PRM-024 · Route B golden-unit restriction (positive algebraic integers in Q(√5), abc = 1)

Premise

Register corrected — R81 (D20, S327j). The PI ruled this correction to the register row (Trackers/RULING_R81_S327_PREMISE_REGISTER.md). Statement now (PRM-024/r1, appendix_e_vacuum_selector.tex:117, quality HOUSE_TRANSCRIBED): Let Jvac = diag(a, b, c) with a, b, c > 0 algebraic integers in ℚ(√5) and abc = 1.. Paper 2:521-527 restates the restriction without "positive"; App E:117 controls. The register row the drafting lane quotes below is the row as it was supplied to the lane, before R81; it is kept in the register's history (R29: nothing is overwritten).

Register kind: ADOPTED_PREMISE. Located suite statement:

Let $\vac=\diag(a,b,c)$ with $a,b,c>0$ algebraic integers in $\mathbb{Q}(\sqrt5)$ and $abc=1$.

Appendix E · Route B: golden-field units + DET-7 · PDF p.165 · appendix_e_vacuum_selector.tex:117.

What kind of premise this is

ADOPTED_PREMISE is the register kind. Appendix E calls the golden-field hypothesis Route B’s residual input. Algebraic integrality, positivity and unit product are all retained: the proof uses them to express the three entries as powers of the positive fundamental unit with zero total exponent. H2 remains a separate equality input. No assertion of total positivity in both field embeddings is added to the source. Source status words: Route~B's residual input is the golden-field hypothesis (appendix_e_vacuum_selector.tex:144).

Where the suite invokes it

Located invocations in the read excerpts, not an exhaustive full-source-tree census. Pages refer to saved Rev32.7 page-delimited text, not PDF-image authentication.

Arguments that rest on it

LIB2-020 — CONFIRM. The theorem’s stated domain is positive algebraic integers in Q(sqrt5), with abc=1; H2 is then separately imposed. Confirm this arithmetic restriction, not the CKM convention. Step locator: appendix_e_vacuum_selector.tex:117 (CONDITIONAL).

LIB2-001 — ADD. The gap-pair classification uses exponent triples obtained from this positive golden-unit domain. Its zero-sum exponent condition comes from abc=1; the field restriction supplies the unit-power reduction. Step locator: appendix_e_vacuum_selector.tex:121 (LOAD_BEARING).

What breaks without it

LIB2-020 and LIB2-001: the proof uses invertibility in the stated integer ring and positivity to obtain the exponent triple (appendix_e_vacuum_selector.tex:121); the classification then works in that domain. Removing the domain removes that printed reduction. NOT ASSESSED — no general classification for an unrestricted field is supplied.

The field-free Route A is a different sufficient route, not a discharge of Route B’s field hypothesis. Ordered physical frame selection remains loaded (appendix_e_vacuum_selector.tex:509).

Registrar sync

Registrar state after R76, R78 and R82 (S327j, master 0e1418ad6501767d). Declared today: LIB2-001, LIB2-020. Declared at S327i from A1634 after the house read each passage (R76): LIB2-001. The lane's lines below describe the master as it was supplied to the lane (d0aa26db154ff43f), before these changes.

Declared today in the supplied material, Registrar master d0aa26db154ff43f: LIB2-020. Draft dispositions — ADD: LIB2-001; CONFIRM: LIB2-020. LIB2-047 is the determinant identity at the working vacuum; its unit product is not proof of dependence on the golden-field restriction. LIB2-296 concerns the superseded singular-vacuum chain, not this positive unit triple. The golden-unit holonomy statement at appendix_x_zero_parameter_input_ledger.tex:958 concerns a different recipient/holonomy argument. Its supersession does not retire this arithmetic hypothesis. Paper 2:521–527 omits the positivity word in its shorter restatement; the complete positive domain at Appendix E:117 controls this note.

Tier and what this does not show

A premise is not a result, and invocation count is not importance. An ADD remains a proposal until house verification under R76; a DISPUTE goes to the PI. No record or premise is upgraded, downgraded, or edited. Unlocated dependencies remain unknown.

Sources

TeX: appendix_e_vacuum_selector.tex:117; appendix_e_vacuum_selector.tex:121; appendix_e_vacuum_selector.tex:144; appendix_e_vacuum_selector.tex:509; appendix_x_zero_parameter_input_ledger.tex:958; p2_mass_hierarchy_resolvent_quintics.tex:524. Registrar: LIB2-001, LIB2-020, LIB2-047, LIB2-296. Premise: PRM-024.