# Query geometry: direct error, nuisance closure, and finite knowledge September 10, 2026. Analytic framework for the five concepts proposed in `research/concepts_2026_09_10/CONCEPTS.md`. These are elementary consequences of explicit models, not claims of historical priority. Complex observation spaces are treated as real Hilbert spaces when the requested answer is real. All norms, source classes, and allowed joint parameter sets are part of an experiment; none is supplied by the terminology below. ## 1. Direct-query transfer theorem Let W be positive definite, X an exact finite augmented observation matrix, G=X*WX >= f I > 0, and Q a real-linear map from coefficient space into the requested real answer space. Q may discard nuisance coordinates and take the real parts of arithmetic coefficients. Write T=G^-1 X*W, L=QT. Suppose every actual explanation satisfies y = X theta + h(theta,z) + e, q(theta)=Q theta, (theta,z) in A, ||e||_W <= eta(theta), after exact known centering. A includes the joint incidence of source, timing, drift, and model uncertainty. Unrestricted nuisance must be exactly represented in X and annihilated by Q, or removed by a valid preceding map. The vector h includes the complete infinite tail where appropriate. For a proposed coefficient vector theta_tilde with verified normal residual r = G theta_tilde - X*W y, ||r||_2 <= rho, the decoded error satisfies the exact identity Q theta_tilde - q(theta) = L h(theta,z) + L e + Q G^-1 r. (1) Consequently, for any unit real answer direction v, an upper bound on its signed error is sup_(theta,z in A) + sup_theta eta(theta) ||L* v||_(W^-1) + rho ||(Q G^-1)* v||_2. (2) The analogous expression with -v is the lower bound. The adjoints here are real Hilbert adjoints. One may instead keep theta-dependent bounds until forming the final outer answer set, which can be sharper. The family of all these half-spaces encloses every decoded answer error. Replacing a joint image by a Minkowski sum of separately bounded images is safe but can lose correlations. It requires no probabilistic independence. **Proof.** Substitute the model in the residual identity and multiply by Q G^-1. Taking a real inner product gives (2) by the definition of a support function and the two dual-norm inequalities. No full-state error bound is required before evaluating the requested query. QED. If Q returns n^2 times the real part of arithmetic coordinate n, then ||L|| <= n^2/sqrt(f) and ||Q G^-1|| <= n^2/f. If the complete decoded tail is bounded by A_n and structured disturbances have direct query bounds K_n, the usual sufficient integer gate becomes A_n + K_n + n^2 [eta/sqrt(f) + rho/f] < 1/2. (3) K_n replaces the corresponding reading-norm disturbance allowance; it is not added a second time. A failed sufficient gate is not an ambiguity witness. Every new data vector still requires its own verified residual. For a general nonlinear pipeline P and any sound outer model C(theta) containing P(y) for each explanation of y, the exact answer set satisfies {q(theta): theta explains y} subset {q(theta): P(y) in C(theta)}. (4) This follows by substitution and composes by transitivity. A singleton outer set determines a discrete query. For continuous answers, the radius of a supplied enclosure controls an estimator's error; half the diameter alone is not automatically an enclosing radius for vector answers. ### Restrictions and relative errors For a restriction R, the original weighted error ball has image R E. The least original squared cost of r in range(R) is min_(Re=r) ||e||_W^2 = r* (R W^-1 R*)^dagger r. (5) Whitening and the minimum-norm solution of a linear system prove (5). It is invalid to silently assign an unrelated spherical noise model to retained coordinates. If the original radius is eta ||u||, all direct query bounds above remain relative to ||u|| unless an absolute source amplitude bound is supplied. Competing explanations have separate source norms and may choose different permitted clock/model parameters. ## 2. Response spectra and contact order For an arbitrary complex scalar decoder L_c, let a=L_c sin(omega x) and b=L_c cos(omega x). The exact arbitrary-phase response obeys sup_phi |a cos(phi)+b sin(phi)|^2 = lambda_max [[|a|^2, Re(a conjugate(b))], [Re(a conjugate(b)), |b|^2]]. (6) Expansion and the Rayleigh principle prove this identity. For the real query ell=Re L_c, the exact answer is instead sup_phi |ell sin(omega x) cos(phi) + ell cos(omega x) sin(phi)| = sqrt((ell sin(omega x))^2+(ell cos(omega x))^2). (7) The complex bound remains safe for the real query, but can be conservative. Frequency maximization is a separate continuum problem: a finite frequency grid alone does not establish its supremum. Complete Taylor remainders or interval subdivisions with certified between-sample bounds do. **Definition (error contact order).