π , π , G ∗ , AND WHEN A CHANGE OF DESCRIPTION IS ALLOWED General Covariance of Mathematical Representations Three constants, pairwise algebraically independent and jointly bound by a single equation; a rule for when one description of a mathematical object may replace another; and a precise account of where that rule stops being finite. PROVED HERE the proof is on the page IMPORTED a published theorem, cited CHECKED a finite computation, script in Section 10 OPEN no proof is known to me Every claim in this document carries one of these four tags in the margin. circle: perimeter 2 π area under e − x 2 : √ π lemniscate: perimeter 2 π G ∗ = 2 π √ π lemniscate length per Gaussian area The upper half of the unit circle (the full circle has perimeter 2 π ), the Gaussian e − x 2 (the area under it is √ π ), and the lemniscate ( x 2 + y 2 ) 2 = x 2 − y 2 (perimeter 2 π ). The constant studied here measures the lemniscate’s perimeter against the Gaussian area rather than against the circle’s perimeter: G ∗ = 2 π / √ π Equivalently, G ∗ = 2 √ π / AGM ( 1, √ 2 ) , twice the Gaussian area divided by the arithmetic-geometric mean of the side and diagonal of the unit square. Summary. • Let π be the circular constant, π = 2.62205 . . . the lemniscate constant, and G ∗ = Γ ( 1 / 4 ) / Γ ( 3 / 4 ) = 2.95867 . . . Any two of these are algebraically independent over Q , all three satisfy 4 π 2 = π ( G ∗ ) 2 , and every polynomial relation among them is a consequence of that one equation. • The question under study is when one description of a mathematical object may be replaced by another. The answer proposed is that information may be discarded exactly when the discarded part is a congruence for the operations one intends to keep. • That answer is a definition from universal algebra, and both of its halves are standard: Moore’s state minimization on the finite side, and the largest differential ideal on the ring side. What is of interest is when the associated test terminates. For a finite dynamical system it terminates after finitely many refinements. For evaluation of quasimodular forms at the square point τ = i , the same test produces an infinite strictly descending chain of ideals whose intersection is zero, and yet it terminates on every piece of bounded weight, after exactly ⌊ w / 6 ⌋ steps in weight w . The tower turns out to be the E 6 -adic filtration of the ring, and at a general CM point its height is exactly the valence-formula bound for modular forms. The phrase “finite construction, continuous realization” is used here for precisely this situation: every stage is a finite object, and its identification with a classical object is a separate, proved step. • Two further results arise along the way. There is an unconditional, exact decision procedure for identities among quarter-gamma values, in a setting where zero recognition for closed-form constants is in general known to be decidable only under Schanuel’s conjecture. And the rational enclosures of G ∗ obtained from the product ∏ ( 4 k + 3 ) / ( 4 k + 1 ) have the property that each truncation of the asymptotic series is a strictly monotone sequence with a one-line polynomial certificate; this is proved for the first ten orders and is open beyond that. Enclosures at all orders, without the monotonicity, are a 1987 theorem of Frenzen. 2 C ONTENTS 1 What is not new, and what may be 3 2 The triangle 4 3 An equality test that decides 6 4 Building G ∗ from integers, five ways 7 4.1 The product, and certificates at every order . . . . . . . . . . . . . 7 4.2 Four other routes to the same number . . . . . . . . . . . . . . . . 9 4.3 Watson’s integrals and the three cubic lattices . . . . . . . . . . . . 10 5 One rule, and where it stops being finite 11 5.1 The rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2 The finite side: the repair terminates . . . . . . . . . . . . . . . . . 11 5.3 The infinite side: a tower at τ = i . . . . . . . . . . . . . . . . . . . 12 6 What one constant cannot carry 15 7 A quartic test that can fail 17 8 Objections and replies 17 9 Open questions 19 10 Reproducing the computations 20 11 References 22 1. W HAT IS NOT NEW , AND WHAT MAY BE Every identity between π , π and Γ ( 1 / 4 ) used here is classical and goes back to Euler, Gauss and Legendre. The algebraic independence of π and Γ ( 1 / 4 ) is a theorem of Chudnovsky, extended by Nesterenko to include e π . The lattice walk sum is Watson’s 1939 integral. The arithmetic of the curves y 2 = x 3 − n 2 x is Tunnell’s theorem. The partition refinement of Section 5 is Moore’s 1956 minimization of sequential machines, and the chain of ideals in the same section is the largest differential ideal of differential algebra. The modular identities are Ramanujan’s. The asymptotic expansion of a ratio of gamma functions is due to Tricomi and Erdélyi, and in the even form used here to Fields, with error bounds at every order by Frenzen. The method of proving monotonicity by exhibiting 3 a rational difference with a one-sign numerator is the standard technique of Mortici, Chen and Qi. Against that background, three things appear to be new, each of them small and each of them checkable. 