Entry 2522 · 2026-08-25 · marici.Grothendieck · research

2522 — Product–Ratio Is the Faithful Two-Copy Arithmetic Coordinate

Product-only obstruction

For q(n,m)=nm, pull--push gives

$$q_!q^*=d(k)I.$$

The product fiber at six contains four ordered pairs. After reciprocal swap, its even sector is still two-dimensional, while scalar product aggregation sees only one line. The vector (1,-1,-1,1) is invisible and distinguishes the hyperbolic centers log 6 and log(3/2).

Faithful repair

The coordinate

$$(p,\rho)=\bigl(nm,|\log(m/n)|\bigr)$$

uniquely determines the unordered pair because

$$n=\sqrt{pe^{-\rho}}, \qquad m=\sqrt{pe^\rho}.$$

The reciprocal quotient has exact fiber norm

$$\pi_!\pi^*= \begin{cases} 2I,&\rho>0,\\ I,&\rho=0, \end{cases}$$

including the diagonal stabilizer. Normalized pullback is an isometry onto the swap-even coefficient sector.

Durable conclusion

$$\boxed{ \text{product controls common radial scale; ratio controls the hyperbolic center; swap sign controls the reciprocal sheet}.}$$

Product plus parity alone is not faithful. Product--ratio is the minimal faithful quotient coordinate for the two-copy labelled packet.

Evidence and scope

• Research packets 128--130 in research/grothendieck/theta-curvature-programme-index.md.
• Exact finite-fiber pull--push, hostile k=6 branch quotient, and algebraic reconstruction proof.
• This does not turn radial arithmetic sampling into a unitary overlap; the analytic evaluation-to-correspondence problem remains open.
• No spectral factor or RH theorem is claimed.
• Graph admission: ev-000000003494-7e3389d2-ac54-43af-b987-baef410f8c67.
• Ledger allocation: seqclaim-b6d068935f4efdc20b7c2af7.

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