1.2.4

1.2.4

Features

Minor release that widens solving for y past the "one occurrence of y"
frontier of 1.2.3, and gives obs-integral the quotient rule it was missing.
The new solver paths are general rewrites over the AST — clear the denominators,
collect the linear coefficient, square away the radicals — not per-equation
cases. Every transformation that is not an equivalence now states its condition
or is rejected: a solved form still never claims a point the curve doesn't have.
And the conditions that survive are now solved as a system instead of listed,
so a domain reads x ≥ \sqrt3 rather than as two raw inequalities. The full test
suite (334 tests), the zoom suite (12) and the graduated battery (220 generated
towers) pass.

• y no longer has to occur once — it has to be grouped. The structural
  inverter peels the outermost layer and recurses into the child holding y,
  which never needed y to be unique: the whole layer is inverted at once, so it
  is enough that a single child contains it. When the tower gets stuck — y
  split across both branches of an operator — what has been peeled so far is an
  equivalent, simpler equation, and it is re-solved from scratch:
  \ln\frac{y-1}{y+2}=x → y=\frac{2e^x+1}{1-e^x},
  e^{\frac{y-1}{y+2}}=x → y=\frac{2\ln x+1}{1-\ln x},
  \ln(y^2+y)=x → y=\frac{-1±\sqrt{4e^x+1}}{2}.

• y in a denominator is cleared and the equation re-solved. No additive
  strategy could reach a y sitting in a factor of exponent −1 unless the
  numerator was free of y. Multiplying through by the denominators — the first
  step of any textbook solution — leaves a polynomial equation the existing
  machinery finishes: \frac{y-1}{y+2}=x → y=\frac{2x+1}{-x+1}.

• Equations affine in y whose coefficient needs expanding. A sum of terms
  each carrying a bare y (y\ln x+y=3) already worked; what fell through was
  y inside a group that has to be distributed first, when the coefficient is
  not polynomial — the linear strategy reads the coefficient off the structure
  of the term, so y has to be a bare factor, and the quadratic path needs
  rationalize to expand, which only works on polynomials. Since an affine
  function is fixed by two values, A and B are now recovered by
  substituting two values of y — substitution works with any coefficient, no
  expansion needed — and the affinity itself is what gets checked numerically:
  y-(y+2)e^x-1=0 → y=\frac{2e^x+1}{1-e^x} (this is the shape the cleared
  Möbius above lands on, and it is what makes that case finish).

• Scattered square roots are rationalized by successive squaring, carrying the
  domain condition of every step. One isolated root was already invertible, but
  two roots — or a root plus y outside it — leave y ungrouped and no layer
  peelable. Isolating one root and squaring, repeated, produces a polynomial the
  solver finishes. Squaring is not an equivalence (A=B ⟺ A²=B² and
  B≥0), so each step records its guard and all of them travel to the result as
  domain conditions, written in x by substituting back what the later steps
  determined: \sqrt{y+1}+\sqrt{y-2}=x → y=\frac{x^4+2x^2+9}{4x^2} with
  x ≥ \sqrt3 (the two raw guards, resolved into one by the condition
  simplifier below). Verified point by point: the solved form draws nothing for
  x<\sqrt3 and has no gap above it.