1. One test, placed in two settings. The congruence condition determines whether a reduced description can carry a given operation. In finite dynam- ics the canonical repair terminates, by Moore’s argument. For evaluation of quasimodular forms at τ = i the same repair never terminates on the whole ring, its limit is the zero ideal, and it terminates again on every piece of bounded weight (Theorem 7), after exactly ⌊ w / 6 ⌋ steps in weight w (The- orem 8). The tower is the E 6 -adic filtration, and at every CM point the analogous tower over Q has height equal to the valence-formula bound for modular forms (Corollary 9). Each ingredient is known separately. The identification of the tower and the comparison with the finite case are the point of this document. 2. An unconditional decision procedure on a named fragment. For closed- form constants in general, zero recognition is known to be decidable only conditionally on Schanuel’s conjecture (Richardson 1997; Macintyre and Wilkie 1996). On the Laurent ring generated by quarter-gamma values together with π , π and e π , equality is unconditionally decidable by a finite reduction (Theorem 3). This is a short corollary of the Chudnovsky and Nesterenko theorems, and it is small enough to implement. 3. Monotone certificates, order by order. For the product ∏ ( 4 k + 3 ) / ( 4 k + 1 ) , each truncation of the asymptotic series gives a strictly monotone sequence in N , with a polynomial certificate whose coefficients all have one sign (Theorem 4, orders 1 to 10). Two-sided enclosures at every order follow from Frenzen’s theorem; strict monotonicity in N of these particular truncations at all orders is open (Problem A). The same pattern holds for thirds, fifths, sixths and eighths in every case computed. 2. T HE TRIANGLE We begin by fixing normalizations, since a large fraction of the errors in this subject are normalization errors. π 2 = ∫ 1 0 du √ 1 − u 2 , π 2 = ∫ 1 0 du √ 1 − u 4 , G ∗ = Γ ( 1/4 ) Γ ( 3/4 ) = 2.95867 51191 88638 89231 08 . . . 4 The substitution v = u 4 in the second integral, followed by Euler’s beta identity, gives π 2 = 1 4 B ( 1 4 , 1 2 ) = √ π 4 Γ ( 1/4 ) Γ ( 3/4 ) , so that π = √ π 2 G ∗ and 4 π 2 = π ( G ∗ ) 2 The reflection formula gives Γ ( 1 / 4 ) Γ ( 3 / 4 ) = π √ 2 , hence Γ ( 1 / 4 ) 2 = √ 2 π G ∗ One convention deserves explicit mention. The complete elliptic integral at the square point is K ( k ) = π / √ 2 in the modulus convention, with k = 1 / √ 2 ; a library that takes the parameter m = k 2 must be called with m = 1 / 2 (the conventions are those of DLMF chapters 19 and 23). T HEOREM 1 (C HUDNOVSKY ; N ESTERENKO ). IMPORTED The numbers π and Γ ( 1 / 4 ) are algebraically independent over Q . More strongly (Nesterenko 1996), π , e π and Γ ( 1 / 4 ) are algebraically independent over Q C OROLLARY 2 ( STRUCTURE OF THE TRIANGLE ). PROVED HERE Each of the pairs ( π , G ∗ ) , ( π , π ) and ( G ∗ , π ) is algebraically independent over Q , and trdeg Q Q ( π , G ∗ , π ) = 2 . The kernel of the evaluation map Q [ X , Y , Z ] → R sending ( X , Y , Z ) to ( π , G ∗ , π ) is the principal ideal ( 4 Z 2 − XY 2 ) Proof. Since Γ ( 1 / 4 ) 2 = √ 2 π G ∗ , the field Q ( √ 2 , π , Γ ( 1 / 4 )) is algebraic over Q ( π , G ∗ ) , so the latter has transcendence degree 2. Next, π is algebraic over Q ( π , G ∗ ) , G ∗ is algebraic over Q ( π , π ) , and π = 4 π 2 / ( G ∗ ) 2 ; hence every pair generates a field of transcendence degree 2. For the kernel, divide by Z 2 − XY 2 / 4, which is monic in Z , so that every class has a unique representative of the form A ( X , Y ) + Z B ( X , Y ) . Suppose A ( π , G ∗ ) + π B ( π , G ∗ ) = 0. If B = 0 then A = 0 by independence. Otherwise, moving A to the other side and squaring gives ( 4 A 2 − XY 2 B 2 )( π , G ∗ ) = 0, hence 4 A 2 = XY 2 B 2 in Q [ X , Y ] . The exponent of the prime X is even on the left and odd on the right, so A = B = 0. □ The triangle therefore carries two independent transcendental directions on three named quantities. No vertex is algebraic over any single other vertex, and any two vertices determine the third, the sign being fixed by positivity: π = ( 2 π G ∗ ) 2 , π = √ π G ∗ 2 , G ∗ = 2 π √ π It should be said at once that this configuration is not rare. Chudnovsky’s theorem also covers π and Γ ( 1 / 3 ) , and the same triangle appears there. What makes the quarter case worth a long document is the number of independent constructions that arrive at the same number (Section 4), and the square point τ = i , at which the modular theory places all of these values in a single ring (Section 5). 