# OP1 Bridge V1: Scalar Classical iid/effective-iid Finite-Grid Fisher Certificate Theorem

Status: theorem-development proof draft for OP1 Bridge V1. Pro proof-draft review returned `op1_bridge_v1_proof_draft_ready_for_formal_review`, with conditional path `op1_bridge_v1_ready_for_accepted_language_after_formal_review` for the narrow finite-grid scalar classical iid/effective-iid Fisher certificate claim. Formal metrology/probability review returned `op1_bridge_v1_clear_after_minor_text_fixes`; this revision applies those text fixes for the narrow V1a finite-grid certificate theorem. The bridge-specific schema/verifier and V1a/V1b fixture layer are implemented in `cct-public-replication/schemas/op1_bridge_v1_certificate.schema.json`, `cct-public-replication/theorem_verifiers/verify_op1_bridge_v1.py`, and `cct-public-replication/examples/op1_bridge_v1_certificate_cases.json`.

Controlling lineage:

- Accepted algebraic OP1 theorem note: `open-theorem-working-notes/op1-regime-local-rfh-theorem-note-2026-05-20.md`
- Accepted algebraic theorem-status review: `external-reviews/op1-formal-theorem-status-review-2026-06-10.md`
- Bridge target: `open-theorem-working-notes/op1-metrology-probability-bridge-target-2026-06-11.md`
- Bridge V1 assumption-freeze note: `open-theorem-working-notes/op1-bridge-v1-regular-classical-iid-lan-sufficient-condition-2026-06-11.md`
- Bridge V1 assumption-freeze Pro confirmation: `external-reviews/op1-bridge-v1-assumption-freeze-confirmation-pro-review-2026-06-11.md`
- Bridge V1 proof-draft Pro review: `external-reviews/op1-bridge-v1-proof-draft-pro-review-2026-06-11.md`
- Bridge V1 formal metrology/probability review: `external-reviews/op1-bridge-v1-formal-metrology-probability-specialist-review-2026-06-12.md`

## Scope

Bridge V1 is a finite-grid sufficient-condition theorem. It shows when a scalar classical iid / effectively-iid metrology experiment supplies the structural OP1 certificate:

```text
R(B) = A(B) + D(B),
A(B2) >= A(B1)(B2/B1)^(-p),
D(B2) >= D(B1) >= 0.
```

It does not prove statistical interval coverage, QFI-resource-class coverage, nuisance-parameter projection, robust/nonregular risk, adaptive-design routing, hardware evidence, OP2 command-effect language, Layer-3 support, or Tau-X capability.

## Definitions

Let:

```text
B_1 < ... < B_n,  n >= 2,
```

be a positive scored grid declared before OP1 scoring. Let:

```text
Theta subset R
theta0 in Theta
H > 0
```

be declared before scoring, with:

```text
theta0 + h / sqrt(J_eff(B_i, theta0)) in Theta
```

for every scored `B_i` and every `|h| <= H`.

For each scored `B_i`, the statistical experiment is one of:

```text
P_{theta,B_i} = product law for n(B_i) iid observations,
```

or:

```text
P_{theta,B_i} is conditionally product after declared conditioning,
```

or it has a frozen effective-iid certificate with declared conditioning variables, declared `n_eff`, or deterministic certified `J_eff` values/bounds derived from a frozen law, no post-hoc conditioning choice, and no long-memory, block-dependence, random-stopping, or conditioning-selection route hidden inside the base V1 certificate.

The scalar target is:

```text
g: Theta -> R,
```

where `g` is differentiable on the declared local set and:

```text
0 < |g'(theta0)| < infinity.
```

V1 uses one coordinate convention:

```text
J_eff(B, theta0) = classical Fisher information for theta at theta0.
```

The target-risk normalization, including the contribution of `g'(theta0)` when applicable, is carried by `q(B, theta0, H)`. Target-coordinate Fisher information is not used in V1 unless a later route explicitly reparameterizes to `eta=g(theta)` with a declared local inverse branch and restates the local alternatives in the `eta` coordinate. There are no nuisance parameters in V1, except fixed known constants outside `theta` that are not estimated, profiled, selected, or conditioned on after the fact.

