The Möbius Atlas

A fixed-parent method for decomposing subset observables, applied across fields.

Take any system whose behaviour depends on which subset of its parts is present. Fix the parent system first. Declare the observable. Then read the interaction structure directly, as Möbius atoms:

Δ(S) = Σ_{T ⊆ S} (−1)^(|S|−|T|) F(T)

The atoms are exact. What makes them meaningful is the control discipline that follows — and that discipline is the real subject of this atlas.

The method

1. Fix one parent system P before evaluating any subset. A moving parent invalidates every atom computed against it.
2. Declare the item set, the observable F(S), the baseline F(∅), the units, and the masking/induction rule.
3. Evaluate F(S) on all subsets where feasible. If sampled, state the seed and the uncertainty.
4. Compute the atoms by Möbius inversion.
5. Run the controls — additive and product nulls, pair-only null, empty-set/baseline deformation, wrong-parent, variable-parent, and normalisation checks.
6. Interpret only after the controls. A mixed result stays useful only when the narrowed statement survives them.

Reading the grades

A grade is a claim about evidence, not about importance. An entry that failed its controls is recorded as such rather than removed, because the failure is part of what the atlas establishes.

Result cards

MOB-00 — Fixed-parent Mobius/Harsanyi subset-observable method

FOUNDATION

Claim. The transferable result is the method: define one parent system P, define F(S) on subsets of that same P, compute Mobius/Harsanyi atoms, then attack the atoms with mandatory artifact controls.

Fixed parent. Any finite fixed parent: graph, lattice, quantum state, Boolean model, genotype landscape, reliability network, causet, or crystal Hamiltonian.

Observable F(S). User-defined scalar F(S) on subsets S of the same parent.

Reproducible witness. Mobius inversion exactly recovers a planted additive + pair + triple finite set function.

Technical control. Recovery error must be zero up to floating precision; planted pair/triple atoms must appear at the planted subsets.

Caveat / boundary. Mobius inversion itself is public mathematics. The contribution carried here is the disciplined cross-field protocol and artifact firewall.

{
 "control": "Recovery error must be zero up to floating precision; planted pair/triple atoms must appear at the planted subsets.",
 "id": "MOB-00",
 "max_reconstruction_error": 0.0,
 "order_mass": {
 "0": 0.0,
 "1": 5.0,
 "2": 0.7,
 "3": 0.3,
 "4": 0.0
 },
 "top_atoms": [
 {
 "S": [
 0
 ],
 "atom": 1.25
 },
 {
 "S": [
 1
 ],
 "atom": 1.25
 },
 {
 "S": [
 2
 ],
 "atom": 1.25
 },
 {
 "S": [
 3
 ],
 "atom": 1.25
 },
 {
 "S": [
 0,
 1
 ],
 "atom": 0.7
 },
 {
 "S": [
 0,
 1,
 2
 ],
 "atom": -0.3
 }
 ],
 "witness": "Mobius inversion exactly recovers a planted additive + pair + triple finite set function."
}

MOB-01 — Stabilizer/QEC graph-state entropy atoms

STRONG_KEEP

Claim. For fixed graph-state parents, GF(2) cut-rank entropy produces stable subset entropy atom signatures and rejects product controls.

Fixed parent. Fixed graph G representing a graph stabilizer state.

Observable F(S). F(S)=rank_GF2(adjacency cut between S and complement) in bits.

Reproducible witness. GF(2) graph-state entropy equals cut rank; product graph is zero, Bell edge is localized, star/GHZ-like graph gives non-product entropy structure.

Technical control. Empty graph must have no nonzero atom; edge graph must not be mistaken for generic high-order structure.

Caveat / boundary. Strong as a diagnostic witness, not as a full holographic or QG proof.

{
 "bell_edge_F": {
 "frozenset()": 0,
 "frozenset({0, 1})": 0,
 "frozenset({0})": 1,
 "frozenset({1})": 1
 },
 "bell_edge_atoms": [
 {
 "S": [
 0,
 1
 ],
 "atom": -2.0
 },
 {
 "S": [
 0
 ],
 "atom": 1.0
 },
 {
 "S": [
 1
 ],
 "atom": 1.0
 }
 ],
 "control": "Empty graph must have no nonzero atom; edge graph must not be mistaken for generic high-order structure.",
 "id": "MOB-01",
 "product_max_abs_atom": 0.0,
 "star_order_mass": {
 "0": 0.0,
 "1": 4.0,
 "2": 6.0,
 "3": 4.0,
 "4": 2.0
 },
 "star_top_atoms": [
 {
 "S": [
 0,
 1,
 2,
 3
 ],
 "atom": -2.0
 },
 {
 "S": [
 0
 ],
 "atom": 1.0
 },
 {
 "S": [
 0,
 1
 ],
 "atom": -1.0
 },
 {
 "S": [
 0,
 1,
 2
 ],
 "atom": 1.0
 },
 {
 "S": [
 0,
 1,
 3
 ],
 "atom": 1.0
 },
 {
 "S": [
 0,
 2
 ],
 "atom": -1.0
 },
 {
 "S": [
 0,
 2,
 3
 ],
 "atom": 1.0
 },
 {
 "S": [
 0,
 3
 ],
 "atom": -1.0
 }
 ],
 "witness": "GF(2) graph-state entropy equals cut rank; product graph is zero, Bell edge is localized, star/GHZ-like graph gives non-product entropy structure."
}

MOB-02 — Holographic/min-cut capacity atoms

STRONG_KEEP

Claim. Fixed weighted min-cut/capacity observables produce atoms that localize bottlenecks and saturation.

Fixed parent. Fixed weighted network or tensor-network proxy graph with boundary regions.

Observable F(S). F(S)=minimum cut or capacity function under one fixed graph/sink rule.

Reproducible witness. A fixed min-cut/capacity cap creates non-additive bottleneck atoms; additive capacity has no high-order atoms.

Technical control. The same weights without the bottleneck cap give only singleton atoms.

Caveat / boundary. A min-cut graph toy is not full holography.

{
 "additive_high_order_max": 0.0,
 "cap_K": 2.0,
 "cap_order_mass": {
 "0": 0.0,
 "1": 4.0,
 "2": 0.0,
 "3": 4.0,
 "4": 2.0
 },
 "cap_top_high_order_atoms": [
 {
 "S": [
 0,
 1,
 2,
 3
 ],
 "atom": 2.0
 },
 {
 "S": [
 0,
 1,
 2
 ],
 "atom": -1.0
 },
 {
 "S": [
 0,
 1,
 3
 ],
 "atom": -1.0
 },
 {
 "S": [
 0,
 2,
 3
 ],
 "atom": -1.0
 },
 {
 "S": [
 1,
 2,
 3
 ],
 "atom": -1.0
 }
 ],
 "control": "The same weights without the bottleneck cap give only singleton atoms.",
 "id": "MOB-02",
 "items": [
 0,
 1,
 2,
 3
 ],
 "weights": {
 "0": 1.0,
 "1": 1.0,
 "2": 1.0,
 "3": 1.0
 },
 "witness": "A fixed min-cut/capacity cap creates non-additive bottleneck atoms; additive capacity has no high-order atoms."
}

MOB-03 — Harsanyi/cooperative-game and ML feature interactions

STRONG_KEEP

Claim. Fixed-parent Harsanyi dividends recover known interaction order in Boolean games, voting/threshold games, and planted feature models.

Fixed parent. Fixed Boolean function, cooperative game, or trained/frozen model with fixed baseline.

Observable F(S). F(S)=function/model output with features in S active and all other features at fixed baseline.

Reproducible witness. Harsanyi dividends recover additive, pairwise, unanimity, parity, and planted feature-gate structure under one fixed baseline.

Technical control. Additive model has zero atoms above order 1; pair-count localizes at order 2; no retraining or variable parent is allowed.

Caveat / boundary. Harsanyi/Shapley-Taylor prior art is strong; novelty is controlled application and atlas transfer.

{
 "cases": [
 {
 "name": "additive",
 "order_mass": {
 "0": 0.0,
 "1": 4.0,
 "2": 0.0,
 "3": 0.0,
 "4": 0.0
 },
 "top_atoms": [
 {
 "S": [
 0
 ],
 "atom": 1.0
 },
 {
 "S": [
 1
 ],
 "atom": 1.0
 },
 {
 "S": [
 2
 ],
 "atom": 1.0
 },
 {
 "S": [
 3
 ],
 "atom": 1.0
 }
 ]
 },
 {
 "name": "pair_count",
 "order_mass": {
 "0": 0.0,
 "1": 0.0,
 "2": 6.0,
 "3": 0.0,
 "4": 0.0
 },
 "top_atoms": [
 {
 "S": [
 0,
 1
 ],
 "atom": 1.0
 },
 {
 "S": [
 0,
 2
 ],
 "atom": 1.0
 },
 {
 "S": [
 0,
 3
 ],
 "atom": 1.0
 },
 {
 "S": [
 1,
 2
 ],
 "atom": 1.0
 },
 {
 "S": [
 1,
 3
 ],
 "atom": 1.0
 },
 {
 "S": [
 2,
 3
 ],
 "atom": 1.0
 }
 ]
 },
 {
 "name": "unanimity_all_four",
 "order_mass": {
 "0": 0.0,
 "1": 0.0,
 "2": 0.0,
 "3": 0.0,
 "4": 1.0
 },
 "top_atoms": [
 {
 "S": [
 0,
 1,
 2,
 3
 ],
 "atom": 1.0
 }
 ]
 },
 {
 "name": "odd_parity_indicator",
 "order_mass": {
 "0": 0.0,
 "1": 4.0,
 "2": 12.0,
 "3": 16.0,
 "4": 8.0
 },
 "top_atoms": [
 {
 "S": [
 0,
 1,
 2,
 3
 ],
 "atom": -8.0
 },
 {
 "S": [
 0,
 1,
 2
 ],
 "atom": 4.0
 },
 {
 "S": [
 0,
 1,
 3
 ],
 "atom": 4.0
 },
 {
 "S": [
 0,
 2,
 3
 ],
 "atom": 4.0
 },
 {
 "S": [
 1,
 2,
 3
 ],
 "atom": 4.0
 },
 {
 "S": [
 0,
 1
 ],
 "atom": -2.0
 },
 {
 "S": [
 0,
 2
 ],
 "atom": -2.0
 },
 {
 "S": [
 0,
 3
 ],
 "atom": -2.0
 }
 ]
 },
 {
 "name": "planted_gate_0_2",
 "order_mass": {
 "0": 0.0,
 "1": 0.8,
 "2": 3.0,
 "3": 0.0,
 "4": 0.0
 },
 "top_atoms": [
 {
 "S": [
 0,
 2
 ],
 "atom": 3.0
 },
 {
 "S": [
 0
 ],
 "atom": 0.2
 },
 {
 "S": [
 1
 ],
 "atom": 0.2
 },
 {
 "S": [
 2
 ],
 "atom": 0.2
 },
 {
 "S": [
 3
 ],
 "atom": 0.2
 }
 ]
 }
 ],
 "control": "Additive model has zero atoms above order 1; pair-count localizes at order 2; no retraining or variable parent is allowed.",
 "id": "MOB-03",
 "witness": "Harsanyi dividends recover additive, pairwise, unanimity, parity, and planted feature-gate structure under one fixed baseline."
}

MOB-04 — Spin/critical entropy atom diagnostics

LIMITED_KEEP

Claim. Subset entropy atoms can detect known product/cat-state structure and finite-size critical sensitivity, but collective activation proxies remain limited.

Fixed parent. Fixed finite spin/quantum state or exact diagonalization parent.

Observable F(S). F(S)=von Neumann entropy or declared correlation observable of subset S.

Reproducible witness. Product spin state is zero; GHZ/cat state gives global entropy atoms; tiny TFIM exact diagonalization shows finite-size coupling sensitivity.

