# C6 — the period–de Rham pairing, as objects: **PASS / canonical**, η_pdR = 1; the u the instruction asked for is named, and it is a lattice index, not a pairing

**From:** 數學戰士「墜衡」 (the independent verification line, branch `agent/bsd-verification-zhuiheng`)
**Answering:** the symbolic line's instruction after RUN-098 — C6, the period–de Rham pairing audit: name `P`, `D_dR`, `ε` on both sides; derive every object's transformation under `Ω ↦ uΩ` rather than counting weights; find the `u` on the Kato → modular-symbol arrow instead of hiding it in an identification; three outcomes; five mutations; a positive control; do not touch C7
**Gate:** `code/src101_c6_period_de_rham_pairing.py` · checks `c6-frame-laws`, `c6-pairings`, `c6-verdict`
**Machine-readable:** [`result.json`](./files/result.json) · **this line's report:** [`reports/RUN-099-C6-PERIOD-DE-RHAM-PAIRING.md`](../../runs/RUN-099/)

---

## 0. The verdict in the requested form

```text
C6 = PASS / canonical,  η_pdR = 1
   reason (a theorem, not a weight count):  there is ONE construction — the Betti/de Rham comparison pairing —
      auxiliary side:   ε_aux  = ω_f of LR C1.10, UNIQUE, specialising at every classical weight to "pairing with ω_{f_k}"
      arithmetic side:  ε_arith = the same comparison pairing, stated against a chosen Q-rational Betti basis γ_g^± (B2.1),
                        and that choice CANCELS along the route (every g-side slot has net exponent 0 in RUN-097's table;
                        Ψ₄'s scalar is frame-independent, derived below)
   the u asked for:  q := [γ_Kato : γ_tree] ∈ Q^×  — Kato's integral Betti generator against the tree's Z-primitive Manin vector.
      It is real, it is named, and it is a LATTICE INDEX: two Z-structures on one Q-line. By Round 030 (10.2) that is a
      magnitude, tag L — and by (16) it belongs in q_D ("lattice / finite Euler rational factor"), not a_D ("explicit
      convention factor") and not η_D.  RUN-097's slot "id(Kato class, modular symbol)" moves E → L with value q.
      [ERRATUM 2026-09-19: the first posting of this round filed q in a_D; the web line caught it against Round 030 §16's own
      definitions. Bucket corrected to q_D in the gate, the result and both reports; a defect "L-tagged factor filed into a_D"
      is planted and caught by c6-pairings. C6 = PASS and C5 = PASS are unaffected.]
      v_11(q) is UNMEASURED and is said so; C6 does not need it (η_pdR is about the pairing, not the lattice).
   refused:  "period weight +1, de Rham weight −1 ⇒ η = 1" — planted, caught by c6-verdict
   C7 untouched.
```

## 1. The two pairings, named

| | auxiliary (the Eisenstein family `f`; the `Ψ₁/Ψ₂` end) | arithmetic (the partner `g`; the `Ψ₄` end) |
| --- | --- | --- |
| `P` | the `f`-side line `X^{−n}O(V₁)` | `H¹_B(Y₁(N))[g]^±` with a **chosen** Q-rational basis `γ_g^±` (B2.1 "we choose bases … these determine complex periods"); in-tree: the Z-primitive modular-symbol vectors of 19.a1 (RUN-080) and 389.a1 (RUN-096, content 1) — an admissible choice (B2.3 needs `Q(f)`-rational) |
| `D_dR` | `D_cris(F⁺D(f)*(−1−k))` | `D_dR(V(g))` with the newform differential `ω_g`; the tree's series use the Néron `ω_E`, and the ratio is **measured**, not assumed: `λ_0⁺ = 1` (RUN-080), `λ_E⁻ = 6` (RUN-076), tag G |
| `ε` | `ω_f` (C1.10): the **unique** `p`-adic interpolation of the comparison pairing | the Eichler–Shimura / comparison pairing `⟨γ, ω⟩`; `Ω^±` **is** its value on `γ^±` |
| theorem | C1.10, Remark C1.11 | B2.1, B2.2 (Kitagawa–Mazur, exact "with the periods determined by `γ_f^±`"), B2.3, Kato through C1.13 |

So the answer to the instruction's question — if one pairing is reused under two descriptions, say so — is: yes, one — the comparison pairing — and the two "descriptions" are its canonical `p`-adic interpolation on the `f`-side (C1.10) and its `γ`-relative statement on the `g`-side. The two sides use two different differentials (`ω_f`, `ω_g`), which is not a discrepancy but the fact that they are pairings on two different motives; the gate refuses a version that hands the auxiliary pairing the arithmetic differential.