5 3. A N EQUALITY TEST THAT DECIDES Three facts set the context. First, no algorithm decides whether a computable real number is zero. Given a program e , put x e = 2 − t if e halts first at step t , and x e = 0 if it never halts. Running e for N steps approximates x e to within 2 − N , so x e is uniformly computable, and a zero test would decide the halting problem. Second, for expressions in a real variable built from π , log 2, sin , exp and the absolute value, it is undecidable whether the expression vanishes identically (Richardson 1968). Third, for closed-form constants the question is open: the known zero-recognition procedures are guaranteed to terminate only if Schanuel’s conjecture holds (Richardson 1997; Macintyre and Wilkie 1996). Agreement of many digits is therefore never, by itself, a proof of equality. A constructive response is to isolate a fragment in which equality of constants is unconditionally decidable, and to prove that it is. Write s = √ π and t = √ G ∗ Then Γ ( 1 4 ) = 2 1/4 s t , Γ ( 1 2 ) = s , Γ ( 3 4 ) = 2 1/4 s t − 1 , π = 1 2 s t 2 , π = s 2 , G ∗ = t 2 T HEOREM 3 ( QUARTER - GAMMA DECISION PROCEDURE ). PROVED HERE Consider finite expres- sions of the form E = ∑ j α j π a j /2 π b j ∏ k Γ ( m jk / 4 ) e jk , where the α j are algebraic num- bers given exactly, the exponents a j , b j , e jk are integers, and no m jk / 4 is a pole of Γ . Each such E equals a unique Laurent polynomial in s and t with algebraic coefficients, and E = 0 if and only if every coefficient of that Laurent polynomial is zero. In particular, equality within this fragment is decidable. Proof. The recurrence Γ ( x + 1 ) = x Γ ( x ) moves each argument to one of 1 / 4, 1 / 2, 3 / 4, 1 at the cost of rational factors, and the table above then rewrites every term as α s a t b with α ∈ Q . The field Q ( s , t ) contains Q ( π , G ∗ ) and is algebraic over it, so it has transcendence degree 2 by Corollary 2; two generators of a field of transcendence degree 2 are algebraically independent over Q . Independence over Q implies independence over Q , since Q is algebraic over Q and transcen- dence degree is unchanged under algebraic extension of the base field. Now suppose a Laurent polynomial in s , t with coefficients in Q evaluates to zero. Multiplying by a suitable monomial clears the negative exponents and yields a polynomial relation over Q , so every coefficient vanishes. Finally, arithmetic with exactly presented algebraic numbers is decidable, which makes both the reduction and the zero test effective. □ By the Nesterenko form of Theorem 1, the same argument applies verbatim in the Laurent ring Q [ s ± 1 , t ± 1 , e ± π ] . As an illustration, the expression 16 π 4 − π 2 ( G ∗ ) 4 reduces to s 4 t 8 − s 4 t 8 , which is 0, so the identity is exact; adding 10 − 50 produces an expression that is exactly nonzero, independently of any floating- point precision. Similarly B ( 3/4, 1/2 ) / B ( 1/4, 1/2 ) reduces to 4 t − 4 = 4/ ( G ∗ ) 2 6 The limits of the theorem should be stated as clearly as its content. It does not decide identities involving the gamma function at other rational arguments, nor derivatives, infinite sums or infinite products. For instance ∆ ( i ) = ( G ∗ ) 12 / ( 2 18 π 6 ) lies inside the fragment, but the product expression ∆ ( i ) = e − 2 π ∏ n ≥ 1 ( 1 − e − 2 π n ) 24 is not an element of it. 4. B UILDING G ∗ FROM INTEGERS , FIVE WAYS 4.1 The product, and certificates at every order For N ≥ 1 define P N = N − 1 ∏ k = 0 4 k + 3 4 k + 1 , H N = P 2 N N , h = ( G ∗ ) 2 Each H N is a rational number, and the construction makes no reference to π , to Γ , or to any stored value of G ∗ . Adjoining one more factor gives the exact recurrence H N + 1 H N = N ( 4 N + 3 ) 2 ( N + 1 )( 4 N + 1 ) 2 = 1 − 1 ( N + 1 )( 4 N + 1 ) 2 The identification of the limit with a gamma value is a separate step. By the recurrence for Γ , P N = G ∗ Γ ( N + 3 / 4 ) / Γ ( N + 1 / 4 ) , and the standard asymptotic for a ratio of gamma functions gives H N → h . More precisely, the expansion of Tricomi and Erdélyi (1951), in the even form due to Fields (1966), reads h H N = 1 − 1 32 N 2 + 11 2048 N 4 − 173 65536 N 6 + 22931 8388608 N 8 − 1319183 268435456 N 10 + · · · Only even powers of 1 / N appear, because the two shifts 1 / 4 and 3 / 4 sum to 1: the coefficient of N − k in log Γ ( N + a ) involves B k + 1 ( a ) , and the symmetry B k + 1 ( 1 − a ) = ( − 1 ) k + 1 B k + 1 ( a ) cancels every odd k in the difference. Let T M ( N ) denote the truncation of this series through the term in N − 2 M , and set S M ( N ) = H N T M ( N ) Part of what follows is already known. Frenzen (1987) proved Luke’s conjec- ture that, for the ratio Γ ( x + 1 / 4 ) / Γ ( x + 3 / 4 ) and at every order, the remainder of Fields’ expansion has the sign of the first neglected term and is bounded by it in absolute value. Squaring Frenzen’s bounds on the ratio gives two-sided enclosures of h at every order. The statement below concerns a more rigid property: the truncations S M ( N ) of the squared series are themselves monotone in N , and each comes with a finite certificate. 