Hadamard Proof

This repo is the live work surface for finding a Hadamard matrix of order 668.

A Hadamard matrix is a square ±1 matrix whose rows are mutually orthogonal. This repo exists to produce an exact matrix, not just a good-looking search state.

Goal

The main goal is to construct and verify a real Hadamard matrix of order 668.

The target split is:

• 428 is the validation rung
• 668 is the active frontier rung
• the final deliverable is a square CSV matrix with entries in {+1, -1}

If a candidate does not pass exact verification, it does not count.

Where We Are

The repo already has:

• a runnable search lane
• a second widened GS/SDS search lane
• an exact verifier
• saved run states
• defect-report tooling
• continuity files so the work can be resumed without rebuilding from zero

Current preserved artifact ladder:

• order 428
  • runs/order_428_lane01_upgrade_seed67_final.json
  • best_score = 5536
  • max_shift_violation = 20
• order 668
  • Williamson lane baseline
    • runs/order_668_lane01_upgrade_seed67_long_final.json
    • best_score = 13216
    • periodic_score = 13152
    • row_sum_penalty = 64
    • max_shift_violation = 24
  • GS/SDS signature (17,17,9,3) baseline
    • runs/order_668_gs_17-17-9-3_baseline_5440_final.json
    • best_score = 5440
    • max_shift_violation = 16
  • exact polish
    • runs/order_668_gs_17-17-9-3_exact_polish_4032_final.json
    • best_score = 4032
    • max_shift_violation = 12
  • coupled capped repair
    • runs/order_668_gs_17-17-9-3_coupled_cap_3456_final.json
    • best_score = 3456
    • max_shift_violation = 12
  • heavier coupled capped repair
    • runs/order_668_gs_17-17-9-3_coupled_cap_3456_repair_3328_final.json
    • best_score = 3328
    • max_shift_violation = 12
  • bounded PB/ILP exporter rung from the 3328 basin
    • runs/order_668_gs_17-17-9-3_coupled_cap_3456_repair_3328_final_pb_ilp.json
    • runs/order_668_gs_17-17-9-3_coupled_cap_3456_repair_3328_final_pb_ilp_cpsat.py
    • runs/order_668_gs_17-17-9-3_coupled_cap_3456_repair_3328_final_pb_ilp_tier1.lp
    • runs/order_668_gs_17-17-9-3_coupled_cap_3456_repair_3328_final_pb_ilp_tier1.opb
    • runs/order_668_gs_17-17-9-3_coupled_cap_3456_repair_3328_final_pb_ilp_tier2.lp
    • runs/order_668_gs_17-17-9-3_coupled_cap_3456_repair_3328_final_pb_ilp_tier2.opb
    • runs/order_668_gs_17-17-9-3_coupled_cap_3456_repair_3328_final_pb_ilp_tier3.lp
    • runs/order_668_gs_17-17-9-3_coupled_cap_3456_repair_3328_final_pb_ilp_tier3.opb
  • corrected PB/ILP ring and hybrid descent
    • runs/order_668_gs_17-17-9-3_pb_ilp_ring2.json
    • runs/order_668_gs_17-17-9-3_pb_ilp_ring2_cpsat.py
    • runs/order_668_gs_17-17-9-3_pb_ilp_ring2_sqcap_4928_final.json
    • runs/order_668_gs_17-17-9-3_pb_ilp_ring2_sqcap_bestscore_3392_final.json
    • runs/order_668_gs_17-17-9-3_pb_ilp_ring2_sqcap_bestscore_2880_final.json
  • root-2880 ring-2 continuation
    • runs/order_668_gs_17-17-9-3_root2880_ring2_pb_ilp.json
    • runs/order_668_gs_17-17-9-3_root2880_ring2_pb_ilp_cpsat.py
    • runs/order_668_gs_17-17-9-3_root2880_ring2_sqcap_K4_4032_final.json
    • runs/order_668_gs_17-17-9-3_root2880_ring2_sqcap_bestscore_2752_final.json
    • best_score = 2752
    • max_shift_violation = 8
  • root-2752 blended surrogate continuation
    • runs/order_668_gs_17-17-9-3_root2752_portfolio_v5_summary.json
    • blended floor escape from that floor:
      • runs/order_668_gs_17-17-9-3_root2752_v5blend_floor_bestscore_2688_final.json
    • best_score = 2688
    • max_shift_violation = 12
  • root-2688 meta-blended continuation
    • runs/order_668_gs_17-17-9-3_root2688_portfolio_v2_summary.json
    • meta-blended floor escape from that floor:
      • runs/order_668_gs_17-17-9-3_root2688_metablend_floor_bestscore_2496_512slack_final.json
    • best_score = 2496
    • max_shift_violation = 8
  • root-2496 portfolio harness validation
    • run_ring_portfolio.py
    • runs/order_668_gs_17-17-9-3_root2496_portfolio_v1_summary.json
    • portfolio winner: mixed_18_18
    • promoted winner touches the 2496 floor but does not beat it
      • runs/order_668_gs_17-17-9-3_root2496_portfolio_v1_mixed_18_18_sqcap_K4_4736_bestscore_2496_final.json

