# Provenance, Evidence Boundaries, and Corrections

Prepared on 18 October 2026 for the *Mathematics of Sound* publication collection.

## 1. What “complete collection” means here

This package consolidates the research material available in the surviving workspace and the restored research journal. It includes the English publication text, a ninety-entry English research catalogue, mathematical appendices, code, exact and approximate computational records, audits, and the historical files from which the new text was prepared. It is complete with respect to those recovered materials. It is **not** a claim that every previously created temporary file has survived, or that missing data have been silently regenerated.

The original historical sources remain unedited in `evidence/history/`. This is important because an earlier candidate may be incorrect, a later audit may narrow its domain, and a later branch may reinterpret rather than replace an earlier result. The English catalogue reports the corrected scope. The historical record allows a reader to inspect how that scope was established.

This is an AI-assisted technical research report. “Proved in the notes” means an inspectable mathematical argument is supplied; “internally checked” means a separate check within this research process. Neither expression denotes external peer review, certification by a formal proof assistant, or established priority over the literature. No proof or disproof of the general Chowla cosine conjecture is claimed.

## 2. Surviving source families

The following counts describe the historical source directories before addition of publication-specific files. Manifest files and historical audit files are included in the counts.

| Archive location | Scope | Files | Provenance |
|---|---|---:|---|
| `history/previous_sources/` | Unnumbered groundwork and approaches 01–40 | 100 | Extracted from the preserved journal; every recorded byte count and SHA-256 hash matches |
| `history/recovered/` | Approaches 41–50 | 7 | Four reconstructed mathematical texts, recovery notice, recreated checker, and its output |
| `history/round5/` | Approaches 51–70 | 40 | Surviving separate notes, code, result files, audits, index, and manifest |
| `history/round6/` | Approaches 71–90 | 43 | Surviving separate notes, code, result files, audits, illustration assets, index, and manifest |
| `history/Chowla_Research_Journal.txt` | Earlier consolidated journal with embedded sources | 1 | Preserved original byte sequence |

The journal is 1,030,916 bytes. Its SHA-256 is:

```text
9d81250bfbe7dff8b90078e76e2eabfb41b6897812051aa3bdc4435bf1eb0721
```

For publication preparation, the catalogue pass recalculated the hashes and byte counts of all 100 source files against the embedded journal registry. All 100 matched. It also compared the 39 listed non-manifest files in the round-five manifest and the 42 listed non-manifest files in the round-six manifest; no mismatches were found. These are **integrity checks**, not fresh mathematical experiment runs. A hash proves that a version is identifiable, not that its mathematics is correct.

The archive-wide inventory and final manifest describe the finished package. The machine-readable [catalogue source map](catalogue_sources.json) connects each numbered English entry to its source paths, byte counts, and source hashes. The [history inventory](history_inventory.json) inventories the copied historical material more broadly.

## 3. What was lost in approaches 41–50

The texts in `history/recovered/` explicitly state that an earlier environment failure removed the original working files. They were reconstructed from surviving research statements. A reconstructed argument is available for scrutiny, but it is not the original source file, original execution log, or original certificate data.

The most consequential gap concerns **approach 42**, the proposed single-frequency local trap at \(A_0=\{1,4,5,6\}\). The prior research reported 1,980 exact rational witnesses for replacement frequencies below 500, together with several Bernstein positivity certificates. Their original witness list and certificate files were lost. The surviving reconstruction gives the polynomial formulas, approximate location information, the reported earlier bounds, and a full analytic proof for every replacement frequency at least 500. It does not itself restore the finite witnesses. Therefore:

- The infinite replacement tail is available as a complete reconstructed proof.
- The prior finite-check success is preserved as a historical report.
- The full local-trap theorem is not presented in the English catalogue as freshly machine-reproduced evidence.
- Any future regeneration must be dated and stored as a new verification record, rather than replacing the provenance notice.

Other reconstructed branches describe earlier finite experiments whose original outputs are absent. Their general mathematical arguments are printed again, where available. In **approach 50**, a new integer checker and result file were actually recreated and survive; they verify the explicit denominator-81 and denominator-128 certificates. Those surviving outputs are different in status from the lost unsuccessful trial vectors at other denominators. Failure of one trial vector would not prove that a different certificate at the same denominator is impossible in any event.

Sources: [recovery notice](history/recovered/README_RECOVERY.txt), [biology reconstruction](history/recovered/biology_41_43_reconstructed.txt), and [rounding reconstruction](history/recovered/quantization_50_reconstructed.txt).

