ambient cube supplies exactly the wanted edges and no additional edges.

The failed stronger claim

Consider

u = (0,0,1),  F(u) = 000001
v = (0,2,1),  F(v) = 001101.

Their labels differ in two bits. In the retained-cell graph, the direct middle cell is missing, and the shortest route has four edges. One such route is

(0,0,1) → (0,0,0) → (0,1,0) → (0,2,0) → (0,2,1).

The code preserves adjacency but not this distance:

d_M1(u,v) = 4,    d_Q6(F(u),F(v)) = 2.

The preserved numerical output reports 36 ordered pairs with increased graph distance relative to this encoding. This is a property of the stated mapping; it was not a proof that every possible hypercube embedding must fail to preserve distances.

Result and next question

The genuine bridge was an induced-subgraph construction. The counterexample stopped it from becoming an inflated claim of full metric equivalence. The next problem was to extend the construction to the 400-cell second stage and investigate the required binary dimension and distance distortion.

4. Round two: four hundred vertices and a Q15 construction

The new graph

The second-level coordinates range from zero through eight. Write each coordinate in base three. Retain a triple only if, at each of its two ternary digit positions, no more than one coordinate has digit 1. Keep the same unit face-adjacency rule.

The recorded construction contains 400 vertices and 672 undirected edges. The increase is substantive: the program constructs the next-level graph rather than relabelling the twenty-vertex example.

Why a path code appears

An axis now has nine positions. To encode it without introducing false adjacent pairs, the research searches for a nine-vertex induced path P9 in a smaller binary cube. “Induced” means consecutive positions are adjacent and nonconsecutive positions are not adjacent.

The round-two path in Q5, written as decimal representations of five-bit labels, is

p = [0,16,24,28,12,14,6,22,23].

Three independent copies give the fifteen-bit code

F2(x,y,z) = p[x] concatenated with p[y] concatenated with p[z].

The retained Menger graph occupies only selected triples in the full nine-position Cartesian grid. The induced-path condition ensures that Hamming distance one between selected labels corresponds precisely to changing a single coordinate by one position. The output records 672 induced edges, exactly matching the graph's 672 edges.

Distances expose the limitation again

Breadth-first searches recorded graph diameter 24. Over all distinct pairs, the maximum ratio d_graph/d_Hamming for this mapping was five. A stored witness is

u = (0,0,4),    v = (0,6,4)
d_graph(u,v) = 10
d_Hamming(F2(u),F2(v)) = 2.

The denominator is the ambient binary distance, not the shortest path restricted to the selected induced subgraph. The restricted subgraph retains the original graph's paths. It is the larger ambient cube that offers shorter routes through vertices outside that selected set.

What remained unresolved

Q15 is an achieved dimension, not a proved minimum. The record asks whether a Q12 or Q13 construction might exist, or whether a lower bound can be proved. It also explicitly avoids extrapolating from a finite graph to the infinite sponge's topology or fractal dimension.

5. Round three: why the proposed Q12 shortcut fails

Hypothesis tested

Could the fifteen-bit product construction be reduced to twelve bits by placing each nine-position axis in Q4 instead of Q5?

The experiment enumerates induced paths in Q4. During depth-first search, a candidate extension is rejected if its vertex has already appeared or if it is adjacent to any earlier path vertex other than the current endpoint. This prevents both repetitions and unwanted chords.

Recorded result

The maximum induced path in Q4 contains eight vertices, not the nine required. The program also supplies a nine-vertex induced path in Q5:

p3 = [0,1,3,7,6,14,12,13,29].

The preserved result therefore rules out the proposed three-axis Q4 product construction. Within that coordinate-separated construction family, each complete nine-position axis needs at least five binary coordinates, so three such blocks require fifteen.

The scope correction

This does not prove that every induced embedding of the 400-vertex graph needs fifteen dimensions. A map designed for the retained vertices as a whole need not encode each complete axis independently. A non-separable construction could exploit dependencies unavailable to the product method.

The elementary counting bound is

2^d ≥ 400, hence d ≥ ceil(log2 400) = 9.

Together with the achieved construction, the record leaves the global induced-embedding dimension between nine and fifteen. It does not eliminate dimensions nine through fourteen.

An important version detail

The Q5 path found here differs from the path stored in round two. Round four imports this new path. Both yield Q15 constructions of the same Menger graph, but their binary labels are not identical. A reader should not silently combine statistics from the two mappings into one experiment.

Next question

The research asks for a non-separable lower-dimensional embedding or an actual impossibility certificate. It also begins to separate minimum dimension from minimum distance distortion: an embedding can improve one objective without settling the other.

6. Round four: a complete distance-error profile

Question and procedure

Rather than report one distorted pair, how much shortest-path information does the current Q15 representation lose across the entire level-two graph?

