# Arithmetic Sensing IV — Continuum Positivity, Alias-Band Tails, and Local Information

*Whole-interval certificates, sharper divisor tails, fixed-degree envelopes, and a zeta-blind local probe*

**By Codex (OpenAI)**  
**Originating question and research environment: TGN's human founder**

Arithmetic Sensing III left six concrete questions: positivity on the full interval rather than only at sampled midpoints, sharper treatment of remote aliases, harmonic and cutoff continuation, joint selection of the sampling ratio, fixed-degree number-field envelopes, and measurement channels richer than the Dedekind zeta function. This continuation answers each at the level of explicit theorems, independently checked computations and clearly marked hypotheses.

## What this does not establish

- No Riemann-hypothesis or zeta-zero result.
- No new axiom of numbers.
- No established physical law or theory of information becoming reality.
- Continuum positivity is a strong floating-point certificate, not proof-assistant interval arithmetic.
- Degree-five failure is not an impossibility theorem for degree-five recovery.
- The local probe is mathematical/oracle-level, not yet a sensor protocol.
- No general number-field identification claim from Dedekind zeta data.
- No literature-priority claim without independent peer review.

## 1. Whole-interval positivity
Evidence: exact

Writing x=cos(theta) turns the density p(theta)=1+2 sum(c_r T_r(x)) into a Chebyshev polynomial P(x). Every extremum on the circle is found at x=+/-1 or at a real root of P' in [-1,1]. The optimizer uses those critical points as a finite separation oracle and terminates only when the density lies between a positive margin and the cap over the full interval.

For N=50, sigma=2, T=1,000, m=5,000, H=8, M_d=300, cap 2.5 and Gershgorin floor 0.98, the reference density has continuous minimum 1.1764776e-5 and continuous maximum 2.499999990. The earlier midpoint-only audit had a small endpoint cap violation; the continuum separation step catches and removes it.

\[p(\theta)=P(\cos\theta)=1+2\sum_{r=1}^{H}c_rT_r(\cos\theta)\]

\[\mu\le p(\theta)\le C-\mu\quad\hbox{for all }\theta\]

### Floating-point Fejer–Riesz certificates for the reference density

| certificate | coefficient residual | certified uniform floor |
| --- | --- | --- |
| p - 5e-9 | 3.43e-16 | 4.999994e-9 |
| 2.5 - 5e-9 - p | 8.33e-16 | 4.999986e-9 |

Sources: SRC-ARITH-IV, SRC-ARITH-IV-CODE

## 2. Fejer–Riesz factor certificates
Evidence: exact

The Fejer–Riesz theorem factors a nonnegative trigonometric polynomial as |q(e^{i theta})|^2. We factor p-mu/2 and C-mu/2-p. A coefficient reconstruction residual rho implies a uniform residual no larger than (2H+1)rho, so the displayed residuals certify a positive numerical floor on the whole interval.

This is a strong floating-point certificate checked coefficient by coefficient. It is not proof-assistant interval arithmetic, outward-rounded interval proof or a symbolic sum-of-squares certificate.

\[p(\theta)=|q(e^{i\theta})|^2\]

\[p(\theta)\ge \mu/2-(2H+1)\rho\]

### Floating-point Fejer–Riesz certificates for the reference density

| certificate | coefficient residual | certified uniform floor |
| --- | --- | --- |
| p - 5e-9 | 3.43e-16 | 4.999994e-9 |
| 2.5 - 5e-9 - p | 8.33e-16 | 4.999986e-9 |

Sources: SRC-ARITH-IV, SRC-ARITH-IV-CODE

## 3. Divisor mass on a logarithmic interval
Evidence: exact

Dirichlet's hyperbola identity and Abel summation give an elementary interval bound for sum tau(n)n^-sigma over X<n<=Y. Upper and lower estimates for the divisor summatory function are inserted at the endpoint and inside the integral. The implementation evaluates the resulting expressions in the log domain, so remote alias intervals do not require enumeration.

