research/CLAIM_LEDGER.md

# ÆQORIA — claim and evidence ledger

All page numbers below are the printed mathematical page labels in the PDF, not the physical PDF page index.
“Proof supplied” is a manuscript status, not independent peer review or historical-priority certification.

| Result | Location | Proof/evidence boundary |
|---|---|---|
| Sharp observation law | 1.2; p. 2 | Main synthesis; proofs distributed across the geometric, frame, and statistical chapters. |
| Fixed-soul quotient metric | 2.2; p. 5 | Explicit minimization proof; numerical quotient-minimization checks. Quotient curvature uses O'Neill. |
| Full normal holonomy | 2.4; p. 6 | Curvature span plus the Ambrose–Singer theorem; qualitative group does not imply quantitative conditioning. |
| Uniform quadratic transport bound | 3.2; p. 9 | Two pure-gauge endpoints and the second parameter derivative; valid for all loop lengths. |
| Uniform weak-parameter area expansion | 3.3; p. 9 | Analytic remainder carries the factor lambda(1-lambda); not inferred from a fixed-parameter expansion. |
| Exact circle writer | 4.1; p. 13 | Exact ODE gauge reduction; independent numerical integration checks. Finite-length global optimality is not asserted. |
| Sharp scalar constant | 4.2; p. 14 | Area upper bound and constructive circle lower bound; uniform as either parameter endpoint is approached. |
| Cap-frame optimum and rigidity | 5.1; p. 15 | Elementary trace and positive-semidefinite equality arguments; related tight-frame framework is established. |
| Determinant and inverse-trace optima | 5.2; p. 16 | Schur-complement optimization; same simplex-rim optimizer. |
| Cap-optimal exterior sensor | 5.4; p. 17 | Pair singular-value products and transverse trace bound; restricted to d >= 3. |
| Exact scalar-resource bridge | 6.2; p. 19 | Compact endpoint set, stabilizer action, and cap optimizer. Does not assume entire cap reachability. |
| No uniform conditioning floor | 6.4; p. 20 | Explicit same-soul family at fixed finite loop budget. Does not exclude bounds using positive normal-curvature nondegeneracy. |
| Sharp deterministic inverse radius | 7.1; p. 22 | Two indistinguishable least-singular-direction targets; calibrated exact frame and bounded aggregate error. |
| Finite scalar-history realization | 8.2; p. 25 | Direct Gram algebra on the cyclic subspace; classical realization framework, included as a self-contained interface. |
| Noisy Gram calibration bound | 9.1; p. 27 | Common polar-gauge alignment and relative positive-matrix perturbation; numerically checked. |
| Exact adaptive Gaussian risk | 10.1; p. 30 | Explicit hypothesis-independent comparison kernel; fixed n, simple hypotheses, independent reset noise. |
| Sharp sample-length-noise constant | 10.2; p. 31 | Exact Gaussian risk combined with sharp geometric scalar asymptotic; integer rounding kept separate. |
| Multiplicative finite-radius calibration | 11.2; p. 34 | Totally geodesic fibers and uniform local convex neighborhoods; detector noise remains a separate assumption. |

## Verification status

142/142 listed checks passed: 10 exact-symbolic, 7 finite algebra, 116 numerical, and 9 high-precision checks. The algebra category includes floating-point checks.

The exact symbolic tests establish selected identities. Numerical tests support selected finite instances, including metric minimization, circle integration, frame and sensor spectra, noisy calibration, and scalar realization. None exhausts all loops or replaces the general analytic proofs.

## Priority boundary

The proposed contribution is the specific quantitative integration of a fixed-soul geometric family, sharp local scalar control, exact resource identities, and the declared statistical experiment. Classical soul theory, holonomy, loop-length norms, tight-frame methods, realization theory, and comparison of experiments are acknowledged. Worldwide priority has not been certified.

## Completion boundary

The manuscript, source files, reference code, test record, examples, and compact theorem memory are delivered. Independent review, proof-assistant verification, physical experiments, arbitrary finite-length circle optimality, and general unknown-manifold reconstruction are not claimed.