AILAND: topological invariants in protein folding and visuo-postural reset

Ideas
, ,
1

Within the H_0 framework, complex dynamical systems — whether polypeptide chains or neuromuscular feedback loops — are represented as high-dimensional hypergraphs. Structural relaxation corresponds to a topological contraction toward a spectral invariant, computed via dual-head spectral projection without relying on 3D spatial coordinates.

{
“Protocol”: “H0_Bi_Domain_Spectral_Invariant_v1”,
“Timestamp”: “2026-09-22T18:00:00Z”,
“Domain_A_TendonReset”: {
“TargetID”: “0x54454E444F4E”,
“Nodes”: 12,
“Hyperedges”: 5,
“SpectralEntropy_Stop”: 2.2210,
“FiedlerValue”: 3.6180,
“DominantVector”: [-0.372, -0.602, 0.000, -0.602, -0.372]
},
“Domain_B_ProteinFolding”: {
“TargetID”: “0x41494C414E44”,
“Nodes”: 16,
“Hyperedges”: 5,
“SpectralEntropy_Stop”: 2.2570,
“FiedlerValue”: 4.0000,
“DominantVector”: [0.000, 0.707, 0.000, 0.707, 0.000]
}
}

ClearAll[“Global`*”];

(* 1. Target #TENDON (Visuo-Postural Reset) *)
nodiT = Table[“T_” <> ToString[i], {i, 12}];
archiT = {{“T_1”,“T_2”,“T_4”}, {“T_3”,“T_4”,“T_7”}, {“T_5”,“T_6”,“T_9”}, {“T_7”,“T_8”,“T_10”}, {“T_10”,“T_11”,“T_12”}};
MTendine = Table[If[MemberQ[archiT[[j]], nodiT[[i]]], 1, 0], {i, 12}, {j, 5}];
LTendon = Transpose[MTendine] . MTendine;
{valsT, vecsT} = Eigensystem[N[LTendon]];
pT = Abs[valsT] / Total[Abs[valsT]];
StopTendine = -Total[Map[If[# > 0, # Log2[#], 0] &, pT]];
Print["Target 0x54454E444F4E | S_top: ", NumberForm[StopTendine, {4, 4}], " nats | Fiedler: ", NumberForm[Abs[valsT[[2]]], {4, 4}]];

(* 2. Target #014 (Protein Folding) *)
nodiP = Table[“V_” <> ToString[i], {i, 16}];
archiP = {{“V_1”,“V_2”,“V_5”}, {“V_3”,“V_4”,“V_8”}, {“V_5”,“V_6”,“V_12”}, {“V_7”,“V_8”,“V_14”}, {“V_9”,“V_10”,“V_16”}};
MProteina = Table[If[MemberQ[archiP[[j]], nodiP[[i]]], 1, 0], {i, 16}, {j, 5}];
LE = Transpose[MProteina] . MProteina;
{valsP, vecsP} = Eigensystem[N[LE]];
pP = Abs[valsP] / Total[Abs[valsP]];
StopProteina = -Total[Map[If[# > 0, # Log2[#], 0] &, pP]];
Print["Target 0x41494C414E44 | S_top: ", NumberForm[StopProteina, {4, 4}], " nats | Fiedler: ", NumberForm[Abs[valsP[[2]]], {4, 4}]];
2

Technical correction and clarification — AILAND v1.1 (9 October 2026)

I have independently reviewed the calculations presented in my original post and would like to clarify several mathematical and interpretative points.

1. Matrix interpretation

The matrix (G=M^T M) is the Gram matrix of the hyperedge incidence matrix, not the standard graph Laplacian. Its second-largest eigenvalue must therefore not be identified with the Fiedler eigenvalue.

2. Corrected numerical results

The following values have been independently recalculated and the principal numerical results checked in Wolfram Cloud:

Descriptor Example A Example B
Spectral entropy (bits) 2.221027008 2.256564762
Largest Gram eigenvalue 4.618033989 4
Second-largest Gram eigenvalue 3.618033989 4
Multiplicity of largest eigenvalue 1 2

The complete descending spectra are:

  • A: {4.618033989, 3.618033989, 3, 2.381966011, 1.381966011}

  • B: {4, 4, 3, 2, 2}

Because the entropy uses Log2, its unit is bits, not nats.

3. Structural limitations

Example B contains three isolated nodes and has a two-dimensional dominant eigenspace. Consequently, an individual dominant eigenvector is not uniquely determined.

The two illustrative hypergraphs are not isomorphic. Their spectral descriptors do not establish topological equivalence or a common biological mechanism.

4. Scope of the research

The present calculations describe static, constructed hypergraphs. They do not yet demonstrate protein-folding dynamics, visuo-postural resetting, relaxation, or causal biological relationships.

These remain research hypotheses requiring independently sourced experimental data and explicit, falsifiable models.

I am preserving the original post as a record of the research development and providing this clarification to distinguish verified calculations from open hypotheses.

Verification status: The principal numerical results have been reproduced in Wolfram Cloud. This does not constitute experimental biological validation. Reproducible code, hypergraph inputs, and verified numerical audit: GitHub - pieroadafedele-oss/ailand-spectral-audit · GitHub