The SymC Foundations repository contains the core theoretical works underlying the Symmetry of Criticality (SymC) research program. All titles are a continuous work in progress open to outside scrutiny and perspective.
The current SymC framework is generator-first. It studies stability boundaries, damping, spectral structure, exceptional points, and inheritance mechanisms while requiring each proposed stability coordinate to be derived from the dynamics of the system being studied.
SymC does not currently assert that a single scalar damping ratio universally governs quantum, biological, cosmological, chemical, or informational systems.
Domain-specific applications are maintained separately.
- Repository map:
REPOSITORY_GUIDE.md - Scientific and evidence governance:
GOVERNANCE.md - Contribution and correction policy:
CONTRIBUTING.md - Citation metadata:
CITATION.cff - Security reporting:
SECURITY.md - Community conduct:
CODE_OF_CONDUCT.md - Intended GitHub-hosting settings:
REPOSITORY_SETTINGS.md - Active prospective substrate-inheritance program:
substrate-inheritancebranch
For scientific use, cite the specific paper, dataset, code artifact, DOI, and version whenever available rather than citing the repository generically.
This README and later explicit contracts, reproducibility records, and active investigation records govern the present interpretation of the Foundations program. Historical PDFs, older README language, release notes, and repository metadata may lag the active work and are OUTDATED wherever they conflict with the current generator-first, carrier-resolved, prospective framework. Historical artifacts remain preserved for provenance rather than silently rewritten.
A dedicated substrate-inheritance branch hosts the prospective Foundations-level substrate-inheritance program and its computational validation layer.
The investigation is ongoing and aims to develop computational evidence for or against substrate-inheritance claims. It formally distinguishes substrate influence from substrate inheritance. Influence means a defined substrate property demonstrably changes child or coupled-system behavior. Inheritance additionally requires independently characterized parent structure, explicit carrier correspondence, a parent-to-child mapping frozen before target reveal, successful prospective child prediction, intervention/counterfactual confirmation, and specificity against scrambled or generic controls.
The program retains a coupled modal + scalar + conglomeration description. Scalars such as mechanical χ are used only where the governing dynamics license that reduction and remain attached to identified modes or subspaces. Collective coupling, memory, bath structure, non-normal carrier geometry, and electronic versus phononic channel identity are retained rather than forced into one scalar.
The first nine computational method-validation layers and the fail-closed real-system ingestion layer are closed as successful synthetic/mathematical/software validation. A hardened reference validation at commit 9a1f357e73a27e532c755649568dde8af0b229cd passed 62 adversarial/unit tests in GitHub Actions run 33292821080 under the pinned validation environment. It includes a ten-case ground-truth battery; same-spectrum modal scrambles; coupling rewiring; analytic intervention cross-checks; single-scalar nonidentifiability; coordinate invariance; near-degenerate subspace robustness; finite-bath recurrence; finite-depth versus analytic semi-infinite substrate embedding; weakened-carrier discriminability; non-normal/biorthogonal carrier geometry; a separate electronic block-Hamiltonian inheritance channel; and a provenance-enforcing real-system mechanical adapter.
These computations are not physical inheritance evidence and do not set physical-system thresholds. The active branch freezes CORRESPONDENCE_PROTOCOL_v0.2.json and REAL_SYSTEM_INPUT_SCHEMA_v0.2.json before first real-system inheritance ingestion. The adapter computes carrier and participation information only and explicitly does not assign an inheritance label, damping coefficient, or χ.
Existing Na/Cu work remains development evidence; CO/Cu(111) is reserved for prospective application after its already-frozen upstream chemistry gates; H/Ru(0001) remains a contrast/limit system.
Older documents that imply broader inheritance, universal χ control, or unqualified cross-domain transfer are OUTDATED wherever they conflict with the current generator-first and prospective-inheritance framework. They remain available as research-history artifacts.
| Title | Description | |
|---|---|---|
| SymC Postulate (v3) | Historical foundational statement of the χ = 1 hypothesis and exceptional-point framing. Current use of χ is restricted by the generator and domain-licensing rules described below. | |
| SymC Noughts | Historical introduction of substrate noughts and the inheritance hypothesis. Current inheritance claims are governed by the active prospective program above; broader or less-qualified inheritance language in this artifact is OUTDATED where it conflicts with that program. | |
| Closing Critical Gaps (v3) | Historical particle-sector extension involving electron, quark, and neutrino scales. Current interpretation distinguishes spectral-width ratios from mechanically licensed χ unless an appropriate generator is independently derived. | PDF / Supps |
| SymC AIF | Explores connections between SymC, predictive processing, and active inference. Domain-specific stability claims require their own generator and empirical validation. |
| Title | Description | |
|---|---|---|
| SymC Lindblad (v4) | Develops open-quantum-system and exceptional-point geometry. Current interpretation requires defectiveness of the relevant full generator for an EP claim; simple amplitude damping is not automatically a finite-frequency critical-damping EP. | PDF / Supps |
| SymC and the QFT (v2) | Explores relaxation and width-scale structure in quantum field settings. Such ratios are not automatically identified with mechanical χ without a licensed dynamical reduction. | |
| SymC Neutrinos | Studies matter-induced dephasing effects in neutrino flavor oscillations with exact vacuum unitarity. |
For a passive positive-curvature second-order mode,
[ \ddot q + \gamma \dot q + \kappa q = 0, \qquad \kappa > 0, ]
define
[ \Omega = \sqrt{\kappa}, \qquad \chi = \frac{\gamma}{2\Omega}. ]
The characteristic roots are
-\frac{\gamma}{2} \pm \sqrt{\frac{\gamma^2}{4}-\Omega^2}. ]
Within this licensed dynamical class:
- 0 ≤ χ < 1 corresponds to underdamped oscillatory decay.