Let `delta_B` be a fixed estimator rule or fixed estimator-selection policy. The complete mapping:

```text
B -> delta_B
```

including preprocessing, hyperparameters, and any allowed data-dependent subroutine, is a single measurable rule/policy frozen before scoring. Outcome-dependent stopping, adaptive design, post-hoc estimator switching, projection selection, or outcome-conditioned continuation is outside V1 unless it is included in that single frozen rule, the estimator/risk is defined on every outcome path, and all resources are charged or routed through a separate adaptive-policy theorem. Projection status in V1 is:

```text
projection_status = none_for_scalar_V1.
```

Define the LAN-local / local-uniform squared risk:

```text
R_H(B; theta0)
= sup_{|h| <= H}
  E_{theta0 + h / sqrt(J_eff(B, theta0))}
  [ (delta_B - g(theta0 + h / sqrt(J_eff(B, theta0))))^2 ].
```

This is the finite-grid risk object being scored. LAN may motivate candidate `J_eff`, `q`, and risk laws, but the theorem engine is the finite-grid certificate itself: exact risk certification, certified monotone remainder absorption, or the separate direct all-pair `R` replacement route.

## V1a Theorem: Fisher Certificate Implies OP1 Bridge V1

Fix one predeclared:

```text
0 < p <= 1.
```

Assume the following regular experiment conditions hold on the declared finite grid and local set:

```text
classical scalar experiment with Theta subset R;
iid product law or explicitly certified effective-iid law;
DQM/LAN or equivalent local regularity on the declared local set;
finite second moments and uniform integrability for the declared squared risk;
scalar target regularity and identifiability;
actual finite-grid risk certificate, not only asymptotic LAN approximation.
```

Assume that, for every scored grid point `B_i`, there are finite values:

```text
0 < J_eff(B_i, theta0) < infinity,
0 < q(B_i, theta0, H) < infinity,
0 <= D(B_i, theta0, H) < infinity,
0 < R_H(B_i; theta0) < infinity,
```

and the exact finite-grid certificate:

```text
R_H(B_i; theta0)
= q(B_i, theta0, H) / J_eff(B_i, theta0) + D(B_i, theta0, H).
```

The `J_eff` value or bound used for scoring at each grid point is predeclared, or is a deterministic certified value/bound derived from a frozen law for the declared full experiment before OP1 scoring. `J_eff` may not be fit, rescaled, or selected after seeing `R_H`, `q`, `D`, `alpha`, or route outcomes.

Assume that, for every scored pair `B_i < B_j`,

```text
J_eff(B_j, theta0) <= J_eff(B_i, theta0) (B_j/B_i)^p,
q(B_j, theta0, H) >= q(B_i, theta0, H),
D(B_j, theta0, H) >= D(B_i, theta0, H).
```

Assume also that `q` is independently declared or certified before OP1 scoring, or derived from a fixed estimator-risk theorem independent of the OP1 pass decision. It may not be chosen after selecting `D` to force the decomposition. `D` is same-process, nonnegative, and monotone; it may be zero, but it may not be a negative covariance, cancellation term, decreasing back-action term, or post-hoc residual.

The disturbance certificate has the auditable content:

```text
D_same_process = true
D_nonnegative = true
D_monotone_all_pair = true
D_not_posthoc_residual = true
D_no_negative_or_canceling_terms = true
```

Define:

```text
A(B_i) = q(B_i, theta0, H) / J_eff(B_i, theta0),
Delta(B_i) = sqrt(R_H(B_i; theta0)),
alpha[B_i,B_j]
= - log(Delta(B_j)/Delta(B_i)) / log(B_j/B_i).
```

The positivity assumption `R_H(B_i; theta0) > 0` for every scored `B_i` makes `Delta` and `alpha` well-defined.

Then, for every scored pair `B_i < B_j`,

```text
alpha[B_i,B_j] <= p/2 <= 1/2.
```

### Lemma 1: Fisher Growth And Non-improving q Give The OP1 A-envelope

Let:

```text
r = B_j / B_i > 1.
```

The all-pair Fisher growth certificate gives:

```text
J_eff(B_j, theta0) <= J_eff(B_i, theta0) r^p.
```

Since `q(B_j, theta0, H) >= q(B_i, theta0, H)` and all quantities are positive:

```text
A(B_j)
= q(B_j, theta0, H) / J_eff(B_j, theta0)
>= q(B_i, theta0, H) / (J_eff(B_i, theta0) r^p)
= A(B_i) r^(-p).
```

Thus the estimator-improvement component satisfies the all-pair OP1 lower power envelope.

### Lemma 2: Exact Risk Certificate Supplies The OP1 Structural Risk

By assumption:

```text
R_H(B_i; theta0) = A(B_i) + D(B_i, theta0, H)
```

for every scored `B_i`, with:

```text
A(B_i) > 0,
D(B_i, theta0, H) >= 0,
D(B_j, theta0, H) >= D(B_i, theta0, H).
```

Therefore the finite-grid risk `R_H` is an OP1 structural `R=A+D` certificate on the scored grid.