## 4. Corrections that the English publication must preserve

### 4.1. Finite and unrestricted frequency universes differ

Approach 13’s relaxed mixed game has an unrestricted-domain value that cannot be asserted unchanged for every finite cutoff. The later clarification gives the finite-domain calculation separately. The catalogue cites both records and uses the corrected scope. A finite optimum elsewhere in the project likewise remains a theorem about that specified universe, not every positive integer frequency.

Source: [game-domain clarification](history/previous_sources/round3/clarification_game13.md).

### 4.2. A nonnegative contact need not have exactly quadratic order

At a zero of a nonnegative analytic real function, the vanishing order is at least two and may be higher. Approach 23’s boundary argument uses the weaker correct condition. The earlier wording was corrected and the correction retained in its audit.

Source: [analysis audit](history/previous_sources/round2/audit_21_23.md).

### 4.3. Rational-factor counting has a factor of \(q^2\) on both sides

For

\[
H=2m+\sum_{a\in A}(z^a+z^{-a}),\qquad
q^2H=\sum_j|R_j|^2,
\]

with Gaussian-integer coefficient polynomials \(R_j\), the total energy is \(2mq^2\), while the sum of positive Fourier coefficients on the left is \(Nq^2\). The corrected estimate is

\[
N\le 2m^2q^2-m.
\]

Its scale is \(\sqrt N/q\) for the induced bound on \(m\), not \(\sqrt N/q^2\). The English discussion uses this normalization. Historical wording about a previously too-weak estimate remains in the reconstruction for traceability.

Sources: [original denominator lemma](history/previous_sources/round2/geometry_24_26.md), [rounding reconstruction](history/recovered/quantization_50_reconstructed.txt), and [corrected denominator function](history/round5/67_denominator_function.txt).

### 4.4. Changing the upward shift changes the certificate question

For the seven-frequency example \(\{1,2,3,5,6,7,8\}\), denominator one is impossible at \(m=2\), but an explicit certificate exists at \(m=3\). The smaller-shift obstruction is not a counterexample to a statement that explicitly requires an additional positivity margin. The catalogue also distinguishes existence of one rational factor from existence of a finite sum of rational squared factors.

Sources: [phase/dilation distinctions](history/previous_sources/round3/phase_dilation_40.md) and [shift distinction](history/round5/67_denominator_function.txt).

### 4.5. A failed detector is not a small true minimum

Several branches prove upper bounds on the effectiveness of a particular lower certificate: fixed-degree moments, residue averages, ordinary-majority conditioning, fixed quantiles, zero-threshold Fermi–Dirac weights, or a specified envelope. These do not prove corresponding upper bounds on \(K(A)\). Consecutive frequencies are repeatedly useful precisely because they have a large true negative excursion that a weak detector misses.

Similarly, the finite minimum \(7/20\) in approach 87 concerns its Riesz-product certificate normalized by \(\sqrt N\) over subsets of \(\{1,\ldots,18\}\). It is not the unrestricted minimum of that certificate and not the extremal Chowla constant. General positivity of that raw detector was not separately proved in the source.

Source: [Riesz-product scope](history/round6/87_positive_riesz_product.txt).

### 4.6. Approximation is not the conjectured growth theorem

Approaches 08, 56, 62, and 83 provide ways to preserve or approximate a minimum. Their bounds do not prove that the minimum is at least a fixed multiple of \(\sqrt N\). The Chebyshev improvement concerns a sufficient polynomial degree in this project’s approximation argument. Without a complete literature comparison it must not be advertised as a new optimal method or a faster best-known algorithm. Its proof uses the unknown edge to exhibit a candidate polynomial; the moment optimization defining the certificate does not require that edge as an input.

Sources: [moment hierarchy](history/round5/62_moment_hankel_hierarchy.txt), [Chebyshev filter](history/round6/83_chebyshev_ground_edge_filter.txt), and [filter audit](history/round6/AUDIT_83_QUANTUM.txt).

### 4.7. Sampling does not certify the unsampled minimum without an error argument

A rational witness proves a lower bound for depth at one point. An approximate grid minimum is not by itself a continuous minimum. The grid example in approach 90 retains exact mean square and sufficient interpolation information while missing a much deeper excursion between nodes. The static plot is illustrative; the analytic identity and exact finite checks establish its stated claim.

