Jose-dev/graphkind-universos

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Abstract

We classify the operators under which the color refinement (1-WL) profile of a graph is invariant, inside a parametric family of observation universes (f, ι) — f compresses neighborhood counts and ι is an involution. The engine proposed and verified that the color refinement partition is complement-invariant (call it T4; the conjecture mechanism was human-built), verified it on 2,131,019 exhaustive graphs (n≤7, 100.0000%) plus 19,407 adversarial ones, and a written proof was produced (injectivity of the neighborhood-multiset subtraction). Parameterizing the universe, the data — not the authors — selected the count law T4 ⟺ f(1) ≠ f(2) (34/34 and 102/102), and further probing produced a characterization:

$$T4(\iota ,f)\text{ holds}⟺\iota \in S_n\cdot K\text{and}(\iota \text{ changes the pair}⟹f(1)\mathrm{\ne }f(2)),K=\{\mathrm{i}\mathrm{d},\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{\_}\mathrm{d}\mathrm{i}\mathrm{r},\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{p},\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{\_}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{p}\}\cong {\mathbb{Z}}_2\times {\mathbb{Z}}_2\mathrm{.}T4(\backslash iota,\; f)\backslash \; \backslash text\{holds\}\; \backslash iff\; \backslash iota\; \backslash in\; S\_n\; \backslash cdot\; K\; \backslash quad\backslash text\{and\}\backslash quad\; \backslash bigl(\backslash iota\backslash \; \backslash text\{changes\; the\; pair\}\; \backslash implies\; f(1)\; \backslash neq\; f(2)\backslash bigr),\; \backslash qquad\; K\; =\; \backslash \{\backslash mathrm\{id\},\backslash \; \backslash mathrm\{comp\backslash \_dir\},\backslash \; \backslash mathrm\{transp\},\backslash \; \backslash mathrm\{comp\backslash \_transp\}\backslash \}\; \backslash cong\; \backslash mathbb\{Z\}\_2\; \backslash times\; \backslash mathbb\{Z\}\_2\; .$$

The forward direction is proved (count mechanism); the converse is verified exhaustively in n=4 (96 exact survivors, zero outside), targeted in n=5 (6,424 witnesses), and on real data (molecules 100/100, CST 94/94). Changing the type of the invariant — partition → orbit → distribution → observer — T4 dies in a deterministic Z₃ universe, dies under process randomness, survives under object randomness, and is inherited by the whole k-FWL hierarchy (≡ (k+1)-WL; 51/51). Every claim is reproducible with one command, each verification carrying an assert with the exact number. Limits are declared; five of our own bugs were caught by controls and are documented, not deleted.

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Contributions

1. T4, proposed and verified by the engine: CR_k(G) = CR_k(Ḡ) as partitions (not merely equal multisets of class sizes) — with a written proof.
2. The count law f(1) ≠ f(2), selected by the data after two human hypotheses were refuted by the table.
3. The characterization Sₙ·K with (⇐) proved and (⇒) verified; the missing piece — uniformity — identified by falsification.
4. The type change: T4 across deterministic, dynamic, probabilistic and observer universes.
5. Method: freeze-before-interpret, negative controls, and five own bugs caught and recorded.

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1. The theorem (T4)

Let G = (V, E) be a finite simple graph, Ḡ its complement, and CR_k(G) the partition of V induced by round k of color refinement with initial colors f(type, degree).

Theorem (EXP-086). For every G and every k ≥ 0: $$CR_k(G) = CR_k(\bar G)$$ as partitions of the same vertex set V (color names differ; the classes are identical).

Corollary (T4). The multisets of class sizes of the stable partitions of CR(G) and CR(Ḡ) coincide, and they coincide at every round.

Real data: Q₃ and its complement, vertices colored by WL class — the partition is identical (T4).

Real data: the WL refinement round by round (k=0…3) — G and its complement keep the same classes at every level.