### Lemma 3: OP1 Algebraic Envelope Applies

By Lemma 1 and Lemma 2, the accepted OP1 algebraic theorem applies to `R_H`. For `r=B_j/B_i`:

```text
R_H(B_j; theta0)
= A(B_j) + D(B_j, theta0, H)
>= A(B_i) r^(-p) + D(B_i, theta0, H)
>= A(B_i) r^(-p) + D(B_i, theta0, H) r^(-p)
= R_H(B_i; theta0) r^(-p).
```

Taking square roots gives:

```text
Delta(B_j) / Delta(B_i) >= r^(-p/2).
```

Taking logarithms yields:

```text
alpha[B_i,B_j] <= p/2.
```

Since `0 < p <= 1`, this gives `alpha[B_i,B_j] <= 1/2`.

## V1b Proposition: Direct All-Pair Risk Replacement Route

The direct replacement route is separate from the regular Fisher certificate bridge. A record that satisfies this proposition but lacks the regular iid/effective-iid Fisher assumptions may pass only:

```text
op1_bridge_direct_all_pair_R_replacement_pass
```

not:

```text
op1_bridge_regular_classical_iid_lan_certificate_pass.
```

Suppose a declared finite-grid local-uniform risk satisfies, for one predeclared `0 < p <= 1`, every scored pair `B_i < B_j`, and `r=B_j/B_i`:

```text
R_H(B_j; theta0) >= R_H(B_i; theta0) r^(-p).
```

Then:

```text
Delta(B_j) / Delta(B_i) >= r^(-p/2),
```

and therefore:

```text
alpha[B_i,B_j] <= p/2 <= 1/2.
```

This route is not vague sandwich language. The certificate must state the all-pair finite-grid risk bound directly, including the propagated slope constant.

## Remainder Handling

If a LAN or finite-sample calculation gives:

```text
R_H(B; theta0)
= q(B, theta0, H)/J_eff(B, theta0) + D(B, theta0, H) + e(B, theta0, H),
```

then Bridge V1 requires one of:

```text
1. e is absorbed into a certified nonnegative monotone D term;
2. the record supplies the V1b direct all-pair risk replacement certificate;
3. the route defers as op1_bridge_defer_lan_remainder_uncertified.
```

A generic small-asymptotic LAN remainder is not a finite-grid OP1 bridge certificate.

## Denominator And Post-selection Rule

Bridge V1 requires:

```text
B is predeclared and not reverse-engineered from achieved risk;
resource denominator is complete before scoring;
post-selection is absent, or R_H is computed for the declared full experiment
with estimator/risk defined on failures, aborts, and stopped paths, and failures /
success probability are denominator-charged by a fixed pre-scoring rule;
setup, calibration, compute, synchronization, side information, and prior tables
are charged or routed away.
```

Retained-record-only scoring is outside V1 unless failures and success probability are included in both the full-experiment risk definition and the denominator policy. Aborted or stopped paths must either be part of the same risk object or route away from V1.

## Boundary Counterexamples

The following cases route outside the regular Fisher bridge:

```text
pointwise or superefficient risk selected at theta0;
biased shrinkage or hidden-prior side information;
post-hoc q fit from observed R_H and selected D;
post-hoc J_eff fit or rescaling after observing R_H, q, D, alpha, or route outcome;
target-coordinate mismatch between J_eff and theta-coordinate local alternatives;
decreasing finite-window q from estimator-efficiency maturation;
global J_eff <= C B^p but local all-pair FI spike;
uncertified LAN remainder crossover;
effective-iid conditioning selected after seeing slope behavior;
vector parameter or nuisance direction smuggled through a scalar target;
flat, nonsmooth, or boundary target functional;
local alternatives leaving Theta;
theta0 or H selected after inspecting slope, q, D, or route outcome;
outcome-dependent stopping with uncharged denominator;
post-selection conditional FI inflation;
QFI/classical-FI mismatch or unattained QFI;
long-memory dependence without effective-iid certificate;
heavy-tailed or nonregular risk;
negative or canceling disturbance term.
```

## Formal Review Questions

The formal review should decide:

```text
1. Is the V1a theorem sufficient for accepted OP1 Bridge V1 language after review?
2. Should V1b direct all-pair risk replacement stay in this theorem object or be held for a later schema route?
3. Are the local-uniform risk and finite-grid exact certificate assumptions tight enough?
4. Are any DQM/LAN, uniform-integrability, or finite-moment assumptions missing from the proof statement?
5. Should a machine-readable bridge certificate schema be added before accepted bridge language?
6. Which counterexamples should become executable bridge fixtures first?
```

## Public / Private Boundary

This proof draft is public-safe. It contains theorem objects, route labels, and synthetic certificate logic only. It contains no material recipes, sample sources, hardware constants, bench values, optical layouts, RF/EM geometries, waveform searches, calibration tolerances, protected notebooks, mission implementation values, or recipe-level experimental advantage.