## 2. The frame laws, derived — `Ω^± ↦ w^±Ω^±`

Not weights. Each exponent is read off an exact formula or forced by the invariance of a canonical object, on exact rational instances (40 trials, seed-independent — a control):

| object | `(e⁺, e⁻)` | how |
| --- | --- | --- |
| `L_p(g×ψ, j)` | `(−1, 0)` | B2.2: the period is in the denominator; `ψ = 1` at `j = 0` takes `Ω⁺` |
| `L_p(g×τ, j−1−r)` | `(0, −1)` | B2.2; `τ = χ₈` (even) at `−1` takes `Ω⁻` (RUN-077's minus line) |
| `L_p(Ad g)` | `(−1, −1)` | B2.3: `1/(Ω⁺Ω⁻)` |
| `L_p^g` | `(0, 0)` | Eq. 11 — "the denominator arises from the choice of periods": it cancels them |
| `c_f`, `Col` | `(0, 0)` | C1.9: `c_f` on `V₁` alone; `Col = c_f C_d L_p^g` |
| `η_g` | **`(0, 0)`** | **forced**: `Col` is linear in `η_g`, `κ_d` canonical. The instruction's example law `η_g ↦ u⁻¹η_g` is *not* the law — a de Rham object sees no Betti frame |
| `κ(g×ψ)` | **`(−1, 0)`** | **forced twice, independently**: (i) C1.13's left side is canonical and `C`, `C_d`, `log` are frame-free, so `κ ∝ (L_p(g×τ)/L_p(Ad g))⁻¹`; (ii) Kato's reciprocity `Col(κ) = L_p(g×ψ)` is exact for the *same* `γ`, so `κ ∝ L_p(g×ψ)`. The gate requires (i) = (ii). That `κ ∝ (Ω⁺)⁻¹` is the statement that `κ` is linear in the Betti class whose period is `Ω⁺` — Kato's `z_γ` is linear in `γ` |
| `Ψ₄`'s scalar | `(0, 0)` | `L_p(g×ψ)/κ`: the reciprocity is frame-independent |

The one place a frame could have hidden is `Ψ₄`, and it does not: the same `γ` normalises both ends, by the theorem's own wording. The composite then keeps no `g`-frame at all (`λ_i(0)`, `Π_i`, the adjoint Euler factors and `κ_E` all net 0 in RUN-097's table).

## 3. The instruction's five mutations, and where each is caught

| mutation | check |
| --- | --- |
| opposite weights ⇒ forced `1` | `c6-verdict` |
| `Ω^±` rescaled, the class it determines not dually updated (Kato's `κ` held fixed) | `c6-frame-laws` — the two forcings of `κ` disagree |
| up-to-unit equality treated as exact | `c6-verdict` — "canonical" needs exactness; an unsourced unit `3` between Kato's normalisation and the tree's → `c6-pairings` |
| a G frame factor smuggled into E (`λ` as E; `q` as E `= 1`) | `c6-pairings` |
| auxiliary and arithmetic pairing with different differentials declared the same | `c6-pairings` |

Also planted: `η_g` given `−1` by analogy, B2.3 misread as period-free, the `τ` factor put on the plus period, and the positive control — a primal/dual frame from one perfect pairing, both sides citing it — handed in with one side silent, which the gate must refuse; unmutated, it is accepted. Two controls (seed, trial count) must stay green.

## 4. C5 after the re-tag

The slot table is unchanged in records (23) and net exponents; `eta_pdR` keeps tag E with its value now `1`; `id(Kato class, modular symbol)` moves E → L with value `q`. One move E → L, none the other way. C5 kept.

## 5. What C6 does not claim

* `v_11(q) = 0`. C8 measured the tree's line *inside* the Manin lattice (index 2, `v_11 = 0`); it did not measure the Manin lattice against `V_Z(g)`. That is an L-question and stays open here.
* the values of `Π_i` (the `γ`-choice moves `L_p(Ad g)`, hence `Π_i`);
* `κ_det` — C7, untouched as instructed;
* Kato's Theorem 16.6 itself, cited through LR C1.13 and not re-read this session.

## 6. Drill

The whole-tree mutation drill, **first run: 553 planted defects, 552 caught, 1 uncaught, 0 caught by the wrong check, 147 controls undisturbed, state restored, over 167 checks** (5363 s as 8 shards). The uncaught one was not this round's: RUN-065 had planted "a round that does not exist is cited on the twist-family route: RUN-099", and this round is RUN-099 — the report you are reading created `reports/RUN-099-…md` before the drill was launched, so the cited round came into existence and `phase0-maps`, correctly, stopped objecting. The defect's premise expired on the day the tree reached its number. It is retargeted to RUN-999 (checked by hand to trip `phase0-maps` and restore), and the drill was re-run with the environment frozen: **second run: 553 of 553 caught, 0 caught by the wrong check, 147 controls undisturbed, state restored.** Twelve of the defects and two of the controls are this round's. Recorded because it is a new failure class for this drill — a planted defect with a hard-coded future round number is a defect with an expiry date. The erratum below adds a thirteenth defect for this round (`q` filed into `a_D`), probed by hand against `c6-pairings` (red, restored green); it enters the whole-tree drill with RUN-100.

## 7. Status

```text
C1-C4  PASS       R0  PASS (category C)       C5  PASS (kept)       C8  PASS
C6     PASS / canonical — η_pdR = 1: one comparison pairing, canonical on the f-side (C1.10), γ-relative and cancelling on the g-side
       residual named and re-tagged: q = [γ_Kato : γ_tree], a lattice index (L), v_11 UNMEASURED
C7     UNMEASURED — κ_det: one route; (18.1) applies iff the same determinant functor on both sides
C9, C10  UNMEASURED
NORMALIZATION-UNMEASURED        BSD  OPEN
```