Datasets:

PureOne
/

EVE-Obstruction-Tomography

# 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.

	# 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.