Proof (sketch). Base. In Ḡ, deg_Ḡ(v) = n−1−deg_G(v); since d ↦ n−1−d is a bijection of degrees, the relation "same color" is the same in G and Ḡ. Step. Assume CR_k(G) = CR_k(Ḡ) =: P_k. For v ∈ V, the closed neighborhood in Ḡ is N_Ḡ[v] = V ∖ N_G(v), and its round-k color multiset is

$$M_k^{\mathrm{\prime }}(v)=T_k-(M_k(v)-[{\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{o}\mathrm{r}}_k(v)]),T_k:=\text{total color multiset of }{P}_k\mathrm{.}M\text{'}\_k(v)\; =\; T\_k\; -\; \backslash bigl(M\_k(v)\; -\; [\backslash mathrm\{color\}\_k(v)]\backslash bigr),\; \backslash qquad\; T\_k\; :=\; \backslash text\{total\; color\; multiset\; of\; \}\; P\_k\; .$$

Since T_k is fixed, the map M_k(v) ↦ M'_k(v) is injective: u, v have the same closed-neighborhood multiset in G iff they do in Ḡ. Because color_{k+1}(v) is a deterministic function of (color_k(v), M_k(v)), the relation "same color" is preserved, hence CR_{k+1}(G) = CR_{k+1}(Ḡ). ∎

Status: written proof (one page), verified on 2,131,019 exhaustive graphs (n≤7), 19,407 adversarial, and 688/688 compatibility pairs; external review pending. Full text: paper/PRUEBA-T4.md · cycle: paper/TEOREMA-UNIVERSOS.md.

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2. The count law

In the standard universe with an involution that changes the connected pair:

$$T4\text{ holds}⟺f(1)\mathrm{\ne }f(2)\mathrm{.}T4\backslash \; \backslash text\{holds\}\; \backslash iff\; f(1)\; \backslash neq\; f(2).$$

Mechanism. In the complement, a vertex with 1 neighbor in a class gets |c|−1 (which may be 2): if f collapses 1 and 2, the duality dies. What f does with 3, 4, 5… is irrelevant. Verified 34/34 (17 functions × 2 involutions, n≤6) and shielded 102/102 (formal definition of dual-global, 6 involutions × 17 functions).

Real data: the 34 universes (17 functions × 2 involutions) — T4 holds exactly when f(1) ≠ f(2).

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3. The characterization

Forward direction (proved). For ι = σ∘s∘c:

case	mechanism	condition
σ (relabel)	WL is isomorphism-invariant; the coloring transports bijectively	holds for every f
s (global in/out swap)	counts swap: (in_D,out_D) → (out_D,in_D); the profile maps by (f(a),f(b)) → (f(b),f(a))	holds for every f
c (uniform complement)	counts map `m →	D

The non-obvious piece is uniformity: preserving the pair is not enough if the action is not uniform (flip_inc0 preserves the pair and dies 0/24). Outside Sₙ·K, everything dies.

Converse (verified).

camera	result
n=4, induced space, exhaustive (528 iotas × 4096 digraphs)	96 exact survivors (= 24·4), zero outside
n=5, targeted (8,160 iotas)	7,680 with valid witness; the 480 without are uniform
outside the space (10,000 random affine)	10,000/10,000 with witness
witness existence (6,424 non-uniform)	6,424/6,424 valid witnesses
real data	molecules 100/100 · CST 94/94

Real data: of the 528 induced iotas at n=4, exactly the 96 uniform ones survive (= 24·4).

Autonomous search. Given a language of 11 primitives + 3 combinators (~400 recipes) and a rate/MDL criterion, the engine found that the best valid witness is the cycle (99.63%, complexity 1) — better than the human construction (97.8%). No valid recipe reaches 100%.