** For a calibrated family with actual answer q(z), a fixed reported estimate map P, base z0, and direction v, put E(h)=P(y(z0+h v))-q(z0+h v). The contact order is the first positive integer k with E^(k)(0) nonzero, componentwise for a vector answer. If the true source/query is fixed during calibration, q is constant and this reduces to the decoded-response definition in the concept note. Ignoring the varying true query can otherwise give a spurious contact order. If E^(j)(0)=0 for 1<=j= sum_m d_14(m) (log m)^2/m^2. (10) Termwise arguments are justified by the complete convergent D2 series. There is no uncharged slope-offset mixed term: (10) is uniform in a and the source difference is split exactly at F(a t). Affine composition preserves degree-11 nuisance and shifts the arbitrary sinusoidal phase. The effective sinusoidal frequency depends on slope, not offset. I independently rechecked the rational consequences for both saved 320- and 384-bit reconstructions, without rewriting either predecessor. With the unchanged slope radius 1e-14 and offset radius 3e-5, the complete errors are below 0.444378507 (multi) and 0.493361414 (outer). The inherited old joint clock allowance is valid for slope alone because zero offset is in its allowed box. Adding (10) is therefore conservative, not a dropped clock component. These are conditional mathematical guarantees under the declared source, drift, noise, mismatch and per-data residual premises. This audit validates their deduction; the actual-matrix reconstruction remains a separate numerical premise with its existing replay route. **A bounded symmetry defect.** For a real query ell, suppose the same involution J obeys ||ell J-ell||_(W->R)<=epsilon_J. The first offset derivative v=F'(a t) is J-odd, so 2 ell(v)=-(ell J-ell)(v). Since ||v||_W<=D1, where D1=sum_m d_14(m)log(m)/m^2, Taylor's theorem gives |ell[F(a t+b)-F(a t)]| <= epsilon_J D1 |b|/2 + ||ell|| D2 b^2/2. (10a) More generally a declared derivative-symmetry defect ||Jv+v||_W<=sigma_1 adds ||ell|| sigma_1 |b|/2. These are uniform statements only if their premises hold for every admitted slope and source. If ell_exact J=ell_exact and ||ell-ell_exact||<=epsilon_L, J-isometry gives epsilon_J<=2 epsilon_L. The lemma concerns the source response. It does not cure nonzero leakage of an unbounded nuisance through an approximate decoder: exact nuisance removal or a per-data residual remains necessary. A solver's small residual also does not establish a physical clock or symmetry model. This gives a continuous error budget between exact symmetry and a declared asymmetry. ### Independent audit of the new complete slope proof The separate [clock development](../arithmetic/PROOFS.md) uses direct real query channels through m=100, a zero-extended second-difference bound for 101<=m<=10^12, and a complete zeta-derivative tail for all m>10^12. I checked the mathematical argument and producer `clock_response.py` independently: the 0.2 time increment gives |1-z|=2|sin(log(m/i)/10)|; the stated concave sine interval covers all arithmetic and polynomial normal channels; both zero-extension boundary terms are included; and the remote derivative mass covers every later alias. Its complete positive envelope permits independent actual source coefficients. The inverse transfer and the stronger comparison C=2, and allow g_e(x)=x+e x^d for e in any nonempty open interval. Then (P_p)_G = span{x^s: s in S_(p,d)}, S_(p,d)={k+(d-1)r: 0<=r<=k<=p}. (12) **Proof.** Expanding (x+e x^d)^k gives binomial(k,r)e^r x^(k+(d-1)r). For each k, evaluating at k+1 distinct admitted e values and inverting the Vandermonde matrix isolates all k+1 monomials. The same expansion supplies the reverse inclusion. QED. For d=2 these exponents are all 0,...,2p. For d=3,p=1 they are {0,1,3}, so P_(dp) would be an incorrect generalization. On sampled points, the dimension is the rank of this sparse monomial evaluation matrix. It is not always min(m,|S|): at {-1,0,1}, x and x^3 coincide. On m distinct strictly positive points it is min(m,|S|). To see this, a nonzero sum of s distinct nonnegative monomials has at most s-1 distinct positive zeros: divide by the smallest power, differentiate, and use Rolle's theorem inductively. Every square minor with sorted positive points is therefore nonsingular. Ordinary P_(2p) has the usual min(m,2p+1) rank on any distinct real sample points. If instead all coefficients of a degree-d clock g_a(x)=a_0+a_1 x+...