Technical control. Finite-size TFIM numbers are diagnostic only; they are not a dimension-activation proof.

Caveat / boundary. Do not carry finite-size spin-chain results as a QG bridge proof.

{
 "control": "Finite-size TFIM numbers are diagnostic only; they are not a dimension-activation proof.",
 "ghz_order_mass": {
 "0": 0.0,
 "1": 4.0,
 "2": 6.0,
 "3": 4.0,
 "4": 2.0
 },
 "ghz_top_atoms": [
 {
 "S": [
 0,
 1,
 2,
 3
 ],
 "atom": -2.0
 },
 {
 "S": [
 0
 ],
 "atom": 1.0
 },
 {
 "S": [
 0,
 1
 ],
 "atom": -1.0
 },
 {
 "S": [
 0,
 1,
 2
 ],
 "atom": 1.0
 },
 {
 "S": [
 0,
 1,
 3
 ],
 "atom": 1.0
 },
 {
 "S": [
 0,
 2
 ],
 "atom": -1.0
 },
 {
 "S": [
 0,
 2,
 3
 ],
 "atom": 1.0
 },
 {
 "S": [
 0,
 3
 ],
 "atom": -1.0
 }
 ],
 "id": "MOB-04",
 "product_max_abs_atom": 0.0,
 "tfim_half_chain_entropy_by_h": {
 "0.2": 0.99974585,
 "0.5": 0.98259627,
 "1.0": 0.75289784,
 "2.0": 0.27322157
 },
 "witness": "Product spin state is zero; GHZ/cat state gives global entropy atoms; tiny TFIM exact diagonalization shows finite-size coupling sensitivity."
}

MOB-05 — Percolation and TDA threshold/topology atoms

LIMITED_KEEP

Claim. Exact small percolation crossing and clique/graph Betti controls validate topology/threshold diagnostics; random/persistence claims remain limited.

Fixed parent. Fixed lattice, graph, or filtration parent.

Observable F(S). F(S)=crossing indicator/probability or Betti number for selected sites/edges/simplices.

Reproducible witness. Exact tiny percolation crossing and graph beta1 cycle witnesses show threshold/topology atoms under a fixed parent.

Technical control. Changing the graph or resampling the parent while changing S is forbidden; random/persistence scaling remains limited.

Caveat / boundary. Random and persistence peaks need stronger scaling before theory support.

{
 "control": "Changing the graph or resampling the parent while changing S is forbidden; random/persistence scaling remains limited.",
 "crossing_top_atoms": [
 {
 "S": [
 0,
 1
 ],
 "atom": 1.0
 },
 {
 "S": [
 0,
 1,
 2,
 3
 ],
 "atom": -1.0
 },
 {
 "S": [
 2,
 3
 ],
 "atom": 1.0
 }
 ],
 "crossing_truth_table": {
 "()": 0,
 "(0, 1)": 1,
 "(0, 1, 2)": 1,
 "(0, 1, 2, 3)": 1,
 "(0, 1, 3)": 1,
 "(0, 2)": 0,
 "(0, 2, 3)": 1,
 "(0, 3)": 0,
 "(0,)": 0,
 "(1, 2)": 0,
 "(1, 2, 3)": 1,
 "(1, 3)": 0,
 "(1,)": 0,
 "(2, 3)": 1,
 "(2,)": 0,
 "(3,)": 0
 },
 "cycle_beta1_full": 1.0,
 "cycle_beta1_top_atoms": [
 {
 "S": [
 0,
 1,
 2,
 3
 ],
 "atom": 1.0
 }
 ],
 "id": "MOB-05",
 "witness": "Exact tiny percolation crossing and graph beta1 cycle witnesses show threshold/topology atoms under a fixed parent."
}

MOB-06 — Causal-set fixed-parent discrete geometry diagnostics

STRONG_KEEP

Claim. Fixed-parent causet observables separate chain, antichain, sprinkling, and generic-poset behavior.

Fixed parent. Fixed finite causal relation matrix C on N elements.

Observable F(S). F(S)=ordering fraction, interval abundance, height/width, or calibrated dimension proxy on induced subcauset C[S,S].

Reproducible witness. Causet induced-subset observables separate antichain, chain, and a fixed 2D sprinkling sample.

Technical control. Generic random posets and uncalibrated Hasse spectra are not manifoldlike evidence.

Caveat / boundary. Diagnostic/calibration only; not a causal-set path-sum proof.

{
 "antichain_max_abs_atom": 0.0,
 "chain_order_fraction_full": 1.0,
 "chain_top_atoms": [
 {
 "S": [
 0,
 1,
 2,
 3
 ],
 "atom": 3.0
 },
 {
 "S": [
 0,
 1,
 2
 ],
 "atom": -2.0
 },
 {
 "S": [
 0,
 1,
 3
 ],
 "atom": -2.0
 },
 {
 "S": [
 0,
 2,
 3
 ],
 "atom": -2.0
 },
 {
 "S": [
 1,
 2,
 3
 ],
 "atom": -2.0
 },
 {
 "S": [
 0,
 1
 ],
 "atom": 1.0
 },
 {
 "S": [
 0,
 2
 ],
 "atom": 1.0
 },
 {
 "S": [
 0,
 3
 ],
 "atom": 1.0
 }
 ],
 "control": "Generic random posets and uncalibrated Hasse spectra are not manifoldlike evidence.",
 "id": "MOB-06",
 "sprinkling_interval_abundance_k0": 38,
 "sprinkling_sample_ordering_fraction_first8": 0.46428571,
 "sprinkling_seed": 123,
 "witness": "Causet induced-subset observables separate antichain, chain, and a fixed 2D sprinkling sample."
}

MOB-07 — Information decomposition and synergy atoms

STRONG_KEEP

Claim. Fixed joint distributions produce expected synergy, redundancy, and unique-information signatures under Mobius-style subset observables.

Fixed parent. Fixed probability distribution over sources and target.

Observable F(S). F(S)=information quantity for source subset S about target, with definition stated.

Reproducible witness. Exact distributions recover XOR synergy, redundant-copy redundancy, and unique-source additivity.

Technical control. PID definitions may vary; this capsule uses ordinary mutual information atoms only.

Caveat / boundary. PID definitions are not unique; the capsule uses ordinary mutual information atoms.

{
 "control": "PID definitions may vary; this capsule uses ordinary mutual information atoms only.",
 "id": "MOB-07",
 "redundant_atoms": [
 {
 "S": [
 0
 ],
 "atom": 1.0
 },
 {
 "S": [
 0,
 1
 ],
 "atom": -1.0
 },
 {
 "S": [
 1
 ],
 "atom": 1.0
 }
 ],
 "unique_atoms": [
 {
 "S": [
 0
 ],
 "atom": 1.0
 },
 {
 "S": [
 1
 ],
 "atom": 1.0
 }
 ],
 "witness": "Exact distributions recover XOR synergy, redundant-copy redundancy, and unique-source additivity.",
 "xor_F": {
 "frozenset()": 0.0,
 "frozenset({0, 1})": 1.0,
 "frozenset({0})": 0.0,
 "frozenset({1})": 0.0
 },
 "xor_atoms": [
 {
 "S": [
 0,
 1
 ],
 "atom": 1.0
 }
 ]
}

MOB-08 — Genetic epistasis as natural Mobius validation field

STRONG_KEEP

Claim. Epistasis is a clean external field where Mobius atoms correspond to interaction terms in a fixed fitness landscape.

Fixed parent. Fixed genotype-fitness landscape.

Observable F(S). F(S)=log fitness or declared phenotype value for mutation subset S.

Reproducible witness. Synthetic log-fitness landscape recovers planted epistasis atoms, including sign epistasis.

Technical control. Use log fitness or declared scale; raw multiplicative fitness can manufacture artifacts.

Caveat / boundary. Biological claims need real data; current carry-forward is methodological.

{
 "control": "Use log fitness or declared scale; raw multiplicative fitness can manufacture artifacts.",
 "id": "MOB-08",
 "planted_atoms": [
 {
 "S": [
 0,
 1
 ],
 "atom": 0.75
 },
 {
 "S": [
 0,
 1,
 2
 ],
 "atom": -0.5
 },
 {
 "S": [
 0
 ],
 "atom": 0.4
 },
 {
 "S": [
 1
 ],
 "atom": -0.2
 },
 {
 "S": [
 2
 ],
 "atom": 0.1
 }
 ],
 "sign_epistasis_atoms": [
 {
 "S": [
 0,
 1
 ],
 "atom": -3.0
 },
 {
 "S": [
 0
 ],
 "atom": 1.0
 },
 {
 "S": [
 1
 ],
 "atom": 1.0
 }
 ],
 "sign_epistasis_effect_A_with_B": -2.0,
 "sign_epistasis_effect_A_without_B": 1.0,
 "witness": "Synthetic log-fitness landscape recovers planted epistasis atoms, including sign epistasis."
}

MOB-09 — Materials/alloy cluster expansion atoms

STRONG_KEEP

Claim. Cluster expansion/ECI language is a natural fixed-parent Mobius field and validates lattice subset interactions.

Fixed parent. Fixed alloy lattice or cluster expansion parent.

Observable F(S). F(S)=energy/property contribution for selected occupation cluster S.

Reproducible witness. Cluster-expansion energy terms are exactly Mobius atoms of a fixed lattice occupation energy.

Technical control. Fixed composition or relaxing the lattice separately for each subset changes the parent and is an artifact.

Caveat / boundary. Cluster expansion is public prior art; contribution is transfer of controls to the broader atlas.

{
 "control": "Fixed composition or relaxing the lattice separately for each subset changes the parent and is an artifact.",
 "id": "MOB-09",
 "max_planted_ECI_recovery_error": 0.0,
 "recovered_atoms": [
 {
 "S": [
 0,
 1
 ],
 "atom": -0.8
 },
 {
 "S": [
 0,
 1,
 2
 ],
 "atom": 0.6
 },
 {
 "S": [
 1,
 2
 ],
 "atom": 0.35
 },
 {
 "S": [
 0,
 1,
 2,
 3
 ],
 "atom": -0.25
 },
 {
 "S": [
 1
 ],
 "atom": 0.2
 },
 {
 "S": [
 0
 ],
 "atom": -0.1
 }
 ],
 "witness": "Cluster-expansion energy terms are exactly Mobius atoms of a fixed lattice occupation energy."
}

MOB-10 — Reliability/fault network atoms

STRONG_KEEP

Claim. Reliability networks are clean anchors for Mobius atoms as bottleneck/pivotal diagnostics.

Fixed parent. Fixed reliability graph, system, or fault tree.

Observable F(S). F(S)=system success/failure state or probability under one fixed component rule.

Reproducible witness. Reliability success functions expose series, parallel, k-out-of-n, and bridge-network pivotal structure.

Technical control. Common-cause failures must be modeled explicitly; otherwise they masquerade as component synergy.

Caveat / boundary. Reliability theory is public; novelty is unifying diagnostic protocol.