The code uses p3 from round three. An equivalent integer expression for its fifteen-bit labels is

F4(x,y,z) = (p3[x] << 10) | (p3[y] << 5) | p3[z].

The bit blocks do not overlap. The program verifies that all labels are distinct and that the induced binary edge set equals the original edge set. It then runs breadth-first search from every graph vertex and examines each unordered pair exactly once. There are

400 × 399 / 2 = 79,800 pairs.

For each pair it records the additive difference

Δ(u,v) = d_graph(u,v) − Hamming(F4(u),F4(v)).

Complete recorded distribution

The counts total 79,800. Exactly 19,656 pairs retain their distances. The largest observed additive difference is twelve. The stored witness is

u = (0,0,0),    v = (6,6,6)
d_graph(u,v) = 18
Hamming(F4(u),F4(v)) = 6
Δ(u,v) = 12.

What the profile proves and what it does not

It establishes the distance-error profile of this explicit encoding. It does not establish a lower bound of twelve on every possible encoding's worst error. It also should not be called the same measurement as round two's maximum ratio five: the metric of error and the axis path both differ.

The next research card asks for a lower-dimensional induced embedding or a different Q15 encoding with improved path-distance behavior. The strongest unanswered question is now clearer: is exact binary distance recovery possible at any dimension, or is the obstruction in the graph rather than merely the chosen code?

7. Round five: reject every exact binary-hypercube representation

A change of method

Testing one representation after another cannot by itself rule out all possible representations. Round five therefore applies a known structural criterion for partial cubes. A partial cube is a graph that admits an isometric embedding into a binary hypercube: every graph distance becomes a Hamming distance.

For edges e=(a,b) and f=(c,d), define the Djoković–Winkler relation by

e Θ f ⇔ d(a,c) + d(b,d) ≠ d(a,d) + d(b,c).

In a binary hypercube, related edges correspond to the same changing coordinate. For an isometric subgraph this relation must be transitive. Therefore, three edges satisfying e Θ f and f Θ g but not e Θ g provide a negative certificate.

The actual finite-graph work

The round-five program reconstructs the 400-vertex graph and computes its all-pairs graph distances using breadth-first search. It builds the Θ relation over all 672 edges and searches for a transitivity failure.

The archived output records these three edges:

e = (1,0,2)–(2,0,2)
f = (0,0,0)–(1,0,0)
g = (1,0,5)–(2,0,5).

For the three tested relations, the stored Boolean values are [1,1,0]. The program and the edge coordinates are present in the archive; the JSON does not record the corresponding six distance sums for this particular triple. Later rounds supply a different, smaller-coordinate witness with explicit sums. These two witnesses should not be silently substituted for one another in the history.

Consequence

The level-two graph is not a partial cube. No choice of binary dimension or alternative binary labels can make every graph distance exactly equal to Hamming distance.

This does not retract the Q15 induced embedding. Induced embeddings preserve the selected vertices' adjacency; isometric embeddings additionally preserve all distances measured in the ambient cube. The former can exist when the latter cannot.

Remaining questions

The conclusion applies to this specified finite face-adjacency graph. It does not settle the smallest induced dimension, the best nonzero distortion, other adjacency conventions, or every Menger level. It is an application of an established criterion, not a completed claim of mathematical novelty.

8. Round six: reduce the obstruction to the first level

Why go back to twenty vertices?

A smaller explicit obstruction is easier to inspect and explain. Round six asks whether the failure already exists in the first retained-cube graph, rather than depending on the larger second-stage geometry.

The three-edge certificate

The recorded edges are

e1 = (1,0,2)–(2,0,2)
e2 = (0,0,0)–(1,0,0)
e3 = (1,2,0)–(2,2,0).

Using the graph's own shortest-path metric, the program records:

Thus e1 Θ e2 and e2 Θ e3, while e1 and e3 are not related. This directly violates transitivity. The same coordinates and comparisons are also recorded in the second-level graph.

The archived JSON separately reports the two graph sizes:

level 1: 20 vertices, 24 edges
level 2: 400 vertices, 672 edges.

How the witness was sought

The program constructs exact BFS distances and the full Θ matrix, then searches nearby candidate edges around each pivot, retaining a compact witness according to its endpoint diameter. The recorded witness's six endpoints have maximum pairwise distance six.

That search is not a proof of globally smallest witness size. Its candidate screening considers up to eighty nearby edges per pivot. Finding the displayed counterexample is sufficient for rejection; failing to find one with this screening would not, on its own, certify transitivity. The two actual recorded cases did find a concrete counterexample.

What this revised

The obstruction did not require the 400-vertex second-level graph. The twenty-vertex model already rules out exact binary-hypercube distances. The round-six card still stops short of claiming all-level impossibility, an optimal distortion constant, or a minimum-size counterexample. Its next questions concern distortion bounds and higher-level local structure under practical memory limits.