So the current state is:

• the widened GS/SDS lane is now the active frontier surface
• the search is improving through exact defect repair, not just generic annealing
• the corrected PB/ILP ring plus exact-local hybrid has now pushed the local basin from 3328 down to 2880
• the root-2880 ring-2 exact-to-local continuation has now pushed the local floor from 2880 down to 2752
• the blended surrogate upgrade then pushed the local floor from 2752 down to 2688
• the next meta-blended floor escape pushed the local floor from 2688 down to 2496
• the portfolio winner has widened with the floor:
  • mixed_6_6 at 2752
  • mixed_12_12 at 2688
  • mixed_18_18 at 2496
• the work is not solved
• the main remaining obstruction is still coupled SDS defect cancellation on the 668 frontier rung

Process

Right now the build is working inside an order-4n construction surface:

• 428 = 4 * 107
• 668 = 4 * 167

The repo now carries two frontier lanes:

• Williamson / four-circulant lane
  • useful as a preserved baseline and defect reference
• Goethals-Seidel / SDS lane
  • four length-n ±1 sequences with exact row sums
  • current active signature for 668: (17,17,9,3)
  • equivalent SDS parameters: (167; 75,75,79,82; 144)
  • uses checkpoint resume, exact one-swap polish, and coupled capped repair
  • now exports bounded PB/ILP instances over compressed monitored shift expressions
  • now supports exporter-time hard-capped guard shifts so seed quality can be shaped before late local repair
  • now uses a second-ring exact model plus hybrid handoff back into bounded exact local repair
  • now supports a fixed-floor ring portfolio harness so multiple exact ring families can be tested from the same basin and ranked globally before promotion
  • now supports blended promotion and floor escapes that combine SDS shift focus with explicit indicator-Fourier frequencies
  • measures both periodic defect and SDS defect structure
  • keeps exact CSV verification as the final gate

This is the current live constructive surface, not a claim that the whole problem is exhausted by one family.

Where We Are Going

The next movement for this repo is:

• validate the toolchain cleanly on 428
• keep the stronger 668 GS/SDS basin as the active live lane
• push coupled defect cancellation on the dominant SDS shift pairs
• keep the exact repair layer under a hard score cap, but allow blended floor-escape slack when a frozen floor has clearly formed
• use staged guard-cap seed shaping at the PB/ILP export level before forcing square-drop on the active winner family
• use the corrected PB/ILP ring and hybrid exact-local cycle as the current exact rung
• keep iterating from the 2496 floor instead of restarting from the older 2752, 2880, or 3328 surfaces
• use the ring portfolio harness from the same frozen floor, then promote only the top 1-2 candidates into hybrid/local repair
• treat mixed_18_18 as the current default portfolio winner until another ring family beats it on true global score
• widen or tighten that exact export schedule before broadening the family
• tune staged guard-cap schedules at the exporter/portfolio layer so leakage is blocked before the seed shape is already wrong
• widen to broader construction families if this one stalls

The repo is therefore meant to show:

• what the target is
• what the current live state is
• what the present construction process is
• what the next mathematical step should be

Exact Gate

Every real candidate has to survive:

python3 verify_hadamard.py path/to/candidate.csv

That verifier is the final acceptance condition for this repo.

