# n51a — the drainage dichotomy: proofs Setting: translation-invariant quadratic drainage. A lattice vacuum with couplings k_e = 1 + dk_e; consumption c_i at sites; a per-site sum rule A dk = -c (A = incidence, unsigned or signed); drainage = the minimizer of a positive-semidefinite quadratic cost Q(dk) = (1/2) dk^T G dk with G translation-invariant and positive-definite on edge space. Local observables are functionals of the couplings in a bounded neighborhood. Throughout, the exact solution of the constrained minimum gives the response kernel: dk = -G^{-1} A^T W^+ c with W = A G^{-1} A^T, and the configuration energy E(c) = (1/2) c^T W^+ c. Everything below follows from the symbol structure of W. --- ## Theorem 1 (bipartite half: no long-range linear response) On a bipartite lattice with the UNSIGNED sum rule, let O be any local observable that is even under the sublattice swap (cell-averaged / parity-neutral). Then the connected response of O to a distant source of strength F decomposes as: (i) a linear-in-F part that is TIDAL — suppressed by one gradient of the field envelope relative to the potential — plus terms decaying on the microscopic correlation length; (ii) a quadratic-in-F part proportional to the SQUARED envelope, which is the leading long-range response. In particular no observable in this class exhibits a long-range clock shift linear in the source: gravitational redshift is impossible. **Proof.** (1) *The only long-range channel sits at the staggered point.* The first-order field response in Fourier space is dk^(1)(q) = -Ghat(q)^{-1} Ahat(q)^dagger What(q)^{-1} chat(q), with What(q) = Ahat(q)^dagger Ghat(q)^{-1} Ahat(q). Since Ghat(q) is positive-definite and bounded, What(q) vanishes iff Ahat(q) = 0. For the unsigned incidence, Ahat_a(q) = 1 + e^{-i q_a}, which vanishes iff every component q_a = pi. Hence What has a unique zero at the staggered point q* = (pi, ..., pi), with quadratic dispersion (measured: n41, n46b; follows from analyticity and the parabolic minimum). All other q give What(q) = O(1): their contribution to dk^(1) is short-ranged (analytic symbol => exponential/power decay on the lattice scale). Thus the long-range part of dk^(1) is a slowly-varying ENVELOPE modulating the staggered wave e^{i q* r}. (2) *Cell averaging places a filter zero exactly on the pole.* A parity-neutral local observable applied to the field acts, at linear order, as a translation-invariant filter f(q) on dk^(1). Evenness under the sublattice swap forces f to be symmetric between the two sites of a cell; the elementary such filter is the cell average, whose symbol carries the factor (1 + e^{-i q_a})/2 per axis — vanishing AT q = q*, i.e., exactly on the unique pole of the long-range channel. The product filter x pole is therefore finite, and the surviving term near q* is O(q - q*) x (envelope spectrum): one extra power of (q - q*) converts, in real space, to one GRADIENT of the envelope. The linear long-range response is tidal, not potential-like. (The remaining smooth-channel linear response, from q near 0 where What is regular, decays on the correlation length — it is the short-ranged piece.) (3) *The quadratic response demodulates.* At second order the observable receives (dk^(1))^2-type terms; squaring the staggered wave gives e^{2 i q* r} = 1 (2 q* = reciprocal vector): the envelope SQUARED appears at q ~ 0, passes the cell filter unsuppressed, and inherits the full long range of the Gauss sector. Hence the leading long-range response of any parity-neutral observable is quadratic in the source — exactly what n47, n47b and n50 measured (clock shifts ~ envelope^2, exponents 2.0; second-order metric channel ratio 4.1). QED. **Corollary (why the force survives).** The force is not a parity- neutral observable: matter is parity-CHARGED, and the pair energy reads the signed staggered channel directly through E = (1/2) c^T W^+ c at the pole. The same pole that the clocks cannot see is the one the force lives on. One structure, both measurements. --- ## Theorem 2 (signed half: like sources repel; clocks read flux) With the SIGNED (divergence) sum rule and ANY positive-definite translation-invariant quadratic cost: (i) the interaction energy of two like sources at separation d is E_int(d) = f^2 K(d), where K is the kernel of W^+ — a positive- definite function, strictly decreasing at long range whenever What(q) has its unique zero at q = 0 with quadratic dispersion. Decreasing energy with separation = REPULSION. No choice of PSD cost can flip this: the sign is fixed by positive-definiteness. (ii) any local observable is a functional of nearby couplings — the FLUX dk — while the potential lambda = -W^+ c is a nonlocal integral of the flux. Local clocks therefore track dk (in 1D, a step function between sources), never lambda (the tent): a linear clock channel exists (the field is smooth; no parity filter) but it is field-coupled, not potential-coupled. **Proof.