budget concept

Equation

Book equation 11.4.

\[\text{directions resolved}(k)=\begin{cases}1\ \text{or}\ 2,& k<d\\[2pt] \operatorname{rank}P_C,& k\ge d\end{cases}\qquad \text{cost}=2d\ \text{consumer calls per operating point}.\]

Conditions

• The budget is the bits, rows, calls, seconds, or dollars a consumer may spend. It bounds what of the read operator can be measured and used, and changes the operator only when it changes the consumer.
• Comparisons are made at matched budget. A verdict at a fixed budget can invert under budget matching, and the ledger carries the case.
• For a probe the budget is consumer calls, and recovery of the read operator is a cliff at the full dimension of the space.

Conditions are curated in entries.toml rather than read from a record.

Ledger

• measures. GO-4 [replicated]. Fixed-budget verdicts invert under budget-matched observation, per the wavelength mechanism. geometric-observation/claims/LEDGER.md:66.

First stated

Volume 14, chapter 4 for the triple and chapter 7 for cost, geometric-observation/chapters/ch07_cost.md, DOI 10.5281/zenodo.21776291.

Measurements

Failures and corrections

none

Invariance envelope

Survived.

• OD:budget, change of the observation budget B at fixed consumer and output metric. Claim: A single-direction probe along coordinate e_i at size rho is reported identifiable at budget B exactly when P_ii > B^2 / rho^2, and a ladder of sizes brackets P_ii between B^2 / r_hi^2 and B^2 / r_lo^2 (article Theorem 2(b), Corollary 3). experiments/DISCOVERY-TRACK.md, D1 record; experiments/OD/D1/grade.json axis. Witness: exact in every evaluator of all fifteen World A cells (three worlds, five budgets, 20 evaluators each) on the run seed, as in every pilot. Absorbed by: .
• OD:budget, change of the observation budget B at fixed consumer and output metric. Claim: Where lambda_min rho^2 < B^2 < lambda_max rho^2 fails the oracle is constant in every direction; where it holds it need not be non-constant in finite samples. experiments/OD/D1/grade.json, B2 entries. Witness: constant in all 20 evaluators at n = 5, B = 2 (no crossing, B^2 = lambda_max rho^2) and at n = 8 positive definite, B = 0.25 (a crossing by theory so thin that 960 random directions in each of 20 evaluators all answered distinguishable). Absorbed by: .
• OD:observer-family, change of observer within a declared family: read operators of declared spectrum, coordinate subsets, coarsenings, and learned consumers’ recovered read operators. Claim: For a positive definite read operator with spectrum in [a, b] the observational and classical window exponents differ by at most log(b / a) / (2 (T - t0)) (article Proposition 1, lean/GET/Horizon.lean). experiments/DISCOVERY-TRACK.md, D3 record; experiments/OD/D3/grade.json exact E1. Witness: 3,200 of 3,200 windows on Lorenz-63 and Lorenz-96 under an anisotropic observer with b / a = 16 and 4; the worst ratio to the bound 0.96 in the pilot, so the bound is nearly attained. Absorbed by: .
• OD:budget, change of the observation budget B at fixed consumer and output metric. Claim: The horizon T_O(B) is non-decreasing in B, lies between the Euclidean horizons at B / sqrt(b) and B / sqrt(a) for a positive definite observer, and is never earlier than the Euclidean horizon for a projection. experiments/OD/D3/grade.json exact H1. Witness: 8,960 of 8,960 checks. Absorbed by: .
• OD:observer-family, change of observer within a declared family: read operators of declared spectrum, coordinate subsets, coarsenings, and learned consumers’ recovered read operators. Claim: An observer’s horizon offset from the full reader at budget B equals the first-passage time Delta_O(B) = E_s[inf{t >= s : L(t) - L(s) + g_O(t) >= log B} - inf{t >= s : L(t) - L(s) >= log B}] of its read-fraction process along the leading Lyapunov direction, computed on an independent long trajectory (the horizon offset law). experiments/DISCOVERY-TRACK.md, D3v2 record; experiments/OD/D3v2/grade.json. Witness: all 22 cells (eleven observers of Lorenz-63 and Lorenz-96 at B = 1000 and 10000) within 0.259 time units, largest error 0.149; signs and the Lorenz-63 order exact; the symmetric site blocks of Lorenz-96 equal within 0.07; anisotropic offsets inside the D3 bracket; the constant-fraction approximation -E[log f_O] / lambda_1, refuted by the first probe, wrong by 0.09 to 1.0 on the same rows. Absorbed by: . Revision: the law replaced the constant-fraction statement before any pilot; recorded in the registration’s Section 7.