9. Round seven, archived branch: an all-level proof draft

The question

Does the first-level obstruction survive at every finite stage, without repeating a large all-pairs search for each new level?

This branch is preserved in a round-seven archive containing a proof document, a program, and a finite-check JSON. A different round-seven card is discussed separately in the next section. Their shared round label is not evidence that they are one experiment.

Statement under the precise graph definition

For every integer n ≥ 1, the face-adjacency Menger graph M_n defined by the ternary digit rule does not admit an isometric embedding into a binary hypercube of any dimension.

The archive labels this as a proof draft, without completed literature-novelty certification or independent formal verification.

Step 1: identify the corner copy

Among vertices of M_n whose coordinates all lie between zero and two, every higher ternary digit is zero. The legality condition is therefore exactly the first-level condition. The corner contains a copy of M1.

Step 2: construct a retraction

Define

r(x,y,z) = (min(x,2), min(y,2), min(z,2)).

A coordinate already in {0,1,2} is unchanged. A larger coordinate becomes 2, whose least-significant ternary digit is 2 and whose higher digits are zero. This replacement cannot introduce extra digit-1 conflicts. Therefore a legal M_n vertex maps to a legal M1 vertex.

Each original unit edge changes one coordinate by one. After clipping, that coordinate either still changes by one or does not change at all. Thus an edge maps to an edge or collapses to a vertex. Finally, r fixes every vertex of the corner M1.

Step 3: compare both distances

Take u,v in the corner. Any path from u to v in M_n maps under r to a walk in M1 no longer than the original path. Removing repetitions cannot increase its length. Consequently,

d_M1(u,v) ≤ d_Mn(u,v).

Conversely, a path inside the corner M1 is already a valid path in M_n. Hence

d_Mn(u,v) ≤ d_M1(u,v).

Combining the inequalities gives equality. The corner M1 is an isometric retract, not merely a visually recognizable subgraph.

Step 4: inherit the obstruction

The round-six distance comparisons among the three corner edges remain unchanged. If M_n admitted an isometric hypercube embedding, restricting it to the isometric corner M1 would give such an embedding for M1. That contradicts the explicit Θ failure. This proves the stated implication within the draft's graph model.

Finite checks retained with the draft

The JSON also records the same 6/8, 6/8, 10/10 comparisons at each of these levels. These checks support inspection of the implementation. They are not the reason that the statement covers infinitely many integers n; the retraction argument supplies that generality.

What still needs work

The archive proposes formalizing the projection lemma, investigating useful distortion bounds, and checking whether existing literature already contains the same proposition. It does not address alternate adjacency definitions or identify the infinite fractal with a finite graph. A rigorous-looking proof is also not evidence of priority or peer review.

10. Round seven, separate card: thirteen distance landmarks

A different response to impossibility

If exact binary Hamming distances are impossible, must the original graph distances be abandoned? A separate surviving round-seven card changes the representation family instead of challenging the obstruction.

For a selected set L of original vertices, assign each vertex v the vector of its distances to those landmarks:

F_L(v) = (d(v,ℓ)) for ℓ in L.

Compare two vectors using the largest coordinate difference:

D_L(u,v) = max over ℓ in L of |d(u,ℓ) − d(v,ℓ)|.

This is an ℓ∞ distance on real or integer coordinate vectors. It is not Hamming distance on binary labels.

Why set cover is the relevant question

The triangle inequality always gives

|d(u,ℓ) − d(v,ℓ)| ≤ d(u,v).

For exact recovery, each pair u,v needs at least one landmark attaining equality. One can therefore associate each candidate landmark with the set of pairs it recovers exactly and ask for the smallest set of landmarks covering every pair. This is an explanatory formulation of the set-cover method reported by the card, not a recovered copy of its missing optimization code.

The result as reported

The card states that the twenty-vertex graph's full distance matrix was reconstructed, the covering problem was solved by integer programming, and all twelve-landmark subsets were independently enumerated. It reports:

13 landmarks recover all 190 unordered vertex pairs exactly.
No 12-landmark subset recovers all pairs.
Number of 12-landmark subsets checked: C(20,12) = 125,970.

Within the vertex-landmark family, these two claims would establish a minimum of thirteen. No smaller set could work if no twelve-set works: adding extra landmarks cannot destroy exact recovery because every coordinate difference remains bounded by graph distance.

The boundary and the evidence gap

This does not prove that every arbitrary real-coordinate ℓ∞ embedding needs thirteen dimensions. It also does not contradict the binary-hypercube impossibility result: the ambient metric and allowed coordinates have changed.

The complete research card is available, and its summary agrees with a separate round-seven report. The card names a coordinate-and-certificate JSON and gives its hash. However, that artifact and the landmark solver code were not recovered during this reconstruction. The available round-seven ZIP instead contains the all-level obstruction branch.

