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Constants Ledger (One-Page Map)

Research · Research Note

Constants Ledger (One-Page Map)

Main Paper (PDF)Proof Appendix (PDF)Rigidity and ReconstructionResearch Library

This page is a single entry point for the constants that appear in the structural chain and in the certification pipeline.

The goal is that a referee (or your own future self) can read one table and know:

  • what each constant means,
  • what it depends on,
  • where it is produced by tools,
  • where it lives in payloads.

A. Primitivity, mixing, and connector bounds

ConstantDefinitionDepends onProduced byStored in
\(L_{\mathrm{mix}}\)primitivity mixing time (a length where all vertex-to-vertex walks exist with positive weight support)base graph \(G\)compute_mixing_time.py (or pipeline stage)pa.json
\(L_0\)fixed-length closing bound for BCB connectors\(L_{\mathrm{mix}}\), graph sizerun_pipeline.py (PA stage)pa.json
\(\beta_{\min},\beta_{\max}\)exact min/max positive entry bounds for \(P(u_*)\)\(u_*\), templatesrun_pipeline.py (Doeblin stage)doeblin.json

B. Doeblin block and projective contraction

ConstantDefinitionDepends onProduced byStored in
\(B\)\(B=P(u_*)\) (Doeblin block matrix)\(u_*\), templatesrun_pipeline.py (Doeblin stage)doeblin.json (B_matrix)
\(\eta\)simplex coordinate floor after applying \(B\) and renormalizing (safe lower bound)\(\beta_{\min},\beta_{\max},d\)run_pipeline.py (Doeblin stage)doeblin.json (eta_coord_lower_bound)
\(\Delta\)Hilbert projective diameter upper bound for \(B\)\(\beta_{\max}/\beta_{\min}\)run_pipeline.py (Doeblin stage)doeblin.json (hilbert_diameter_upper_bound)
\(\tau\)Birkhoff contraction upper bound for \(B\) (projective contraction coefficient)\(\Delta\)run_pipeline.py (Doeblin stage)doeblin.json (birkhoff_contraction_upper_bound)

C. Inequality engine, pruning, and explicit gap

ConstantDefinitionDepends onProduced byStored in
\(c_{\mathrm{target}}\)target multiplicative separation used in edge-augmented inequalitieschosen parameterrun_prune_loop.pyg_min.json, prune_loop_payload.json
\(\Lambda_{\mathrm{up}}\)certified periodic upper bound on the minimizing exponent (computed from nucleus word)nucleus word \(w_*\), templatescompute_lambda_upper.py (Collatz–Wielandt bracket)embedded in g_min.json (lambda_upper, details)
\(\varepsilon_{\mathrm{gap}}\)explicit slack parameter used to keep strictness robust under floating tolerancechosen parameterrun_prune_loop.pyg_min.json
\(g_{\min}\)explicit additive lower bound for the strict cycle gap on the pruned complement\(c_{\mathrm{target}},\Lambda_{\mathrm{up}},\varepsilon_{\mathrm{gap}}\)run_prune_loop.pyg_min.json (g_min_gap)

D. Regime classification and return selection

Constant / TagDefinitionDepends onProduced byStored in
regime_tagP (periodic-core) or A (aperiodic-core), used to route which claims are possiblenucleus SCC structurecheck_scg_regime.py (or pipeline stage)scg_regime.json
obstruction_testPASS means no aperiodic-core obstruction detected; FAIL is expected in Regime A examples and signals the obstruction is presentsame as abovecheck_scg_regime.py (or pipeline stage)scg_regime.json
return boundbounded-gap return to \(u_*\) within the kept edge set, used by selection principlekept edges, \(u_*\)run_pipeline.py (PR return stage)pr_return.json

E. Where to look first

  • If a verifier fails, check the payload file named in the verifier output and confirm the constant exists with the expected key.
  • If a constant exists but does not agree numerically, check the tool that produced it and confirm it is using the same word/edge set.
  • If a constant is missing, the correct fix is: update the producing tool to emit it and update the schema doc to declare it.

Books by Drew Higgins