• OD:simulation-resolution, change of the simulation resolution N at fixed observation budget B, for dynamical systems. Claim: The horizon offset law holds across a change of dynamical system, from Lorenz-63 (exponent 0.906) to Lorenz-96 at N = 40 (exponent 1.699). experiments/OD/D3v2/grade.json O1 by world. Witness: Lorenz-96 errors 0.001 to 0.138 over four observers and two budgets, Lorenz-63 0.014 to 0.149 over seven. Absorbed by: .
• OD:observer-family, change of observer within a declared family: read operators of declared spectrum, coordinate subsets, coarsenings, and learned consumers’ recovered read operators. Claim: Second version, the rehabilitation of D1: mixed probes at one radius, where the sphere crosses the ellipsoid substantially, recover the read operator up to the pencil s P + (1 - s)(B^2 / rho^2) I, below-threshold eigenvalues included, with Frobenius, above-threshold and below-threshold medians at most REC = 0.39 of chance at 960 queries, REC fixed before the pilot as 1.5 times the pilot’s largest graded ratio. experiments/DISCOVERY-TRACK.md, D1v2 record; experiments/OD/D1v2/grade.json. Witness: all six well-crossed cells on a second fresh seed: Frobenius 0.045 to 0.212 of chance, above-threshold 0.049 to 0.303, below-threshold 0.022 to 0.146; single-direction verdicts and brackets exact in all fifteen World A cells; the pencil’s end nearer than the truth in both positive definite cells; the cell that decided D1 at 0.303 again, 0.087 under the bar. Absorbed by: . Revision: none; the 0.30 of chance for the above-threshold eigenvalues at 960 queries in the positive definite world is a measured property of the analytic-centre estimator, stable across seeds.
• OD:closure-geometry, . Claim: Third version: for a filtered two-dimensional flow with the spectral cutoff as consumer and budget, at closure rank 64 and above, the subfilter modes of largest read distortion (sensitivity times energy, the read operator probed blind on the solver) close the resolved tendency better than the same number of energy-ranked modes, pooled at every in-scope rank by a declared margin, behind in no cell by more than a declared tolerance, with a smaller margin below the scope and a margin stable between n = 128 and 192. experiments/DISCOVERY-TRACK.md, D7v3 record; experiments/OD/D7v3/grade.json. Witness: 48 in-scope rank cells on three fresh fields at n = 128 and 192: the read-distortion closure ahead of the energy closure pooled by 2.8 percent at rank 64 and 10.0 at rank 128 (bar 2), behind in 6 cells by at most 3.6 (bar 5), ahead by -0.2 and 0.4 percent below the scope; the in-scope margin 6.3 percent at n = 128 and 6.6 at n = 192 (bar 2.0 on the change); every structured closure below random; the eigen-direction closure behind energy in 40 of 48 cells; the read operator’s effective rank the same at both resolutions. Absorbed by: . Revision: none; the declared arm of D7 (the leading eigen-directions) is refuted in all three registrations and the read distortion is the ranking quantity inside the rank scope.
• OD:observer-choice, . Claim: Third version: for a linear system with candidate sensors, the placement reaching a required count of directions identifiable at budget B on the exact window Gramian with the fewest sensors, found exactly where the exhaustive search is checkable (ties among smallest sets broken by the m-th eigenvalue) and by a pruned greedy beyond, with feasibility decided at the sensor cap, never needs more sensors than the energy ranking, a random ordering or D6’s greedy, needs strictly fewer than the energy ranking in a registered fraction of the cells where placement matters, and leaves a longer forecast horizon than the energy placement at the same count. experiments/DISCOVERY-TRACK.md, D6v3 record; experiments/OD/D6v3/grade.json. Witness: 54 feasible cells on five worlds with fresh seeds (one count proved unreachable under the cap of 12 and recorded): the selector equal to the exhaustive minimum in all 50 checkable cells; strictly fewer sensors than the energy ranking in 10 of 28 discriminating cells (bar 0.25) and never more; never more than a random ordering; never more than D6’s greedy and fewer in six cells; the horizon 12.3 percent longer than the energy placement’s pooled on 14 discriminating growth cells (bar 10) and 19.5 longer than random, never shorter by more than 1.0 percent (tolerance 5). Absorbed by: . Revision: none; D6’s two misses were greedy’s, D6v2’s two were definitions, D6v3 has none.