- χ = 1 is the critical-damping boundary where the two characteristic roots coalesce.
- χ > 1 corresponds to overdamped monotone decay.
- In the standard companion-matrix realization, the repeated root at χ = 1 is defective and therefore has the algebraic structure of an EP2.
- For fixed restoring scale Ω, χ = 1 gives the fastest asymptotic nonoscillatory relaxation within the critical/overdamped branch.
These statements are exact consequences of the specified second-order generator. They are not automatically transferable to systems with different generators.
The current framework follows a simple principle:
No boundary without a generator; no coordinate without a licensed domain.
Different dynamical classes require different coordinates.
For
[ \ddot q+\gamma\dot q+\Omega^2 q=0, ]
the mechanical damping ratio χ is licensed and χ = 1 is the critical-damping boundary.
For
[ \ddot q+\gamma\dot q-\omega_b^2 q=0, ]
a useful normalized friction coordinate is
[ \alpha_b=\frac{\gamma}{2\omega_b}, ]
but α_b = 1 is not a critical-damping exceptional point. The roots remain distinct with one stable and one unstable direction.
The flat-ΛCDM density-growth equation contains an inverted restoring term. Its normalized balance coordinate may satisfy
[ \alpha_\delta=1 \iff q=0 ]
within flat ΛCDM, but this is a balance/kinematic synchronization, not a mechanical critical-damping EP of the density-growth generator.
Ratios such as
[ \Gamma/(2M) ]
are treated as spectral-width coordinates unless an appropriate mechanical or equivalent dynamical generator is independently demonstrated.
When dissipation contains memory, the characteristic object is generally
[ D(s)=s^2+s\widetilde K(s)+\Omega_0^2. ]
A repeated pole requires
[ D(s_)=0, \qquad D'(s_)=0. ]
The ordinary χ = 1 boundary is recovered only under an appropriate local/Markovian reduction.
An exceptional-point claim requires coalescence and defectiveness of the relevant open-system generator. A scalar second-order rewriting alone does not establish defectiveness of the full microscopic or Liouvillian generator.
The following distinctions are part of the current SymC framework:
- The positive-curvature mechanical χ = 1 boundary is an exact dynamical result.
- The normalized underdamped poles of the canonical scalar generator trace a well-defined stability arc in the complex plane.
- Earlier operator-first inheritance results remain bounded to their registered passive-mechanical test domain; the active substrate-inheritance program is a broader prospective investigation and has not established a universal inheritance law.
- Preferential natural occupancy near χ = 1 remains an open empirical question requiring an independently justified population or control null.
- A universal adaptive band such as 0.8 ≤ χ ≤ 1.0 is not a current general claim.
- A universal maximum of information efficiency near χ = 1 is not a current supported claim for the proxies previously tested.
- Chemical barrier coordinates, cosmological balance coordinates, particle spectral-width ratios, and mechanical χ are not pooled as though they were the same physical observable.
- SymC is not presently asserted as a universal law of nature.
Negative results, retractions, failed prospective criteria, influence-only outcomes, nonidentifiability, and domain restrictions are retained as part of the research record rather than removed or reclassified as successes.
Several PDFs in this repository are versioned research artifacts produced before the current generator-first domain-licensing and prospective-inheritance frameworks were completed. They may therefore contain broader language or hypotheses that have since been narrowed, reclassified, retracted, or superseded.
Such broader language is OUTDATED wherever it conflicts with a later explicit correction, retraction, domain license, reproducibility record, or the active substrate-inheritance contract.
The files are retained unchanged to preserve the research history. Historical statements are not prospective evidence simply because they anticipated a later hypothesis.
- Preserve the foundational SymC research record.
- Maintain stable, versioned paper releases.
- Host supplementary derivations and supporting materials.
- Separate exact mathematical results from hypotheses and empirical claims.
- Record revisions, negative results, and scope corrections transparently.
- Develop prospective computational tests of substrate influence and inheritance.
- Provide a common theoretical base for independently tested domain-level applications.
Zenodo releases: https://zenodo.org/communities/symc-universe/
ResearchGate: https://www.researchgate.net/profile/Nate-Christensen-2
Published stability-architecture work:
Exceptional-point stability boundaries from quantum dissipation to cosmological acceleration
Scientific Reports (2026)
DOI: 10.1038/s41598-026-56887-7
This repository and its contents are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0) unless otherwise stated in a specific file.
Reuse, redistribution, and adaptation are permitted with appropriate attribution. See the LICENSE file for details.
When citing a research work, use the DOI and version associated with that work where available.
Nate Christensen
Independent Researcher
SymC Universe Project