Route-APIA v1 (leg 312): arbitrary-precision re-measurement of leg 178 and leg 176

Branch leg/312-apia-v1. Agent: LEG (heavy, construction-eligible). Gate answer: YES. Data: writeup/data/p2_route_apia_v1.json. Runner: experiments/p2_route_apia_v1.py. Evidence/figure: experiments/p2_route_apia_v1_evidence.py, fig73. Novelty: writeup/novelty/leg_312.md.

0. The gate, verbatim (pre-committed by the DM)

Do the two pre-registered re-measurements (leg 178, leg 176), run at arbitrary precision, change the banked headline of leg 178 (the 3.1e+03 contamination) or leg 176 (the lost N=1024 row), magnitudes, not booleans?

Answer: YES, both change.

1. Territory and what this leg builds

solver/interval_mp.py (NEW): a decimal.Decimal-backed rigorous interval type MPInterval, mirroring solver/interval.py's directed-rounding design but using Decimal's native Context(rounding=ROUND_FLOOR/ROUND_CEILING) instead of float64's np.nextafter one-ulp workaround. Two sections:

• Section 1 (leg 178): + - * /, isum/dot, and Taylor-series transcendentals (dsin, dcos, dtan_ratio, mp_pi) with proved remainder bounds, all rigorous (endpoints, not point values).
• Section 2 (leg 176): banded Cholesky, banded triangular inverse, banded matmul, flagged non-rigorous (high working precision, not interval-enclosed), exploiting x_gram's exact bandwidth-4 structure.

test_interval_mp.py (NEW): 13-test adversarial battery against fractions.Fraction ground truth, extending test_interval_stress.py's pattern. It caught three real bugs before either re-measurement ran on top of them, see §4.

Not built: a general arbitrary-precision library. writeup/novelty/leg_312.md names the seven existing ones (INTLAB, MPFI, mpmath's iv, Arb/FLINT, kv, Boost.Interval, Julia's IntervalArithmetic.jl) and states why none is importable (requirements.txt dependency ban) and why the scope is deliberately two named numbers, not their general problem.

2. Leg 178's re-measurement

2.1 Configuration, read off the banked runner verbatim

experiments/p2_route_wes_v1_space.py: N_QUAD_CHECK = 128, GRADE_DEPTHS = (12, 24, 48, 96); the banked reading is wes_coercivity_gap_exact(128, "T2_egm", "E", 4.0, n_grade=96), i.e. class T2_egm, weight family E (E_egm), gamma=4. n_unif defaults to max(64, 4*128) = 512. This is the exact call this leg re-measures: chosen by leg 178's own pre-registration, not by this leg.

2.2 Where the precision loss lives

wes_form_matrices_constrained forms F_m = sum_k V[k,m] sin(k*theta) POINTWISE, at each of the (here) 7,824 quadrature nodes, before any weight or inner product is applied: V is the exactly-constrained integer basis (wes_constrained_basis, leg 178's own "central instrument finding"), whose columns cancel F_m to order theta^(2p+1) by construction. At theta ~ 3e-31 (the innermost node of the n_grade=96 grading), the cancellation needs digits float64's 53-bit mantissa does not have: the pre-cancellation terms are O(theta), the roundoff floor is O(eps*theta) ~ 1e-47, and the true post-cancellation signal is O(theta^3) ~ 1e-92, swamped by seven orders of magnitude before it is even that small.

2.3 The fix, scoped

Only the 1,640 of 7,824 nodes with min(theta, pi-theta) < 1e-3 are re-derived (the rest are far enough from the endpoints that float64's sin(k*theta) is already accurate: verified, not assumed, by leaving them untouched and confirming the float64-reproduction check in §2.4 matches the banked reading bit-for-bit). At each unsafe node:

1. sin(theta), cos(theta) via an ADAPTIVE-term Taylor series at 120 working decimal digits (_mp_sin_cos in experiments/p2_route_apia_v1.py): adaptive because at theta ~ 3e-31 a SINGLE term already saturates 120 digits of precision (the next term is ~theta^3, i.e. ~1e-92 relative to the first), while the outer end of the graded region (theta ~ a few e-3) needs ~15-20 terms. solver/interval_mp.py's own dsin/dcos deliberately use a FIXED, generous term count (3*prec+30) for simplicity and a uniform correctness proof; this leg's local adaptive version is a performance optimisation on top of that already-tested machinery, not a change to it.
2. sin(k*theta), cos(k*theta) for k=1..128 via the angle-addition recurrence (sk, ck = sk*c1+ck*s1, ck*c1-sk*s1), O(k) Decimal multiplications per node instead of re-deriving each k from a fresh series.
3. F_m, LF_m (the linearisation contraction), HF_m recomputed as exact integer-weighted Decimal sums over V's (at most 4) nonzero entries per column.
4. Converted back to float64 and substituted into the SAME G, B, contamination, and Rayleigh-quotient pipeline wes_form_matrices_constrained/ wes_coercivity_gap_exact use (mirrored, not imported, since that pipeline does not expose the intermediate F this leg needs to patch: solver/ energy_coercivity.py itself is read-only, unedited).