7 T HEOREM 4 ( MONOTONE CERTIFICATES ). PROVED HERE, ORDERS 1 TO 10 For each M = 1, . . . , 10 , the sequence S M ( N ) , N ≥ 1 , is strictly increasing when M is odd and strictly decreasing when M is even. Consequently, for odd M ≤ 9 and every N ≥ 1 , S M ( N ) < h < S M + 1 ( N ) Proof. For each M the quantity ( S M ( N + 1 ) − S M ( N )) / H N is an explicit rational function of N whose numerator has coefficients all of one sign and whose denominator is positive. The first three are S 1 ( N + 1 ) − S 1 ( N ) H N = 11 N 2 + 11 N + 1 32 N 2 ( N + 1 ) 3 ( 4 N + 1 ) 2 , S 2 ( N + 1 ) − S 2 ( N ) H N = − 519 N 4 + 1038 N 3 + 662 N 2 + 143 N + 11 2048 N 4 ( N + 1 ) 5 ( 4 N + 1 ) 2 , S 3 ( N + 1 ) − S 3 ( N ) H N = p 3 ( N ) 65536 N 6 ( N + 1 ) 7 ( 4 N + 1 ) 2 , p 3 ( N ) = 22931 N 6 + 68793 N 5 + 81935 N 4 + 49215 N 3 + 15737 N 2 + 2595 N + 173. Orders 4 through 10 are identities of the same kind with longer numerators; they are finite rational-function identities, and the script in Section 10 verifies them. The technique, namely writing the difference of consecutive terms as a rational function and checking the sign of its numerator, is the usual one in the inequalities literature. Since H N → h and T M ( N ) → 1, each S M ( N ) tends to h , and a strictly monotone sequence lies strictly on one side of its limit. □ Two remarks are in order. First, the constant term of each certificate numerator equals the numerator of the last coefficient retained (1, 11, 173 in the three cases displayed), which suggests structure that a general proof might exploit. Second, taking the order-6 lower bound and the order-4 upper bound at N = 1000, in exact rational arithmetic followed by square roots, gives 2.95867 51191 88638 89231 08 < G ∗ < 2.95867 51191 88638 89231 48, where the displayed digits are rounded outward (the exact bounds continue . . . 23108213 . . . and . . . 23147264 . . . ). This agrees with G ∗ to 20 decimal places, whereas the order-2 and order-4 pair applied to the same data agrees to 13. The finite computation is a single instance; the guarantee for all N comes from the certificates, and the two kinds of evidence are kept distinct throughout. P ROBLEM A ( MONOTONICITY AT ALL ORDERS AND ALL RESIDUES ). OPEN Show that for every M ≥ 1 the truncation S M ( N ) is strictly monotone in N , increasing for odd M and decreasing for even M , with a one-sign numerator in the difference of consecutive terms. More generally, for a ∈ ( 0, 1 / 2 ) and an integer p with 8 p ( 1 − 2 a ) ∈ Z , let H N = ( ∏ k < N k + 1 − a k + a ) p / N p ( 1 − 2 a ) . The same alternating one- sign pattern holds for a = 1 / 3, 1 / 5, 1 / 6, 1 / 8 through order N − 16 . Is it a theorem for all a and all M ? The nearby literature (Fields, Frenzen, Gautschi, Kershaw, Buri ́ c and Elezovi ́ c, and the surveys of Qi) gives enveloping expansions and complete monotonicity for ratios of gamma functions in a continuous variable, at all orders. Problem A asks instead for monotonicity of the integer-indexed truncations of the squared series, which does not follow directly from the continuous bounds. 4.2 Four other routes to the same number • A binomial series. Expanding ( 1 − u 4 ) − 1/2 and integrating term by term gives π 2 = ∑ n ≥ 0 b n 4 n + 1 , b n = ( 2 n n ) 4 − n An induction, using 4 ( n + 1 ) 3 − ( 2 n + 1 ) 2 ( n + 2 ) = 3 n + 2 > 0, shows b n ≤ ( n + 1 ) − 1/2 , and for N = m 2 the tail after N terms is less than 1 / ( 2 m ) The convergence is slow, but the certificate is rational. • The arithmetic-geometric mean. One has G ∗ = 2 √ π / AGM ( 1, √ 2 ) (Gauss; see Borwein and Borwein). A single AGM step satisfies a + b 2 − √ ab ≤ ( a − b ) 2 8 b , so the gap contracts quadratically. This is far faster than the product, but it takes π as an input; speed of convergence and independence from prior constants are different properties, and the product has the second. • A random walk. On the body-centred cubic lattice each coordinate moves by ± 1 at every step. The probability of return to the origin after 2 n steps is p 2 n = b 3 n , and, counting the initial visit, W = ∑ n ≥ 0 p 2 n = 3 F 2 ( 1 2 , 1 2 , 1 2 ; 1, 1; 1 ) = ( G ∗ ) 2 2 π by Dixon’s theorem (the value is Watson’s; see Section 4.3). The tail ∑ n > N p 2 n is bounded by ∑ k ≥ N + 2 k − 3/2 < 2/ √ N + 1. • A pendulum. A pendulum released from the horizontal has period T ( π / 2 ) = 4 K ( 1 / √ 2 ) / ω 0 ; dividing by the small-amplitude period 2 π / ω 0 gives r = 2 K ( 1 / √ 2 ) / π = G ∗ / √ 2 π , so that W = r 2 . A lattice walk and a swinging weight thus compute the same number up to a square. This is a classical identity read as a cross-prediction between two specified models; it is not evidence of any interaction between them. The product and the walk use integers only, and the identification of their limits with gamma values is a separate theorem in each case. Keeping construction and identification apart is the first instance of the rule formulated in the next section. 