# n55 — pre-registration: the derived clock (before any code)

Source: the Einstein memorandum (n55_einstein_memorandum.md). Claims to
be tested, converted to gates. The load-bearing new objects:
- M_ad(x) = 2 sum_{n!=0} |<n|d_x psi0>|^2/(E_n - E_0): for the Gaussian
  vacuum this is exactly (hbar = 1)
    M_ad = (1/4) sum_{ij} (X_ij)^2 / (w_i w_j (w_i + w_j)^3),
  X = eigenbasis representation of dK = sum_e (Delta_x dk*_e) P_e —
  the same Daleckii-Krein machinery as G with ONE extra resolvent
  weight 1/(w_i + w_j). (Derivation: two-phonon matrix elements of the
  Gaussian state; the FS pullback is the same sum with (w_i+w_j)^2.)
- V(x) = gamma_drain * C(x), C the information cost; embedding A:
  gamma_drain = const (a = 0); embedding B: gamma_drain ~ sqrt(k_local)
  (a = 1/2). No third embedding.
- omega^2(x) = V''(x)/M_ad(x); s_op and gamma_toy = 0.5/s_op - 1 as in
  n53/n54.

## Gates
- G1 (anchor): two-site closed form: for the relative mode of a 2-site
  system, M_ad must equal (d omega/d k)^2/(8 omega^3) analytically;
  match to 1e-10.
- G2 (THE LOAD-BEARING POWER): uniform k -> k(1+eps), eps = 0.05, 0.02:
  M_ad ratio = (1+eps)^{-5/2} within 1e-3. KILL: any other exponent
  buries the memorandum (referee's own words).
- G3 (the number, uniform limit — arithmetic once G2 holds, still
  measured): s_op(A) = 0.250 +- 1% -> gamma = 1; s_op(B) = 0.500 +- 1%
  -> gamma = 0. The identity gamma = (1-2a)/(1+2a) checked at both
  points.
- G3b (bump, reported not gated): the n54 trap+background architecture
  with the derived omega; tidal contamination expected at these scale
  separations, reported.
- G4 (EP as identity): trap widths/dipole geometries: uniform-limit
  responses identical to 1e-3 (they must be — same exponents); bump
  version reported.
- G6 (Stage-B consistency, reported): derived omega near a real
  staggered source (F = 0.15, 0.3): F-exponent and coefficient,
  compared with n54 Stage B.
- G5 (the Mach gate — thermal inertia): DEFERRED to n55b with its own
  registration: the finite-temperature adiabatic mass requires the
  thermal Bures/response derivation, and this program has been caught
  twice guessing semantics; we will not guess a formula. Deferred, not
  dropped.

## Declared limitations
- gamma_drain is an input in the toy; n55 converts the unknown
  embedding into one exponent a with a laboratory handle (the n48 knee
  across potentials/materials). The parent bath framework argues a = 0;
  arguing is not deriving; the derivation of a is the program's true
  Stage C endpoint.
- Static substrate, 1D, harmonic clocks only, as before.

## RESULTS (same day; n55_derived_clock.py) — ALL GATES PASS
- G1 anchor: 3.0e-16.
- G2, THE LOAD-BEARING POWER: M_ad ratio matches (1+eps)^{-5/2} to
  8.8e-7 (eps=0.05) and 3.6e-7 (eps=0.02). The adiabatic inertia IS the
  gap-weighted fidelity metric with the predicted five-halves scaling.
  The memorandum survives its own kill clause.
- G3, THE NUMBER: embedding A (state-independent monitor, a=0):
  s_op = 0.25002 -> gamma_toy = +0.9999. GENERAL RELATIVITY, nothing
  tuned. Embedding B (dressed monitor, a=1/2): s_op = 0.50002 ->
  gamma_toy = -0.0000 (Nordstrom). The identity
  gamma = (1-2a)/(1+2a) is verified as arithmetic at both points.
  Clocks slow in the well in both embeddings: the D3 sign is derivably
  right at last.
- G3b (reported): bump version s_op = 0.415 — tidal-contaminated at
  these scale separations, as declared; the uniform limit carries the
  gate.
- G6 (reported): near-source F-exponent 1.261 — the matter-clock linear
  channel persists in the derived clock, with the same contamination
  n54 flagged.
STANDING CLAIM after n55, stated with its condition: the drainage
theory's weak-field PPN gamma equals (1-2a)/(1+2a), where a is the
scaling dimension of the vacuum's monitoring rate. gamma = 1 — general
relativity — if and only if the ledger's clock does not care what it
prices (a = 0), which is the parent bath architecture's natural
reading. Cassini's bound reads |a| < 5.8e-6. The cross-prediction
stands: the n48 decoherence knee must be identical across materials
and across gravitational potentials to ~6e-6 per unit delta-k/k, or
gamma is not 1 and the theory dies by the solar system. What remains
for Stage C proper: DERIVE a in the parent framework (large-N bath),
with its own registration. EP status: an identity of scaling
dimensions — charge (k^-2), inertia (k^{-5/2}), and the metric's two
sectors all descend from one geometric object and one rate.