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4. From conjecture to theorem candidate

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5. The type change

universe	invariant	T4	evidence
standard	partition	✅ holds (proved)	2,131,019/2,131,019
deterministic Z₃	orbit (periods)	❌ dies	max 52.4% (maj); 6 rules
random R1 (async WL)	distribution	❌ dies	204/1096 = 18.6%
random R2 (G(n,p) vs G(n,1−p))	distribution	✅ holds	exactly (complement bijects ensembles)
random R3 (random involutions)	measure	threshold = measure	0/5000 uniform vs 96/528 induced
k-FWL hierarchy (≡ (k+1)-WL)	partition of k-tuples	✅ inherited	51/51 in k=1,2,3
observer cascade (n≤7)	—	—	26 → 0 (k=1 → k=2)
Rook vs Shrikhande	—	—	k=1 no, k=2 no, k=3 yes

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6. Method

• Freeze before interpreting: every experiment freezes results_frozen.json first.
• Negative controls: the check that caught the false "proof" (52/104 uniform failures) and the wrongly built Shrikhande (is_isomorphic = True).
• Five own bugs caught and recorded (not deleted):

#	bug	fix
1	multiset of sizes (weak: C6 and 2·C3 share it)	colors
2	comparing colors across graphs	compare partitions
3	indices in the initial coloring	degrees + adjacencies
4	separated multisets (weak k-WL)	correlated pairs (= k-FWL)
5	wrongly built Shrikhande (it was the Rook twice)	is_isomorphic

The method is part of the result.

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7. Reproduce

git clone https://github.com/cripto-bot/graphkind-universos-v2
cd graphkind-universos-v2
python3 -m venv .venv && .venv/bin/pip install -r requirements.txt
bash reproduce.sh            # V1–V12 (~10-15 min)
bash reproduce.sh --rapido   # V1, V3, V5, V8, V10, V11 (~3 min)

Every verification carries an assert with the exact number from the paper: if anything does not match, it fails. CI runs the fast subset on every push and the full suite on demand.

#	verification	assert
V1	the law f(1)≠f(2) (17 f × 2 involutions, n≤6)	34/34
V2	characterization n=4 exhaustive	96 and 48
V3	T4_k complement-invariance (k-FWL)	51/51
V4	observer cascade n≤7	26 → 0
V5	Rook vs Shrikhande (k-FWL; 3-WL std does NOT)	k*=3
V6	deterministic Z₃	does not hold
V7	random R1	204/1096
V8	R3 measure dependence	96/528
V9	autonomous search	cycle 99.63%
V10	hash certificate (12-hex == 64-hex)	no collisions
V11	convergence certificate (stable == n+2)	OK
V12	classification constant across graphs (n=4)	OK

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7b. Corrections (2026-09-13)

Independent verification (EXP-159 of the lab) found a naming error and two certificates were added:

1. k-FWL, not "k-WL" (D-005). The kernel implemented in codigo/wl.py is the correlated one (folklore, k-FWL), equivalent to (k+1)-WL. The plain name "k-WL" is the standard variant (position- separated multisets), which is strictly weaker: standard 3-WL does NOT separate Rook from Shrikhande; 3-FWL (≡ 4-WL) does. The code, docstrings and prints now say k-FWL (≡ (k+1)-WL). V3/V5 are unchanged numerically.
2. V10 — hash certificate: the 12-hex hash does not alter any partition (12-hex partition == 64-hex partition over 74 graphs × 17 f).
3. V11 — convergence certificate: the refinement reaches its stable partition within the n+2 cap (stable == fixed n+2 rounds over 1098 graphs × 17 f); wl_sym/wl_k_colors now stop at convergence.
4. V12 — classification scope: clasificar_involucion classifies on one graph; V12 verifies that the class is constant across all graphs n=4 for the tested (ι, f). The universal statement remains the open direction of the paper.

The numerical results of the paper are unchanged; the naming and the certificates are the corrections.

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7c. The universe arc (v2, EXP-162 → 169)

The v2 adds the universe arc (multilayer, matrix, hypergraphs, arities, dual), with its freezes in resultados/ and V13–V16 in codigo/verificaciones_v2.py:

#	verification	assert
V13	multilayer: total holds, per-layer dies in the canonical variant, channels hold	4160/4160 · failures > 0 · 8320/8320
V14	3-uniform hypergraphs: T4 holds	6042/6042 (n≤5 + n=6 sample)
V15	per-arity dual: partition + saving	1024/1024 · 5120→3392 edges
V16	mixed arities: per-arity holds (repair)	1024/1024
V17	complement quotient (T4's price)	52 graphs · 24 pairs (SG-01 n≤8: 6,168)
V18	IR: IR_1 does not separate Rook/Shri, IR_2 does	i*=2
V19	SG-04 certificate: n=10 exhaustive, no failures	12,005,168 · 0 collisions

The multilayer coherence law (EXP-162/163): total holds, per-layer dies in the canonical variant.