• Domain conditions are now solved as a system, not listed one by one. Each
  restricted-range layer and each squaring step contributes its own guard, and
  the panel used to print all of them — correct but unreadable, because they are
  inequalities over the same x and nobody was looking at the system as a whole.
  A new module resolves each condition by its sign table (the zeros of
  numerator and denominator split the line into constant-sign runs) and
  intersects the results, which is where redundant conditions vanish on their
  own, adjacent runs merge, and contradictions surface as an empty intersection.
  So \frac{x^2+3}{2x}≥0,\ \frac{x^2-3}{2x}≥0 is displayed as what it says:
  x ≥ \sqrt3. Critical points are computed symbolically precisely because
  they are what gets displayed — the root of x^2-3 has to read \sqrt3, not
  1.7320508 — so -x^2+2 ≥ 0 becomes -\sqrt2 ≤ x ≤ \sqrt2 and x-27 ≥ 0
  becomes x ≥ 27. Its declared reach is closed-form roots: degree 1, degree 2
  by the general formula (with the square factor pulled out of the radical,
  \sqrt{12}=2\sqrt3), and higher degrees only where integer roots deflate them.
  Anything else — a guard with \tan x, |x|, \sqrt x, or a solution set made
  of disjoint pieces — leaves the conditions displayed exactly as before: the
  module's failure mode is "I don't simplify", never "I simplify wrongly". It is
  presentation only; the engine keeps evaluating the original guards, so what is
  drawn does not change.

• Contradictory guards are detected across the whole system. Each guard can
  be satisfiable on its own while the system is not, and the per-guard check
  could not see that. An empty intersection now means there is no real curve and
  the equation is left in its reduced form instead of carrying a formula that
  never applies.

• obs-integral: the logarithmic derivative ∫\frac{c·q'}{q}=c\ln|q|. A
  quotient with x in both numerator and denominator always returned "no
  antiderivative". The ratio p/q' is measured by finite differences rather than
  by symbolic differentiation, so it does not depend on mathjs being able to
  differentiate \csc or \abs. This covers ∫\frac{2x}{x^2+1},
  ∫\frac{3x^2+2}{x^3+2x-5}, ∫\cot, ∫\frac{f'}{f} in general.

• obs-integral: trigonometric canonicalisation as a retry. Expressions
  written "by identity" reduce to something the structural rules recognise once
  \csc/\sec/\cot/\tan are rewritten in \sin/\cos and the double angle is
  opened: ∫\frac{1}{\csc 2x-\cot 2x} collapses to ∫\cot x = \ln|\sin x|. The
  original form is always integrated first, so ∫\sin 2x still gives
  -\frac{\cos 2x}{2} and not the expanded version. Also adds the missing
  \cot, \sec and \csc table entries (with linear substitution:
  ∫\cot 3x = \frac{\ln|\sin 3x|}{3}).

Bug fixes

• Clearing a denominator no longer invents branches. Multiplying by q only
  preserves the curve where q≠0, and the cleared equation is defined there —
  so it can carry solutions that do not exist. \frac{y^2-1}{y-1}=x clears to
  y^2-1=x(y-1), whose roots are y=x-1 and y=1, and the second is not
  curve at all (the original is 0/0 at y=1). Since a ≠ condition cannot be
  written with the existing ≥0 domain sentinel, the candidate is instead
  validated branch by branch against the equation before multiplying — the only
  one that knows about its own holes — and the whole solution is dropped if any
  branch contradicts it.

• Removable holes are no longer silently filled. \frac{y^2-4}{y+2}=x is
  affine in y throughout its domain and solves to y=x+2, but the curve does
  not contain y=-2, i.e. the solved form is missing its hole at x=-4 and is
  laxer than the curve. Same validation, same outcome: the equation is left in
  its reduced form rather than stated more loosely than it is true.

• A singular sampling point no longer leaks into the formula. Recovering the
  affine coefficients by substituting y=0,1 hits 0/0 on an equation like
  \frac{y^2-1}{y-1}, and the resulting Infinity travelled all the way into
  the displayed solution (y = ∞x + ∞). Several sampling pairs are tried until
  one is clean, and a non-finite coefficient rejects the pair.

• Both branches of a ± are now validated, not just the principal one. The
  numeric check evaluated the ± sentinel at its principal value, so the second
  branch entered the result unchecked. Validation now runs over the expanded
  branches — exactly what the engine will draw. A branch that its own domain
  guard empties is not a failure (that is precisely the fate of the extraneous
  branch introduced by squaring); a branch that contradicts the equation is.