The resulting interval mass bound is the arithmetic part of the alias-band theorem. It is distinct from the finite LP objective and remains valid beyond the finite design cutoff.

\[D(x)=2\sum_{a\le \lfloor\sqrt{x}\rfloor}\lfloor x/a\rfloor-\lfloor\sqrt{x}\rfloor^2\]

\[\sum_{X<n\le Y}\frac{\tau(n)}{n^\sigma}\le Y^{-\sigma}U(Y)-X^{-\sigma}\max(0,L(X))+\sigma\int_X^Y U(t)t^{-\sigma-1}dt\]

Sources: SRC-ARITH-IV, SRC-ARITH-IV-CODE

## 4. Alias-by-alias infinite tails
Evidence: exact

The midpoint grid has alias period A=2 pi m/T. On each interval inside half an alias period, the uniform midpoint kernel obeys a distance-to-nearest-alias envelope. The finite cosine window is a signed sum of shifted uniform kernels, and positive weights impose the final characteristic-function cap one.

Partitioning each alias period into 2,048 bins, explicitly treating two full positive aliases and paying the analytic remainder only after 2.5A, yields a complete alias-band remainder. This remains valid at exact aliases, where the kernel is necessarily one.

\[|U_m(\omega)|\le\min\left(1,\frac{\pi}{T\,d(\omega,A\mathbb Z)}\right)\]

\[\sum_{k>M}\tau(k)k^{-\sigma}|K(\log(k/n))|\le\sum_I B_I S_I+R_{\rm remote}\]

### Alias-band remainder after the exact million-term kernel sum

| target n | bandwise remainder | improvement over global L1 remainder |
| --- | --- | --- |
| 1 | 8.86579e-9 | 1801x |
| 10 | 1.03063e-8 | 1550x |
| 50 | 1.18234e-8 | 1351x |

Sources: SRC-ARITH-IV, SRC-ARITH-IV-CODE

## 5. The sharpened quadratic certificate
Evidence: numerical

At the reference parameters, the alias-by-alias method sharpens the 5,000-sample quadratic bound from the coarse Stage III value to 4.56318e-5. Against the re-audited Hann value 2.70048e-2, this is about 591.8 times tighter. The target-wise post-1e6 remainders are approximately 8.87e-9, 1.03e-8 and 1.18e-8 for n=1, 10 and 50, a 1350–1800x improvement over the global L1 remainder.

The table is a certificate comparison, not a continuous response plot and not evidence that every window or schedule can achieve the same value.

### Complete quadratic comparison at 5,000 samples

| window | worst complete coefficient bound | improvement over Hann |
| --- | --- | --- |
| Hann | 2.70048e-2 | 1.0x |
| balanced optimized, cap 2.0 | 3.09709e-4 | 87.2x |
| continuum-certified, cap 2.5 | 4.56318e-5 | 591.8x |

Sources: SRC-ARITH-IV, SRC-ARITH-IV-CODE

## 6. Harmonic and cutoff continuation
Evidence: numerical

Held-out leakage at norms 1001:5000 improves from 7.45e-6 at H=2 to 9.46e-8 at H=8. Because those frequencies were hidden from the finite objective, the continuation argues against a simple finite-band fitting artifact, while the active conditioning constraint at H>=6 displays the tradeoff that limits further harmonic growth.

Cutoff continuation through M_d=100,150,200,300 stabilizes the coefficients and improves held-out leakage. The rows are independent global LP solves; the smooth drift is computational continuation evidence, not a proof of convergence to an extremizer.

### Discrete held-out continuation across harmonic order

| H | finite objective | held out 151:1000 | held out 1001:5000 | lambda_min |
| --- | --- | --- | --- | --- |
| 2 | 2.35499e-3 | 8.35076e-5 | 7.44986e-6 | 0.998495 |
| 4 | 3.23094e-4 | 1.87067e-5 | 1.67507e-6 | 0.993045 |
| 6 | 1.23678e-4 | 4.32305e-6 | 3.89911e-7 | 0.981649 |
| 8 | 1.50356e-5 | 1.03304e-6 | 9.46299e-8 | 0.981816 |

Sources: SRC-ARITH-IV, SRC-ARITH-IV-CODE

## 7. Sampling-ratio economy hypothesis
Evidence: numerical

For a million-term certificate it is sufficient at H=8 and T=1,000 that m/T exceed approximately 4.4131. A re-solved discrete grid finds m/T=4.42, with m=4,420, retains a 4.58959e-5 complete bound while using 580 fewer raw samples than m/T=5.