Boundary measured.

• OD:observer-family, change of observer within a declared family: read operators of declared spectrum, coordinate subsets, coarsenings, and learned consumers’ recovered read operators. Claim: Mixed probes at one radius, where the sphere crosses the ellipsoid substantially, recover the read operator up to the pencil s P + (1 - s)(B^2 / rho^2) I, below-threshold eigenvalues included, with Frobenius, above-threshold and below-threshold medians at most 0.30 of chance at 960 queries. experiments/DISCOVERY-TRACK.md, D1 record; experiments/OD/D1/grade.json mixed. Boundary: held in five of six well-crossed cells (Frobenius 0.045 to 0.111 of chance, above 0.055 to 0.111, below 0.037 to 0.172); in the positive definite world at B = 1 the Frobenius ratio 0.198 and the below ratio 0.068 held and the above ratio was 0.303 against 0.30. Witness: the tolerance was fixed as the pilot’s own maximum in that same cell (0.299) rounded up to two decimals, so it carried no margin for a fresh seed; the estimator, the crossing and the recovery are unchanged between pilot and run. Absorbed by: declaration. Revision: none for the claim; for later gates, fix a tolerance as a declared multiple of the pilot’s maximum.
• OD:observer-family, change of observer within a declared family: read operators of declared spectrum, coordinate subsets, coarsenings, and learned consumers’ recovered read operators. Claim: At fixed budget the horizon depends on the observer where no symmetry of the flow relates the observers (Lorenz-63, x against z) and not where one does (Lorenz-96, one block of ten sites against the next). experiments/DISCOVERY-TRACK.md, D3 record; experiments/OD/D3/grade.json H2. Boundary: x later than z for the same perturbation in 77, 78, 70 and 64 percent of pairs at B = 10, 100, 1000, 10000 with sign-test p 0.0000, 0.0000, 0.0022 and 0.033 against the registered 0.01 at every budget; the Lorenz-96 control at p 0.11 to 0.86. Witness: the paired effect weakens as the budget grows and the perturbation aligns with the leading direction for longer, so a fixed per-budget level over the ladder was the wrong registration, not the wrong claim; the pilot at the same budget had 77 percent and p below 0.0001. Absorbed by: declaration. Revision: none for the claim; for later gates, declare the level per budget or grade the ladder as a whole.
• OD:budget, change of the observation budget B at fixed consumer and output metric. Claim: Coarsening the observer’s resolution changes hub polarity monotonically in the resolution, and a resolution of a tenth of the nearest-neighbour distance changes little (exploratory, reported). experiments/OD/D2/grade.json B1. Boundary: monotone in all four sweeps; at a tenth of the neighbour distance 12 percent of Gaussian points and 6 percent of Wikipedia points change polarity under the isotropic observer, at a quarter 24 and 79 percent, against 4 to 7 percent under the anisotropic observer. Witness: the finite budget is a first-order effect on real embeddings under an isotropic reader; not graded, as declared. Absorbed by: declaration.
• OD:observer-choice, . Claim: For a linear system with candidate sensors, the placement that reaches a required count of directions identifiable at budget B (Theorem 2’s count on the exact window Gramian) with the fewest sensors is found by a greedy selection on the Gramian’s spectrum, never needs more sensors than the energy ranking or a random ordering, is within a factor of the exhaustive optimum, and leaves a longer forecast horizon than the energy and random placements at the same count. experiments/DISCOVERY-TRACK.md, D6 record; experiments/OD/D6/grade.json. Boundary: 55 feasible cells on five worlds (a diffusion chain, the Lorenz-96 Jacobian, a shallow-water grid, an unseen advection-diffusion chain, an unseen Lorenz-96 at F = 10): greedy within 1.33 of the exhaustive optimum in all 50 checkable cells and equal in 45; strictly fewer sensors than the energy ranking in 16 of 27 discriminating cells, never more than a random ordering, and one sensor more than the energy ranking in one unseen cell where greedy missed the optimum; the forecast horizon 10.7 percent longer than the energy placement’s and 17.8 percent longer than random pooled on 14 discriminating growth cells, never shorter; greedy at the sensor cap of 12 without the count in one Lorenz-96 cell. Witness: the count of identifiable directions at a budget is the placement quantity: where any placement works, every selector ties; where placement matters, the Gramian’s spectrum finds fewer sensors than the energy of the readings does, and the sensors it picks leave the forecast error in the slower directions; the one cell the energy ranking won is a cell greedy lost to the exhaustive optimum. Absorbed by: declaration. Revision: the selector, not the claim: a one-step lookahead or the exhaustive optimum where feasible would test the spectrum without greedy’s slack; not registered.