Certification: _mp_sin_cos's plain recurrence is spot-checked against solver/interval_mp.py's RIGOROUS MPInterval dsin/dcos on 6 representative unsafe nodes; the plain value lands inside the rigorous interval at all 6 (interval widths ~1e-120/~1e-123), spot_check_vs_rigorous_mpinterval.all_contained = true in the JSON.

2.4 Result

The float64 re-derivation matching the banked reading bit-for-bit (row 2 of the table) confirms this leg's mirror of wes_form_matrices_constrained is faithful before any patch is applied. The MP-patched gap, +0.4999999875, agrees with the clean n_grade=48 reading (+0.49999975) to five significant figures and with every other clean depth's +0.5 ceiling.

Caveat on the "contamination" number after patching: the contamination diagnostic is (eps*Fabs)^2*pw / F^2*pw, a bound on FLOAT64's OWN roundoff relative to the signal. Applied to the MP-patched F (now near its true, correctly-tiny value rather than float64's spuriously large garbage), the same formula's denominator shrinks, so the ratio goes UP: this is an artifact of reusing a float64-specific diagnostic on a corrected numerator, not evidence that the MP-patched value is itself contaminated (see the rigorous interval spot-check in §2.3, and the gap's agreement with the clean-depth ladder). The honest reading of "does arbitrary precision change the contamination number" is: yes, but that specific diagnostic formula was never designed to answer "is the arbitrary- precision result trustworthy", the gap comparison is.

3. Leg 176's re-measurement

3.1 Configuration

experiments/p2_route_h2c_v1_construction.py: rect_sigma(N, bordered=True, gram=True, lo_mode=0) at N=1024 (SIG_N's own top rung), cross-checked at N=512 (REL_N's own top rung, "the float-reliable window").

3.2 Where the precision loss lives, and the substitution

rect_sigma whitens the bordered operator A via Gch @ A @ Gdih, where Gch = sqrt(Gc), Gdih = sqrt(Gd)^{-1} are computed by origin_h2_certificate._sym_sqrt: a dense np.linalg.eigh-based SYMMETRIC matrix square root. Gd, Gc are bordered_gram(x_gram(...)): block-diagonal, with x_gram(N) = I + J^4 exact and banded (bandwidth 4), condition number reaching ~1e13 at N=1024.

Mathematical substitution, not an approximation: for SPD G = R^T R (ANY valid factorisation, a Cholesky factor works exactly as well as the symmetric square root, since two factorisations of the same inner product differ only by a unitary, which does not change singular values), ||Ax||_G = ||Rx||_2. So the singular values of Rc @ A @ Rd^{-1} (Cholesky factors) equal those of sqrt(Gc) @ A @ sqrt(Gd)^{-1} (symmetric square roots). This leg computes Rd = banded_ cholesky(Gx_d, ..., bw=4), Zd = banded_triangular_inverse(Rd, ...) (O(N^2*bw), not O(N^3)), and applies Rc via banded_matmul, never forming a dense eigendecomposition of the ill-conditioned Gram matrix at all.

Since Gd, Gc are EXACTLY block-diagonal (the border amplitude carries an uncoupled weight of 1), their Cholesky factors and inverses are block-diagonal too: the dense (Nr+1)x(nd+1) solve degenerates to one O(N^2*bw) banded triangular inverse plus one O(N^2) dense border-row product (the border row itself, border_row, is dense by construction).

A = A_re + i*A_im splits cleanly: A_re is the REAL tridiagonal l0_plus block, A_im carries the (purely imaginary) symmetry-mode column and border row. Every Decimal matrix operation is therefore on REAL matrices, no complex Decimal arithmetic anywhere; only the final (by-then well-conditioned) SVD, done in float64, recombines W = W_re + i*W_im.

Working precision: 60 decimal digits (L176_PREC), chosen generously above the ~13 decimal digits of condition-number-driven loss _sym_sqrt suffers at N=1024.

3.3 Result

The banked N=1024 reading is HIGHER than the banked N=512 reading: a direct violation of rect_sigma's own documented monotone-non-increasing shape (nested trial spaces). The MP-patched N=1024 reading, 0.09079113, sits BELOW the N=512 value, restoring the expected shape, and is close in magnitude to the banked value (0.16% relative change): a real but small correction, unlike leg 178's sign flip.

The N=512 cross-check between float64 and MP agrees to 5e-7 absolute (0.09080465 vs 0.09080414), confirming this leg's banded-Cholesky substitution reproduces the "reliable window" faithfully before trusting its N=1024 output.

