Pure Time Theory – Chapter V - Temporal Mathematics: Cadenced Geometry, the Generated Zeta Spectrum, and the Conjectural Signature of a Derived Frame

Notation used below: J is the imaginary unit; (+) and (-) are carried composition and carried difference of displacements; (.) is the carried scalar action; D_g is the geometric oriented-difference operator and D its static read; Half is the unique half-displacement; ScaleAddr is the generated additive scale address; Read is the static read map. This chapter develops Temporal Mathematics as a self-contained generative foundation. Its primitive mathematical commitments are a no-gap carried continuum, serial composition, one simple oriented non-branching production, invariant cadence, and invariant non-degenerate thrust. The conical stratification is not postulated. It is generated from the null advance word, the continuous phase cell, and finite successor attachment: the null word produces one apex, every non-null advance word carries one co-level section, a fixed phase prefix generates one ray, and canonical homology changes only the advance word. The absence of a non-null section at null depth excludes a cylindrical base. A complete cadence returns to the same phase while thrust places the return at a distinct carried position. This produces a unique successor and an intrinsic order before counting. Finite closure segments generate N_TM, followed by Z_TM, Q_TM, exact quotient-residual reads, and potential rather than actual infinity. Homologous axial and radial cells give a constant slope from the apex, straight generators, homothetic sections, and a direction-independent scale relation. Composition of scale words supplies multiplication, reverse homology supplies inversion, and serial displacement supplies addition. Exact finite division gives the unique half-displacement and carried midpoint; the real and complex read grammars are introduced only afterward. The finite spectral construction is generated before any mode coordinate is evaluated. Invariant non-degenerate thrust first produces a non-null successor section. Its non-null radial attachment, together with the unique half-displacement, yields a canonical modal seed class before any role delta or complex coordinate exists. Any non-null role delta can later refine the seed provenance, but is not required for the existence of the modal grammar. For each closure ratio r_n = 1/(n+1), the geometric modal law produces one homologous position A_n = Hr_n(v). The first new readable relation between adjacent positions is their oriented chord, and reapplying the same relational operation generates the finite delta tower D_g A_n = A_n (-) Aₙ₊₁, D_g^(k+1) A_n = D_g^k A_n (-) D_g^k Aₙ₊₁. Repeated unique halving then produces the difference-apex endpoint Q_K = SUMₖ₌₀K-1 Half^(k+1)( D_g^k A_0 ), where SUM denotes carried composition. This object contains no s, sigma, gamma, logarithm, exponential, zero, or analytic function. The endpoint is not a sum chosen because its read is G_K. The binary complement-exchange continuation has, after K finite unfoldings, the unique normal form E(A) ->_K Q_K (+) R_K^geom. Here Q_K is the part already generated by K exact exchange-and-halving steps, and R_K^geom is the still-unfolded geometric continuation. Thus the apex and its residual are produced together before their static reads G_K and R_K. The alternating signs and finite coefficients are theorems of this geometry. Expanding the oriented differences gives D_g^k A_0 = SUMₙ₌₀ᵏ (-1)^n C(k,n) (.) A_n, so Q_K = SUMₙ₌₀K-1 (-1)^n BK,n (.) A_n, BK,n = SUMₖ₌ₙK-1 2^(-k-1) C(k,n), where C(k,n) is the binomial coefficient. The sign records endpoint orientation under repeated differencing; it is not imported from a role parity or a separately assumed alternating history. The factors 2^(-k-1) are the reads of successive unique half-displacements. Only after this finite geometry exists is its composable homothety-phase law classified statically. Order-rigidity supplies unique coefficients sigma and gamma and the paired read chi_s(r) = exp( s * ScaleAddr(r) ), s = sigma + J gamma. In conventional notation, chi_s(r) = r^s, hence Read(A_n) = (n+1)^(-s). This classification assigns coordinates to every generated modal law but produces no particular mode. A finite mode appears only through the geometric closure Q_K = 0. Its exact static read is Read(Q_K) = G_K(s) = SUMₖ₌₀K-1 2^(-k-1) D^k a_0(s), a_n(s) = (n+1)^(-s). Thus the complex roots are the downstream coordinates of generated modal laws whose difference-apex path closes; the geometry produces the condition and the decimals are calculated as its solutions. The developed alternating polygon and its midpoint triangle are proved equivalent to the delta-tower endpoint; they are presentations of the construction, not the provenance of its alternation. The marked static orbit of a classified law is a similarity helix Z_s(u) = e^(-sigma u) R-gamma u Z_0, H_s(u) = ( u, Z_s(u) ), with