• OD:singular-transition, . Claim: Approaching a Burgers shock, the share of a carried perturbation’s energy that a spectral reader of budget B still sees falls on a curve in B over the front wavenumber that converges in resolution and is the same across initial conditions and resolutions (a scaling collapse tighter than the unscaled null), and the reader’s alarm precedes the classical gradient alarm while a regular two-dimensional flow does not fire it. experiments/DISCOVERY-TRACK.md, D5 record; experiments/OD/D5/grade.json. Boundary: converges: twelve resolution pairs at N = 256 against 512 differ by 0.03 to 0.13 against 0.21; collapses where a front forms: median bin IQR 0.085 and 0.048 against 0.13, tighter than the null by 0.54 and 0.48 at nu = 0 and 0.005, and 0.87 at nu = 0.02 where viscosity holds the front; the observational alarm fired once in seventeen trajectories and 0.10 after the classical alarm; both controls fired at B = 2 and one at B = 4 and 8, four of eight control cells. Witness: the reader’s share of a perturbation falls when the flow’s scale crosses the reader’s budget, after the gradient has grown and on regular flows too; shock formation is one way for the flow’s scale to cross it, not the only one, and a budgeted reader sees it later than the gradient does; the classical extrapolation of the inverse gradient is exact for inviscid Burgers and nothing leads it. Absorbed by: declaration. Revision: none registered: the signature is observer-relative and lawful (the collapse) but not early and not specific; two boundaries recorded, the highest viscosity and the low-budget alarm on regular flows.
• OD:closure-geometry, . Claim: For a filtered two-dimensional flow with the spectral cutoff as consumer and budget, the read operator of the resolved tendency with respect to the subfilter state, probed blind on the solver, has leading eigen-directions on which a closure predicts the resolved tendency better than the same number of energy-ranked modes; and the modes of largest read distortion (sensitivity times energy) close better than the energy ranking by a pooled margin at every rank. experiments/DISCOVERY-TRACK.md, D7 record; experiments/OD/D7/grade.json. Boundary: 96 rank cells on three fields at n = 96 and 128, cutoffs 8 and 16, ranks 16 to 128: the eigen-direction closure behind the energy closure in 87 cells, median 15 percent, up to 53; the read-distortion ranking ahead of energy pooled by 0.4, 0.7, 1.4 and 6.1 percent at ranks 16, 32, 64 and 128 against a margin of 1, never behind by more than 7.1 against a tolerance of 13; the margin 0.016 at n = 96 and 0.026 at n = 128; every structured closure below random; the full-rank linear read leaving 3.2 percent (median) to 9.2 percent of the subgrid term at cutoff 8 and nothing at 16. Witness: the leading eigen-directions of the read operator rank subfilter modes by the resolved dynamics’ response alone, and the flow’s energy is elsewhere; the read distortion, response times energy, is the quantity that ranks, and it beats energy by an amount that grows with rank and cutoff, which is where a closure keeps enough modes for sensitivity to separate them. Absorbed by: declaration. Revision: none registered; the track’s declared order ends with this gate.
• OD:singular-transition, . Claim: Second version, the rehabilitation of D5: in periodic Burgers at viscosity at most 0.005 the read fraction of carried perturbations under a spectral reader of budget B converges in resolution and collapses on B over the front wavenumber max|u_x| / max|u| tighter than on the unscaled null by a declared ratio, and the same scaling does not organise the read fractions of a regular two-dimensional flow. experiments/DISCOVERY-TRACK.md, D5v2 record; experiments/OD/D5v2/grade.json. Boundary: 24 trajectories on four fresh initial conditions at N = 256 and 512 and two controls at 64 squared: the scaled collapse at 0.48 and 0.39 of the unscaled null inside the scope (bar 0.75), median bin IQR 0.063 and 0.033 (bar 0.13); the controls at 1.02 (bar at least 0.75); the resolution pairs within 0.04 to 0.08 on three initial conditions and at 0.18 on the fourth against 0.16, at budgets 2 and 4 near 0.9 t* where the front is sharpest; outside the scope nu = 0.02 at 0.64. Witness: the collapse on B over the front wavenumber is the forming front’s: the same construction on a regular two-dimensional flow organises nothing; the instrument’s convergence in resolution is not uniform across initial conditions at the smallest budgets near the shock. Absorbed by: declaration. Revision: the convergence bar on f_B(t) / f_B(0) at budgets 8 and above, or up to 0.8 t*, would test the claim on the part of the instrument that converges; not registered.