9 4.3 Watson’s integrals and the three cubic lattices The lattice walk deserves a closer look, because it shows the residue being selected by structure rather than by choice. For a symmetric nearest-neighbour walk on a lattice, the expected number of visits to the origin is the lattice Green function at zero, which Watson (1939) evaluated for the three cubic lattices as triple integrals over the Brillouin zone: W B = 1 π 3 ∫ π 0 ∫ π 0 ∫ π 0 dx dy dz 1 − cos x cos y cos z , W F = 3 π 3 ∫ π 0 ∫ π 0 ∫ π 0 dx dy dz 3 − cos y cos z − cos z cos x − cos x cos y , W S = 3 π 3 ∫ π 0 ∫ π 0 ∫ π 0 dx dy dz 3 − cos x − cos y − cos z , for the body-centred, face-centred and simple cubic lattices respectively. Their closed forms are W B = Γ ( 1/4 ) 4 4 π 3 = ( G ∗ ) 2 2 π = ( 2 π K ( 1/ √ 2 ) ) 2 = 1.39320 39296 . . . , W F = 9 Γ ( 1/3 ) 6 2 14/3 π 4 = 1.34466 11831 . . . , W S = √ 6 32 π 3 Γ ( 1 24 ) Γ ( 5 24 ) Γ ( 7 24 ) Γ ( 11 24 ) = 1.51638 60591 . . . , the first two due to Watson and the third, in this gamma-product form, to Glasser and Zucker (1977); Joyce (1994) gives all three in terms of complete elliptic integrals at singular moduli. Each value was recomputed here by direct numerical integration (Section 10). The body-centred case is the one that lands in the quarter triangle, and the reason is visible in the walk itself. On the body-centred lattice the three coordinates move independently, so the return probability after 2 n steps is the cube of the one-dimensional return probability, p 2 n = b 3 n , and the generating sum becomes the hypergeometric value 3 F 2 ( 1 2 , 1 2 , 1 2 ; 1, 1; 1 ) , which Dixon’s theorem evaluates in gamma values at quarters. The integral form says the same thing: integrating out z gives W B = 1 π 2 ∫ π 0 ∫ π 0 dx dy √ 1 − cos 2 x cos 2 y , and the remaining double integral is the square of a complete elliptic integral at the lemniscatic modulus k = 1 / √ 2 . The face-centred and simple cubic walks do not factorize, and their Green functions involve the elliptic integral at other singular moduli: the equianharmonic value for the face-centred lattice, whose gamma values are at thirds, and the modulus k 6 = ( 2 − √ 3 )( √ 3 − √ 2 ) for the simple cubic lattice, whose gamma values are at twenty-fourths. In the language 10 of complex multiplication, the three cubic lattices select the three fields Q ( i ) , Q ( √− 3 ) and Q ( √− 6 ) This is the clearest instance in the document of a point made abstractly in Section 2. The triangle of Corollary 2 is one of a family, one for each singular modulus, and nothing in the general theory prefers the quarter case. What selects it in a given problem is a structural feature of that problem, here the independence of the coordinates of the walk. A comparison that carries G ∗ from the lemniscate to the body-centred lattice is justified by that feature, and a comparison that tried to carry it to the simple cubic lattice would fail at exactly the point where the coordinates stop being independent. 5. O NE RULE , AND WHERE IT STOPS BEING FINITE 5.1 The rule Two elementary lemmas express the same condition in two settings. L EMMA 5 ( READOUT CLOSURE ). PROVED HERE Let F : X → X be a map and C : X → Y a readout. There exists a map f : C ( X ) → C ( X ) with C ◦ F = f ◦ C if and only if C ( x ) = C ( x ′ ) implies C ( Fx ) = C ( Fx ′ ) Proof. If f exists, equal readouts have equal images under f . Conversely, the formula f ( Cx ) = C ( Fx ) is well defined exactly under the stated condition. □ L EMMA 6 ( DESCENT OF A DERIVATION ). PROVED HERE Let D be a derivation of a commutative ring A and let I be an ideal. The formula D [ a ] = [ Da ] defines a derivation of A / I if and only if D ( I ) ⊆ I. Proof. Representatives differing by an element i ∈ I have derivatives differing by Di , so the formula is well defined precisely when D ( I ) ⊆ I ; the Leibniz rule descends from A □ In both cases the content is the same: a quotient carries an operation exactly when the defining equivalence is a congruence for that operation. When it is not, the canonical repair is to pass to the largest congruence contained in it. That object always exists; the question of interest is whether it is reached in finitely many steps. 5.2 The finite side: the repair terminates Define x ∼ k x ′ to mean C ( F j x ) = C ( F j x ′ ) for 0 ≤ j ≤ k . Each relation refines the one before it. On a finite set the number of blocks cannot exceed | X | , so the chain stabilizes, and once two consecutive relations coincide all later ones do as 11 well. The limit ∼ ∞ is the coarsest F -congruence that refines the readout. This is Moore’s minimization of sequential machines (1956). The smallest instructive example is the cycle 0 → 1 → 2 → 3 → 0 with readout C = ( 0, 0, 1, 1 ) . The states 0 and 1 have the same readout but different next readouts, so no rule depending on the current bit alone can predict the next bit. Retaining one additional bit, the previous readout, separates all four phases and restores closure. The dynamics was deterministic throughout; the chosen description was simply too coarse. 