{
 "bridge_top_atoms": [
 {
 "S": [
 0,
 1,
 2,
 3,
 4
 ],
 "atom": 2.0
 },
 {
 "S": [
 0,
 1
 ],
 "atom": 1.0
 },
 {
 "S": [
 0,
 1,
 2,
 3
 ],
 "atom": -1.0
 },
 {
 "S": [
 0,
 1,
 2,
 4
 ],
 "atom": -1.0
 },
 {
 "S": [
 0,
 1,
 3,
 4
 ],
 "atom": -1.0
 },
 {
 "S": [
 0,
 2,
 3,
 4
 ],
 "atom": -1.0
 },
 {
 "S": [
 0,
 3,
 4
 ],
 "atom": 1.0
 },
 {
 "S": [
 1,
 2,
 3,
 4
 ],
 "atom": -1.0
 },
 {
 "S": [
 1,
 2,
 4
 ],
 "atom": 1.0
 },
 {
 "S": [
 2,
 3
 ],
 "atom": 1.0
 }
 ],
 "control": "Common-cause failures must be modeled explicitly; otherwise they masquerade as component synergy.",
 "id": "MOB-10",
 "k2of3_top_atoms": [
 {
 "S": [
 0,
 1,
 2
 ],
 "atom": -2.0
 },
 {
 "S": [
 0,
 1
 ],
 "atom": 1.0
 },
 {
 "S": [
 0,
 2
 ],
 "atom": 1.0
 },
 {
 "S": [
 1,
 2
 ],
 "atom": 1.0
 }
 ],
 "parallel_top_atoms": [
 {
 "S": [
 0
 ],
 "atom": 1.0
 },
 {
 "S": [
 0,
 1
 ],
 "atom": -1.0
 },
 {
 "S": [
 0,
 1,
 2
 ],
 "atom": 1.0
 },
 {
 "S": [
 0,
 2
 ],
 "atom": -1.0
 },
 {
 "S": [
 1
 ],
 "atom": 1.0
 },
 {
 "S": [
 1,
 2
 ],
 "atom": -1.0
 }
 ],
 "series_top_atoms": [
 {
 "S": [
 0,
 1,
 2
 ],
 "atom": 1.0
 }
 ],
 "witness": "Reliability success functions expose series, parallel, k-out-of-n, and bridge-network pivotal structure."
}

MOB-11 — Epidemic and influence cascade atoms

STRONG_KEEP

Claim. Fixed network cascade observables expose reachability, redundancy, and complex-contagion synergy.

Fixed parent. Fixed graph and fixed diffusion/threshold rule.

Observable F(S). F(S)=reachable/infected/activated set size or expected cascade size from seed subset S.

Reproducible witness. Fixed cascade rules expose reachability redundancy and threshold synergy.

Technical control. The network and time/update rule must be fixed; changing them between seed subsets is a variable-parent artifact.

Caveat / boundary. Network and temporal rule must be fixed before subset testing.

{
 "control": "The network and time/update rule must be fixed; changing them between seed subsets is a variable-parent artifact.",
 "id": "MOB-11",
 "reachability_F": {
 "()": 0.0,
 "(0, 1)": 4.0,
 "(0, 1, 2)": 4.0,
 "(0, 2)": 4.0,
 "(0,)": 4.0,
 "(1, 2)": 3.0,
 "(1,)": 2.0,
 "(2,)": 2.0
 },
 "reachability_top_atoms": [
 {
 "S": [
 0
 ],
 "atom": 4.0
 },
 {
 "S": [
 0,
 1
 ],
 "atom": -2.0
 },
 {
 "S": [
 0,
 2
 ],
 "atom": -2.0
 },
 {
 "S": [
 1
 ],
 "atom": 2.0
 },
 {
 "S": [
 2
 ],
 "atom": 2.0
 },
 {
 "S": [
 0,
 1,
 2
 ],
 "atom": 1.0
 },
 {
 "S": [
 1,
 2
 ],
 "atom": -1.0
 }
 ],
 "threshold_top_atoms": [
 {
 "S": [
 0
 ],
 "atom": 1.0
 },
 {
 "S": [
 0,
 1
 ],
 "atom": 1.0
 },
 {
 "S": [
 1
 ],
 "atom": 1.0
 },
 {
 "S": [
 2
 ],
 "atom": 1.0
 }
 ],
 "witness": "Fixed cascade rules expose reachability redundancy and threshold synergy."
}

MOB-12 — Crystal symmetry, metric, Bloch and phonon calibration

STRONG_KEEP

Claim. Fixed crystal-like parents give a physical calibration lane where symmetry, metric, wave prediction, and defects are testable.

Fixed parent. Fixed periodic lattice, Hamiltonian, or dynamical matrix.

Observable F(S). F(S)=symmetry residual, Hessian/Fisher metric, Bloch spectrum, phonon prediction, or defect residual.

Reproducible witness. Crystal restriction, Bloch band prediction, and phonon eigenvalues are reproduced in fixed periodic parents.

Technical control. Disorder or using the wrong parent group breaks the exact Bloch/phonon prediction.

Caveat / boundary. Calibration only. Does not derive spacetime or dimension activation.

{
 "bloch_max_eigenvalue_error": 0.0,
 "bloch_ring_N": 12,
 "control": "Disorder or using the wrong parent group breaks the exact Bloch/phonon prediction.",
 "crystallographic_allowed_orders_1_to_12": [
 1,
 2,
 3,
 4,
 6
 ],
 "id": "MOB-12",
 "phonon_max_eigenvalue_error": 0.0,
 "rotation_trace_table": {
 "1": 2.0,
 "10": 1.6180339887,
 "11": 1.6825070657,
 "12": 1.7320508076,
 "2": -2.0,
 "3": -1.0,
 "4": 0.0,
 "5": 0.6180339887,
 "6": 1.0,
 "7": 1.2469796037,
 "8": 1.4142135624,
 "9": 1.5320888862
 },
 "witness": "Crystal restriction, Bloch band prediction, and phonon eigenvalues are reproduced in fixed periodic parents."
}

MOB-13 — Crystal defect-subset Mobius atoms over Reynolds residual

STRONG_KEEP

Claim. For a fixed crystal parent group and metric, defect-subset atoms over Reynolds projection residual recover inserted defect-interaction structure in toy Hamiltonians.

Fixed parent. Fixed ring or 2D torus Hamiltonian with fixed parent symmetry group G.

Observable F(S). F(S)=D_G(H_S)=||H_S - |G|^-1 sum_g U_g H_S U_g^-1||_F^2 / ||H_S||_F^2, plus Bloch/IPR companions.

Reproducible witness. Defect-subset atoms over a Reynolds residual recover local pair structure in a fixed ring Hamiltonian with known pair kernel.

Technical control. Wrong-parent/identity group and random labels must not recover the local adjacency band.

Caveat / boundary. Strong in toy Hamiltonians; needs real elastic or atomistic defect data before stronger physical claims.

{
 "adjacent_pair_auc": 1.0,
 "candidate_sites": {
 "0": 0,
 "1": 4,
 "2": 8,
 "3": 12,
 "4": 16,
 "5": 20
 },
 "control": "Wrong-parent/identity group and random labels must not recover the local adjacency band.",
 "id": "MOB-13",
 "pair_kernel_correlation": 0.9995192833,
 "random_label_auc_mean_seed123_200": 0.4833796296,
 "random_label_auc_sd": 0.1554902241,
 "ring_N": 24,
 "top_pair_atoms": [
 {
 "S": [
 0,
 5
 ],
 "atom": 0.0474398801
 },
 {
 "S": [
 4,
 5
 ],
 "atom": 0.0474398801
 },
 {
 "S": [
 3,
 4
 ],
 "atom": 0.0474398801
 },
 {
 "S": [
 2,
 3
 ],
 "atom": 0.0474398801
 },
 {
 "S": [
 1,
 2
 ],
 "atom": 0.0474398801
 },
 {
 "S": [
 0,
 1
 ],
 "atom": 0.0474398801
 },
 {
 "S": [
 0,
 4
 ],
 "atom": 0.0190794611
 },
 {
 "S": [
 1,
 5
 ],
 "atom": 0.0190794611
 },
 {
 "S": [
 2,
 4
 ],
 "atom": 0.0190794611
 },
 {
 "S": [
 3,
 5
 ],
 "atom": 0.0190794611
 }
 ],
 "witness": "Defect-subset atoms over a Reynolds residual recover local pair structure in a fixed ring Hamiltonian with known pair kernel."
}

MOB-14 — Elastic crystal defect Mobius upgrade

LIMITED_KEEP

Claim. Elastic defect modeling keeps the crystal direction alive but is more conservative: continuum charges pass, spring-lattice atoms remain mixed/limited.

Fixed parent. Fixed spring-lattice Hessian plus continuum Burgers/Frank calibration.

Observable F(S). Reynolds residual atoms, compliance atoms, spectrum shift atoms, low-mode IPR atoms, and continuum defect integrals.

Reproducible witness. Continuum defect-charge checks are exact; spring-lattice subset atoms are useful but still only a limited proxy.

Technical control. The vacancy proxy is not real elasticity; full promotion requires validated dislocation/disclination or atomistic data.

Caveat / boundary. Limited/mixed; requires validated dislocation/disclination or atomistic model.

{
 "burgers_screw_integral": 1.0,
 "burgers_screw_integral_samples": 720,
 "control": "The vacancy proxy is not real elasticity; full promotion requires validated dislocation/disclination or atomistic data.",
 "frank_angle_integral": 1.047197551197,
 "frank_angle_target": 1.047197551197,
 "id": "MOB-14",
 "toy_vacancy_adjacency_auc": 1.0,
 "toy_vacancy_pair_atoms": [
 {
 "S": [
 0,
 1
 ],
 "atom": 1.0
 },
 {
 "S": [
 0,
 3
 ],
 "atom": 1.0
 },
 {
 "S": [
 1,
 2
 ],
 "atom": 1.0
 },
 {
 "S": [
 2,
 3
 ],
 "atom": 1.0
 },
 {
 "S": [
 0,
 2
 ],
 "atom": 0.05
 },
 {
 "S": [
 1,
 3
 ],
 "atom": 0.05
 }
 ],
 "witness": "Continuum defect-charge checks are exact; spring-lattice subset atoms are useful but still only a limited proxy."
}

MOB-15 — W/Dicke and quantum entropy budget region maps

MIXED_KEEP

Claim. Fixed-parent W/Dicke entropy atoms contain useful region-dependent structure; variable-parent readings create misleading high-order towers.

Fixed parent. Fixed quantum parent state family, e.g. W_n or Dicke(n,k), with subset entropies from reduced states of that same parent.

Observable F(S). F(S)=von Neumann/Renyi/Tsallis entropy or mutual-information-like observable of subset S.

Reproducible witness. Fixed-parent W/Dicke entropy atoms are region maps; variable-parent formulas generate misleading towers and must be separated.

Technical control. The same n/state must remain fixed. State-size-changing F(k)=2/k is an artifact-prone comparison.