• OD:closure-geometry, . Claim: Second version, the rehabilitation of D7: for a filtered two-dimensional flow with the spectral cutoff as consumer and budget, at closure rank 64 and above, the subfilter modes of largest read distortion (sensitivity times energy, the read operator probed blind on the solver) close the resolved tendency better than the same number of energy-ranked modes, pooled at every in-scope rank by a declared margin, behind in no cell by more than a declared tolerance, with a smaller margin below the scope and a margin stable in resolution. experiments/DISCOVERY-TRACK.md, D7v2 record; experiments/OD/D7v2/grade.json. Boundary: 48 in-scope rank cells on three fresh fields at n = 96 and 128, cutoffs 8 and 16, ranks 64 and 128: the read-distortion closure ahead of the energy closure pooled by 2.1 and 8.1 percent (bar 2), behind in 9 cells by at most 14.1 (bar 15), ahead by 0.0 and -0.5 percent at ranks 16 and 32 below the scope; the in-scope margin 6.5 percent at n = 96 and 3.7 at n = 128 against an allowed change of 2.0; every structured closure below random; the eigen-direction closure behind energy in 43 of 48 cells by a median of 19 percent; the full-rank linear read leaving 7.6 percent (median) to 10.5 of the subgrid term at cutoff 8. Witness: sensitivity separates subfilter modes only where the closure has a choice among modes of comparable energy, at rank 64 and above, and there it is worth 2 to 8 percent of the closure error; below that rank both rankings keep the same modes; the margin’s size fell with resolution on this draw where it rose on D7’s, and at this ladder it is not a number. Absorbed by: declaration. Revision: a finer resolution ladder (n = 128 against 192) or the cutoff scaled with the resolution would say whether the margin settles; not registered.
• OD:observer-choice, . Claim: Second version, the rehabilitation of D6: for a linear system with candidate sensors, the placement reaching a required count of directions identifiable at budget B on the exact window Gramian with the fewest sensors, found exactly where the exhaustive search is checkable and by a pruned greedy beyond, never needs more sensors than the energy ranking, a random ordering or D6’s greedy, needs strictly fewer than the energy ranking in a registered fraction of the cells where placement matters, and leaves a longer forecast horizon than the energy placement at the same count. experiments/DISCOVERY-TRACK.md, D6v2 record; experiments/OD/D6v2/grade.json. Boundary: 55 feasible cells on five worlds with fresh seeds: the selector equal to the exhaustive minimum in all 50 checkable cells; strictly fewer sensors than the energy ranking in 18 of 30 discriminating cells (bar 0.30) and never more, never more than a random ordering, never more than D6’s greedy and fewer in four cells; the horizon 16.3 percent longer than the energy placement’s pooled on 16 discriminating growth cells and shorter in one cell by 10.3 percent against a tolerance of 10.0; one count unreachable under the cap of 12 sensors by exhaustive proof while all 20 sensors reach it. Witness: the spectrum places sensors: once the search’s slack is removed the energy ranking wins nowhere; the two misses are definitions, a feasibility test that ignores the cap and a smallest set that is not unique, whose first member in index order gives up the horizon a tie-break on the m-th eigenvalue would keep. Absorbed by: declaration. Revision: feasibility defined at the cap and the exact search breaking ties by the m-th eigenvalue: two definitions, not a claim; not registered.
• OD:singular-transition, . Claim: Third version: D5v2’s claim with convergence in resolution measured on f_B(t) / f_B(0) at budgets 8 and above, the quantity the collapse uses at the budgets the coarser grid resolves. experiments/DISCOVERY-TRACK.md, D5v3 record; experiments/OD/D5v3/grade.json. Boundary: 24 trajectories on four fresh initial conditions at N = 256 and 512 and two controls: convergence 0.014 to 0.062 against 0.10 on every initial condition; the scaled collapse at 0.41 and 0.34 of the unscaled null inside the scope (bar 0.70) and 0.65 outside it; the two controls pooled at 0.57 against a bar of 0.70 and a fail clause of 0.50, each control alone at 1.00. Witness: the transition converges on the quantity the collapse uses; the pooled control bar measured the separation between the two controls’ front wavenumbers, not the organisation within them, and the within-control comparison the claim needs is exactly one on both. Absorbed by: declaration. Revision: the control bar taken per control: a definition, not a claim; not registered.

