# n68 — pre-registration: gravity and confinement as two budget regimes
# (user's conjecture, made precise; before any code)

**The hypothesis.** One drainage, two regimes. Sub-critical charges:
flux dilutes over spheres — 1/r^2, gravity (measured, n46/n46b).
SUPER-critical charges: the per-link budget bound (|dk| <= 1, measured
to matter in n57) forbids dilution near the source; the flux is forced
into a SATURATED TUBE of constant cross-section between opposite
charges — constant tension, LINEAR potential: confinement. The model
already contains the phenomenological trio of the strong force:
emergent neutrality (only "singlets" allowed on compact space, n40/n41
C3), two charge classes (parity, n49), and now possibly flux tubes.
If the gates pass, the registered claim is STRUCTURAL: a single
conserved-drainage vacuum exhibits both gravitational (dilute) and
confining (saturated) phases — NOT a claim about QCD's scales, which
sit 38 orders away and are not addressed here.

## Toy measurement (2D L=24, bounded info-metric drainage, n57b machinery)
Opposite-parity pair on the axis, separation d = 3, 5, 7, 9 (odd).
Two regimes: F_sub = 2 (no clamping expected), F_super = 96.

## Gates
- T1 (control): F_sub: V(d) fits the 2D lattice Coulomb (log-like)
  as in n46 (fracRMS < 2%), no clamped edges.
- T2 (THE TUBE): F_super: a connected set of saturated edges
  concentrated in the strip between the charges (strip fraction > 3x
  the far-field fraction), growing with d.
- T3 (THE STRING): F_super: V(d) markedly more linear than Coulomb —
  linear-fit residual < half the Coulomb-fit residual over d = 3..9;
  report the tension (slope) and its constancy.
- T4 (tension scale): tension ~ (bound magnitude) x (tube width) x
  (metric scale): report the measured tension against the saturated
  edge cost — consistency, not a gate.
- KILL: no tube forms (exhaustion stays radial around each charge) or
  V remains Coulombic while clamped — the two-regime unification is
  dead and the conjecture is buried with a measurement, not a shrug.

## Honesty clauses
- 2D, L = 24, monotone active-set solver (n57 caveat).
- "Confinement" here = linear potential + flux tube in THIS vacuum;
  no color, no non-abelian structure, no QCD scale claim. The
  registered novelty: dilution-vs-saturation as a two-phase structure
  of one conserved drainage.
- Finite lattice: d <= 9 on L = 24; wrap effects reported.

## RESULTS (same day; runs in the session record)
- Implementation error owned first: the original run added a jellium
  background that an opposite-parity pair does not need (staggered
  charge already zero) and that flooded the lattice at large F. The
  jellium-free rerun is the record.
- T1 PASS: F = 2 — Coulomb at 0.17%, zero clamping, near-zero tension.
  The gravitational phase, clean.
- T2 PASS: F = 24 — the saturated set concentrates in the strip
  between the charges at 10-20x the far-field fraction and GROWS with
  separation (24 -> 57 edges): a flux tube that lengthens like a
  string. The far field stays essentially empty (1.6-3.6%).
- T3 DIRECTION CONFIRMED, registered threshold not met: at F = 48 the
  linear fit overtakes the Coulomb fit (16.0% vs 22.5%; gate wanted
  < 0.5x) and the tension rises steeply with F (+4.6e-4 -> +1.9e-2 ->
  +7.2e-2). At L = 24 the string regime (tube length >> core size)
  is only marginally reachable; n68b at larger L is the registered
  completion of the gate.
VERDICT (superseded same day — see below): the two-phase structure is
real; the first gates passed in the transitional window.

## THE KILL (same day; adversarial pass) — HEADSTONE 25
Two blades, one analytic and one numerical, and both landed:
- ANALYTIC: in d >= 2 a per-link bound CANNOT confine at long range.
  Beyond r_core ~ F/(2 pi b), the diluted flux density F/(2 pi r)
  falls below the bound, the unconstrained Coulomb solution becomes
  feasible, hence optimal: the potential is ASYMPTOTICALLY CoulomB.
  A bound produces saturated CORES (an obstacle problem), not strings.
  True confinement needs dilution to be energetically punished (the
  dual-Meissner mechanism), not merely capped.
- NUMERICAL, confirming: at F = 16, short range (d = 3-7) prefers
  linear (4.0% vs 10.4%) — the transitional window n68 mistook for a
  string — while long range (d = 7-11) returns decisively to Coulomb
  (0.87% vs 7.74%), V flattens, the tube's middle empties (0.25 ->
  0.00) and the clamped count DECREASES with separation. At F = 48 the
  "tube" thins as it lengthens (0.50 -> 0.29): a true string keeps its
  cross-section.
The "flux tube" was two exhausted cores holding hands; the "string
tension" was the crossover between merged and separated cores.
GRAVITY AND CONFINEMENT AS TWO BUDGET REGIMES IS DEAD as an asymptotic
statement — buried with a mechanism, an analytic argument, and the
measurement that confirmed it, one day after passing its first gates.
What survives, honestly: the saturated-core phase itself (n57b) with
its modified SHORT-range interaction — real physics of this vacuum,
just not the strong force. Prediction 12 is re-stamped a headstone.