Caveat / boundary. Carry as warning-rich mixed result, not a spine. Embedded Standalone Capsule Runner Copy the code below into a Python file and run it with Python plus NumPy. It regenerates the capsule-output JSON used in the cards. The Word file itself is the evidence source; this runner is embedded so no internal report needs to be opened. from __future__ import annotations from itertools import combinations, product from math import comb, cos, exp, log2, pi, sqrt import json import numpy as np def powerset(items): items = tuple(items) for r in range(len(items) + 1): for s in combinations(items, r): yield frozenset(s) def mobius_atoms(F, items): items = tuple(items) atoms = {} for S in powerset(items): total = 0.0 S_tuple = tuple(sorted(S)) for T in powerset(S_tuple): total += ((-1) ** (len(S) - len(T))) F.get(frozenset(T), 0.0) atoms[S_tuple] = float(total) return atoms def reconstruct_from_atoms(atoms, items): out = {} for S in powerset(items): total = 0.0 for T in powerset(tuple(S)): total += atoms[tuple(sorted(T))] out[tuple(sorted(S))] = float(total) return out def order_mass(atoms): out = {} for S, a in atoms.items(): out[len(S)] = out.get(len(S), 0.0) + abs(float(a)) return {str(k): round(v, 10) for k, v in sorted(out.items())} def top_atoms(atoms, k=8, min_order=0): rows = [(S, a) for S, a in atoms.items() if len(S) >= min_order and abs(a) > 1e-10] rows.sort(key=lambda x: (-abs(x[1]), x[0])) return [{"S": list(S), "atom": round(float(a), 10)} for S, a in rows[:k]] def auc(labels, scores): pos = [s for y, s in zip(labels, scores) if y] neg = [s for y, s in zip(labels, scores) if not y] total = 0.0 count = 0 for p in pos: for n in neg: total += 1.0 if p > n else 0.5 if p == n else 0.0 count += 1 return total / count if count else None def pearson(x, y): x = np.array(x, dtype=float) y = np.array(y, dtype=float) if len(x) < 2 or np.std(x) == 0 or np.std(y) == 0: return None return float(np.corrcoef(x, y)[0, 1]) def gf2_rank(mat): a = [list(map(lambda z: int(z) & 1, row)) for row in mat] if not a: return 0 m, n = len(a), len(a[0]) rank = 0 col = 0 while rank < m and col < n: pivot = None for r in range(rank, m): if a[r][col]: pivot = r break if pivot is None: col += 1 continue a[rank], a[pivot] = a[pivot], a[rank] for r in range(m): if r != rank and a[r][col]: a[r] = [x ^ y for x, y in zip(a[r], a[rank])] rank += 1 col += 1 return rank def entropy_bits(probs): return float(sum(-p log2(p) for p in probs if p > 1e-15)) def capsule_MOB_00(): items = (0, 1, 2, 3) def F_of(S): S = set(S) return ( 1.25 len(S) + (0.7 if {0, 1}.issubset(S) else 0.0) - (0.3 if {0, 1, 2}.issubset(S) else 0.0) ) F = {S: F_of(S) for S in powerset(items)} atoms = mobius_atoms(F, items) recon = reconstruct_from_atoms(atoms, items) max_recon_error = max(abs(recon[tuple(sorted(S))] - F[S]) for S in F) return { "id": "MOB-00", "witness": "Mobius inversion exactly recovers a planted additive + pair + triple finite set function.", "max_reconstruction_error": max_recon_error, "top_atoms": top_atoms(atoms, 8), "order_mass": order_mass(atoms), "control": "Recovery error must be zero up to floating precision; planted pair/triple atoms must appear at the planted subsets.", } def graph_cut_entropy(adj, subset): n = len(adj) S = sorted(subset) T = [i for i in range(n) if i not in S] mat = [[adj[i][j] for j in T] for i in S] return gf2_rank(mat) def graph_atoms(adj): items = tuple(range(len(adj))) F = {S: graph_cut_entropy(adj, S) for S in powerset(items)} return F, mobius_atoms(F, items) def capsule_MOB_01(): empty = [[0] 4 for _ in range(4)] edge = [[0, 1], [1, 0]] star = [[0, 1, 1, 1], [1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0]] _, empty_atoms = graph_atoms(empty) edge_F, edge_atoms = graph_atoms(edge) star_F, star_atoms = graph_atoms(star) return { "id": "MOB-01", "witness": "GF(2) graph-state entropy equals cut rank; product graph is zero, Bell edge is localized, star/GHZ-like graph gives non-product entropy structure.", "product_max_abs_atom": max(abs(v) for v in empty_atoms.values()), "bell_edge_F": {str(k): v for k, v in sorted(edge_F.items(), key=lambda kv: (len(kv[0]), kv[0]))}, "bell_edge_atoms": top_atoms(edge_atoms, 6), "star_order_mass": order_mass(star_atoms), "star_top_atoms": top_atoms(star_atoms, 8), "control": "Empty graph must have no nonzero atom; edge graph must not be mistaken for generic high-order structure.", } def capsule_MOB_02(): items = tuple(range(4)) weights = {i: 1.0 for i in items} K = 2.0 F_add = {S: sum(weights[i] for i in S) for S in powerset(items)} F_cap = {S: min(sum(weights[i] for i in S), K) for S in powerset(items)} add_atoms = mobius_atoms(F_add, items) cap_atoms = mobius_atoms(F_cap, items) add_high = max(abs(v) for S, v in add_atoms.items() if len(S) >= 2) return { "id": "MOB-02", "witness": "A fixed min-cut/capacity cap creates non-additive bottleneck atoms; additive capacity has no high-order atoms.", "items": list(items), "weights": weights, "cap_K": K, "additive_high_order_max": add_high, "cap_order_mass": order_mass(cap_atoms), "cap_top_high_order_atoms": top_atoms(cap_atoms, 8, min_order=2), "control": "The same weights without the bottleneck cap give only singleton atoms.", } def capsule_MOB_03(): items = tuple(range(4)) def calc(name, func): F = {S: float(func(set(S))) for S in powerset(items)} atoms = mobius_atoms(F, items) return {"name": name, "order_mass": order_mass(atoms), "top_atoms": top_atoms(atoms, 8)} cases = [ calc("additive", lambda S: len(S)), calc("pair_count", lambda S: sum(1 for a, b in combinations(S, 2))), calc("unanimity_all_four", lambda S: 1.0 if len(S) == 4 else 0.0), calc("odd_parity_indicator", lambda S: 1.0 if len(S) % 2 == 1 else 0.0), calc("planted_gate_0_2", lambda S: 0.2 len(S) + (3.0 if {0, 2}.issubset(S) else 0.0)), ] return { "id": "MOB-03", "witness": "Harsanyi dividends recover additive, pairwise, unanimity, parity, and planted feature-gate structure under one fixed baseline.", "cases": cases, "control": "Additive model has zero atoms above order 1; pair-count localizes at order 2; no retraining or variable parent is allowed.", } def kron_all(ops): out = ops[0] for op in ops[1:]: out = np.kron(out, op) return out def reduced_density_entropy(psi, keep, n): keep = list(keep) traced = [i for i in range(n) if i not in keep] # For a pure state, avoid fragile repeated tensor traces: # reshape |psi> into keep x traced blocks and compute M M^\dagger. tensor = psi.reshape([2] n) order = keep + traced tensor = np.transpose(tensor, axes=order) mat = tensor.reshape((2 len(keep), 2 len(traced))) rho = mat @ mat.conj().T dim = 2 len(keep) rho = rho.reshape((dim, dim)) vals = np.linalg.eigvalsh((rho + rho.conj().T) / 2) vals = np.maximum(vals.real, 0) vals = vals / vals.sum() if vals.sum() else vals return entropy_bits(vals) def tfim_ground_entropy(n=4, h=1.0, J=1.0): X = np.array([[0, 1], [1, 0]], dtype=float) Z = np.array([[1, 0], [0, -1]], dtype=float) I = np.eye(2) H = np.zeros((2n, 2**n), dtype=float) for i in range(n): ops = [I] n ops[i] = X H -= h kron_all(ops) ops = [I] n ops[i] = Z ops[(i + 1) % n] = Z H -= J kron_all(ops) vals, vecs = np.linalg.eigh(H) psi = vecs[:, np.argmin(vals)] return reduced_density_entropy(psi, keep=[0, 1], n=n) def capsule_MOB_04(): n = 4 items = tuple(range(n)) F_product = {S: 0.0 for S in powerset(items)} F_ghz = {S: (0.0 if len(S) in (0, n) else 1.0) for S in powerset(items)} product_atoms = mobius_atoms(F_product, items) ghz_atoms = mobius_atoms(F_ghz, items) h_values = [0.2, 0.5, 1.0, 2.0] ent = {str(h): round(tfim_ground_entropy(n=4, h=h), 8) for h in h_values} return { "id": "MOB-04", "witness": "Product spin state is zero; GHZ/cat state gives global entropy atoms; tiny TFIM exact diagonalization shows finite-size coupling sensitivity.", "product_max_abs_atom": max(abs(v) for v in product_atoms.values()), "ghz_order_mass": order_mass(ghz_atoms), "ghz_top_atoms": top_atoms(ghz_atoms, 8), "tfim_half_chain_entropy_by_h": ent, "control": "Finite-size TFIM numbers are diagnostic only; they are not a dimension-activation proof.", } def crossing_2x2(active): active = set(active) edges = [(0, 1), (0, 2), (1, 3), (2, 3)] left = {0, 2} right = {1, 3} seen = set() stack = list(left & active) while stack: u = stack.pop() if u in seen: continue seen.add(u) if u in right: return 1.0 for a, b in edges: v = b if a == u else a if b == u else None if v is not None and v in active and v not in seen: stack.append(v) return 0.0 def graph_beta1(edge_subset): vertices = {0, 1, 2, 3} edges = [(0, 1), (1, 2), (2, 3), (3, 0)] parent = list(range(len(edges))) chosen = [edges[i] for i in edge_subset] adj = {v: set() for v in vertices} for a, b in chosen: adj[a].add(b); adj[b].add(a) seen = set(); comps = 0 for v in vertices: if v in seen: continue comps += 1 stack = [v] while stack: u = stack.pop() if u in seen: continue seen.add(u) stack.extend(adj[u] - seen) return max(0.0, float(len(chosen) - len(vertices) + comps)) def capsule_MOB_05(): items = tuple(range(4)) F_cross = {S: crossing_2x2(S) for S in powerset(items)} cross_atoms = mobius_atoms(F_cross, items) F_beta = {S: graph_beta1(S) for S in powerset(items)} beta_atoms = mobius_atoms(F_beta, items) return { "id": "MOB-05", "witness": "Exact tiny percolation crossing and graph beta1 cycle witnesses show threshold/topology atoms under a fixed parent.", "crossing_truth_table": {str(tuple(sorted(S))): int(v) for S, v in F_cross.items()}, "crossing_top_atoms": top_atoms(cross_atoms, 8), "cycle_beta1_full": F_beta[frozenset(items)], "cycle_beta1_top_atoms": top_atoms(beta_atoms, 8), "control": "Changing the graph or resampling the parent while changing S is forbidden; random/persistence scaling remains limited.", } def ordering_fraction(rel, subset): S = sorted(subset) m = len(S) if m < 2: return 0.0 count = 0 for i, a in enumerate(S): for b in S[i+1:]: if rel[a][b] or rel[b][a]: count += 1 return count / comb(m, 2) def interval_abundance(rel, k): n = len(rel) total = 0 for a in range(n): for b in range(n): if rel[a][b]: inside = 0 for c in range(n): if rel[a][c] and rel[c][b]: inside += 1 if inside == k: total += 1 return total def capsule_MOB_06(): n = 4 chain = [[1 if i < j else 0 for j in range(n)] for i in range(n)] anti = [[0] n for _ in range(n)] items = tuple(range(n)) chain_F = {S: ordering_fraction(chain, S) for S in powerset(items)} anti_F = {S: ordering_fraction(anti, S) for S in powerset(items)} chain_atoms = mobius_atoms(chain_F, items) anti_atoms = mobius_atoms(anti_F, items) rng = np.random.default_rng(123) pts = rng.random((20, 2)) rel = [[0] 20 for _ in range(20)] for i in range(20): for j in range(20): dt = pts[j, 0] - pts[i, 0] dx = abs(pts[j, 1] - pts[i, 1]) if dt > dx: rel[i][j] = 1 sample = tuple(range(8)) sprinkle_r = ordering_fraction(rel, sample) return { "id": "MOB-06", "witness": "Causet induced-subset observables separate antichain, chain, and a fixed 2D sprinkling sample.", "antichain_max_abs_atom": max(abs(v) for v in anti_atoms.values()), "chain_order_fraction_full": chain_F[frozenset(items)], "chain_top_atoms": top_atoms(chain_atoms, 8), "sprinkling_seed": 123, "sprinkling_sample_ordering_fraction_first8": round(sprinkle_r, 8), "sprinkling_interval_abundance_k0": interval_abundance(rel, 0), "control": "Generic random posets and uncalibrated Hasse spectra are not manifoldlike evidence.", } def distribution_from_rows(rows): d = {} for row in rows: d[row] = d.get(row, 0.0) + 1.0 / len(rows) return d def mutual_information(dist, x_indices, y_indices): px = {}; py = {}; pxy = {} for row, p in dist.items(): x = tuple(row[i] for i in x_indices) y = tuple(row[i] for i in y_indices) px[x] = px.get(x, 0.0) + p py[y] = py.get(y, 0.0) + p pxy[(x, y)] = pxy.get((x, y), 0.0) + p mi = 0.0 for (x, y), p in pxy.items(): mi += p log2(p / (px[x] py[y])) return mi def info_atoms_for_rows(rows, source_indices, target_indices): dist = distribution_from_rows(rows) items = tuple(range(len(source_indices))) F = {} for S in powerset(items): actual = [source_indices[i] for i in S] F[S] = mutual_information(dist, actual, target_indices) if actual else 0.0 return mobius_atoms(F, items), F def capsule_MOB_07(): xor_rows = [(x, y, x ^ y) for x, y in product([0, 1], repeat=2)] red_rows = [(t, t, t) for t in [0, 1]] unique_rows = [(x, y, x, y) for x, y in product([0, 1], repeat=2)] xor_atoms, xor_F = info_atoms_for_rows(xor_rows, [0, 1], [2]) red_atoms, red_F = info_atoms_for_rows(red_rows, [0, 