Failed, with witness.

• OD:singular-transition, . Claim: Fourth version: D5v3’s claim with the control bar taken per control, each two-dimensional control’s read fractions collapsing no more tightly on B over its own front wavenumber than on B alone. experiments/DISCOVERY-TRACK.md, D5v4 record; experiments/OD/D5v4/grade.json. Boundary: both controls under the fail line: scaled over unscaled median bin IQR 0.56 (gradient up a third, front wavenumber 3.2 to 4.1) and 0.22 (gradient up two percent, 1.8 to 2.2), against a bar of 0.75 and a fail clause of 0.5; the Burgers worlds at 0.44 and 0.46 on the same draw; convergence 0.012 to 0.095 against 0.11 and the collapse bars held. Witness: the unscaled null takes one value per budget, so each of its bins holds a budget’s whole time series; dividing by any scale that drifts in time spreads that series across bins and lowers the within-bin scatter by construction; the collapse measure is not specific to the forming front, and the readings of D5v2 and D5v3 that it is are withdrawn. Absorbed by: declaration. Revision: a null that is not degenerate, a wrong scale with the same time dependence; a change of the null and so of the claim; not registered.

Machine checked

lean/DataMiningAsObservation/ProbeCliff.lean, theorems centralDiff_affine, centralDiff_basis, exists_blind_direction, indistinguishable, budget_cliff, at observation-data-mining f3914f0; what the check covers is stated in the book’s appendix C.

Used in

Data Mining as Observation primer S, chapters 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13.

Related

observer; water-filling; budget cliff; coverage.

See also

Book equations stated beside the entry’s terms, not defining it: 1.1, 0.13.

Ledger rows that cite the entry’s records without naming it: OT-3, GO-12.

Sources-table rows that share a record with the entry without naming it: chapter 4 section 4.2, chapter 11 section 11.7, chapter 13 section 13.6.

Status

Generated 2026-09-10 by encyclopedia/generate.py; book at observation-data-mining f3914f0; the commit of every record is listed in the encyclopedia’s provenance.