+a_d x^d vary jointly over an open subset of R^(d+1), then (P_p)_G=P_(dp). Indeed the coefficient of each parameter monomial in g_a(x)^p is a nonzero multiple of x^s for some 0<=s<=dp, and every such s can be written as a sum of p integers in {0,...,d}. The span of values of a vector-valued polynomial on an open set contains each coefficient: a linear functional annihilating all values gives a scalar polynomial zero on an open set, whose every coefficient is zero. Finite-dimensional duality proves the assertion. This full family is different from (12). **Bounded-leakage alternative.** Suppose L annihilates P_p, x is the declared sample vector, |e|<=h, v(x)=sum_(k=0)^p c_k x^k, and sum_k |c_k|<=B. Then |L[v(x+e x^d)]| <= B max_(k<=p) sum_(r: 0<=r<=k, k+(d-1)r>p) binomial(k,r) h^r |L x^(k+(d-1)r)|. (13) Expansion, exact annihilation, the triangle inequality, and l1/linfinity duality prove (13). For a fixed e the exact worst l1 coefficient effect is B max_k |L[(x+e x^d)^k]|. Replacing its signed polynomial response by the termwise sum in (13) is conservative. Other coefficient norms use their own dual support function. A fixed h and finite B give a finite budget even when exact nuisance closure would consume useful source directions. The first surviving exponent in e also gives a nuisance contact order. An amplitude norm must be declared in a specified basis; it is not invariant under arbitrary polynomial reparameterization. ### An exact three-reading experiment: closed certification gap This independently checkable example turns the closure/leakage choice into a complete result. At x=(-1,0,1), observe y_j=q x_j^2 + c (x_j+e x_j^2) + b + noise_j, q in {0,1}, |c|<=B, |e|<=h, b arbitrary real, ||noise||_2<=eta. (14) Each explanation has its own (c,e,b). Let ell=(1/2,-1,1/2); then ell(1)=ell(x)=0, ell(x^2)=1, and ||ell||_2=sqrt(3/2). Thus |ell(y)-q| <= Bh+eta sqrt(3/2). (15) The exact distance between the two response sets is delta=max(0,1-2Bh) sqrt(2/3). (16) **Proof of the lower bound.** Apply ell to any unequal-label center difference: its magnitude is at least max(0,1-2Bh). Cauchy--Schwarz and ||ell||=sqrt(3/2) give (16). For Bh<1/2, take c=B in both explanations, e=h for q=0 and e=-h for q=1, and b=-(2/3)(q+c e). Their difference is (1-2Bh)(x^2-2/3), whose norm is (1-2Bh)sqrt(2/3). For Bh>=1/2 take c=B and e=+/-1/(2B), with equal b, producing identical centers. The degenerate B=0 case belongs to the first construction. QED. Therefore recovery is possible exactly when eta0, and it fails at equality because the closest pair is attained. If delta=0 it fails even at zero noise. The sufficient decoder (15) meets the exact threshold. At B=2,h=1/10 this gives radius squared 3/50, and the checker constructs the exact rational common midpoint and its two error vectors. At unrestricted c and any h>0, the same construction gives a noiseless actual ambiguity. Here a general linear-removal warning has been upgraded to a true model-specific impossibility witness. ## 4. Observational completion and a terminating strict-gap procedure The actual quadratic source is a real quadratic field, without a bound on its discriminant. Give every prime an arbitrary symbol chi(p) in {-1,0,1}, extend completely multiplicatively, and set a_chi(n)=sum_(d divides n) chi(d). These formal coefficients satisfy 0<=a_chi(n)<=d_2(n). The existing [finite-prefix theorem](../../five_paths_2026_09/path4_realizability/PROOF.md) proves that every finite prime-symbol assignment is realized by infinitely many actual real quadratic fields. Its arithmetic ingredients are the quadratic splitting laws, CRT, and Dirichlet's theorem for a residue coprime to its modulus. We do not reinterpret arbitrary d_14 envelopes as number fields. Let F be any fixed finite collection of readings at Re(s)=2, in a positive mass-one weighted norm. Two series with matching coefficients through N differ in this norm by at most T_N=sum_(n>N) d_2(n)/n^2 <= 2(log N+2)/N. (17) The coefficient difference is at most d_2(n), not twice this number, because both coefficients lie