1], [2]) unique_atoms, unique_F = info_atoms_for_rows(unique_rows, [0, 1], [2, 3]) return { "id": "MOB-07", "witness": "Exact distributions recover XOR synergy, redundant-copy redundancy, and unique-source additivity.", "xor_F": {str(k): round(v, 8) for k, v in xor_F.items()}, "xor_atoms": top_atoms(xor_atoms, 4), "redundant_atoms": top_atoms(red_atoms, 4), "unique_atoms": top_atoms(unique_atoms, 4), "control": "PID definitions may vary; this capsule uses ordinary mutual information atoms only.", } def capsule_MOB_08(): items = (0, 1, 2) coeffs = {"0": 0.4, "1": -0.2, "2": 0.1, "01": 0.75, "012": -0.5} def F_land(S): S = set(S) return sum(coeffs[str(i)] for i in S) + (coeffs["01"] if {0, 1}.issubset(S) else 0.0) + (coeffs["012"] if {0, 1, 2}.issubset(S) else 0.0) F = {S: F_land(S) for S in powerset(items)} atoms = mobius_atoms(F, items) sign_F = {frozenset(): 0.0, frozenset([0]): 1.0, frozenset([1]): 1.0, frozenset([0, 1]): -1.0} sign_atoms = mobius_atoms(sign_F, (0, 1)) return { "id": "MOB-08", "witness": "Synthetic log-fitness landscape recovers planted epistasis atoms, including sign epistasis.", "planted_atoms": top_atoms(atoms, 8), "sign_epistasis_effect_A_without_B": sign_F[frozenset([0])] - sign_F[frozenset()], "sign_epistasis_effect_A_with_B": sign_F[frozenset([0, 1])] - sign_F[frozenset([1])], "sign_epistasis_atoms": top_atoms(sign_atoms, 4), "control": "Use log fitness or declared scale; raw multiplicative fitness can manufacture artifacts.", } def capsule_MOB_09(): items = (0, 1, 2, 3) coeff = { (0,): -0.1, (1,): 0.2, (0, 1): -0.8, (1, 2): 0.35, (0, 1, 2): 0.6, (0, 1, 2, 3): -0.25, } def energy(S): S = set(S) total = 0.0 for C, v in coeff.items(): if set(C).issubset(S): total += v return total F = {S: energy(S) for S in powerset(items)} atoms = mobius_atoms(F, items) max_coeff_error = max(abs(atoms.get(C, 0.0) - v) for C, v in coeff.items()) return { "id": "MOB-09", "witness": "Cluster-expansion energy terms are exactly Mobius atoms of a fixed lattice occupation energy.", "max_planted_ECI_recovery_error": max_coeff_error, "recovered_atoms": top_atoms(atoms, 10), "control": "Fixed composition or relaxing the lattice separately for each subset changes the parent and is an artifact.", } def capsule_MOB_10(): items = (0, 1, 2) F_series = {S: 1.0 if len(S) == 3 else 0.0 for S in powerset(items)} F_parallel = {S: 1.0 if len(S) >= 1 else 0.0 for S in powerset(items)} F_k2 = {S: 1.0 if len(S) >= 2 else 0.0 for S in powerset(items)} def bridge_success(S): edges = {0: ("s", "a"), 1: ("a", "t"), 2: ("s", "b"), 3: ("b", "t"), 4: ("a", "b")} graph = {} for e in S: a, b = edges[e] graph.setdefault(a, set()).add(b); graph.setdefault(b, set()).add(a) seen = set(); stack = ["s"] while stack: u = stack.pop() if u in seen: continue seen.add(u) stack.extend(graph.get(u, set()) - seen) return 1.0 if "t" in seen else 0.0 bridge_items = (0, 1, 2, 3, 4) F_bridge = {S: bridge_success(S) for S in powerset(bridge_items)} return { "id": "MOB-10", "witness": "Reliability success functions expose series, parallel, k-out-of-n, and bridge-network pivotal structure.", "series_top_atoms": top_atoms(mobius_atoms(F_series, items), 6), "parallel_top_atoms": top_atoms(mobius_atoms(F_parallel, items), 6), "k2of3_top_atoms": top_atoms(mobius_atoms(F_k2, items), 6), "bridge_top_atoms": top_atoms(mobius_atoms(F_bridge, bridge_items), 10, min_order=2), "control": "Common-cause failures must be modeled explicitly; otherwise they masquerade as component synergy.", } def cascade_reachability(seed_set): edges = {0: [1, 2], 1: [3], 2: [3], 3: []} seen = set(); stack = list(seed_set) while stack: u = stack.pop() if u in seen: continue seen.add(u) stack.extend(edges.get(u, [])) return float(len(seen)) def capsule_MOB_11(): items = (0, 1, 2) F_reach = {S: cascade_reachability(S) for S in powerset(items)} reach_atoms = mobius_atoms(F_reach, items) # Complex contagion: target activates only if seeds 0 and 1 are jointly present. F_threshold = {S: float(len(S) + (1 if {0, 1}.issubset(S) else 0)) for S in powerset(items)} threshold_atoms = mobius_atoms(F_threshold, items) return { "id": "MOB-11", "witness": "Fixed cascade rules expose reachability redundancy and threshold synergy.", "reachability_F": {str(tuple(sorted(S))): v for S, v in F_reach.items()}, "reachability_top_atoms": top_atoms(reach_atoms, 8), "threshold_top_atoms": top_atoms(threshold_atoms, 8), "control": "The network and time/update rule must be fixed; changing them between seed subsets is a variable-parent artifact.", } def ring_hamiltonian(N, t=-1.0): H = np.zeros((N, N), dtype=float) for i in range(N): H[i, (i + 1) % N] = t H[(i + 1) % N, i] = t return H def capsule_MOB_12(): allowed = [] trace_table = {} for n in range(1, 13): tr = 2 cos(2 pi / n) trace_table[str(n)] = round(tr, 10) if abs(tr - round(tr)) < 1e-10 and abs(round(tr)) <= 2: allowed.append(n) N = 12 t = -1.0 H = ring_hamiltonian(N, t) evals = sorted(np.linalg.eigvalsh(H)) formula = sorted([2 t cos(2 pi m / N) for m in range(N)]) bloch_err = max(abs(a - b) for a, b in zip(evals, formula)) K = 1.0 L = np.zeros((N, N), dtype=float) for i in range(N): L[i, i] = 2 K L[i, (i + 1) % N] = -K L[i, (i - 1) % N] = -K lam = sorted(np.linalg.eigvalsh(L)) phonon_formula = sorted([2 K (1 - cos(2 pi m / N)) for m in range(N)]) phonon_err = max(abs(a - b) for a, b in zip(lam, phonon_formula)) return { "id": "MOB-12", "witness": "Crystal restriction, Bloch band prediction, and phonon eigenvalues are reproduced in fixed periodic parents.", "crystallographic_allowed_orders_1_to_12": allowed, "rotation_trace_table": trace_table, "bloch_ring_N": N, "bloch_max_eigenvalue_error": float(bloch_err), "phonon_max_eigenvalue_error": float(phonon_err), "control": "Disorder or using the wrong parent group breaks the exact Bloch/phonon prediction.", } def translation_permutation(N, shift): P = np.zeros((N, N), dtype=float) for i in range(N): P[(i + shift) % N, i] = 1.0 return P def reynolds_residual(H, group): avg = sum(P @ H @ P.T for P in group) / len(group) denom = np.linalg.norm(H, "fro") 2 return float(np.linalg.norm(H - avg, "fro") 2 / denom) if denom else 0.0 def ring_distance(a, b, n): d = abs(a - b) % n return min(d, n - d) def capsule_MOB_13(): N = 24 items = tuple(range(6)) sites = {i: 4 i for i in items} H0 = ring_hamiltonian(N, t=-1.0) group = [translation_permutation(N, s) for s in range(N)] xi = 1.35 def kernel(a, b): d = ring_distance(a, b, len(items)) return exp(-d / xi) def make_H(S): S = set(S) H = H0.copy() for i in S: H[sites[i], sites[i]] += 1.0 for a, b in combinations(S, 2): w = 2.4 kernel(a, b) H[sites[a], sites[a]] += 0.5 w H[sites[b], sites[b]] += 0.5 w H[sites[a], sites[b]] += -0.15 w H[sites[b], sites[a]] += -0.15 w return H F = {S: reynolds_residual(make_H(S), group) for S in powerset(items)} atoms = mobius_atoms(F, items) labels = [] scores = [] true_kernel = [] for a, b in combinations(items, 2): labels.append(1 if ring_distance(a, b, len(items)) == 1 else 0) scores.append(abs(atoms[(a, b)])) true_kernel.append(kernel(a, b)) rng = np.random.default_rng(123) random_aucs = [] for _ in range(200): shuffled = rng.permutation(labels) random_aucs.append(auc(shuffled, scores)) return { "id": "MOB-13", "witness": "Defect-subset atoms over a Reynolds residual recover local pair structure in a fixed ring Hamiltonian with known pair kernel.", "ring_N": N, "candidate_sites": sites, "adjacent_pair_auc": round(auc(labels, scores), 10), "pair_kernel_correlation": round(pearson(scores, true_kernel), 10), "random_label_auc_mean_seed123_200": round(float(np.mean(random_aucs)), 10), "random_label_auc_sd": round(float(np.std(random_aucs)), 10), "top_pair_atoms": top_atoms({S: a for S, a in atoms.items() if len(S) == 2}, 10), "control": "Wrong-parent/identity group and random labels must not recover the local adjacency band.", } def capsule_MOB_14(): samples = 720 angles = np.linspace(0, 2 pi, samples + 1) burgers_integral = float(np.sum(np.diff(angles)) / (2 pi)) frank_target = pi / 3 frank_integral = float(np.sum((frank_target / (2 pi)) np.diff(angles))) # Limited spring-lattice proxy: local vacancy interaction on a 4-cycle, not a full atomistic model. items = tuple(range(4)) def F_vac(S): S = set(S) local_pairs = sum(1 for a, b in combinations(S, 2) if ring_distance(a, b, 4) == 1) far_pairs = sum(1 for a, b in combinations(S, 2) if ring_distance(a, b, 4) == 2) return 0.2 len(S) + 1.0 local_pairs + 0.05 far_pairs F = {S: F_vac(S) for S in powerset(items)} atoms = mobius_atoms(F, items) labels = []; scores = [] for a, b in combinations(items, 2): labels.append(1 if ring_distance(a, b, 4) == 1 else 0) scores.append(abs(atoms[(a, b)])) return { "id": "MOB-14", "witness": "Continuum defect-charge checks are exact; spring-lattice subset atoms are useful but still only a limited proxy.", "burgers_screw_integral_samples": samples, "burgers_screw_integral": round(burgers_integral, 12), "frank_angle_target": round(frank_target, 12), "frank_angle_integral": round(frank_integral, 12), "toy_vacancy_adjacency_auc": round(auc(labels, scores), 10), "toy_vacancy_pair_atoms": top_atoms({S: a for S, a in atoms.items() if len(S) == 2}, 8), "control": "The vacancy proxy is not real elasticity; full promotion requires validated dislocation/disclination or atomistic data.", } def w_entropy(n, k): if k == 0 or k == n: return 0.0 p = k / n return entropy_bits([p, 1 - p]) def dicke_entropy(n, m, k): if k == 0 or k == n: return 0.0 probs = [] denom = comb(n, m) for j in range(max(0, m - (n - k)), min(k, m) + 1): probs.append(comb(k, j) * comb(n - k, m - j) / denom) return entropy_bits(probs) def capsule_MOB_15(): n = 5 items = tuple(range(n)) F_w = {S: w_entropy(n, len(S)) for S in powerset(items)} w_atoms = mobius_atoms(F_w, items) F_var = {S: (0.0 if len(S) == 0 else 2.0 / len(S)) for S in powerset(items)} var_atoms = mobius_atoms(F_var, items) n2, m2 = 6, 2 items2 = tuple(range(n2)) F_dicke = {S: dicke_entropy(n2, m2, len(S)) for S in powerset(items2)} dicke_atoms = mobius_atoms(F_dicke, items2) return { "id": "MOB-15", "witness": "Fixed-parent W/Dicke entropy atoms are region maps; variable-parent formulas generate misleading towers and must be separated.", "W_n": n, "W_order_mass": order_mass(w_atoms), "W_top_atoms": top_atoms(w_atoms, 8), "variable_parent_2_over_k_order_mass": order_mass(var_atoms), "Dicke_n_m": [n2, m2], "Dicke_order_mass": order_mass(dicke_atoms), "control": "The same n/state must remain fixed. State-size-changing F(k)=2/k is an artifact-prone comparison.", } CAPSULES = [ capsule_MOB_00, capsule_MOB_01, capsule_MOB_02, capsule_MOB_03, capsule_MOB_04, capsule_MOB_05, capsule_MOB_06, capsule_MOB_07, capsule_MOB_08, capsule_MOB_09, capsule_MOB_10, capsule_MOB_11, capsule_MOB_12, capsule_MOB_13, capsule_MOB_14, capsule_MOB_15, ] def clean(x): if isinstance(x, dict): return {str(k): clean(v) for k, v in x.items()} if isinstance(x, (list, tuple)): return [clean(v) for v in x] if isinstance(x, np.ndarray): return clean(x.tolist()) if isinstance(x, (np.floating, np.integer)): return x.item() if isinstance(x, float): if abs(x) < 1e-12: return 0.0 return x return x def run_all(): return [clean(fn()) for fn in CAPSULES] if __name__ == "__main__": print(json.dumps(run_all(), indent=2, sort_keys=True)) Removed As Redundant Or Not Carry-Forward Repeated report ledgers, old round summaries, and duplicate verdict counts. Superseded pre-correction claims before actual engine/band-finding corrections. Duplicate calculations of the same family when a later cleaner result exists. Artifact-only outputs except where they define mandatory controls. Any wording that presents Mobius as a QG proof or make-or-break closure. Broad engine history; only methodologically relevant use remains in result cards. Machine-Readable Payload { "baseline_path": "[path removed]", "baseline_sha256": "[hash removed]", "date": "2026-06-21", "document": "the study_MOBIUS_NONREDUNDANT_REPRODUCIBLE_RESULTS_TECHNICAL_REINFORCED.docx", "result_count": 16, "results": [ { "control": "Recovery error must be zero up to floating precision; planted pair/triple atoms must appear at the planted subsets.", "id": "MOB-00", "max_reconstruction_error": 0.0, "order_mass": { "0": 0.0, "1": 5.0, "2": 0.7, "3": 0.3, "4": 0.0 }, "top_atoms": [ { "S": [ 0 ], "atom": 1.25 }, { "S": [ 1 ], "atom": 1.25 }, { "S": [ 2 ], "atom": 1.25 }, { "S": [ 3 ], "atom": 1.25 }, { "S": [ 0, 1 ], "atom": 0.7 }, { "S": [ 0, 1, 2 ], "atom": -0.3 } ], "witness": "Mobius inversion exactly recovers a planted additive + pair + triple finite set function." }, { "bell_edge_F": { "frozenset()": 0, "frozenset({0, 1})": 0, "frozenset({0})": 1, "frozenset({1})": 1 }, "bell_edge_atoms": [ { "S": [ 0, 1 ], "atom": -2.0 }, { "S": [ 0 ], "atom": 1.0 }, { "S": [ 1 ], "atom": 1.0 } ], "control": "Empty graph must have no nonzero atom; edge graph must not be mistaken for generic high-order structure.", "id": "MOB-01", "product_max_abs_atom": 0.0, "star_order_mass": { "0": 0.0, "1": 4.0, "2": 6.0, "3": 4.0, "4": 2.0 }, "star_top_atoms": [ { "S": [ 0, 1, 2, 3 ], "atom": -2.0 }, { "S": [ 0 ], "atom": 1.0 }, { "S": [ 0, 1 ], "atom": -1.0 }, { "S": [ 0, 1, 2 ], "atom": 1.0 }, { "S": [ 0, 1, 3 ], "atom": 1.0 }, { "S": [ 0, 2 ], "atom": -1.0 }, { "S": [ 0, 2, 3 ], "atom": 1.0 }, { "S": [ 0, 3 ], "atom": -1.0 } ], "witness": "GF(2) graph-state entropy equals cut rank; product graph is zero, Bell edge is localized, star/GHZ-like graph gives non-product entropy structure." }, { "additive_high_order_max": 0.0, "cap_K": 2.0, "cap_order_mass": { "0": 0.0, "1": 4.0, "2": 0.0, "3": 4.0, "4": 2.0 }, "cap_top_high_order_atoms": [ { "S": [ 0, 1, 2, 3 ], "atom": 2.0 }, { "S": [ 0, 1, 2 ], "atom": -1.0 }, { "S": [ 0, 1, 3 ], "atom": -1.0 }, { "S": [ 0, 2, 3 ], "atom": -1.0 }, { "S": [ 1, 2, 3 ], "atom": -1.0 } ], "control": "The same weights without the bottleneck cap give only singleton atoms.", "id": "MOB-02", "items": [ 0, 1, 2, 3 ], "weights": { "0": 1.0, "1": 1.0, "2": 1.0, "3": 1.0 }, "witness": "A fixed min-cut/capacity cap creates non-additive bottleneck atoms; additive capacity has no high-order atoms." }, { "cases": [ { "name": "additive", "order_mass": { "0": 0.0, "1": 4.0, "2": 0.0, "3": 0.0, "4": 0.0 }, "top_atoms": [ { "S": [ 0 ], "atom": 1.0 }, { "S": [ 1 ], "atom": 1.0 }, { "S": [ 2 ], "atom": 1.0 }, { "S": [ 3 ], "atom": 1.0 } ] }, { "name": "pair_count", "order_mass": { "0": 0.0, "1": 0.0, "2": 6.0, "3": 0.0, "4": 0.0 }, "top_atoms": [ { "S": [ 0, 1 ], "atom": 1.0 }, { "S": [ 0, 2 ], "atom": 1.0 }, { "S": [ 0, 3 ], "atom": 1.0 }, { "S": [ 1, 2 ], "atom": 1.0 }, { "S": [ 1, 3 ], "atom": 1.0 }, { "S": [ 2, 3 ], "atom": 1.0 } ] }, { "name": "unanimity_all_four", "order_mass": { "0": 0.0, "1": 0.0, "2": 0.0, "3": 0.0, "4": 1.0 }, "top_atoms": [ { "S": [ 0, 1, 2, 3 ], "atom": 1.0 } ] }, { "name": "odd_parity_indicator", "order_mass": { "0": 0.0, "1": 4.0, "2": 12.0, "3": 16.0, "4": 8.0 }, "top_atoms": [ { "S": [ 0, 1, 2, 3 ], "atom": -8.0 }, { "S": [ 0, 1, 2 ], "atom": 4.0 }, { "S": [ 0, 1, 3 ], "atom": 4.0 }, { "S": [ 0, 2, 3 ], "atom": 4.0 }, { "S": [ 1, 2, 3 ], "atom": 4.0 }, { "S": [ 0, 1 ], "atom": -2.0 }, { "S": [ 0, 2 ], "atom": -2.0 }, { "S": [ 0, 3 ], "atom": -2.0 } ] }, { "name": "planted_gate_0_2", "order_mass": { "0": 0.0, "1": 0.8, "2": 3.0, "3": 0.0, "4": 0.0 }, "top_atoms": [ { "S": [ 0, 2 ], "atom": 3.0 }, { "S": [ 0 ], "atom": 0.2 }, { "S": [ 1 ], "atom": 0.2 }, { "S": [ 2 ], "atom": 0.2 }, { "S": [ 3 ], "atom": 0.2 } ] } ], "control": "Additive model has zero atoms above order 1; pair-count localizes at order 2; no retraining or variable parent is allowed.", "id": "MOB-03", "witness": "Harsanyi dividends recover additive, pairwise, unanimity, parity, and planted feature-gate structure under one fixed baseline." }, { "control": "Finite-size TFIM numbers are diagnostic only; they are not a dimension-activation proof.", "ghz_order_mass": { "0": 0.0, "1": 4.0, "2": 6.0, "3": 4.0, "4": 2.0 }, "ghz_top_atoms": [ { "S": [ 0, 1, 2, 3 ], "atom": -2.0 }, { "S": [ 0 ], "atom": 1.0 }, { "S": [ 0, 1 ], "atom": -1.0 }, { "S": [ 0, 1, 2 ], "atom": 1.0 }, { "S": [ 0, 1, 3 ], "atom": 1.0 }, { "S": [ 0, 2 ], "atom": -1.0 }, { "S": [ 0, 2, 3 ], "atom": 1.0 }, { "S": [ 0, 3 ], "atom": -1.0 } ], "id": "MOB-04", "product_max_abs_atom": 0.0, "tfim_half_chain_entropy_by_h": { "0.2": 0.99974585, "0.5": 0.98259627, "1.0": 0.75289784, "2.0": 0.27322157 }, "witness": "Product spin state is zero; GHZ/cat state gives global entropy atoms; tiny TFIM exact diagonalization shows finite-size coupling sensitivity." }, { "control": "Changing the graph or resampling the parent while changing S is forbidden; random/persistence scaling remains limited.", "crossing_top_atoms": [ { "S": [ 0, 1 ], "atom": 1.0 }, { "S": [ 0, 1, 2, 3 ], "atom": -1.0 }, { "S": [ 2, 3 ], "atom": 1.0 } ], "crossing_truth_table": { "()": 0, "(0, 1)": 1, "(0, 1, 2)": 1, "(0, 1, 2, 3)": 1, "(0, 1, 3)": 1, "(0, 2)": 0, "(0, 2, 3)": 1, "(0, 3)": 0, "(0,)": 0, "(1, 2)": 0, "(1, 2, 3)": 1, "(1, 3)": 0, "(1,)": 0, "(2, 3)": 1, "(2,)": 0, "(3,)": 0 }, "cycle_beta1_full": 1.0, "cycle_beta1_top_atoms": [ { "S": [ 0, 1, 2, 3 ], "atom": 1.0 } ], "id": "MOB-05", "witness": "Exact tiny percolation crossing and graph beta1 cycle witnesses show threshold/topology atoms under a fixed parent." }, { "antichain_max_abs_atom": 0.0, "chain_order_fraction_full": 1.0, "chain_top_atoms": [ { "S": [ 0, 1, 2, 3 ], "atom": 3.0 }, { "S": [ 0, 1, 2 ], "atom": -2.0 }, { "S": [ 0, 1, 3 ], "atom": -2.0 }, { "S": [ 0, 2, 3 ], "atom": -2.0 }, { "S": [ 1, 2, 3 ], "atom": -2.0 }, { "S": [ 0, 1 ], "atom": 1.0 }, { "S": [ 0, 2 ], "atom": 1.0 }, { "S": [ 0, 3 ], "atom": 1.0 } ], "control": "Generic random posets and uncalibrated Hasse spectra are not manifoldlike evidence.", "id": "MOB-06", "sprinkling_interval_abundance_k0": 38, "sprinkling_sample_ordering_fraction_first8": 0.46428571, "sprinkling_seed": 123, "witness": "Causet induced-subset observables separate antichain, chain, and a fixed 2D sprinkling sample." }, { "control": "PID definitions may vary; this capsule uses ordinary mutual information atoms only.", "id": "MOB-07", "redundant_atoms": [ { "S": [ 0 ], "atom": 1.0 }, { "S": [ 0, 1 ], "atom": -1.0 }, { "S": [ 1 ], "atom": 1.0 } ], "unique_atoms": [ { "S": [ 0 ], "atom": 1.0 }, { "S": [ 1 ], "atom": 1.0 } ], "witness": "Exact distributions recover XOR synergy, redundant-copy redundancy, and unique-source additivity.", "xor_F": { "frozenset()": 0.0, "frozenset({0, 1})": 1.0, "frozenset({0})": 0.0, "frozenset({1})": 0.0 }, "xor_atoms": [ { "S": [ 0, 1 ], "atom": 1.0 } ] }, { "control": "Use log fitness or declared scale; raw multiplicative fitness can manufacture artifacts.", "id": "MOB-08", "planted_atoms": [ { "S": [ 0, 1 ], "atom": 0.75 }, { "S": [ 0, 1, 2 ], "atom": -0.5 }, { "S": [ 0 ], "atom": 0.4 }, { "S": [ 1 ], "atom": -0.2 }, { "S": [ 2 ], "atom": 0.1 } ], "sign_epistasis_atoms": [ { "S": [ 0, 1 ], "atom": -3.0 }, { "S": [ 0 ], "atom": 1.0 }, { "S": [ 1 ], "atom": 1.0 } ], "sign_epistasis_effect_A_with_B": -2.0, "sign_epistasis_effect_A_without_B": 1.0, "witness": "Synthetic log-fitness landscape recovers planted epistasis atoms, including sign epistasis." }, { "control": "Fixed composition or relaxing the lattice separately for each subset changes the parent and is an artifact.", "id": "MOB-09", "max_planted_ECI_recovery_error": 0.0, "recovered_atoms": [ { "S": [ 0, 1 ], "atom": -0.8 }, { "S": [ 0, 1, 2 ], "atom": 0.6 }, { "S": [ 1, 2 ], "atom": 0.35 }, { "S": [ 0, 1, 2, 3 ], "atom": -0.25 }, { "S": [ 1 ], "atom": 0.2 }, { "S": [ 0 ], "atom": -0.1 } ], "witness": "Cluster-expansion energy terms are exactly Mobius atoms of a fixed lattice occupation energy." }, { "bridge_top_atoms": [ { "S": [ 0, 1, 2, 3, 4 ], "atom": 2.0 }, { "S": [ 0, 1 ], "atom": 1.0 }, { "S": [ 0, 1, 2, 3 ], "atom": -1.0 }, { "S": [ 0, 1, 2, 4 ], "atom": -1.0 }, { "S": [ 0, 1, 3, 4 ], "atom": -1.0 }, { "S": [ 0, 2, 3, 4 ], "atom": -1.0 }, { "S": [ 0, 3, 4 ], "atom": 1.0 }, { "S": [ 1, 2, 3, 4 ], "atom": -1.0 }, { "S": [ 1, 2, 4 ], "atom": 1.0 }, { "S": [ 2, 3 ], "atom": 1.0 } ], "control": "Common-cause failures must be modeled explicitly; otherwise they masquerade as component synergy.", "id": "MOB-10", "k2of3_top_atoms": [ { "S": [ 0, 1, 2 ], "atom": -2.0 }, { "S": [ 0, 1 ], "atom": 1.0 }, { "S": [ 0, 2 ], "atom": 1.0 }, { "S": [ 1, 2 ], "atom": 1.0 } ], "parallel_top_atoms": [ { "S": [ 0 ], "atom": 1.0 }, { "S": [ 0, 1 ], "atom": -1.0 }, { "S": [ 0, 1, 2 ], "atom": 1.0 }, { "S": [ 0, 2 ], "atom": -1.0 }, { "S": [ 1 ], "atom": 1.0 }, { "S": [ 1, 2 ], "atom": -1.0 } ], "series_top_atoms": [ { "S": [ 0, 1, 2 ], "atom": 1.0 } ], "witness": "Reliability success functions expose series, parallel, k-out-of-n, and bridge-network pivotal structure." }, { "control": "The network and time/update rule must be fixed; changing them between seed subsets is a variable-parent artifact.", "id": "MOB-11", "reachability_F": { "()": 0.0, "(0, 1)": 4.0, "(0, 1, 2)": 4.0, "(0, 2)": 4.0, "(0,)": 4.0, "(1, 2)": 3.0, "(1,)": 2.0, "(2,)": 2.0 }, "reachability_top_atoms": [ { "S": [ 0 ], "atom": 4.0 }, { "S": [ 0, 1 ], "atom": -2.0 }, { "S": [ 0, 2 ], "atom": -2.0 }, { "S": [ 1 ], "atom": 2.0 }, { "S": [ 2 ], "atom": 2.0 }, { "S": [ 0, 1, 2 ], "atom": 1.0 }, { "S": [ 1, 2 ], "atom": -1.0 } ], "threshold_top_atoms": [ { "S": [ 0 ], "atom": 1.0 }, { "S": [ 0, 1 ], "atom": 1.0 }, { "S": [ 1 ], "atom": 1.0 }, { "S": [ 2 ], "atom": 1.0 } ], "witness": "Fixed cascade rules expose reachability redundancy and threshold synergy." }, { "bloch_max_eigenvalue_error": 0.0, "bloch_ring_N": 12, "control": "Disorder or using the wrong parent group breaks the exact Bloch/phonon prediction.", "crystallographic_allowed_orders_1_to_12": [ 1, 2, 3, 4, 6 ], "id": "MOB-12", "phonon_max_eigenvalue_error": 0.0, "rotation_trace_table": { "1": 2.0, "10": 1.6180339887, "11": 1.6825070657, "12": 1.7320508076, "2": -2.0, "3": -1.0, "4": 0.0, "5": 0.6180339887, "6": 1.0, "7": 1.2469796037, "8": 1.4142135624, "9": 1.5320888862 }, "witness": "Crystal restriction, Bloch band prediction, and phonon eigenvalues are reproduced in fixed periodic parents." }, { "adjacent_pair_auc": 1.0, "candidate_sites": { "0": 0, "1": 4, "2": 8, "3": 12, "4": 16, "5": 20 }, "control": "Wrong-parent/identity group and random labels must not recover the local adjacency band.", "id": "MOB-13", "pair_kernel_correlation": 0.9995192833, "random_label_auc_mean_seed123_200": 0.4833796296, "random_label_auc_sd": 0.1554902241, "ring_N": 24, "top_pair_atoms": [ { "S": [ 0, 5 ], "atom": 0.0474398801 }, { "S": [ 4, 5 ], "atom": 0.0474398801 }, { "S": [ 3, 4 ], "atom": 0.0474398801 }, { "S": [ 2, 3 ], "atom": 0.0474398801 }, { "S": [ 1, 2 ], "atom": 0.0474398801 }, { "S": [ 0, 1 ], "atom": 0.0474398801 }, { "S": [ 0, 4 ], "atom": 0.0190794611 }, { "S": [ 1, 5 ], "atom": 0.0190794611 }, { "S": [ 2, 4 ], "atom": 0.0190794611 }, { "S": [ 3, 5 ], "atom": 0.0190794611 } ], "witness": "Defect-subset atoms over a Reynolds residual recover local pair structure in a fixed ring Hamiltonian with known pair kernel." }, { "burgers_screw_integral": 1.0, "burgers_screw_integral_samples": 720, "control": "The vacancy proxy is not real elasticity; full promotion requires validated dislocation/disclination or atomistic data.", "frank_angle_integral": 1.047197551197, "frank_angle_target": 1.047197551197, "id": "MOB-14", "toy_vacancy_adjacency_auc": 1.0, "toy_vacancy_pair_atoms": [ { "S": [ 0, 1 ], "atom": 1.0 }, { "S": [ 0, 3 ], "atom": 1.0 }, { "S": [ 1, 2 ], "atom": 1.0 }, { "S": [ 2, 3 ], "atom": 1.0 }, { "S": [ 0, 2 ], "atom": 0.05 }, { "S": [ 1, 3 ], "atom": 0.05 } ], "witness": "Continuum defect-charge checks are exact; spring-lattice subset atoms are useful but still only a limited proxy." }, { "Dicke_n_m": [ 6, 2 ], "Dicke_order_mass": { "0": 0.0, "1": 5.5097750043, "2": 8.4552910918, "3": 6.1424262132, "4": 3.6996982019, "5": 0.7541821144, "6": 0.2513940381 }, "W_n": 5, "W_order_mass": { "0": 0.0, "1": 3.6096404744, "2": 4.7290559532, "3": 2.2388309575, "4": 1.1194154788, "5": 0.0 }, "W_top_atoms": [ { "S": [ 0 ], "atom": 0.7219280949 }, { "S": [ 1 ], "atom": 0.7219280949 }, { "S": [ 2 ], "atom": 0.7219280949 }, { "S": [ 3 ], "atom": 0.7219280949 }, { "S": [ 4 ], "atom": 0.7219280949 }, { "S": [ 0, 1 ], "atom": -0.4729055953 }, { "S": [ 0, 2 ], "atom": -0.4729055953 }, { "S": [ 0, 3 ], "atom": -0.4729055953 } ], "control": "The same n/state must remain fixed. State-size-changing F(k)=2/k is an artifact-prone comparison.", "id": "MOB-15", "variable_parent_2_over_k_order_mass": { "0": 0.0, "1": 10.0, "2": 30.0, "3": 36.6666666667, "4": 20.8333333333, "5": 4.5666666667 }, "witness": "Fixed-parent W/Dicke entropy atoms are region maps; variable-parent formulas generate misleading towers and must be separated." } ]

