# Riemann C1 Cell Set v0 - Explicit-Formula Parity Scaffold > Concrete v0 cell set for > [`FRONT_A_FUNCTIONAL_EQUATION_READING.md`](FRONT_A_FUNCTIONAL_EQUATION_READING.md). > Status: drafted, desk-auditable, unreviewed, unrun. Filed 2026-05-28. This file pins the first reviewable cell set for the Riemann Front-A reading. It does **not** run a smoothed explicit-formula computation, does **not** claim a new explicit formula, and does **not** test RH. Its job is to make the functional-equation reading rejectable before any numerical residual is interpreted. ## Internal Audit Addendum - Linearity Verdict Post-filing desk audit sharpened the status of this cell set: it is soundly constructed, but it appears to operationalize the vacuity rather than escape it. The reason is structural. The explicit formula is linear in the test function. Therefore: - `C1-O` is forced-vacuous on the zero side under the symmetric `+-gamma` window: an odd test function summed over a symmetric multiset is zero before any arithmetic data is loaded. - `C1-E` is a classical explicit-formula consistency check. A clean pass would verify the normalization, data, truncation floor, and implementation; it would not by itself be a Sundog Front-A edge. - `C1-M` is determined by linearity: `h_M = h_E + epsilon * h_O`, so the mixed row is an implementation linearity check, not an independent receipt. This addendum does not delete the cell set. It records the likely disposition: without reviewer rescue, the cell files `R-C1-NEG-A` (Front-A vacuity) or `R-C1-NEG-D` (competitor / standard-treatment dominance), with `R-C1-NEG-B` kept as the useful hygiene guard against identity-zero laundering. Secondary scale note: with `A = 10`, only the first handful of zeros and very small primes carry meaningful numerical mass under the Gaussian. The stated `N_zero = 5000` and prime cutoff `p <= 1,000,000` are conservative inert ceilings for this smoke-scale cell, not evidence that the full window is actively tested. Any future executable version must either right-size the cutoffs to `A` or explicitly label the run as a small-term pipeline smoke. ## 1. Regime And Formula Contract The cell set works with a standard smoothed explicit formula for the completed Riemann zeta function on the critical line. The formula is treated as an imported object, not a Sundog-derived object. **Formula contract.** - Use one Weil / Riemann explicit-formula normalization cited from the lit-pass source list before execution. - Express the formula as a comparison between: - a zero-side sum over ordinates `gamma`; - a prime-side sum over prime powers; - archimedean / gamma-factor terms; - pole / main terms required by the chosen normalization. - The implementation must include a `formula_normalization` manifest field with source, equation number or section, Fourier-transform convention, and all constants. **Pre-execution hold.** This v0 file pins the cell set but does not pin a final equation-number-level normalization. A reviewer or maintainer must fill that normalization before any run. If no such normalization can be fixed without changing the cell after seeing output, file `R-C1-NEG-C` (floor / normalization failure). ## 2. Symmetry Object The admitted symmetry is the functional-equation reflection, rendered on the critical-line ordinate as: ```text R(t) = -t Z2 = {id, R} ``` For any registered test function `h(t)`, define: ```text h_even(t) = (h(t) + h(-t)) / 2 h_odd(t) = (h(t) - h(-t)) / 2 ``` The scaffold does not ask whether the Z2 symmetry exists. It asks whether the receipt separates two categories: ```text forced parity cancellation informative arithmetic residual ``` Forced cancellation may be identity-zero. It must be labeled as such and may not be counted as an empirical residual pass. ## 3. Kernel Family The v0 kernel family is a Gaussian pair with one odd perturbation. Let `A = 10`. ```text h_E(t) = exp(-(t / A)^2) h_O(t) = (t / A) * exp(-(t / A)^2) h_M(t) = h_E(t) + epsilon * h_O(t), epsilon = 0.1 ``` Roles: - `h_E` is the residual-bearing even test. It is the only row allowed to support a bounded operational residual receipt. - `h_O` is the identity-zero negative control. Under a symmetric zero window and a correctly parity-aware formula read, this row is expected to cancel by parity. It is not evidence of signal. - `h_M` is the laundering guard. Its odd component must not improve or rescue the even residual; the mixed row should reduce to the `h_E` residual plus the separately labeled forced cancellation of `epsilon * h_O`. The value `A = 10` is a v0 review default, not an optimized scale. If a reviewer rejects it before execution, replace this file with a new v0.1 cell set. If it is changed after inspecting output, file `R-C1-NEG-C`. ## 4. Cutoffs And Source Data Default v0 cutoffs: | Object | v0 value | Role | | --- | --- | --- | | Zero source | Odlyzko `zeros1` | zero-side ordinates | | Zero window | first `N_zero = 5000` positive ordinates, mirrored to `+-gamma` | symmetric zero-side truncation | | Observed zero ceiling | `T_zero = 5447.861998301` from Probe 01 v1 | registered window read-back | | Prime cutoff | prime