Query geometry: developed concepts and demonstrated consequences

September 10, 2026. This completes the five concepts and two conjectures proposed in the preceding concept note. Both conjectures are now proved in their declared mathematical models. The developments share a direct-query transfer theorem, and combining them gives stronger guarantees with the existing experiments.

The organizing idea is to follow uncertainty through to the answer being requested. Large changes in observations can have small effects on that answer. Symmetry, nuisance structure, arithmetic constraints, and the actual decoder determine which changes matter. These are definitions and results within this project; no historical-priority claim is made.

1. What the two experiments now guarantee

Arithmetic: improved clock tolerances and wider drift families

The clock-rate conjecture requested a tenfold reduction from the earlier complete clock allowance. Applying the actual real query decoder before taking magnitudes proves a reduction exceeding 897 times in both weightings, relative to that specified earlier allowance. This comparison does not assert superiority to every possible optimized slope-only bound. The new proof includes every source coefficient and every later sampling alias, with a complete nonlinear remainder.

A sharper real-query response spectrum then increases the certified drift bandwidth. The following three joint contracts all recover the 49 unknown integers a(2),...,a(50), using the same 8,900 complex readings, degree-11 polynomial nuisance fit, and 61-column augmented decoder:

Every error is strictly below 1/2, so rounding returns the correct integer. These are three alternative joint uncertainty boxes; their largest coordinates cannot be combined into a fourth box without a new check.

All rows permit drift B sin(omega x+phi), where B and phi are real, |B|<=4,000, |omega|<=Omega, and x=t_actual/890. Polynomial nuisance may have arbitrary complex coefficients. Retain sensor radius 4e-5, separate model-mismatch radius 1e-6, digital-centering radius 1e-20, and a verified normal residual at most 1e-30, in the stated norms. The two weightings give different sensor norms and therefore different noise families. An arbitrary complex sinusoidal amplitude requires the earlier complex-modulus bound.

The source class is the full independent integer envelope 0<=a(n)<=d_14(n), a(1)=1, for every positive integer n. Multiplicativity and number-field realizability are not premises of this branch. The bandwidth 9 contract triples the preceding bandwidth of 3. The clock-rate radius 1e-8 is one million times the original radius, while retaining a larger bandwidth and a nonzero offset allowance. All clock quantities are model units.

Three fresh raw data sets include actual clock distortion, complex polynomial coefficients of scale 1e20, nonzero source tails, drift, mismatch, and sensor perturbations. Both weightings recover every integer in all six solves. A 384-bit reconstruction reproduces every saved rational reading and verifies the 320-bit residual enclosures. These examples exercise the decoder; the complete proof establishes the source-uniform guarantee.

See the clock proof, the spectrum proof, joint evidence, and the independent integration review.

Spatial: four times the joint calibration area

The conjecture requested twice the previously certified normalized clock/potential area. The same six rows, 485, 493, 499, 501, 507, 515, now certify four times that area. Both parameter radii increase from 1e-8 to 2e-8, giving the normalized square [-2,2]^2 of area 16 instead of 4.

All 21 target labels, every nonzero two-mode source, and the full admitted time interval [1,2] remain covered. At sensor radius 3e-8 times source norm, the complete relative source error is below 0.000842860553, meeting the requested 0.001 threshold. The fixed spatial potential family and exact source-preparation assumptions remain part of the contract.

The improvement comes chiefly from retaining a tighter consequence of the existing complete kernel remainder: 6e-10 instead of rounding it to 1e-9. The experiment itself is unchanged. Directional inversion further sharpens the result, but a coarser inverse bound also passes this enlarged box. Mixed clock/potential terms and verified time normalization are charged.

Fresh kernel replays, 2,688 directional cells, and 1,280 source/pair records support the full-time proof. All 24 new synthetic packets at enlarged parameter endpoints decode correctly. Their largest observed source error is below 0.000023628221. These are simulations of the declared model.

The sufficient square-family budget passes half-width 2.09 and fails its label inequality at 2.10. This locates a boundary of this certificate, not an actual ambiguity threshold. The supplied decoder accepts half-width 2. See the spatial report and proof.

2. The five concepts now have precise mathematical content

Query response spectrum

For a real query decoder ell, its response to the family sin(omega x+phi) is the phase supremum of |ell sin(omega x+phi)|. In the present symmetric experiment the odd sine channel contributes nothing to the real answer, so that supremum equals |ell cos(omega x)|.

We reconstruct signed even moments through degree 64 and retain a complete order-66 remainder. Interval evaluation covers every frequency in each declared band, including the clock-rate enlargement. This produces an amplitude/frequency tradeoff and the improved joint contracts above. The auxiliary polynomial degree belongs to the proof computation; nuisance removal still has degree 11. The spectrum report distinguishes certified amplitude allowances from actual impossibility.

Query contact order and calibration geometry

Contact order is the first nonzero variation of decoded answer minus true answer along an uncertainty direction. This correction matters when calibration also changes the true query. Vanishing first derivatives imply a quadratic allowance only with a complete second-derivative bound.