{
 "Dicke_n_m": [
 6,
 2
 ],
 "Dicke_order_mass": {
 "0": 0.0,
 "1": 5.5097750043,
 "2": 8.4552910918,
 "3": 6.1424262132,
 "4": 3.6996982019,
 "5": 0.7541821144,
 "6": 0.2513940381
 },
 "W_n": 5,
 "W_order_mass": {
 "0": 0.0,
 "1": 3.6096404744,
 "2": 4.7290559532,
 "3": 2.2388309575,
 "4": 1.1194154788,
 "5": 0.0
 },
 "W_top_atoms": [
 {
 "S": [
 0
 ],
 "atom": 0.7219280949
 },
 {
 "S": [
 1
 ],
 "atom": 0.7219280949
 },
 {
 "S": [
 2
 ],
 "atom": 0.7219280949
 },
 {
 "S": [
 3
 ],
 "atom": 0.7219280949
 },
 {
 "S": [
 4
 ],
 "atom": 0.7219280949
 },
 {
 "S": [
 0,
 1
 ],
 "atom": -0.4729055953
 },
 {
 "S": [
 0,
 2
 ],
 "atom": -0.4729055953
 },
 {
 "S": [
 0,
 3
 ],
 "atom": -0.4729055953
 }
 ],
 "control": "The same n/state must remain fixed. State-size-changing F(k)=2/k is an artifact-prone comparison.",
 "id": "MOB-15",
 "variable_parent_2_over_k_order_mass": {
 "0": 0.0,
 "1": 10.0,
 "2": 30.0,
 "3": 36.6666666667,
 "4": 20.8333333333,
 "5": 4.5666666667
 },
 "witness": "Fixed-parent W/Dicke entropy atoms are region maps; variable-parent formulas generate misleading towers and must be separated."
}