in [0,d_2(n)]. For a full response compared with its own truncation the same bound holds. To prove the last inequality, A(x)=sum_(n<=x)d_2(n)<=x(1+log x), since A(x)=sum_(a<=x)floor(x/a). Partial summation gives -A(N)/N^2+2 integral_N^infinity A(t)/t^3 dt; drop its nonpositive term. For each query determined by finitely many prime symbols, the formal response image in that query class equals the closure of the actual-field response image in that class. One inclusion follows by realizing ever longer prefixes and using (17). For the converse, take a diagonal subsequence of prime assignments. Every fixed prefix stabilizes, and the uniform tail bound gives convergence to the resulting formal response. The image is compact and closed. This proof applies separately to each finite-symbol class, so it does not require the query to be recoverable from observations in advance. A query without a continuous finite-symbol extension, such as an arbitrary property of unbounded discriminants, is not covered by this conclusion. ### Finite strict-gap promise procedure For the algorithmic conclusion, supply a finite query-label rule on the prescribed prime symbols and computable specifications of the finitely many sensor times, positive mass-one weights, and nonnegative noise radius eta. Here computable means that certified rational intervals of any requested accuracy can be produced. These premises hold for the project's rational time grid, computable weights and rational noise contracts. The preceding analytic completion statement does not itself assert an algorithm for arbitrary noncomputable real experimental inputs. Fix N large enough to determine the supplied query label. There are K_N=3^pi(N) possible prefixes, all realized by actual fields. Enclose every truncated response and every unequal-label pair distance by verified intervals [l_ab,u_ab]. Put l_N=min l_ab and u_N=min u_ab; interval arithmetic must cover all eligible pairs, rather than only a floating-point favorite. Then the actual-field separation infimum delta obeys max(0,l_N-2T_N) <= delta <= u_N+2T_N. (18) The lower bound applies to every actual field pair after truncation. For the upper bound, realize each prefix of a minimizing upper pair and use two complete tails. Thus the reported interval is justified by actual objects, even though a limiting optimizer itself need not be a field. Choose interval precision so u_N-l_N tends to zero, refine the certified noise-radius interval as well, and increase N. The computable-input premises allow these refinements. Assume the strict-gap promise 2 eta!=delta. If 2 eta2 eta, proving recovery. If 2 eta>delta, eventually u_N+2T_N<2 eta. The finite-prefix construction then supplies an actual unequal-label pair whose midpoint noise is strictly below eta. These strict comparisons can eventually be verified against the appropriate noise-radius interval endpoints. This proves termination on every input meeting the strict-gap promise. It deliberately gives no guaranteed termination at equality; it is not an unrestricted decision procedure for all noise thresholds. The arithmetic witnesses can be made explicit: build the CRT progression in the cited finite-prefix proof, enumerate its positive candidates, certify primality (trial division is sufficient for termination), and form the positive fundamental discriminant. Dirichlet ensures this search terminates. Complete tails certify the resulting series, and a strict remaining margin permits a rational approximation to the common midpoint if a finite output vector is desired. The theorem supplies no practical uniform bound on how many progression candidates are needed. The computational cost is explicit in the finite search size: storing prefixes takes O(N K_N) scalar coefficients; direct response evaluation takes O(m N K_N), and exhaustive pair comparison O(m K_N^2), with arithmetic bit cost and certified transcendental precision additional. At N=5 there are 27 prefixes and at most 729 ordered pairs; at N=50 there are 14,348,907 prefixes and over 2e14 ordered pairs. Branch-and-bound may reduce work but must preserve valid lower bounds and full coverage. This is a convergent finite decision route, not a claim that naive enumeration is an efficient replacement for the existing directional tail certificates. `check_examples.py` independently constructs all 27 assignments at primes 2,3,5 using exact CRT and trial-division primality, verifies the positive fundamental