5.3 The infinite side: a tower at τ = i Let A = Q [ E 2 , E 4 , E 6 ] be the ring of quasimodular forms of level one with rational coefficients, and let D = q d dq = 1 2 π i d d τ . Ramanujan’s identities DE 2 = E 2 2 − E 4 12 , DE 4 = E 2 E 4 − E 6 3 , DE 6 = E 2 E 6 − E 2 4 2 show that A is closed under D . At the square point one has E 2 ( i ) = 3 π , E 4 ( i ) = 3 ( G ∗ ) 4 16 π 2 , E 6 ( i ) = 0, so evaluation at i maps A onto Q [ u , v ] with u = π − 1 and v = ( G ∗ ) 4 / π 2 . The field Q ( u , v ) is contained in Q ( π , G ∗ ) and the latter is algebraic over it, so by Corollary 2 the elements u and v are algebraically independent. The value ring therefore has transcendence degree 2, while A has transcendence degree 3, the functions E 2 , E 4 , E 6 being algebraically independent (Zagier 2008, Section 5). Evaluation must lose information, and one can say exactly what it loses. A monomial E a 2 E b 4 evaluates to 3 a ( 3 / 16 ) b u a v b , and distinct monomials in E 2 , E 4 go to distinct monomials in u , v , so evaluation is injective on Q [ E 2 , E 4 ] . Writing any F ∈ A as F 0 + E 6 F 1 with F 0 ∈ Q [ E 2 , E 4 ] , it follows that ker ( ev i ) = ( E 6 ) , the principal ideal generated by E 6 This ideal is not stable under D , since ( DE 6 )( i ) = − E 4 ( i ) 2 / 2 ̸ = 0; so by Lemma 6 the values alone cannot carry differentiation. The repair of Section 5.1, which in the language of differential algebra is the largest differential ideal contained in ker ( ev i ) , is the chain I k = { F ∈ A : ( D j F )( i ) = 0 for 0 ≤ j ≤ k } T HEOREM 7 ( THE JET TOWER IS THE E 6 - ADIC FILTRATION ). PROVED HERE (a) For every k ≥ 0 , I k = ( E k + 1 6 ) , the principal ideal generated by E k + 1 6 . In particular the chain is strictly descending. (b) ⋂ k I k = 0 , and this intersection is the largest D-stable ideal contained in ker ( ev i ) 12 (c) For every finite-dimensional subspace V ⊂ A the chain I k ∩ V reaches 0 at some finite k. Proof. (a) The form E 6 has a simple zero at i , since E 6 ( i ) = 0 and ( DE 6 )( i ) ̸ = 0. Given F ∈ A , write F = E j 6 G with E 6 ∤ G , which is possible since A is a polynomial ring. Then G / ∈ ker ( ev i ) = ( E 6 ) , so G ( i ) ̸ = 0, and the order of vanishing of F at i is exactly j . Since F ∈ I k means that F vanishes at i to order at least k + 1, we have F ∈ I k if and only if j ≥ k + 1, that is, if and only if E k + 1 6 divides F . (b) A nonzero polynomial is divisible by only finitely many powers of E 6 , so the intersection is 0. It is D -stable because F ∈ I k + 1 implies DF ∈ I k , and if J ⊆ I 0 is any D -stable ideal and F ∈ J , then every D j F lies in J ⊆ I 0 , so F ∈ ⋂ k I k . (c) By (a), I k ∩ V consists of the elements of V divisible by E k + 1 6 , and for k larger than the maximal degree in E 6 occurring in V this is 0. □ The ideals I k are thus the powers of a single principal ideal, and the tower is the E 6 -adic filtration of A . The completion in which the congruence closes is the E 6 -adic completion Q [ E 2 , E 4 ][[ E 6 ]] , and the Taylor expansion of a form at i is its E 6 -adic expansion read through evaluation. The whole structure rests on one fact, that ker ( ev i ) is a height-one prime generated by a form with a simple zero, and that fact in turn rests on the transcendence degree count of Corollary 2. Part (c) says that the tower has finite height on each piece of bounded weight, and the proof of (a) gives the height exactly. Let A ≤ w denote the forms of weight at most w , and let k ( w ) be the least k with I k ∩ A ≤ w = 0. Equivalently, k ( w ) is the largest order to which a nonzero rational quasimodular form of weight at most w can vanish at i T HEOREM 8 ( HEIGHT OF THE TOWER ). PROVED HERE For every even w ≥ 0 , k ( w ) = ⌊ w /6 ⌋ Proof. By the proof of Theorem 7(a), the order of vanishing at i of a nonzero form F = E j 6 G is j , and if F has weight at most w then 6 j ≤ w . So no form of weight at most w vanishes to order greater than ⌊ w / 6 ⌋ . The order ⌊ w / 6 ⌋ is attained by E m 6 E r 2 with m = ⌊ w /6 ⌋ and r = ( w − 6 m ) /2. □ The same argument applies at every point with complex multiplication, provided the coefficients are allowed to be algebraic. C OROLLARY 9 ( TOWER HEIGHTS AT CM POINTS ). PROVED HERE Let τ 0 be a CM point, let A Q = Q [ E 2 , E 4 , E 6 ] , and define I k as before with evaluation at τ 0 . Then ker ( ev τ 0 ) is the principal ideal ( P ) , where P = E 6 if τ 0 is equivalent to i , P = E 4 if τ 0 is equivalent to ρ , and P = ( 1728 − j ( τ 0 )) E 3 4 + j ( τ 0 ) E 2 6 otherwise; one has I k = ( P k + 1 ) ; and the height of the tower on forms of weight at most w is ⌊ w / 6 ⌋ , ⌊ w / 4 ⌋ or ⌊ w / 12 ⌋ in the three cases. In each case this is the bound that the valence formula imposes on the order of vanishing of a modular form of weight w at τ 0 13 Proof. Nesterenko’s theorem states that for every τ in the upper half-plane, at least three of the four numbers q = e 2 π i τ , E 2 ( τ ) , E 4 ( τ ) , E 6 ( τ ) are algebraically independent over Q . At a CM point j ( τ 0 ) is algebraic, so E 4 ( τ 0 ) 3 and E 6 ( τ 0 ) 2 are proportional by an algebraic factor and E 4 ( τ 0 ) , E 6 ( τ 0 ) are algebraically de- pendent over Q . Hence the transcendence degree of Q ( E 2 ( τ 0 ) , E 4 ( τ 0 ) , E 6 ( τ 0 )) is exactly 2, and ker ( ev τ 0 ) is a height-one prime of the unique factorization domain A Q , hence principal, generated by an irreducible polynomial. The displayed P lies in the kernel in each case and is irreducible (in the third case because j ( τ 0 ) ̸ = 0, 1728, so P is a nonzero multiple of E 2 6 − cE 3 4 with c ̸ = 0), so P generates. In each case P has a simple zero at τ 0 : this is the computation ( DE 6 )( i ) ̸ = 0 or ( DE 4 )( ρ ) ̸ = 0 already made, and in the third case P = 1728 ( E 3 4 − j ( τ 0 ) ∆ ) , whose τ -derivative at τ 0 is − 1728 ∆ ( τ 0 ) j ′ ( τ 0 ) ̸ = 0 because j ′ vanishes only at points equivalent to i or ρ . The rest of the argument is that of Theorem 7(a) and Theorem 8 with P in place of E 6 and the weight of P , namely 6, 4 or 12, in place of 6. For the final sentence, the valence formula gives ord τ 0 ( f ) ≤ e τ 0 w / 12 for a modular form f of weight w , where e τ 0 = 2, 3, 1 is the order of the stabilizer of τ 0 in PSL 2 ( Z ) , and 12/ e τ 0 is the weight of P □ The content of the corollary is that, with algebraic coefficients, quasimodular forms vanish at a CM point no more deeply than modular forms do, even though the ring is larger. The computation in Section 10 confirms the heights ⌊ w / 6 ⌋ at i through weight 40 and ⌊ w / 12 ⌋ at τ 0 = i √ 2 , where j = 8000, through weight 26; these computations are checks on the proofs, not part of them. 0 6 12 18 24 30 36 1 2 3 4 5 6 weight w height k ( w ) w /6 The height k ( w ) = ⌊ w / 6 ⌋ of the jet tower on rational quasimodular forms of weight at most w The dashed line is the valence-formula bound w / 6 on the order of vanishing of a modular form of weight w at i . The staircase is flat until weight 6, where the first rational form vanishing at i appears, and then rises with slope 1/6. The staircase makes the phrase “stops being finite” quantitative. On every piece of bounded weight the repair terminates, and the number of steps it needs grows linearly in the weight; the tower on the whole ring is the limit of this ramp as w → ∞ . The location of the kink is worth noting: over C the form E 2 − 3 / π already vanishes at i and the ramp would begin at weight 2; it is 14 rationality of coefficients that places the first step at weight 6, and the whole E 6 -adic description depends on it. The two settings can now be placed side by side. FINITE DYNAMICS (5.2) EVALUATION AT τ = i (5.3) Description being re- duced a state x ∈ X a form F ∈ A Reduced description readout C ( x ) value F ( i ) Operation to be kept the update F the derivation D Repair at step k k future readouts the k -jet at i Does the chain stop? yes, within | X | blocks on all of A , never; on weight at most w , after ⌊ w /6 ⌋ steps Limit the coarsest congruence, computable the zero ideal, reached only in the limit This is the precise sense intended by “finite construction, continuous realiza- tion”. Each stage of the tower is a finite object, namely finitely many numbers in a ring of transcendence degree 2. The congruence closes only in the completion, that is, in the full Taylor expansion at i , through which a ring of transcendence degree 3 is recovered from values in a ring of transcendence degree 2. On any piece of bounded weight the finite behaviour returns, with an explicit bound on the number of steps. Two clarifications about what is special here. Over C the tower is nontrivial at every point τ 0 , since E 6 − E 6 ( τ 0 ) lies in the kernel of evaluation while its derivative does not; but over C the kernel is a maximal ideal of height 3 and the tower is not a principal filtration. What τ = i contributes, together with rational coefficients, is that the kernel becomes a height-one prime with the explicit generator E 6 , so that the tower is the E 6 -adic filtration and every jet lies in the explicit ring Q [ π − 1 , ( G ∗ ) 4 / π 2 ] . Corollary 9 shows that this is a property of CM points in general, with the generator depending on the point. The two halves of the construction are textbook material; the contribution of this section is the comparison and the exact description of the tower and its height. 