Classifying 0/0 Limits — Why This Classification Matters & What It Unlocks (0C, 0D)

Why bother classifying 0over0 at all? Because real systems do not fail gracefully at indeterminates. We need clear headlines (ZERO / FINITE / INF) and an auditable confidence trace that travels through computations.

0C) Why This Classification Matters

Deterministic headlines.
A single lexicographic rate rule yields unambiguous outcomes; logs only tie-break when p_f = p_g.

Branch-free outputs.
Emit <m_out, a_div>; collapse conservatively:

phi(<m,a>) = m

Alignment a_div is metadata; it never alters the classical magnitude.

Robust near singularities.
Avoid brittle epsilon hacks; downstream logic can threshold on alignment bands for confidence/direction.

Transparent and teachable.
The why (rate order via (Delta_p, Delta_q)) and the how well (alignment band) are visible; print REG=SIDED and REG=OSC when present.

Interoperable and composable.
Same rapidity law across SSM/SSMS keeps composition bounded and stable:

a_mul = tanh(atanh(a1) + atanh(a2))
a_div = tanh(atanh(a_f) - atanh(a_g))
a_(1/x) = -a_x
a := clamp(a, -1+eps_a, +1-eps_a)   # always clamp before any atanh; eps_a ~ 1e-6

Reproducible and auditable.
Manifest-driven knobs (K, window grid, R2->a mapping, eps_a, reference tier a_g) make runs deterministic; one-line SSMS prints provide a traceable confidence record.

0D) What This Unlocks — From 0over0 to a Working Symbolic Standard

Purpose in one line.
A deterministic 0over0 classifier that emits <m_out, a_div> with conservative collapse phi(<m,a>) = m, plus SSMS one-liners for transparent auditing and teaching.

Immediate capabilities.

  • Stable headlines. ZERO / FINITE / INF from a single rule; logs only tie-break when p_f = p_g.
  • Confidence and direction. a_div provides a bounded, composable signal (bands A--/A-/A0/A+/A++); optional DIR+, DIR0, DIR-.
  • Post-hoc safety. Flags SIDED/OSC/MULTI/NOFIT surface tricky tails instead of forcing brittle decisions.
  • Drop-in printing. One line per outcome:
SSMS: {CLASS}@{A-tag} DIV[a_div]@{A-tag} {DIR?} {REG?} {BADGES?}

Where to use it today (examples).

  • Calculus/analysis. Canonicalize 0over0 at removable, essential, or oscillatory points; keep proofs clean while exposing tail quality.
  • Control and robotics. Treat near-singular sensor ratios as <m,a>; gate actions on classical m, not a; use a_div for operator confidence.
  • Numerics/optimizers. Replace ad-hoc eps checks with headlines + bands; retry only on REG=OSC/MULTI or when band < A0.
  • Signals/telecom. Division of noisy streams uses a_div = tanh(atanh(a_f) - atanh(a_g)); print REG=SIDED for asymmetric fades.
  • Physics/chem. Rate-law comparisons at boundaries: headline from rates; publish DIV[...] to quantify evidence.
  • ML and diagnostics. Feature ratios at degenerate points: use bands to decide keep/clip/skip without leaking label info.
  • Finance/econ. Normalize indicator ratios with <m,a>; display MULTI when regimes switch across nested windows.
  • Medicine/earth-risk. Handle physiological or geophysical ratios near zeros; keep SSMS audit trail for analysts/clinicians.

Interoperability (one glance).

Symbol:  x := <m, a>             Collapse: phi(<m,a>) = m
Clamp:   a := clamp(a, -1+eps_a, +1-eps_a)   # eps_a ~ 1e-6
Compose: a_mul = tanh(atanh(a1) + atanh(a2))
         a_div = tanh(atanh(a_f) - atanh(a_g))
         a_(1/x) = -a_x
Priors:  no prior -> a_f = a_g = 0
         from fits -> a := 2*R2 - 1  OR  a := tanh(c*(R2 - 0.5))  # publish c
         reference tier a_g in {0.60, 0.80, 0.90}

Determinism and stability (practical guarantees).

  • Headline stability under refinement. For sufficiently small windows, the class stabilizes; bands typically tighten monotonically.
  • No hidden randomness. Fixed K, grid, mapping, a_g tier -> identical outputs on rerun.
  • Conservative parity. When the classical limit exists, the printed headline matches it after collapse (VAL[m], Z, or INF±).

Adoption recipe (quick start).

1) Choose nested windows (K, log grid).
2) Map fit quality to alignment (a := 2*R2 - 1  OR  contrast with c).
3) Pick reference tier (a_g = 0.60 / 0.80 / 0.90).
4) Emit one-line SSMS with {CLASS} and DIV[a_div] bands; add SIDED/OSC/MULTI when present.
5) Log a manifest (knobs + versions) for replayability.

One-liner examples.

Zero:     SSMS: Z@A0          DIV[0.00]@A0
Finite:   SSMS: VAL[0.400]@A+ DIV[+0.714]@A+ DIR+
Infinity: SSMS: INF-@A--      DIV[-0.910]@A-- DIR-
Oscill.:  SSMS: REG=OSC       DIV[+0.05]@A0 EDGE(p) EDGE(q)

Roadmap (beyond this note).

  • Extend to other indeterminates (0*inf, inf-inf, 0^0, 1^inf, inf^0).
  • Piecewise and saddle analysis (automatic SIDED summaries).
  • Multivariate paths (curve-dependent alignments).
  • Complex-plane alignment (research; not in this release).

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