Paper 06Canonical2026-07-15

Non-Source Projection and Internal Identifiability

The boundary theorem: when an observable tier is provably a non-source projection, and what internal observation can identify

DOI: 10.5281/zenodo.21371451 View on Zenodo

Authority role

Integrates the sequence into a boundary theorem: any model exhibiting an essential non-gauge distinction inside a readout fiber is, relative to that model and target, a non-source projection — with exact deterministic and statistical ceilings on what an internal observer can ever identify.

Summary

Proves that once a model exhibits two physically inequivalent source states sharing the same readout but demanding different answers to a nominated target, that readout is — relative to the model and target — a non-source projection. The dynamical consequences divide sharply: fiber-preserving evolution descends to an autonomous law even when the readout is noninjective, so dynamical closure is evidence of fiber preservation, not fiber triviality; fiber-breaking evolution admits no deterministic instantaneous law. Under explicit factorization hypotheses, no internal probe, statistic, or test at any sample size distinguishes fiber members, and a strong lumpability theorem gives the exact condition for a projected Markov process to remain Markov. The mathematics does not supply a source domain or witness: those must be instantiated by a physical model, and the boundary is marked explicitly.

Notes

Reading notes

The integrating paper of the mathematical sequence — and deliberately a boundary theorem. It works with a supplied quadruple: a source domain, a declared physical equivalence (quotiented out first, so fiber multiplicity is never gauge residue), an exact readout $p: S \to T_1$ , and a nominated target with correct-answer map $a_Q$ .

Theorem — Non-source projection

If two source states share a readout but have different correct answers to the target — an essential fiber distinction — then the readout tier is, relative to that model and target, a non-source projection: it remains the exact quotient $S/\!\sim_p$ , yet no deterministic function of readout data answers the target correctly on all of $S$ , the minimal target completion properly refines $T_1$ , and no gauge or coordinate change removes the distinction.

The dynamical consequences divide sharply. Fiber-preserving evolution descends to a unique autonomous law on $T_1$ even when the readout is noninjective — so "dynamical closure is evidence of fiber preservation, not of fiber triviality," the paper's single most important negative result. Fiber-breaking evolution admits no deterministic law on the instantaneous readout; and the failure establishes only a negative — a completion variable, a distribution, or a memory representation each requires an additional construction.

Two identifiability ceilings follow from two separate hypotheses. If every admissible internal probe factors through the readout, no probe value, derived statistic, or deterministic selector distinguishes members of a common fiber (this is an identifiability theorem — the mechanism is equality of inputs, not Gödelian self-reference). If complete outcome laws factor through the readout, no statistical test at any sample size discriminates either. The finite-state stochastic counterpart is a strong lumpability theorem: the projected Markov process remains Markov for every initial distribution exactly under the block-sum condition on the generator.

Cite this paper

Rodgers, Jeremy. (2026). Non-Source Projection and Internal Identifiability. https://doi.org/10.5281/zenodo.21371451