discriminants and their five-term coefficient prefixes, and finds maximum discriminant 28,920. These small witnesses exercise the constructive route; their finite enumeration does not prove the infinitude input, which remains Dirichlet's theorem. ## 5. Intrinsic ambiguity, certification gap, and computational priorities For explanations theta,theta' let f,f' be their noiseless responses and let their allowed errors be scaled norm balls of radii epsilon r(theta) and epsilon r(theta'). For r+r'>0 the least scale at which the two balls meet is c(theta,theta')=||f-f'||/[r(theta)+r(theta')]. (19) If r(theta)+r(theta')=0 (both radii are nonnegative), both error sets are singletons at every finite scale. Define c=0 when f=f', and c=+infinity otherwise. This includes noiseless rival explanations in the infimum and avoids dividing by zero. Such coincident unequal-query centers are already ambiguous at zero noise; distinct zero-radius centers never meet at any finite noise scale. The triangle inequality proves necessity. Sufficiency follows by choosing the point dividing the segment between centers in the proportions of the two radii. Equal unit radii give the absolute margin half-distance; source-relative noise gives r=||u|| and r'=||u'||. General shaped or correlated error sets require their actual intersection cost instead. Profiling unrestricted nuisance means including its value in each complete explanation before taking the infimum. Formula (19) need not define a metric between query classes: set distances can fail the triangle law. Let r_q be the infimum of (19) over unequal query answers. Below r_q, unequal-answer neighborhoods are disjoint. Above r_q an actual pair gives an ambiguity. At r_q the endpoint depends on attainment and the actual error sets. For continuous answers, one can instead compare pairs whose query distance exceeds a declared tolerance; this is an ambiguity modulus, not an automatic vector estimator-radius formula. A proof supplies r_low<=r_q. A verified actual pair supplies r_q<=r_high. Their difference is the certification gap, provided source class, query, units, uncertainty incidence, noise scaling and endpoint convention agree. A relaxed point that cannot be realized is only an upper bound for the relaxed model. An improved lower certificate with unchanged observations changes knowledge about the experiment, not its intrinsic information. **Exact endpoint and finite convergence example.** Let formal observations be y=sum_(k>=1)b_k/4^k, b_k in {0,1}, query b_1. Let actual objects be the eventually-zero sequences. They are dense in each formal query class. The formal class-0 maximum is 1/12 and the class-1 minimum is 1/4, so their distance is 1/6. No actual class-0 sequence reaches 1/12. Thus all unequal actual pairs have distance strictly greater than 1/6, but actual finite prefixes approach it. Actual closed noise balls remain disjoint at radius 1/12, whereas completed classes have a common midpoint at that radius. For N-bit truncations the exact separation is delta_N=1/6+1/(3*4^N), T_N=1/(3*4^N). (20) The generic interval [delta_N-2T_N,delta_N+2T_N] converges with width 4T_N. The checker enumerates all prefixes through N=8 and reconstructs exact minimizers and tail sums. The analytic geometric-series proof covers all N and the nonattained endpoint; enumeration alone would not. **Decision guidance with mathematical content.** Given a required noise scale eta, if etar_high, extra proof precision cannot rescue that experiment and model; one must alter acquisition, the question, or justified uncertainty. If eta lies between them, either a stronger lower certificate or a better actual witness can settle the issue. For the completion hierarchy (18), refining all interval evaluations far below T_N cannot materially shrink the gap until N or the complete tail bound improves. Conversely increasing N is unhelpful while matrix/interval error dominates. In the exact example (20), the cost and tail are explicit: quadrupling prefix count by increasing N by two reduces T_N by sixteen. The three-reading experiment (14)-(16) has zero certification gap because decoder and actual collision meet. These results give the transfer line a concrete purpose: preserve the geometry of answer-changing perturbations through exact nuisance removal, restrictions, approximations and finite computation, while tracking which remaining uncertainty is experimental and which is only in its proof.