The k\*(L) ladder is flat and corrects EXP-162's k\*=2.

Results: (1) multilayer — the coherence law: total holds (1298/1298, 1404/1404), per-layer dies only in the canonical variant (6.9%/7.4%); the preserving group is exactly the uniform one; the k* ladder is flat (corrects EXP-162's k*=2); (⇐) proved, (⇒) open. (2) matrix — the frontier moves with the observer, not with typing. (3) hypergraphs — prediction registered and correct (1831/1831). (4) arities — the ladder holds; the per-arity transplant fails (the law is of channels). (5) per-arity dual — assimilated: partition 100%, saving 14.37% vs 10.26%.

The frontier moves with the observer, not with typing.

Prediction correct (1831/1831) and the arity transplant that fails.

Per-arity dual: 14.37% saving with guaranteed partition.

The refinement trajectory: T4 and the multilayer divergence.

The arity ladder and the failing transplant.

7d. The Holy Grail (SG-01→03)

• Completeness in the class: the minimal family is {WL, 2-WL, 3-WL} with C_8 = 0 over 76,205,685 pairs (n≤8); leave-one-out: k-WL 3 does the work; the n=16 frontier (Rook/Shri) is separated only by the full family. Freeze: resultados/SG-01_results_frozen.json.

The frontier curves per observer (SG-01/03, EXP-119).

• The price of T4: the complete invariant is not complement-invariant; the quotient G~Ḡ merges 6,168 pairs.
• Individualization: i*=1 at n≤8 (13,597 graphs); i*=2 at Rook/Shrikhande (IR_1 no, IR_2 yes). The IR↔k-WL lattice: at n≤9 the complete ones collapse (vacuous equivalence); anchors A/B, C not observed. Freezes: SG-02, SG-03.

Completeness, T4's price and i\*.

• The multilayer proof ships in paper/PRUEBA-MULTICAPA.md ((⇐) proved, (⇒) open) and the transversal patterns + DECISION_LOG (D-001→D-017) in paper/DECISION_LOG.md.

7e. SG-04: the door (n=10 exhaustive)

• 0 collisions over 12,005,168 exhaustive n=10 graphs (KW3/IR1p, 11.1 h) → the first incompleteness is not at n=10.
• Directed: 9 regular combos n=11–15 (cap 5,000) + 10 Cayley/Paley/Q4 pairs → 0 failures.
• The first known failure remains at n=16 (Rook/Shri): separated by elementary descriptors (SNFL, AUT, CICLOS, LOCAL, HOM2) + KF3/IR2p. Freeze: resultados/SG-04_results_frozen.json (certificate V19).
• Registered correction: geng -d 3 (separated) fails; it is -d3.

8. Repository structure

paper/PAPER.md                   the paper (bilingual abstract, method, results, §12 v2 arc)
paper/TEOREMA-UNIVERSOS.md       the full cycle EXP-099→119
paper/PRUEBA-CARACTERIZACION.md  the (⇐) proof and the status of (⇒)
codigo/universos.py              the law inside the engine (t4_garantizado, oracle)
codigo/wl.py                     symmetric WL + correlated k-FWL (≡ (k+1)-WL)
codigo/verificaciones.py         V1–V12 with asserts of the exact numbers
paper/PRUEBA-MULTICAPA.md        (⇐) proved, (⇒) open (EXP-162)
paper/DECISION_LOG.md            laboratory decisions D-001→D-017
resultados/*.json                original freezes (EXP-102..169 + SG-01..04)
resultados/atlas/*.json          the atlas: 15 universes (EXP-105)
assets/*.svg                     figures
index.html · index.en.html       presentation pages (ES/EN)

