Invariant Preservation Check

Public status boundary. This route is a transformation harness for review. It does not assert accepted physics or proof status.

PROPOSED COMPUTATIONAL MODEL

This page specifies a source-bounded numerical testbed for an authorial research framework. It is not an accepted physical law, a proof of the underlying theory, or experimental validation. Every new equation below is a computational proposal for review. Standard mathematical constructs are identified as such, and failure conditions are part of the model.

Purpose

Invariant Engineering defines a system through its state space, transformations, invariant set, trace function, observer role, and boundary class. This route converts that grammar into a deterministic test harness.

It does not assume that every transformation preserves every quantity. Instead, it asks:

Which invariants are declared for a transformation, what tolerance is allowed, and can the result be reproduced from the trace?

The simulation is domain-general. It may be used for numerical states, graph states, corpus records, or software-state transitions, provided that each invariant is explicitly defined.

Source basis

Canonical source route	Evidence ID	Route	publicText	proposed computational mapping
/02-foundations/invariant-engineering	IPC-S1	/04-mathematics/simulations/invariant-preservation-check	Core invariant families include identity, provenance, transformation, observer, governance, proof/status, software/runtime, and publication/source invariants.	Transformation invariant harness
/02-foundations/observer-monad-theory	IPC-S2	/04-mathematics/simulations/invariant-preservation-check	Observer Monad Theory treats the observer as a law-bearing closure node rather than an external spectator; it should be framed as a formalization target for observer-indexed measurement and coherence, not as a claim that consciousness creates reality.	Transformation invariant harness
/02-foundations/science-of-fabric-reality	IPC-S3-GRAMMAR	/04-mathematics/simulations/invariant-preservation-check	Every physics-facing extension must declare domain, observable, transformation law, invariant, trace, observer, boundary, and failure mode.	Transformation invariant harness
/02-foundations/science-of-fabric-reality	IPC-S3-FAILURE	/04-mathematics/simulations/invariant-preservation-check	Public failure modes include metaphor inflation, physics overreach, proof inflation, project inflation, and totalization.	Transformation invariant harness
/02-foundations/digital-fabrica-theory	IPC-S4	/04-mathematics/simulations/invariant-preservation-check	DFT is an authorial systems architecture for designing digital infrastructure around fabric principles: relation, state, identity, provenance, governance, trace, interoperability, and lawful transformation.	Transformation invariant harness
/09-library/physica-nova	IPC-S5	/04-mathematics/simulations/invariant-preservation-check	PHYSICA should support simulation protocols; a simulation route should include model equations, parameter definitions, discretization choices, stability criteria, error bounds, sensitivity analysis, code version, input data, output data, and reproduction instructions; if numerical claims change after correction, the public page should make the correction visible; proposed extensions should include a standard contact point, recovery limit, new parameter, observable, prediction, error analysis, and failure interpretation.	Transformation invariant harness

Numerical invariant residual

For a scalar invariant (I_j), define

$${\rho }_{j,k}=\frac{|I_j(x_{k+1})-I_j(x_k)|}{s_j+\epsilon },$$

where (s_j) is a declared scale, normally

$$s_j=max(|I_j(x_k)|,s_{j,min}).$$

The invariant passes when

$${\rho }_{j,k}\le {\tau }_j.$$

For vector-valued invariants, replace the absolute value with a declared norm.

For Boolean invariants, record true or false directly rather than manufacturing a numerical residual.

Composite run result

Define the maximum normalized residual

$$R_k=\underset{j\in A(T_k)}{max}\frac{{\rho }_{j,k}}{{\tau }_j},$$

where (A(T_k)) is the set of invariants applicable to transformation (T_k).

The transformation passes when

$$R_k\le 1$$

and all Boolean invariants pass.

This score is a test aggregation rule, not a physical quantity.

Invariant families

The harness supports the current corpus families:

Identity invariants

Examples:

• stable object identifier;
• preserved graph node identity;
• preserved semantic role;
• preserved state dimension when required.

Provenance invariants

Examples:

• source path remains resolvable;
• author and version remain reconstructable;
• content hash matches the recorded artifact;
• transformation has an inspectable parent record.

Transformation invariants

Examples:

• mass or sum preserved by a conservative update;
• norm preserved by an orthogonal transform;
• graph connectivity preserved when required;
• state remains inside an admissible domain.