powers with base prime `p <= 1,000,000` | prime-side truncation | | Prime-power depth | all `k` with `p^k <= 1,000,000` | finite prime-power slate | | Arithmetic precision | at least double precision for v0 smoke; higher precision if reviewer requests | numerical floor input | The zero window is symmetric by construction. The run must never use only positive ordinates for an odd-sector verdict. ## 5. Registered Rows | Row | Test function | Expected parity read | Can support a bounded positive? | | --- | --- | --- | --- | | `C1-E` | `h_E` | even sector, residual-bearing | pipeline consistency only unless reviewer finds non-vacuous audit edge | | `C1-O` | `h_O` | odd sector, identity-zero negative control | no | | `C1-M` | `h_M` | even residual plus labeled odd cancellation | no independent row; linearity check only | Per Probe 01 v1, an identity-zero row is a null / guardrail receipt. It is not a discovery. ## 6. Error Floor And Threshold Rule The residual threshold must be derived before execution from a floor independent of the observed residual. The v0 floor is: ```text tau_floor = 10 * (zero_tail_bound + prime_tail_bound + archimedean_quadrature_bound + source_precision_bound + floating_point_bound) ``` Required components: - **Zero tail bound.** Bound the contribution of omitted zeros `|gamma| > T_zero` using the chosen zero-counting density and the Gaussian decay of the registered `h`. - **Prime tail bound.** Bound omitted prime powers above the cutoff using the chosen Fourier-transform convention and a monotone majorant for the Gaussian transform. - **Archimedean quadrature bound.** Pin a deterministic quadrature tolerance for the gamma-factor integral or term in the chosen normalization. - **Source precision bound.** Propagate Odlyzko `zeros1` source precision through the zero-side sum. - **Floating-point bound.** Pin a conservative roundoff bound or require a higher-precision implementation. If any component cannot be bounded before looking at the residual, file `R-C1-NEG-C`. ## 7. Disposition Branches | Branch | Trigger | Disposition | | --- | --- | --- | | `R-C1-A` | `C1-E` residual <= `tau_floor`, `C1-O` labeled identity-zero, `C1-M` decomposes cleanly, and reviewer accepts the forced-cancellation / residual split as a non-vacuous audit distinction | bounded operational receipt | | `R-C1-NEG-A` | reviewer says the cell is only standard even-test-function bookkeeping | Front-A vacuity | | `R-C1-NEG-B` | `C1-O` or the odd part of `C1-M` is presented as empirical evidence | identity-zero laundering | | `R-C1-NEG-C` | formula normalization, cutoffs, or floor components cannot be fixed pre-run | floor / normalization failure | | `R-C1-NEG-D` | Connes / standard explicit-formula treatments already supply the exact operational distinction claimed | competitor dominance | | `R-C1-FAIL` | `C1-E` residual > `tau_floor` after an admitted run | residual breach in registered cell | No branch may be rescued by changing `A`, `N_zero`, prime cutoff, formula normalization, or floor components after output inspection. ## 8. What Would Count As Non-Vacuous After the linearity audit, the cell is presumed vacuity-boundary unless all three are true: 1. the reviewer accepts that the forced-cancellation / residual split is a real audit distinction, not just decorative language for standard parity; 2. the run can compute `C1-E` against an independently fixed floor; 3. the report keeps `C1-O` and the odd part of `C1-M` in the negative-control bucket rather than counting them as evidence. If these fail, the correct result is not a better cell. It is the named negative. ## 9. Review Questions Ask the external reviewer: 1. Is the selected explicit-formula normalization appropriate for the three registered rows? 2. Is `h_O` an admissible odd perturbation under that normalization, or should the odd negative control be expressed differently? 3. Is `A = 10` a reasonable desk-audit scale, or should a new pre-execution v0.1 cell be filed with a different scale? 4. Are the proposed tail-bound components sufficient to keep the residual floor independent of the observed result? 5. Does this cell add any audit value beyond ordinary explicit-formula bookkeeping? ## 10. Status - Drafted: 2026-05-28. - Internal audit: linearity verdict added 2026-05-28; expected disposition is `R-C1-NEG-A` / `R-C1-NEG-D` unless a reviewer treats the labeling-hygiene distinction as a real audit contribution. - Execution: none. - Review: none. - Formula normalization: not yet equation-number-pinned. - Floor calculator: not written. - Promotion: blocked until review and a pre-run formula/floor fill. ## Cross-References - [`FRONT_A_FUNCTIONAL_EQUATION_READING.md`](FRONT_A_FUNCTIONAL_EQUATION_READING.md) - Front-A reading note this cell set instantiates. - [`receipts/2026-05-28_probe01_pathi_parity_decomposition.md`](receipts/2026-05-28_probe01_pathi_parity_decomposition.md) - negative-control precedent for identity-zero labeling. - [`../RIEMANN_LITPASS_MEMO.md`](../RIEMANN_LITPASS_MEMO.md) - prior-art and competitor boundaries. - [`../SUNDOG_V_RIEMANN.md`](../SUNDOG_V_RIEMANN.md) - main ledger and promotion criteria.