Paper 06Canonical2026-07-15

Non-Source Projection and Internal Identifiability

Exact criteria for non-source projection, dynamical descent, and internal identifiability

DOI: 10.5281/zenodo.21371451 View on Zenodo

Authority role

Integrates the sequence into the non-source projection theorem: an essential non-gauge distinction inside a readout fiber proves non-source projection for the stated model and target, with exact deterministic and statistical identifiability results.

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 a non-source projection for the stated model and target. 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 theorem gives the exact specification a physical model must instantiate: source domain, physical equivalence, readout, target, and essential witness pair.

Notes

Reading notes

The integrating paper of the mathematical sequence proves the non-source projection 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, forming 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. Thus, "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. Representing the missing dependence therefore requires an additional construction, such as a completion variable, a distribution, or a memory representation.

Two identifiability results 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 whose 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