CLAIM_LEDGER.md

Claim ledger

This ledger separates mathematical derivation, numerical checking, historical originality, and physical validation. None substitutes for another.

ID	Exact claim	Location	Evidence	Boundary
C1	A finite mixed-clock cost frame identifies an observable linear model and its preparation metric up to similarity.	Theorem 5; Lemma 4; Propositions 2-3	Full proof; 30 synthetic reconstructions; rational finite-shift certificate	Dimension bound; exact quadratic full preparation; ideal real thresholds and clocks
C2	All singleton limits can coincide while joint constraints differ; unbounded frequencies create arbitrarily close aliases.	Proposition 7; Theorem 8	Explicit rotations; branch and near-alias datasets	Different information interfaces, not equally informed algorithms
C3	Kernel, rank, transition, local generator, and prediction errors admit explicit bounds.	Proposition 9; Corollary 10	Full proof; 120 noise cases	Rank gap, calibrated support errors, principal-log branch and local radius
C4	Principal-cost exponents equal twice the observability depths.	Theorem 11	Full factorization proof; two chain designs	Fixed observable model and metric; distinct scaled times; asymptotic limit
C5a	Detecting the last state of an n-chain requires squared-support precision of power epsilon^(2(n-1)).	Theorem 12	Matching exponent upper/lower arguments	Fixed chain pair, normalized directions, model-dependent constants
C5b	Under independent Gaussian support noise, the two-hypothesis query count has power epsilon^(-4(n-1)).	Theorem 13	KL lower bound and exact Gaussian test; 96,000 trials	Known chain pair; fixed variance and support-oracle assumptions
B1	Full joint feasible subspace already determines dynamics.	Proposition 6	Full proof	Cost values are not essential to exact generator recovery in this class
E1	Cost reconstruction matches a direct exact-kernel baseline.	Experiment section; exact_reconstruction.csv	Numerical tie to rounding accuracy	No acquisition-efficiency or performance advantage
N1	First theorem of its kind in history	No supporting evidence	Unestablished	Do not claim
P1	New physical law or new physical phenomena	No physical validation	Unestablished	The model is stipulated
G1	Arbitrary physical-model reconstruction	Open program	Not proved	No universal theorem

Theorem numbers include the initial definition and propositions. “Full proof” means a written derivation, not proof-assistant verification or external review.

Proof review performed

Checked these potential failure points explicitly:

1. A mixed-clock rank plateau implies inverse-clock invariance by finite-dimensional injectivity.
2. Two irrationally related exponentials determine the continuous group by density, including repeated eigenvalues.
3. The limiting reconstruction is not advertised as a stable finite-precision algorithm.
4. Cost data identify a metric and similarity class, not hidden factors or named microscopic variables.
5. A near-identity clock map alone does not identify the logarithm branch.
6. The short-time leading matrix is full rank by polynomial interpolation and injectivity on each filtration increment.
7. Principal task eigenvectors can change with time scale.
8. The deterministic midpoint oracle works for every normalized query, including adaptive queries.
9. The Gaussian lower bound handles adaptive histories through the chain rule.
10. All rank-gap and model-dependent-constant assumptions remain explicit.

This is internal review. External mathematical review remains needed before treating the claims as independently validated.

Version 2 extension

ID	Claim	Location	Evidence and boundary
C6	All-budget joint linear probes can fail to determine even observable state dimension	Proposition 14	Exact circle/disk construction; standard convexification fact
C7	Quadratic mismatch graphs determine a compact nonlinear flow and cost up to conjugacy	Theorem 15	Written proof; assumes embedding, full preparation and infinite exact data
C8a	Finite-grid Hausdorff bound and sharp square-root noise exponent	Theorem 16	Matching exponent bounds; six grid cases; disk/annulus indistinguishability
C8b	Graph geometry controls map error on an aligned domain	Proposition 17	Written elementary Lipschitz argument
C8c	Anchored signed losses recover the next state and bound drift error	Proposition 18	Written proof; 76,800 losses; exact anchors and shared timing assumption
C9	The oscillator has an explicit delay embedding and no finite linear realization with linear readout	Proposition 19	Strict derivative and Vandermonde arguments; separate ODE integration check
B2	An information-equivalent conditional-response observer ties the quadratic readout	Nonlinear discussion	Losses algebraically encode the response; no information advantage

Internal review checked: compact zero sets; single-valued graph through embedding; complete-flow inverses; dense signed times; cost recovery from nested sublevels; both directed grid distances; all-center midpoint lower bound; common-domain requirement; correct factor four in loss differences; shared timing; monotone delay coordinate; and the explicit linear-readout restriction.

The six new statements have written proofs. No general theorem has proof-assistant verification or external peer review. Their historical originality remains unestablished.