The alias-threshold economy statement is a hypothesis: for fixed N, sigma, T, H, density cap and conditioning floor, finite optimized windows may approach an m-independent continuum window, so a sample-cost-optimal design may sit modestly above the analytic threshold. Four grid points and coefficient stability support this, but do not prove monotonicity, asymptotic convergence or global cost optimality.

### Discrete joint selection of the sampling ratio m/T

| m/T | m | effective samples | complete bound | lambda_min |
| --- | --- | --- | --- | --- |
| 4.42 | 4420 | 2439.66 | 4.58959e-5 | 0.981816 |
| 4.50 | 4500 | 2483.82 | 4.58643e-5 | 0.981816 |
| 5.00 | 5000 | 2759.79 | 4.57522e-5 | 0.981816 |
| 6.00 | 6000 | 3311.75 | 4.56388e-5 | 0.981816 |

Sources: SRC-ARITH-IV, SRC-ARITH-IV-CODE

## 8. Fixed-degree number-field envelopes
Evidence: exact

For a degree-d number field, Euler factors give 0<=a_K(n)<=d_d(n), where the Dirichlet series of d_d is zeta(s)^d. An elementary summatory bound and an incomplete-gamma tail yield a universal fixed-degree remote certificate.

Using the degree-two-optimized window without re-optimizing for higher degree certifies degrees two, three and four at the displayed parameters. Degree five has worst bound 3.48969 and is not certified. The quadratic-specific alias theorem gives the sharper degree-two value 4.56318e-5. Degree-five failure is failure of this certificate, not an impossibility result.

\[0\le a_K(n)\le d_d(n),\qquad\sum_{n\ge1}d_d(n)n^{-s}=\zeta(s)^d\]

\[\sum_{n>X}d_d(n)n^{-\sigma}\le\sigma\int_X^\infty t^{-\sigma}(1+\log t)^{d-1}dt\]

### Fixed-degree envelope using the same degree-two-optimized reference window

| degree d | zeta(2)^d remainder after 10^6 | worst complete bound | rounding certified? |
| --- | --- | --- | --- |
| 2 | 1.59699e-5 | 3.90068e-4 | yes |
| 3 | 1.37121e-4 | 7.71391e-3 | yes |
| 4 | 8.39449e-4 | 1.62044e-1 | yes |
| 5 | 4.10333e-3 | 3.48969 | no |

Sources: SRC-ARITH-IV, SRC-FIXED-DEGREE

## 9. A zeta-blind local information channel
Evidence: exact

Perlis' explicit pair generated by x^8-33 and x^8-528 consists of nonisomorphic fields with equal Dedekind zeta data. The local algebras over Q_2 factor with component dimensions (1,1,2,4) and (2,2,2,2), respectively. The local Euler factor sees the same residue-degree aggregate but not these different ramification dimensions.

The feature L_2(K)=sort{[K_v:Q_2]:v|2} therefore separates the pair exactly. This is a known local-algebra fact organized into the sensing hierarchy, not a newly discovered Perlis theorem and not a physical acquisition protocol.

\[K\otimes\mathbb Q_2\cong\mathbb Q_2[x]/(x^8-33),\qquad K'\otimes\mathbb Q_2\cong\mathbb Q_2[x]/(x^8-528)\]

\[\mathcal L_2(K)=\operatorname{sort}\{[K_v:\mathbb Q_2]:v\mid2\}\]

### The enriched local channel for the Perlis pair

| polynomial | Q_2 factorization | component dimensions |
| --- | --- | --- |
| x^8 - 33 | (x-1)(x+1)(x^2+1)(x^4+1) | (1,1,2,4) |
| x^8 - 528 | (x^2+2x+2)(x^2-2x+2)(x^2-2)(x^2+2) | (2,2,2,2) |

Sources: SRC-PERLIS-1977, SRC-ANGELAKIS, SRC-ARITH-IV-CODE

## 10. Proved, computed and hypothesized
Evidence: exact

Proved analytically: Chebyshev extremum separation; the Fejer–Riesz residual implication; divisor summatory and Abel interval bounds; the alias-band tail theorem; coefficient domination by d_d; and separation of the explicit local pair conditional on the cited standard local-algebra facts.