Observer invariants

Examples:

• reviewer or instrument context declared;
• promotion decision tied to a named gate;
• measurement basis or extraction procedure stored.

Governance and status invariants

Examples:

• no unauthorized status escalation;
• PUBLIC_RECORD does not become PEER_REVIEWED without evidence;
• release operation requires authorization;
• rollback route remains available.

Runtime invariants

Examples:

• no NaN or infinity;
• schema remains valid;
• permissions and anchors remain resolvable;
• build and test result is traceable.

Benchmark transformations

The reference implementation should include:

Transformation	Expected invariant
permutation of a vector	sum and multiset
orthogonal matrix action	Euclidean norm
conservative graph diffusion	weighted sum
reversible serialization	exact content hash after round trip
lossy compression	expected identity or hash failure
unauthorized status mutation	expected governance failure
source deletion	expected provenance failure

Including expected failures is essential. A harness that only tests passing cases cannot demonstrate discrimination.

Property-based protocol

For each registered transformation:

1. generate or load a state satisfying its preconditions;
2. compute pre-transform invariant values;
3. apply the transformation;
4. compute post-transform values;
5. calculate residuals;
6. evaluate Boolean predicates;
7. store the complete trace;
8. reproduce selected failures with a minimized counterexample.

Trace object

A trace should contain:

{
  "runId": "",
  "systemId": "",
  "stateBeforeHash": "",
  "stateAfterHash": "",
  "transformationId": "",
  "transformationVersion": "",
  "parameters": {},
  "seed": 0,
  "observerContext": {},
  "boundaryClass": "",
  "applicableInvariants": [],
  "residuals": {},
  "booleanResults": {},
  "result": "PASS|FAIL|ERROR",
  "counterexample": null
}

Tolerance law

A tolerance must be:

• declared before the run;
• justified by floating-point error, discretization error, measurement uncertainty, or domain policy;
• stored in the trace;
• tested by sensitivity analysis.

Post hoc tolerance inflation is a test failure.

Failure conditions

• invariant_residual_exceeded;
• boolean_invariant_failed;
• precondition_undefined;
• tolerance_missing;
• tolerance_changed_after_run;
• trace_incomplete;
• observer_context_missing;
• status_boundary_violation;
• source_unresolvable;
• counterexample_not_reproducible;
• non_finite_state.

Worked numerical benchmark

Let (x\in\mathbb R^N) and let (Q) be numerically orthogonal.

Transformation:

$$x^{\prime }=Qx.$$

Declared invariant:

$$I(x)=\parallel x{\parallel }_2^2.$$

Residual:

$$\rho =\frac{|\parallel Qx{\parallel }_2^2-\parallel x{\parallel }_2^2|}{\parallel x{\parallel }_2^2+\epsilon }.$$

The test passes when (\rho\le\tau).

A deliberately perturbed matrix

$$\tilde{Q}=Q+\eta A$$

provides a stress case. The harness should detect loss of norm preservation as (\eta) increases.

Structural benchmark

For a publication record, declare:

• title preserved;
• DOI preserved;
• source file resolvable;
• boundary label preserved;
• author changed only when source evidence changes.

A transformation that inserts an author not encoded in the source must fail the provenance invariant.

Acceptance matrix

The page should render a matrix:

Transform	Invariant	Residual or predicate	Tolerance	Result
orthogonal-map	norm	value	value	PASS/FAIL
permutation	sum	value	value	PASS/FAIL
serialization	content hash	true/false	exact	PASS/FAIL
status-mutation	evidence class	true/false	exact	PASS/FAIL

Falsifiability boundary

The harness weakens an invariant claim when:

• a valid counterexample is found;
• preservation depends on hidden state;
• the result changes under reasonable numerical precision;
• the trace cannot reproduce the outcome;
• the declared transformation domain was too broad;
• the tolerance is larger than the effect being claimed.

Passing finite tests does not prove universal invariance. It provides bounded evidence for the tested domain.

Reproducibility

A valid release of the harness requires:

1. a versioned invariant registry;
2. deterministic seeds where applicable;
3. exact transformation versions;
4. machine-readable traces;
5. positive and negative controls;
6. minimized counterexamples;
7. separate numerical and policy tolerances.

See the Researcher Workbench for current formalization status.