** (i) E(c) = (1/2) c^T W^+ c is a positive quadratic form; for c = f(delta_a + delta_b) + neutralizing background, expanding and subtracting the isolated-source terms leaves E_int(d) = f^2 K(d) with K(d) = W^+(a, a+d), a positive-definite function of d (Fourier transform of the nonnegative symbol 1/What). Bochner: K(0) >= |K(d)|; with the unique quadratic zero at q = 0, the long-range behaviour is the Coulomb kernel of the ambient dimension (1D: K(0) - K(d) ~ d/(2S)), strictly decreasing. E_int decreasing in d means work is RELEASED by separation: repulsion. (ii) is definitional once stated: locality of the observable = dependence on finitely many dk_e; lambda requires inverting W, which has unbounded support. Measured: n51 (clock exponents 1.000; shifts step-shaped, 29% misfit to the tent). QED. --- ## Theorem 3 (the dichotomy) In the class above, no architecture yields gravitational-redshift- shaped phenomenology (long-range clock response, linear in the source, tracking the potential, probe-universal) TOGETHER with attraction between like sources: - unsigned/bipartite: attraction holds, but Theorem 1 forbids the linear long-range clock response; - signed: a linear clock channel exists, but Theorem 2(i) forces repulsion and 2(ii) makes the response flux-coupled regardless. The obstruction is structural: gravitational redshift requires the potential to be a LOCAL field (as g_00 is in general relativity), while in flux-conservation drainage the potential is irreducibly the nonlocal integral of the local flux. Within quadratic drainage, Newton and Einstein's clocks cannot coexist. QED (by Theorems 1 and 2). **Scope and sharpness.** "Quadratic" is used twice and both uses are load-bearing: the cost is quadratic (superposition exact) and the observables' response is analyzed to second order (Theorem 1(iii) shows the quadratic term IS the leading long-range one, so higher orders cannot rescue linearity). Escapes not covered by the theorem, named: non-quadratic cost with long-range-restoring nonlinearity; observables non-local at the substrate level; a constraint whose kernel is itself a local potential mode. Each is a different theory, not a repair. ## Numerical gates for the proofs (n51a_verify.py, written after ## the proofs, before the run) - T1a: the odd-in-F (linear) part of the cell-averaged C2 clock response in the bipartite vacuum is short-ranged/tidal: at cells 5..30 from the source its magnitude is < 5% of the even part, and it tracks the envelope GRADIENT better than the envelope (fit comparison). - T1b: the cell filter zero: |1 + e^{-i q}|/2 at q = pi is 0 exactly (trivial) and the measured long-range linear residue scales with the envelope gradient, not the envelope. - T2a: signed case, exact identity E_int(d) = f^2 K(d) with K from the symbol: match to machine precision at all d. - T2b: K(0) - K(d) ~ d/(2 S_signed) with S_signed from n51 (32.4): slope within 2%. ## VERIFICATION RESULTS (same day; n51a_verify.py) - T1a PASS, exceeding the theorem: the linear part of the cell-averaged clock response is ZERO AT MACHINE PRECISION (3.4e-15 vs a quadratic part of 3.2e-4) at every distance, including the nearest cells. In 1D the reason is exact, and sharper than the filter argument: the per-site sum rule dk_{e-1} + dk_e = 0 at every EMPTY site forces the field to be exactly staggered away from sources, so parity-neutral observables are exactly even functions of the source strength — the general theorem's tidal bound is the d>1 statement; in 1D the linear response is identically zero. - T1b VOID for the strongest possible reason: there is no linear residue to characterize (fitting noise against noise, 18.2% = 18.2%). The tidal-vs-envelope discrimination requires d > 1, where empty-site constraints no longer force exact staggering; declared open, not load-bearing (the dichotomy needs only the absence of the linear channel, which is verified in the strongest form). - T2a PASS: E_int(d) = f^2 K(d) exact to 8.4e-12 at every d — the Bochner/repulsion identity confirmed at machine precision. - T2b PASS: fitted 1/(2S) matches the n51 stiffness to 0.004%. Status: Theorems 1-3 verified where they bite; the dichotomy is a theorem with machine-precision numerical witnesses on both halves. ## SCOPE AMENDMENT (2026-07-09, after adversarial referee review + n54) The theorems govern clocks that are STATIC, PARITY-NEUTRAL functionals of nearby couplings — every clock the program had tested (substrate spectral clocks, bond clocks, cell-averaged observables) and every bookkeeping clock (maintenance costs). They do NOT govern clocks built of the theory's own matter: matter is parity-charged (the theorem's own "why the force survives" corollary), a dressing sits at definite sites, and n54 Stage B measures its operational frequency shifting approximately LINEARLY in a source (F-exponent 1.101). The epitaph "Newton and Einstein's clocks cannot coexist" is therefore narrowed to: "...cannot coexist if clocks are parity-neutral substrate observables." Whether matter-built clocks deliver potential-tracking, universal, GR-shaped redshift is open again and hinges on the derivation of the dressing's dynamics (n54 Stage C). Theorems 1-3 stand as proven; their epitaph was wider than their premises.