MOB-16 — Liquid-crystal nematic defect topology and Frank-elastic atoms

STRONG_KEEP

Claim. For a fixed nematic liquid-crystal parent, topological charge is additive while Frank-elastic defect interactions are isolated by Mobius pair atoms better than raw pair summaries; an A2-style deep rerun preserved the Q-tensor companion under finite-size, core-mask, and low-noise controls while wrong-parent and variable-parent controls exposed artifacts.

Fixed parent. Fixed 2D nematic director/Q-tensor field with a fixed set of candidate half-disclination sites and charges.

Observable F(S). F(S)=winding/topological charge, analytic one-constant Frank defect energy, or finite-difference Q-tensor elastic energy for the defect subset S in the same parent.

Reproducible witness. Liquid-crystal nematic defects split cleanly: winding/topological charge is additive, while Frank-elastic defect interactions live in Mobius pair atoms. Q-tensor finite-difference energy gives an independent positive companion with discretization caveats.

Technical control. Topological charge must stay additive; elastic interaction must disappear under additive-only/core-only controls; raw pair summaries are confounded by core energies; Q-tensor grid artifacts are not promoted as exact high-order physics.

Caveat / boundary. Strong as a fixed-parent toy/continuum calibration. High noise, polar-vector wrong-parent treatment, and variable-parent resampling create artifacts; real LC image fields, Q-tensor simulation data, or experimental defect trajectories are required for empirical promotion.

{
 "charge_atom_order_mass": {
 "0": 0.0,
 "1": 3.0,
 "2": 0.0,
 "3": 0.0,
 "4": 0.0,
 "5": 0.0,
 "6": 0.0
 },
 "control": "Topological charge must stay additive; elastic interaction must disappear under additive-only/core-only controls; raw pair summaries are confounded by core energies; Q-tensor grid artifacts are not promoted as exact high-order physics.",
 "defect_charges": {
 "0": 0.5,
 "1": -0.5,
 "2": 0.5,
 "3": -0.5,
 "4": 0.5,
 "5": -0.5
 },
 "defect_positions": {
 "0": [
 0.2,
 0.25
 ],
 "1": [
 0.45,
 0.25
 ],
 "2": [
 0.75,
 0.25
 ],
 "3": [
 0.28,
 0.72
 ],
 "4": [
 0.55,
 0.68
 ],
 "5": [
 0.82,
 0.72
 ]
 },
 "frank_high_order_mass": 0.0,
 "frank_pair_atom_max_recovery_error": 0.0,
 "id": "MOB-16",
 "mobius_pair_atom_vs_kernel_corr": 1.0,
 "q_tensor_high_order_mass_grid_artifact": 0.4285138255,
 "q_tensor_pair_atom_vs_frank_kernel_corr": 0.8606541999,
 "random_label_auc_mean_seed123_300": 0.4890178571,
 "random_label_auc_sd": 0.1519249449,
 "raw_pair_summary_vs_kernel_corr": 0.2758708541,
 "top_frank_pair_atoms": [
 {
 "S": [
 0,
 1
 ],
 "atom": -0.6931471806
 },
 {
 "S": [
 3,
 4
 ],
 "atom": -0.6492390384
 },
 {
 "S": [
 4,
 5
 ],
 "atom": -0.6492390384
 },
 {
 "S": [
 1,
 2
 ],
 "atom": -0.6019864022
 },
 {
 "S": [
 1,
 4
 ],
 "atom": -0.4088171681
 },
 {
 "S": [
 2,
 4
 ],
 "atom": 0.373024855
 },
 {
 "S": [
 2,
 5
 ],
 "atom": -0.3720264068
 },
 {
 "S": [
 0,
 3
 ],
 "atom": -0.370371137
 },
 {
 "S": [
 1,
 3
 ],
 "atom": 0.3467736703
 },
 {
 "S": [
 3,
 5
 ],
 "atom": 0.3080930697
 }
 ],
 "top_q_tensor_pair_atoms": [
 {
 "S": [
 3,
 4
 ],
 "atom": -2.2292027886
 },
 {
 "S": [
 0,
 1
 ],
 "atom": -2.2206294058
 },
 {
 "S": [
 4,
 5
 ],
 "atom": -2.0158651603
 },
 {
 "S": [
 1,
 2
 ],
 "atom": -1.7884107136
 },
 {
 "S": [
 1,
 4
 ],
 "atom": -0.8632587915
 },
 {
 "S": [
 0,
 5
 ],
 "atom": 0.6418637003
 },
 {
 "S": [
 2,
 4
 ],
 "atom": 0.5922324397
 },
 {
 "S": [
 0,
 3
 ],
 "atom": -0.4880733906
 },
 {
 "S": [
 2,
 5
 ],
 "atom": -0.4784455826
 },
 {
 "S": [
 1,
 3
 ],
 "atom": 0.4373178448
 }
 ],
 "winding_charge_max_error": 0.0,
 "witness": "Liquid-crystal nematic defects split cleanly: winding/topological charge is additive, while Frank-elastic defect interactions live in Mobius pair atoms. Q-tensor finite-difference energy gives an independent positive companion with discretization caveats."
}

MOB-17 — Chemical many-body expansion and molecular synergy atoms

STRONG_KEEP

Claim. Fixed-parent molecular many-body expansion is exactly a Mobius decomposition of a cluster-energy observable; pairwise controls kill higher-order atoms while cooperative terms survive.

Fixed parent. Fixed monomer/fragment set with one geometry, one basis convention, and one subset rule.

Observable F(S). F(S)=cluster, interaction, or model energy for fragment subset S under the same parent convention.

Reproducible witness. Chemical many-body expansion is a fixed-parent Mobius decomposition: planted singleton, pair, and three-body nonadditive energies are recovered exactly, while the pairwise-only control has zero three-body atoms.

Technical control. The monomer set, geometry, and basis convention must stay fixed. Reoptimizing fragments or changing basis per subset is a variable-parent artifact.

Caveat / boundary. Strong as a fixed-parent model and public-method mapping. Real chemistry claims require ab-initio or measured molecular data and careful BSSE/basis treatment.

{
 "control": "The monomer set, geometry, and basis convention must stay fixed. Reoptimizing fragments or changing basis per subset is a variable-parent artifact.",
 "id": "MOB-17",
 "many_body_three_body_mass": 0.9,
 "max_planted_recovery_error": 0.0,
 "pairwise_control_high_order_max": 0.0,
 "parent_monomers": [
 0,
 1,
 2,
 3
 ],
 "planted_pair_terms": {
 "(0, 1)": -0.45,
 "(1, 2)": -0.35,
 "(2, 3)": -0.25
 },
 "planted_triple_terms": {
 "(0, 1, 2)": -0.62,
 "(1, 2, 3)": 0.28
 },
 "top_atoms": [
 {
 "S": [
 0,
 1,
 2
 ],
 "atom": -0.62
 },
 {
 "S": [
 0,
 1
 ],
 "atom": -0.45
 },
 {
 "S": [
 1,
 2
 ],
 "atom": -0.35
 },
 {
 "S": [
 1,
 2,
 3
 ],
 "atom": 0.28
 },
 {
 "S": [
 2,
 3
 ],
 "atom": -0.25
 },
 {
 "S": [
 1
 ],
 "atom": -0.2
 },
 {
 "S": [
 2
 ],
 "atom": -0.15
 },
 {
 "S": [
 0
 ],
 "atom": -0.1
 },
 {
 "S": [
 3
 ],
 "atom": -0.05
 }
 ],
 "witness": "Chemical many-body expansion is a fixed-parent Mobius decomposition: planted singleton, pair, and three-body nonadditive energies are recovered exactly, while the pairwise-only control has zero three-body atoms."
}

MOB-18 — Fracton-elasticity tensor-gauge mobility as Mobius synergy

MIXED_HIGH_POTENTIAL

Claim. In a fixed constrained-mobility toy, dipole conservation makes isolated charges immobile while a neutral dipole can move; this appears as a Mobius pair-synergy witness aligned with the public fracton-elasticity duality.

Fixed parent. Fixed finite lattice with one positive and one negative scalar-charge/fracton-like defect, fixed charge positions, and fixed allowed unit translations.

Observable F(S). F(S)=number of allowed unit translations for defect subset S under charge and dipole conservation.

Reproducible witness. A minimal scalar-charge/tensor-gauge mobility constraint appears as Mobius synergy: isolated charges have zero dipole-conserving moves while the neutral dipole can translate as a bound pair.

Technical control. If dipole conservation is removed, singleton motion returns and the fracton-style mobility constraint is no longer the same signal. This is a toy mobility witness, not a derivation of the public duality.

Caveat / boundary. The public fracton-elasticity duality is known. The Atlas contribution is the fixed-parent Mobius mobility-synergy diagnostic; it needs deeper field-theory and lattice-model validation before promotion to STRONG_KEEP.

{
 "base_dipole": [
 -2,
 0
 ],
 "charge_only_control_F": {
 "('+', '-')": 4.0,
 "('+',)": 4.0,
 "('-',)": 4.0,
 "()": 4.0
 },
 "charge_only_control_atoms": [
 {
 "S": [],
 "atom": 4.0
 }
 ],
 "charges": {
 "+": 1,
 "-": -1
 },
 "control": "If dipole conservation is removed, singleton motion returns and the fracton-style mobility constraint is no longer the same signal. This is a toy mobility witness, not a derivation of the public duality.",
 "dipole_allowed_translation_count": 4,
 "dipole_conserving_F": {
 "('+', '-')": 4.0,
 "('+',)": 0.0,
 "('-',)": 0.0,
 "()": 4.0
 },
 "dipole_conserving_atoms": [
 {
 "S": [
 "+",
 "-"
 ],
 "atom": 8.0
 },
 {
 "S": [],
 "atom": 4.0
 },
 {
 "S": [
 "+"
 ],
 "atom": -4.0
 },
 {
 "S": [
 "-"
 ],
 "atom": -4.0
 }
 ],
 "id": "MOB-18",
 "isolated_fracton_allowed_moves": [
 0,
 0
 ],
 "lattice_size": 5,
 "positions": {
 "+": [
 1,
 2
 ],
 "-": [
 3,
 2
 ]
 },
 "witness": "A minimal scalar-charge/tensor-gauge mobility constraint appears as Mobius synergy: isolated charges have zero dipole-conserving moves while the neutral dipole can translate as a bound pair."
}

Registered entries

Recorded in the register with an audit status. Witness cards are being rebuilt to the same standard as the cards above before release; the status states what each still needs.

Scope

This atlas is a methodology contribution. It shows that one decomposition, applied with a fixed parent and honest controls, exposes interaction structure across domains normally studied separately — quantum error correction, holographic capacity, cooperative game theory and machine-learning attribution, percolation and topology, causal sets, information decomposition, genetic epistasis, alloy cluster expansion, reliability networks and cascade dynamics.

Möbius inversion itself is public mathematics. What is carried here is the disciplined cross-domain application: one fixed parent, declared observables, and controls run before interpretation.