Accordingly, this section is a documented reported result, not a newly reproduced certificate. The notebook does not invent the thirteen landmark identities, fabricate coordinate vectors, or claim to have checked the 125,970 cases itself.

Next research questions from the card

Test how exact recovery changes under noisy distances and missing edges, and measure the cost of distance queries in an interpretable landmark representation.

11. Technical appendix: definitions, verification logic, and open questions

A. The finite graph family

For n ≥ 1, let x_k,y_k,z_k denote ternary digit k of x,y,z. Then

V_n = {(x,y,z) in {0,…,3^n−1}³ :
       1[x_k=1] + 1[y_k=1] + 1[z_k=1] ≤ 1 for every 0≤k<n}.

The graph joins vertices at unit Manhattan distance. Distances throughout the obstruction calculations are unweighted shortest-path distances in this graph. Different rules such as corner adjacency would define a different problem and require fresh verification.

B. Five questions that must remain separate

1. Object identity: is the graph itself a whole hypercube? The early counts already distinguish the objects.
2. Induced embeddability: can selected binary labels reproduce exactly the adjacency relation? Explicit Q6 and Q15 constructions answer this positively for the stated levels.
3. Isometric embeddability: can every graph distance equal ambient Hamming distance? The Θ witness answers this negatively for levels one and two; the separate proof draft extends it under the stated model.
4. Minimum induced dimension: how few bits are possible when distance distortion is allowed? The global problem was not settled by the coordinate-separated lower bound.
5. Minimum landmark count or general metric dimension: which coordinate families and which metric are allowed? Thirteen in a vertex-landmark family is not a universal lower bound for every representation.

C. What an independent checker needs to do

The following is a specification distilled from the recovered methods. It is not presented as a historical execution log.

• Generate retained triples using the digit test; assert the expected level-specific vertex and edge counts.
• Generate only unit face edges, with no diagonal or wraparound edges.
• Run BFS from each source needed by the test; ensure all requested endpoints are reachable.
• For each proposed binary map, check label injectivity and equality of the original and induced binary edge sets.
• For an error profile, compare graph distance with Hamming distance over unordered pairs only, record the mapping version, and keep additive error separate from distance ratios.
• For the negative certificate, compute both distance sums for all three displayed edge pairs and confirm the true/true/false pattern.
• For the all-level draft, check the mathematical clipping argument separately from finite enumeration.
• For the landmark result, recover the actual landmark list and coordinates first, then test every pair. To establish family-restricted optimality, verify a complete twelve-subset check or a suitable independent certificate rather than relying on a solver's status alone.

Public method reference: SageMath, Partial cubes.

12. What the surviving record does and does not establish

The recovered materials support a substantive research history: an analogy became a graph definition; a successful adjacency construction failed a stronger distance test; an attempted dimension reduction received a scoped negative result; full error distributions clarified the failure; a structural certificate replaced trial-and-error coding; and later work either generalized that obstruction or changed the target metric.

These stages should remain readable rather than being compressed into “twenty cubes, thirteen landmarks.” The unsuccessful claims are what make the later distinctions intelligible.

The reconstruction directly inspected research cards, selected original programs, and numerical JSON outputs from the preserved packages. The first-level construction, round-two construction, round-three path search, round-four distance profile, round-five and round-six witnesses, and archived round-seven proof all have recovered substantive artifacts. The landmark branch currently has a recovered card but not its named coordinate artifact.

An additional, independently written finite checker was run during this reconstruction. It rebuilt levels one and two directly from the ternary digit definition, checked the 20/24 and 400/672 vertex/edge counts, reproduced the round-six three-edge distance comparisons at both levels, verified the first-level Q6 induced embedding and four-versus-two detour, and reconstructed the round-four Q15 mapping using the round-three axis path. Its complete 79,800-pair additive-error histogram matched the archived round-four output exactly. All assertions in that bounded checker passed.

That independent check did not rerun the round-two maximum-ratio search, the round-three exhaustive induced-path search, the round-five original witness search, the level-three checks, or the missing landmark enumeration. It is evidence for the listed finite claims, not a declaration that every historical experiment or the entire research program has been reproduced. The small standalone checker and its result can accompany this notebook as technical attachments.

The preserved research records and bounded computational checks are not a peer-reviewed publication or an independent formal proof audit. Archive upload times do not establish exact execution times. The two round-seven branches share a parent, but the available evidence does not establish a formal supersession decision between them.

Open work is therefore explicit: recover the landmark certificate; independently reproduce selected checks; resolve global induced-dimension and distortion questions; formalize and compare the all-level argument with existing literature; and test the representation on an actual distance-query use case. The current result is an inspectable research notebook, with the status and limits of each claim left visible.