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9. Limits and open problems

• (⇒) for all n: open. Verified in n≤4 exhaustive, n=5 targeted, and real data; the ∀n argument is the remaining step.
• Sufficiency of f(1)≠f(2) for arbitrary f: verified in the catalogs; general proof open.
• The full affine space n≥5 is intractable (involutions of S₂₀); the tested space is the induced one plus random affine.
• T4: written proof, external review pending; Lean formalization not available in the environment.
• Universes: our own definitions, declared; the space is open.
• Data: ChEMBL (CC-BY 4.0), CodeSearchNet, UniProt, Pfam — cited; fixed samples by seed.

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10. References (selection)

• Weisfeiler, Leman (1968). A reduction of a graph to a canonical form…
• Morgan (1965); Rogers, Hahn (2010). Extended-connectivity fingerprints.
• Dvořák (2010); Dell, Arvind, Larsson (2018). Homomorphisms and 1-WL.
• Babai (2016). Graph isomorphism in quasipolynomial time.
• Xu et al. (2019). How powerful are graph neural networks?
• Morris et al. (2019). Weisfeiler and Leman go neural.
• Zamfirescu (1980). Non-traceable 3-connected planar cubic graphs.
• Lovász (1970). Problem on vertex-transitive graphs.

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11. The laboratory behind this result

This paper is one thread of a 120-experiment laboratory (EXP-000 → EXP-120; 26,384 lines of run.py; 321 KB of frozen results) spanning six arcs. Every number here is anchored to a frozen artifact, and a meta-experiment (EXP-120) verifies the laboratory's key claims: 26/26 hits.

arc	range	what it built
Foundational	000–010	taxonomy collapse 0.5876; molecules 3.00×; parity with Morgan 1.0000 vs 0.9973; granularity k*=21→24
Riemann	011–019	2,001,052 zeros; JS 0.1350 vs 0.0161; Euler bridge 0.16110 ⊃ 1/2π; critical line σ*=0.50000; clean negatives
Arithmetic/Algebra/Physics	020–046	r* emerged (0.8934); discrete WL = Morgan (0.8877 vs 0.8864); structural calculus with no rules; NS-3D verified
Discovery/Ontology	047–079	real code (129 chains / 119 rings / 152 stars); 6/6 protein hypotheses rejected by controls; emergent math (λ=νn², ABC 6/6, cascade √5, Euler limit ~1.2); the tribunal
T4	080–098	68/68 → 19,407 → 2,131,019/2,131,019; written proof; refine_dual 10.3× / 40.5%; conjecture bank
Universes	099–120	this paper: law, characterization, witnesses, autonomous search, Z₃, random, k-FWL
Universes v2	162–169	multilayer (uniform law, group, k* ladder), universe×observer matrix, hypergraphs (prediction), arities, arity dual

Transversal patterns: freeze before interpreting · negative controls (single-pass shuffle is not evidence: it varies 0.05–0.84) · five of our own bugs caught and recorded, not deleted · four types of result: kinds → laws → theorems → universes.

Full map: paper/MAPA-DEL-LABORATORIO.md.

12. The engine: how it works and its measured power

The engine pipeline: data → graph → label₀ → round → STOP → kinds; the example shows rounds 3 → 5 → 5 → 5 up to the stable partition.

Measured power: parity with Morgan (AUC 0.9608 vs 0.9541), compression 3.00× (molecules) and 187.5× (code), refine_dual 10.3×, dual-normalization saving 40.5%, universe oracle 87/87, engine tests 104/104, T4 bank 2.13M.

_{© 2026 Juri (cripto-bot) · preprint · CC-BY-4.0 license (attribution required).
This document claims no priority over open problems: it reports a verified characterization in a bounded domain.
Cite as: Argaña Silguero, J. (2026). GraphKind — Universes v2 [software and data]. Zenodo. 10.5281/zenodo.22747350 (concept DOI; v2.0.0: 10.5281/zenodo.22747351) · see CITATION.cff.}