The arithmetic real query is exactly even in shared offset at every fixed clock slope. Its offset error is therefore quadratic, uniformly in slope. A new symmetry-defect lemma shows precisely how imperfect symmetry restores a linear term. If the decoder's symmetry defect has norm at most epsilon_J, the offset charge includes epsilon_J D1 |b|/2 plus the complete quadratic term. This gives a quantitative calibration requirement for using the symmetry advantage. Joint Jacobian images and their nonlinear remainders define calibration regions without treating correlated parameters as independent errors.

Nuisance closure under uncertain acquisition

For degree-p polynomial nuisance and the clock x -> x+epsilon*x^d over an open epsilon interval, the exact closure has monomial exponents

{k+(d-1)r : 0<=r<=k<=p}.

The quadratic family fills every degree through 2p. Higher-degree single-parameter families can leave gaps; full open degree-d polynomial clock families instead generate every degree through dp. Sampled closure uses the actual evaluation rank, which can be smaller on symmetric grids.

If nuisance amplitudes are unbounded, a fixed linear removal must annihilate the whole closure. If their coefficient norm is bounded, a proved direct leakage allowance can preserve information that exact enlarged removal would discard. Thus the amplitude model can change the appropriate decoder.

Observational completion

For fixed finite readings on Re(s)=2 and queries determined by finitely many prime symbols, formal independent quadratic prime-symbol responses form exactly the closure of actual real-quadratic-field responses. This is proved separately for each query class. It uses the finite-prefix realization theorem and a complete tail bound T_N<=2(log N+2)/N.

The development adds a finite strict-gap decision procedure for computably specified sensor times, weights and noise radius, with a supplied finite query-label rule: certify every truncated unequal-answer distance, then add both complete tails. Increasing the prefix and precision terminates whenever the noise level is strictly below or strictly above the critical radius. On the ambiguity side it can construct actual-field witnesses. Termination of the required prime search uses Dirichlet's theorem, also stated as Theorem 18.1 in MIT's notes.

This is a decision procedure promised to terminate away from equality, with explicit, expensive search cost: 3^pi(N) prefixes and naive quadratic pair comparison. At N=50 this means 14,348,907 prefixes and over 2e14 ordered pairs. No efficient implementation of that full search or guaranteed termination exactly at the threshold is claimed. The actual-field conclusion does not turn the d_14 sensing envelope into a family of number fields.

The certification gap

A sufficient recovery bound and an actual ambiguity witness bound different sides of the information threshold. Their units, source class, and noise model must match. For source-relative noise, collision cost is response distance divided by the sum of the competing source norms. Completion preserves separation infima but can change attainment at the endpoint.

An exact three-reading example closes the entire gap. At x=-1,0,1, observe q*x^2+c*(x+epsilon*x^2)+b, with q in {0,1}, |c|<=B, |epsilon|<=h, and b unrestricted. In the unweighted Euclidean sensor norm, unequal-query responses have exact separation

max(0,1-2Bh)*sqrt(2/3).

The second-difference decoder reaches half this separation as its noise threshold, and an explicit midpoint supplies a matching ambiguity witness. For B=2 and h=1/10 the critical noise radius squared is 3/50; ambiguity occurs at equality. With unbounded c, any h>0 permits noiseless ambiguity. This is a proved obstruction, unlike a failed sufficient error budget.

3. How the concepts fit the transfer theorem

For exact design X, G=X*WX, real-linear query Q, and L=QG^-1X*W, a proposed coefficient vector with normal residual r satisfies the exact identity

decoded query - true query = Lh + Le + QG^-1r.

Here h contains the complete model discrepancy and structured uncertainty, and e contains the allowed sensor error. Taking support bounds on the actual joint image Lh retains the incidence of source and calibration parameters. It need not first bound the entire discrepancy in a single measurement norm. Nuisance closure makes the decomposition legitimate; contact order controls its local behavior; spectra compute specific families; completion and ambiguity witnesses assess its sharpness.

The framework proof also treats nonlinear outer answer sets, composition, the exact transported noise metric after restricting readings, scalar rounding versus vector enclosure, and degenerate zero-noise collision conventions. The arithmetic and spatial consequences above use their own declared models and norms within this framework.

4. Verification boundaries and the next questions

The package includes analytic proofs, exact rational consequence checkers, 320/384-bit clock reconstruction, 320/448-bit signed-spectrum reconstruction, 320/384-bit spatial kernel reconstruction, and actual-data residual checks. The component suites pass 13 arithmetic tests, 13 spatial tests, six spectral corruption controls, five framework corruption controls, and four joint integration controls. Separate agent reviews audited the clock proof, spectrum, and combined budgets. Such reviews and numerical examples support the proof audit; they do not replace its infinite-tail or continuum arguments.

Inherited complete arithmetic-bias premises retain their earlier full enumeration reproduction routes. We rechecked their consequences and source bindings here; we did not repeat every historical enumeration. Model-family membership and physical calibration are assumptions that a solver residual cannot establish. These results do not supply apparatus measurements or a statistical noise distribution.

The next useful work is now narrower: tighten the remote clock-alias tail and direct second-order offset response; seek actual spatial ambiguity witnesses to compare with the sufficient calibration boundary; optimize nuisance degree jointly with a justified amplitude bound; and prune the quadratic-field prefix search while retaining complete directional tails. These would improve precision or expose limits of the present guarantees. The present scope is complete without assuming those future results.

Use the reproduction guide to distinguish artifact integrity, consequence checking, defining-model reconstruction, and expensive inherited tail replays.