Core Files

• williamson_search.py preserved Williamson baseline lane
• goethals_seidel_search.py active GS/SDS construction lane
• exact_sds_local_repair.py bounded coupled exact repair over selected SDS shift sets
• export_bounded_pb_ilp.py bounded PB/ILP exporter for the live GS/SDS basin, with CP-SAT plus install-free LP/OPB output and optional hard-capped guard shifts
• solve_bounded_pb_sqcap.py monitored square-sum exact solver for bounded PB/ILP rings, including guard-shift caps
• run_ring_portfolio.py fixed-floor portfolio runner that exports 4 ring families, ranks them globally, and promotes the top 1-2
• HADAMARD_EXACT_MODEL_TEST_MAP.md visual rung map for the exact-model and hybrid test ladder
• HADAMARD_EXACT_MODEL_PROTOCOL.md stable scan protocol plus the adaptive branching rules
• verify_hadamard.py exact verifier for any candidate CSV
• report_williamson_defects.py defect report for the Williamson lane
• report_goethals_seidel_defects.py defect report for the GS/SDS lane, including SDS and indicator-Fourier reads
• Start_Search.command launches background search runs
• HADAMARD_GS_SDS_PROGRESS_NOTE.md short public progress note for the active 668 ladder
• HADAMARD_CONTINUITY_AND_SEARCH_PLAN.md current search ladder and widening plan
• HADAMARD_SEARCH_MATRIX.md lane map and current posture
• HADAMARD_RESONANCE_KEY_FOR_RICK.md handoff file for the next dedicated Hadamard thread
• HADAMARD_CHATLOG_RECOVERY.md what the recovered thread file did and did not actually contribute

Minimal Run Surface

Validation rung:

python3 williamson_search.py --order 428

Frontier GS rung:

python3 goethals_seidel_search.py --order 668 --target-signature 17,17,9,3

Inspect a stored GS defect state:

python3 report_goethals_seidel_defects.py runs/order_668_gs_17-17-9-3_coupled_cap_3456_repair_3328_final.json

Run a coupled capped repair:

python3 exact_sds_local_repair.py runs/order_668_gs_17-17-9-3_coupled_cap_3456_final.json --focus-top-unique 12 --depth 6 --beam-width 128 --endpoint-limit 20 --swap-pool-limit 256 --score-slack 0

Export a bounded PB/ILP instance from the live 3328 basin:

python3 export_bounded_pb_ilp.py runs/order_668_gs_17-17-9-3_coupled_cap_3456_repair_3328_final.json --focus-top-unique 12 --endpoint-limit 12

Run the default ring portfolio from the 2496 floor:

python3 run_ring_portfolio.py runs/order_668_gs_17-17-9-3_root2688_metablend_floor_bestscore_2496_512slack_final.json --output-prefix runs/order_668_gs_17-17-9-3_root2496_portfolio

Run the monitored square-sum exact rung on a bounded PB/ILP export:

python3 solve_bounded_pb_sqcap.py runs/order_668_gs_17-17-9-3_root2880_ring2_pb_ilp.json --output-prefix runs/order_668_gs_17-17-9-3_root2880_ring2_sqcap --K-values 4,6,8 --translation-fix

Read the exact-model workflow and protocol:

open HADAMARD_EXACT_MODEL_TEST_MAP.md
open HADAMARD_EXACT_MODEL_PROTOCOL.md

Short Handoff

If another node needs the cleanest boot block, use this:

This repo is the live work surface for the Hadamard order 668 problem. 428 is the validation rung. 668 is the frontier rung. The repo carries a runnable Williamson baseline plus a widened GS/SDS construction lane, exact CSV verification, stored run states, and continuity files so the work can continue without amnesia. The live 668 ladder is now 13216 -> 5440 -> 4032 -> 3456 -> 3328 -> 2880 -> 2752, with the active GS signature (17,17,9,3). A result only counts if verify_hadamard.py accepts the final matrix.