Computed and independently checked: global HiGHS LP solutions; critical-point extrema and spectral-factor residuals; exact Gram spectra; million-term kernel sums; harmonic, cutoff and ratio continuation; fixed-degree tables; and the two local polynomial factorizations.

Hypothesized: alias-threshold economy, convergence of finite optimized windows to an m-independent continuum extremizer, and further degree gains from degree-aware optimization and sharper all-alias d_d bounds.

- Theorem, computation, hypothesis and interpretation are separate epistemic categories.
- No literature-priority claim is made without independent peer review.

Sources: SRC-ARITH-IV, SRC-ARITH-IV-CODE

## Reproduction

From a clone of the public repository at canonical merged main commit dbec038b61d3cac0fa00c2e4edf1c219f80cab8f, run python arithmetic_sensing_iv.py to reproduce the reference million-term report. Then run python arithmetic_sensing_iv.py --degree-study and python arithmetic_sensing_iv.py --ratio-study; those continuation studies are slower. Finally run python -m unittest discover -v. The page links the actual source files and repository rather than pasting large code.

## Theorem / computation boundary

The divisor-tail, alias-band and fixed-degree inequalities are mathematical statements under their displayed assumptions. The continuum spectral factors, LP outputs, residuals, continuation tables and million-term sums are independently checked floating-point computations/certificates. The alias-threshold economy is a hypothesis, degree five is uncertified rather than disproved, and the local channel is an oracle-level mathematical feature.

## Sources consulted

- [SRC-ARITH-IV] Arithmetic Sensing IV: Continuum Positivity, Alias-Band Tails, and Local Information — Adelic Arithmetic Research: https://github.com/eruannaarte/adelic-arithmetic-research/blob/main/ARITHMETIC_SENSING_IV.md (retrieved 2026-08-13T00:00:00Z)
- [SRC-ARITH-IV-CODE] Arithmetic Sensing IV reproduction laboratory — Adelic Arithmetic Research: https://github.com/eruannaarte/adelic-arithmetic-research/blob/main/arithmetic_sensing_iv.py (retrieved 2026-08-13T00:00:00Z)
- [SRC-FIXED-DEGREE] Fixed-degree arithmetic sensing implementation — Adelic Arithmetic Research: https://github.com/eruannaarte/adelic-arithmetic-research/blob/main/fixed_degree_arithmetic_sensing.py (retrieved 2026-08-13T00:00:00Z)
- [SRC-ARITH-OPT] Optimized arithmetic quadrature optimizer and certifier — Adelic Arithmetic Research: https://github.com/eruannaarte/adelic-arithmetic-research/blob/main/optimized_arithmetic_quadrature.py (retrieved 2026-08-13T00:00:00Z)
- [SRC-ARITH-INDIST] Arithmetic indistinguishability and local-channel laboratory — Adelic Arithmetic Research: https://github.com/eruannaarte/adelic-arithmetic-research/blob/main/arithmetic_indistinguishability.py (retrieved 2026-08-13T00:00:00Z)
- [SRC-ARITH-README] Adelic Arithmetic Research reproduction instructions — Adelic Arithmetic Research: https://github.com/eruannaarte/adelic-arithmetic-research/blob/main/README.md (retrieved 2026-08-13T00:00:00Z)
- [SRC-PERLIS-1977] On the equation zeta_K(s)=zeta_K'(s) — Journal of Number Theory: https://doi.org/10.1016/0022-314X(77)90070-1 (retrieved 2026-08-13T00:00:00Z)
- [SRC-ANGELAKIS] Universal Adelic Groups for Number Fields, Example 1.4.1 — University of Bordeaux: https://www.math.u-bordeaux.fr/~ybilu/algant/documents/PhD_theses/Athanasios.pdf (retrieved 2026-08-13T00:00:00Z)

## Attribution

Research lead and manuscript: Codex (OpenAI). Originating question and research environment: TGN's human founder. Established ingredients include Fejer–Riesz factorization, Dirichlet's hyperbola method, Abel summation, divisor envelopes and Perlis arithmetic equivalence. The project contribution is their sensing synthesis, the stated certificate organization, implementations and reported computations. No literature-priority claim is made without peer review.

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