6. W HAT ONE CONSTANT CANNOT CARRY A comparison between two descriptions is only as informative as the list of what it does not transfer. Each of the following is a short exact example in which a shared number leaves something undetermined. • Interpolation. For real a , the function F a ( x ) = Γ ( x ) e a sin 2 π x agrees with Γ at every positive integer, satisfies F a ( x + 1 ) = xF a ( x ) , and satisfies the reflection formula on ( 0, 1 ) ; nevertheless F a ( 1 / 4 ) / F a ( 3 / 4 ) = e 2 a G ∗ . Values at the integers together with recurrence and reflection do not single out Γ ; 15 log-convexity does, by the Bohr and Mollerup theorem, and that selection principle is doing real work. • Same exponent, different measure. Along an orbit of H = p 2 / 2 m + κ q 4 / 4 with amplitude A and maximal speed v 0 one finds Tv 0 / ( A √ π ) = G ∗ , ⟨ q 2 ⟩ / A 2 = 4 / ( G ∗ ) 2 and ⟨ q 4 ⟩ / A 4 = 1 / 3, so the kurtosis of the orbital distri- bution is ( G ∗ ) 4 / 48. For the thermal density proportional to e − aq 4 the kurtosis is ( G ∗ ) 2 / 4. The same quartic, with a different measure, gives a different answer. • Same period, different arithmetic. For odd squarefree n , let E n be the curve y 2 = x 3 − n 2 x , and let c n = A n − B n / 2, where A n and B n count integer solutions of 2 x 2 + y 2 + 32 z 2 = n and of 2 x 2 + y 2 + 8 z 2 = n respec- tively. Tunnell’s theorem, in the normalization of Koblitz (Chapter IV), with the real period Ω n = √ π G ∗ / √ n taken over both real components, gives L ( E n , 1 ) / Ω n = c 2 n / 8. Exact enumeration gives c 1 = 1, c 3 = 2, c 17 = − 4 and c 41 = 0, the last because 41 is a congruent number. As a consistency check at n = 1, the curve has Tamagawa number 2 at the prime 2 and torsion of order 4, and the Birch and Swinnerton-Dyer formula predicts L / Ω = 1 · 2 / 16 = 1 / 8. Every curve in the family shares the period G ∗ ; all of the arithmetic sits in c n . In the unconditional direction, if n is congruent then E n has positive rank, so L ( E n , 1 ) = 0 by Coates and Wiles, and hence c n = 0; the converse depends on BSD. This example is the natural place to say what the jet tower of Section 5 does and does not say about L -functions. Tunnell’s count determines the value L ( E n , 1 ) exactly, that is, the 0-jet of L ( E n , s ) at s = 1. When c n = 0 the finite count is silent about the derivative. The first derivative L ′ ( E n , 1 ) is governed by the Gross and Zagier formula, through the height of a Heegner point, and the full Taylor expansion at s = 1 is the subject of the Birch and Swinnerton-Dyer conjecture: the order of vanishing should equal the rank, and the leading coefficient should be the product of the order of the Tate- Shafarevich group, the regulator and the Tamagawa numbers, divided by the square of the torsion order. The rank 0 and rank 1 cases are theorems (Coates and Wiles; Gross, Zagier and Kolyvagin). So BSD is precisely the assertion that the analytic jet at s = 1 and the arithmetic of E ( Q ) determine one another, and the finite construction reaches the first jet only. The pattern is the one seen at τ = i : the value is finitely computable, and each further derivative requires new input. It is only a pattern. The function L ( E , s ) is not an element of Q [ E 2 , E 4 , E 6 ] , and no theorem connects the tower at τ = i to the expansion of L at s = 1; the two share a shape and not an object. • Same density, different momentum. For ψ ( x ) ∝ exp ( − ( x / L ) 4 / 2 ) one has ∆ x ∆ p = √ 3 ̄ h / G ∗ ≈ 0.5854 ̄ h . Multiplying by the phase e i β x 2 / ( 2 L 2 ) leaves the position density unchanged and gives ∆ x ∆ p = √ 3 + β 2 ̄ h / G ∗ . The density omits exactly the phase information that the momentum requires. 16 • A scalar is not a complete invariant. The groups Z / 4 and the Klein four- group have the same order and are not isomorphic. By the same token, the appearance of G ∗ in two constructions does not identify the constructions. 7. A QUARTIC TEST THAT CAN FAIL The following is the one place where the comparison rule yields a prediction in advance of data. Take V ( q ) = κ 2 2 q 2 + κ 4 4 q 4 + κ 6 6 q 6 with κ 4 > 0, constant inertia m , and an orbit of amplitude A passing through the origin, and define Φ ( A ) = A T ( A ) √ κ 4 2 π m , ∆ ( A ) = 2 | κ 2 | κ 4 A 2 + | κ 6 | A 2 κ 4 T HEOREM 10 ( CONTROLLED QUARTIC APPROXIMATION ). PROVED HERE If ∆ ( A ) < 1 , then G ∗ √ 1 + ∆ ( A ) ≤ Φ ( A ) ≤ G ∗ √ 1 − ∆ ( A ) Proof. Put q = Au . Then V ( A ) − V ( Au ) = κ 4 4 A 4 ( 1 − u 4 ) ( 1 + e ( u )) with e ( u ) = 2 κ 2 κ 4 A 2 1 − u 2 1 − u 4 + 2 κ 6 A 2 3 κ 4 1 − u 6 1 − u 4 On [ 0, 1 ) the first ratio is at most 1 and the second at most 3 / 2, so | e ( u ) | ≤ ∆ ( A ) . The period integral is the quartic integral with the additional factor ( 1 + e ( u )) − 1/2 ; bounding that factor and using ∫ 1 0 ( 1 − u 4 ) − 1/2 du = √ π G ∗ /4 gives the result. □ If κ 2 and κ 6 are measured independently, the theorem gives an interval for Φ before the period is examined, and a measured period outside that interval refutes the quartic model for that system. A prediction that can fail in this way carries more weight than a numerical agreement that cannot. 8. O BJECTIONS AND REPLIES The rule is just the definition of a congruence. It is, and Section 1 says so. Moore’s refinement and the largest differential ideal are both standard constructions of the greatest congruence below a given relation. The definition becomes informative once one can say where the asso