generated scale address and marks retained. Its tangent ratio, pitch angle, and one-turn radius law are downstream geometric reads of the appeared mode. The four role histories carry the two radial exponents sigma and 1-sigma and the two phase orientations +gamma and -gamma; centering is equality of the complementary radial laws and never erases role provenance. The first complete three-role first-delta record requires three homologous positions for each of three co-level roles, hence nine incidences. This makes depth nine the first record-size threshold; it does not generate the difference law, the signs, the seed, or the halving coefficients, all of which exist before record completeness. The modal seed is already non-null by non-degenerate thrust and the first radial attachment. A non-null role delta supplies additional role-specific provenance; a null role record does not erase the frame-level modal seed. At depth nine, 512 G_9(s) = 511 - 502*2^(-s) + 466*3^(-s) - 382*4^(-s) + 256*5^(-s) - 130*6^(-s) + 46*7^(-s) - 10*8^(-s) + 9^(-s). It has one simple zero in the certified disk centred at rho1,9 = 0.1221546597316678506450804236219617952246998731913241 + 14.0205296130986500837692575205259443888311171988437585 J. At the published centre, ( Re G_9 , Im G_9 ) ~ ( -4.27333 , -2.73574 ) x 10^(-61), |G_9| ~ 5.07402 x 10^(-61). The two components vanish at the unique simple root certified inside the displayed disk. The short point has residual |G_9| ~ 7.3644 x 10^(-18), while the twenty-one-digit point has residual |G_9| ~ 5.2707 x 10^(-22). The potential alternating read admits the downstream decomposition eta_TM = G_K + R_K, and its odd/even quotient gives eta_TM = (1 - 2^(1-s)) zeta(s) on the stated analytic domain. The Gamma factor used by this continuation is not imported. Finite positive-ray truncations produce one carried target Gamma_TM(s) on Re s > 0; the finite beta identity gives Gauss' limit, the generated Euler boundary gives the reciprocal product, and the product proves non-vanishing. The same construction produces a logarithm of Gamma_TM, its digamma read, and an exact cellwise Bernoulli residual. Thus the remainder, completed exchange factor, and smooth phase consume a derived expression-specific object rather than an undeclared classical function. The two later laws have different ranks. Finite-depth modal production is the geometric event: on a bounded terminal cell, a mode appears at the first depth after which the winding of the generated apex remains equal to the terminal winding. The decreasing geometric-residual bound gives a finite certificate depth, hence irregular production plateaus and saturation on every bounded prefix. This law belongs to generated depth and does not consume the completed exchange phase. The completed exchange identity is a downstream terminal ordering readout. On the centred read, the smooth single-history phase is theta(T) = Im Log Gamma_TM( 1/4 + J T / 2 ) - (T/2) log pi_TM, the paired exchange phase is 2 theta(T), and the residual phase is the lifted argument of the zeta read. Their exact sum gives the terminal count identity 2 pi_TM * TerminalCount(T) = 2 pi_TM + 2 theta(T) + 2 Arg^ zeta( 1/2 + J T ), where Arg^ denotes the lifted (unwound) argument. It orders and counts the open-continuation modes after production; it does not produce the finite geometric mode. Production depth and terminal height remain different read types. Depth K and mode label j remain distinct. Finally, the projection that retains only the invariant total forgets the finite split (G_K, R_K), the generated depth, the incidence record, and the successor transfer. The static-object perimeter, the derived codomain, the finite depth-eight/depth-nine separating fibre, and differential closure jointly establish the framework verdict: the classical ZFC frame is a derived static truncation, and it can neither prove nor refute RH. The truth-bearer of the centering question is carried by source provenance that the quotient does not contain. A census of proof strings inside the derived truncation is therefore not a stronger missing result; it is bookkeeping on the object whose boundary has already been proved from the encompassing TM frame. The certified depth-nine mode refutes the global temporal centering claim. It exhibits no terminal zero of the zeta quotient off the centred line. These two statements concern different objects, and their coexistence is exactly what the codomain theorem explains. The same codomain, fibre, and differential-closure chain yields a general conjectural-signature result. A derived frame can name only objects generated in its codomain. A source coordinate omitted by that codomain cannot appear internally as a named absence; its only native manifestation is the structurally persistent conjectural status of the well-formed boundary sentence whose truth depends on that coordinate. For the derived classical ZFC frame, RH has exactl