Sources: [continuous grid certificate](history/previous_sources/round3/constructions_37_39.md) and [phase-grid counterexample](history/round6/90_quantized_phase_and_boundary_twist.txt).

## 5. Relationship between branches

The ninety entries were produced in successive rounds of hypothesis generation and checking. Their numbering measures bookkeeping, not scientific independence. The following connections are particularly important when describing the work publicly.

| Recurrent mechanism | Representative branches | What changes between versions |
|---|---|---|
| Frequency relations and compactness | 08, 30, 39, 51, 56 | Attainment, descent structure, random torus coding, extremal functions, and effective finite approximation |
| Polynomial spectral filters | 19, 47, 62, 83 | Variable of the polynomial, approximation target, and sufficient degree |
| Positive rational factorization | 26, 31–33, 40, 47–50, 67 | Arithmetic constraints, shifts, finite decision bounds, helper factors, rounding, and denominator estimates |
| Gibbs and information weighting | 03, 17, 35, 46, 63, 66, 82, 87, 88 | Proposed zero information, inverse problems, choice of weight, entropy budget, and saturation |
| Spectral and graph language | 01, 02, 14, 61, 78–83, 90 | Commuting versus free operators, finite compression, boundary conditions, and lower-edge filters |
| Phase information and averaging | 09, 12, 41, 45, 68, 72–74, 86 | Whether the original common-time orbit and signed mean are preserved |
| Positive weighted profiles | 11, 16, 25, 27–29, 44, 70, 89 | Attempts to transfer weighted positivity to distinct equal-amplitude frequencies |

The same counterexample can legitimately illuminate two different physical descriptions, but should not be counted twice as a new mathematical discovery. The identical-intensity example in 72 and 86 is an explicit instance. Standard constructions such as Fejér kernels, Chebyshev polynomials, Riesz products, and tight-binding operators are attributed in the full articles rather than being renamed as original methods.

## 6. Mathematical proof, exact arithmetic, and numerical exploration

The historical code contains more than one evidence type:

1. **General mathematical arguments** establish claims for infinite families or every set satisfying stated hypotheses.
2. **Exact certificates** use integer identities, rational intervals, or polynomial positivity decompositions to certify specified finite statements.
3. **Exhaustive finite searches** establish claims only in the enumerated universe, with their correctness depending on the enumeration and certificate logic.
4. **Floating-point searches and plots** locate candidates or illustrate behavior; they do not themselves certify unrestricted conclusions.

An exact negative value is a valid lower bound for \(K\), even if no global minimum was computed. Conversely, showing positivity of \(k+f_A\) on the entire circle gives an upper bound for \(K\). Both directions must be kept distinct. A proof using a rationally certified value found first by a numerical search is valid because the final certificate no longer depends on the numerical optimizer’s correctness.

A separate publication verification report records the selected scripts actually rerun while assembling the English material. It should be consulted for current environment details and execution results. This provenance note deliberately does not claim that all original experiments were rerun. Re-running a program in a disposable copy is preferable to overwriting an archival result file whose earlier hash and date are part of the evidence.

## 7. Physical and product scope

The negative crest factor is an exact identity for the stated equal-amplitude signal. The lattice operator is an exact auxiliary mathematical model. Quantum oscillator examples show what mean motion and measurement variance do or do not constrain. None of these facts alone demonstrates a new law of acoustics, negative physical energy, a quantum behavior of the Bass Plus application, or improved audible output.

The publication may explain why sound motivates the investigation and why missed peaks or lost phase information matter conceptually. Any future product claim needs its own implemented method, applicable signal assumptions, and relevant measurements. Arbitrary equalizer gains, phases, transient recordings, and noninteger frequencies are not automatically covered by the equal-unit-amplitude, full-period conjecture studied here.

## 8. Recommended language for future citations

Suitable: “The report records ninety tested research branches, including explicit counterexamples to proposed shortcuts and frequency-independent approximation bounds under stated assumptions.”

Suitable: “An AI-assisted internal check found no remaining error in the documented scope; no external peer review or formal verification is claimed.”

Unsuitable: “Ninety new mathematical discoveries,” “Chowla’s conjecture was solved,” “the best-known algorithm was improved,” or “quantum physics now improves the app’s sound.” Those stronger statements have not been established by this collection.

The corrected catalogue and full appendices should be cited for current claims. Historical hypotheses should be quoted as historical hypotheses, with their later correction or rejection attached.