4. Adversarial battery: three bugs found and fixed

test_interval_mp.py's test_pi_matches_known_digits (an EXTERNAL ground-truth comparison (a textbook pi digit string), not an internal self-consistency check) caught three real bugs in interval_mp.py's transcendental path before either re-measurement ran on top of them:

1. _atan_small's term recurrence was missing a (2k-1)/(2k+1) ratio factor.
2. Several steps used Python's bare abs()/unary - on Decimal, both of which silently round to the AMBIENT thread-local decimal context (28-digit default) rather than the module's own prec-digit context: invisibly truncating any value with more than 28 significant digits, no error, no warning. Fixed with Decimal's context-independent .copy_abs()/.copy_negate().
3. mp_pi's Decimal(1) / Decimal(239), computed OUTSIDE any explicit context: 1/239 does not terminate in decimal (unlike 1/5), so this division ALSO rounded silently to 28 digits and was then treated as an EXACT input by a series that assumes exact inputs. Fixed by enclosing 1/239 in a rigorous MPInterval (via mp_div) and generalising _atan_small to accept an interval-valued input, propagating the tiny-but-real uncertainty through the whole series.

All three fixes are in solver/interval_mp.py; test_interval_mp.py (13/13 tests) is the battery that would catch a regression of any of them, plus checks on dsin/dcos against an independent Fraction-based Taylor reference, the exact leg-178-regime theta (3e-31), zero-crossing guards failing closed, and banded_cholesky/banded_triangular_inverse/banded_matmul against exact Fraction arithmetic on random banded SPD matrices.

5. Performance

Both re-measurements were assessed for cost before the long combined run, per CONTINUATION_PROMPT.md's binding rule:

• Leg 178: initial profiling showed solver/interval_mp.py's own fixed-term-count dsin/dcos would cost ~0.1s/node, at ~1,640 unsafe nodes, ~3 minutes, acceptable but improvable. The runner's local adaptive-term series (_mp_sin_cos) cuts this to ~0.007s/node (measured: 25.9s total for 1,640 nodes, 15.8ms/node including the full F/LF/HF dot products over 126 basis columns) by exploiting that theta ~ 3e-31 needs only ONE Taylor term at 120 digits, not the fixed conservative count.
• Leg 176: profiled at N=64/128/256 first (0.14s/1.2s/6.5s) to extrapolate before committing to N=1024; measured N=1024 cost was [CORRECTED 2026-08-12, leg 338: Route-LCB1: originally read 117s; the banked JSON is authoritative and is cited directly below] 100.553s (N1024_seconds in writeup/data/p2_route_apia_v1.json), N=512 cross-check [CORRECTED 2026-08-12, leg 338: originally read 48s] 35.515s (N512_seconds, same file).
• Combined total measured wall time [CORRECTED 2026-08-12, leg 338: this paragraph originally reported 346s (25.9s + 309.0s), a sum that does not itself add up (25.9 + 309.0 = 334.9, not 346), and is non-claim-bearing prose (recorded here rather than re-derived]: 345.478s (total_seconds fields in writeup/data/p2_route_apia_v1.json) leg-178's 36.480s plus leg-176's 308.998s, both fields already in the JSON and unchanged by this correction), well under the 10-minute budget, no long run was needed to answer the gate.

6. Integration note (no dependency change proposed)

solver/interval_mp.py and this leg's runner use only decimal and fractions (stdlib). No addition to requirements.txt is proposed.

Downstream consumers to flag (this leg reads but does not edit any of these; territory is solver/interval_mp.py, test_interval_mp.py, experiments/p2_route_apia_v1.py, writeup/data/p2_route_apia_v1.json, writeup/novelty/leg_312.md, experiments/journal/leg_312.md, plus this leg's own quartet files under writeup/4_p2_lottery/ and experiments/):

• Leg 178's own headline (writeup/data/p2_route_wes_v1_space.json, writeup/4_p2_lottery/{BLOG,TECHNICAL}_P2_WES_V1.md, experiments/journal/ leg_178.md, DIRECTION.md, experiments/journal/leg_272.md): all quote the 3.1e+03 contamination / −230.71 reading as the diagnostic that the instrument (not the mathematics) breaks past a scoped grading depth. This leg's finding SUPPORTS that reading's own interpretation (the −230.71 really was an artifact) rather than contradicting it, no change to leg 178's own YES gate answer is implied, but a reader following the 3.1e+03 number specifically should know it is now independently confirmed as a float64 artifact, not an open question.
• Leg 176's own headline (writeup/data/p2_route_h2cv_v1_postconstruction.json and siblings, writeup/CORRECTIONS.md, writeup/curate_evidence.py, DIRECTION.md): the N=1024 sigma_min row is corrected from 0.09093626 to 0.09079113; any prose that cites the banked N=1024 value specifically (rather than the N<=512 reliable window, which this leg's N=512 cross-check confirms is unaffected) should be updated to the corrected figure or flagged as pending re-derivation.

Neither correction moves a link of the L1→L4 chain: both source legs are explicitly exploratory, not on the critical path, and the corrections are internal-consistency repairs (monotonicity restored, an artifact identified), not new mathematical claims. Clay odds stay ~0.05%.