The Riemann Hypothesis as Harmonic Zero-Line Closure in the Coupling Potential Grammar

The Riemann Hypothesis (RH), one of the seven Clay Mathematics Institute Millennium Prize Problems, conjectures that all non-trivial zeros of the Riemann zeta function (s) have real part equal to 1/2. This companion deposit proposes that RH is the formalism-substrate-resolution-class readout of an underlying substrate-coupling-architecture in which the critical line (s) = 1/2 is the unique axial closure-coordinate zc for arithmetic phase-memory. Operator-canonical-substrate description: the critical line is "a hidden string of zeros holding up prime pebbles, rigid in a beautiful pattern" — substrate-coupling-architecture-substance at the zeta-function-resolution-class. Prime-distribution irregularity is the formalism-substrate-readout of substrate-coupling-junction-routing across number-line-substrate-positions. The "rhythm" of prime-distribution is architectural (substrate-coupling-architecture-pattern), with probabilistic-appearance being measurement-resolution-class-surface-edge artifact per Apex Deposit Corollary 0. 1 ("curves don't exist as architectural substance"). The architectural-mechanism: the zeta function's non-trivial zeros encode the harmonic-substrate-readout that reconstructs the prime-counting function (x) via Riemann's explicit formula. An off-critical-line zero would represent an IE-017 symmetry-breaking event sufficient to produce an unbounded or structurally-non-minimal IE-019 residual in the prime-counting harmonic reconstruction. Conversely, the critical line is the locus at which IE-019 residual is minimized — the closure-axis under the architectural-grammar. The corpus's approach-geometry-of-amplitude-zeros methodology (HLRP #1 Approach Geometry of Amplitude Zeros, Zenodo DOI 10. 5281/zenodo. 18727570 (https: //doi. org/10. 5281/zenodo. 18727570), originally developed for gauge-field amplitudes at the quarter-collinear regime n = 18) applies directly to zeta-zeros at the analytic-number-theory-substrate-resolution-class. Both substrate-resolution-classes share the same architectural-substance: amplitude-zero distribution as substrate-coupling-architecture-substance, with axial closure-coordinate constraints. This work proposes a formal architectural scaffold for RH as harmonic zero-line closure; the conjectures are stated as internal consequences of the proposed grammar. Keywords: Riemann Hypothesis; Riemann zeta function; critical line; non-trivial zeros; prime distribution; analytic number theory; substrate-coupling architecture; HLRP; Hydrogen Lifecycle Research Programme; approach geometry of amplitude zeros; IE-006 feedback gap; IE-009 phase memory; IE-017 symmetry breaking; IE-019 residual; phase-memory closure; harmonic zero-line; architectural formalization